A Systematic Review of Design of Electrodes and Interfaces for Non-Contact and Capacitive Biomedical Measurements: Terminology, Electrical Model, and System Analysis

Abstract

With the advent of ubiquitous healthcare and advancements in textile industry, non-invasive wearable biomedical solutions are becoming an increasingly attractive alternative to in-hospital monitoring, allowing for timely diagnostics and prediction of severe medical conditions. Non-contact biopotential monitoring is particularly promising because non-contact biopotential electrodes can be applied over clothing or embedded in the material without almost any preparation. However, due to the intricacies of capacitive coupling they rely on, the design of such electrodes and their interface with the body plays a key role in achieving measurement repeatability and their widespread utilization in clinical-grade diagnostics. Based on exhaustive investigation of several decades of the literature on non-contact and capacitive biopotential electrodes and electric potential sensors, this study is intended to serve as a state-of-the-art overview of their historical development and design challenges, a collecting point for important research theories and development milestones, a starting point for anyone seeking for a soft head start into this research area, and a remedy for occasional misnomers and conceptual errors identified in the existing papers. The ultimate goal of this comprehensive analysis is to demystify phenomena of non-contact biopotential monitoring and capacitive coupling, systematically reconciliate terminological inconsistencies, and enhance accessibility to the most important findings for future research. To accomplish this, fundamental concepts are thoroughly revisited—from fundamentals of electrochemistry and working principles of capacitors and operational amplifiers to system stability and frequency-domain analysis. With the use of various mathematical tools (Laplace transform, phasors and Fourier analysis, and time-domain differential calculus), discussions on non-contact and capacitive biopotential electrodes, collected from the 1960s onward, are for the first time compiled into a unified, abstracted, bottom-up analysis. The laid-out inspection provides analytical explanation for various aspects of measurement results available in the referenced literature, but also serves an educative purpose by devising a methodological framework that can be easily applied to other similar research fields. Firstly, the differences and similarities between wet, dry, surface-contact, non-contact, capacitive, insulated, on-body, and off-body biopotential electrodes are clarified. For this purpose, equivalent electrical models of various non-invasive biopotential electrodes are analyzed and compared. As a result, a proposal for a revised classification of biopotential electrodes is given. Secondly, instead of using the concept of a purely capacitive biopotential electrode, a test is proposed for assessing the predominant coupling mechanism achieved with an electrode over an insulating layer. Thirdly, a fundamental model of a buffer active non-contact biopotential electrode and its interface with the body is built and generalized, and the proposed test is applied for analyzing the influence of voltage attenuation and phase shifts on signal morphology. Lastly, guidelines for designing the described electrode–body interfaces are proposed, along with a discussion on practical aspects of their implementation.

1. Introduction

1.1. Introduction and Motivation

Cardiovascular diseases and respiratory disorders remain among the leading causes of death worldwide [1,2,3], afflicting more and more people and becoming the key public health concern, especially since the recent COVID-19 pandemic. A hectic sedentary lifestyle, intense stress at work, persistent geopolitical conflicts, and exacerbating climate change severely affect not only physical but also mental health, aggravating depression, anxiety, and sleep disorders. The rapid aging of the ever-growing population [4] and global economic issues burden the often slow-adapting healthcare system, demanding cost-effective personal care solutions and urging the extension of health services to out-of-hospital monitoring in domestic environments [5]. Unobtrusive health tracking in subjects’ everyday surroundings [6] allows for timely diagnostics and prompt intervention, which could prevent further worsening of unrecognized health issues and eventually foresee conditions such as stroke and cardiac arrest. Providing continuous hands-on insight into the state of wellness can significantly speed up the process of preliminary diagnostics and lower medical expenses, especially given the fact that some of the main risk factors associated with health issues are a consequence of unhealthy life habits and physical inactivity. This way, out-of-hospital long-term monitoring solutions have a chance to serve as the first line of defense against the development of chronic diseases and prolonged treatments.

For this purpose, biomedical recordings are widely used as indicators of psychophysiological state. Specifically, biopotential signals that arise from the body-area electric field, created as a result of physiological and electrochemical processes within the human body, can serve as a viable diagnostic tool. For instance, electrocardiogram (ECG) could provide information on abnormalities in heart activity, while electromyogram (EMG) could indicate musculoskeletal disorders and assess ergonomic issues. With the advent of ubiquitous availability of versatile smart gadgets, miniaturization of electronic devices, and rapid development of advanced polymers and textiles, these possibilities extend to a wide portfolio of wireless remote applications for quality-of-life enhancement that facilitate the concepts of telemedicine, e-health, and human-centered healthcare. They can be found in consumer-oriented products such as stress monitors, step counters, and fitness trackers, sports physiology and recreation, rehabilitation treatment, ambient assisted living, fall detection and human–machine interface technology for the disabled and elderly, veterinary medicine, and the monitoring of body vital signs for life-threatening professions, such as military and firefighting.

Since such devices must first and foremost be simple-to-use, affordable, portable, and convenient, non-invasive biomedical measurement methods are preferred over more accurate implantable devices, especially for the purpose of long-term monitoring during activity and movement, when minimal invasiveness and obstruction of the user must be achieved. Many unobtrusive and non-contact methods have been developed, including Superconducting Quantum Interference Device (SQUID) magnetometers, thermography, photoplethysmography (PPG), ballistocardiography (BCG), and radar, which are described and compared in [7,8,9,10,11]. Among them, capacitive sensing, as a cost-efficient portable alternative, is proving to be a beneficial technology in a wide range of applications, such as touch screens, pressure, strain, and temperature sensing [12], dimensional metrology [13], and fluid presence [14] and liquid-level detection [15], environmental [16] and plant monitoring [17,18,19], and biochemical sensing [20]. The history of capacitive sensing is summarized in [10].

Electrophysiological capacitive sensing applications in particular are emerging. Compared to piezoresistive, piezoelectric, and iontronic sensing strategies [21], non-contact capacitive biopotential sensing offers one of the most simple-to-use approaches whilst allowing a small form factor, reusability, low power consumption, and low cost, which makes it especially practical for mobile, wearable, and in-vehicle applications [22]. Also, these sensors do not require direct contact with skin because they are based on capacitive coupling between the electrode and the human body, accomplished over one or more insulating layers that separate the sensor from the skin. This means that such non-contact (also called contactless or indirect-contact) and capacitive biopotential electrodes can be applied over clothing or embedded in the material, maximizing user safety and freedom of movement. Instead of being worn, non-contact and capacitive biopotential electrodes could also be placed on furniture (sofa [23], chair [24,25], wheelchair [26], lavatory seat [27], or bathtub [28]), car seats [22,29,30,31] and safety belts [32], bicycle handles [33], airplane seats [34], or incorporated into a mattress [35]. They are particularly suitable in situations when direct contact with skin is difficult to achieve or undesirable (for instance, measurement on animals [36,37,38] or on burned skin [39], or monitoring infants in neonatal intensive care units [40]). They can even be used underwater [41,42]. Principles of capacitive sensing can be employed in various areas of biomedical measurement, such as BCG [43,44] and blood pressure monitoring [45], as well as in electrocardiography (ECG) and heart rate monitoring [46,47], used as alternative methods to PPG and mechanocardiography (MCG)—ballistocardiography (BCG), seismocardiography (SCG) and gyrocardiography (GCG) [48]. Further applications can be found in electrooculography (EOG) [49], electromyography (EMG) [50,51], and electroencephalography (EEG) [52,53], along with steady-state visually evoked potentials (SSVEPs) and auditory steady-state response (ASSR) [54]. Capacitive sensing can also be utilized for electrosurgery [55], bioimpedance measurement [56,57], electrical impedance tomography (EIT) [58], impedance cardiography (ICG) [59], and impedance plethysmography (IPG) [60,61], as well as respiration [62,63], sleep respiration [64] and sleep apnea [65], sleep position classification [66] and sitting posture monitoring [67], sweat rate measurement [12] and stress level measurement during driving [68]. Another use case is human localization [69,70], along with motion and gesture recognition [71] and emotion recognition [72]. Human activity sensing can be further extended to human body communication [10,73,74,75], along with capacitive intrabody communication (IBC) [75]. Also, applications of capacitive power transfer have gained popularity [76,77].

In this study specifically, non-contact and capacitive sensing of biopotential signals (ECG, EMG, and EEG in particular) will be in focus. The contactless capacitive principle, on which this sensing is based, and the fact that the utilization of non-contact and capacitive biopotential electrodes does not require skin preparation, pave the way for plug-and-play brain–computer interfaces (BCIs) [54,78], body area networks (BANs) [10,79,80,81], multimodal long-term home monitoring [6,26,82,83], and various physical rehabilitation and robotic applications [12,84,85], employed without the need for the on-site presence of trained medical professionals.

In spite of offering ease of use and short preparation time, pervasiveness of non-contact and capacitive biopotential electrodes is limited due to their sensitivity to ambient conditions, environmental noise, displacement, and movement of surrounding objects and people in the vicinity. These phenomena impair repeatability of measurements and significantly reduce signal quality achieved in non-contact and capacitive biopotential recording compared to its commonly used counterparts—surface-contact electrodes that are placed directly on the skin, such as the popular wet Ag/AgCl electrodes. Advanced software-based filtering techniques and machine learning can be utilized to enhance robustness. However, this can affect real-time performance and often demands a posteriori processing algorithms, which could hinder the very idea of designing a health monitoring system that would not need additional on-site technical assistance. In this study, it will be shown that most of the issues with using non-contact and capacitive biopotential electrodes originate from the tradeoffs in their design, as well as from unwanted coupling between the system and the environment. To counteract these challenges and allow for the creation of larger datasets that can later be used for reliable healthcare diagnostics, it is necessary to come up with solutions that will circumvent the weaknesses of capacitive coupling early in the recording stage. In this process, the hardware design of electrodes and their interface with the human body will prove to play a role of the utmost importance.

1.2. Objectives and Contributions

The main objectives of this study are the following:

• This study is based on previously published reviews on non-contact and capacitive biopotential electrodes [86,87,88,89,90,91,92,93,94,95]. Yet, it is composed with the aim of consolidating various equivalent electrical models of the electrode–body interface and clarifying the terminology used in the field of non-contact and capacitive biopotential sensing, which is often inconsistently applied, as this complex matter is discussed from the perspective of various scientific disciplines such as electronics, biomedicine, chemistry, and textile technology. The need for such demystification of erroneously used terms and unclear concepts was intelligibly emphasized in the 2010s by several letters and comments on earlier papers and reviews [96,97,98,99]. Relying on the caveats therein expressed, this study is carried out with the hope of preventing such misinterpretations in the future. As a result of a systematic review of differences and similarities between electrical models of various electrode–body interfaces, a revised classification of biopotential electrodes is proposed. Accordingly, practical limitations of capacitive coupling are addressed and extended to the concept of predominantly capacitive coupling. A test for assessing the predominant coupling mechanism is proposed and employed to examine the parameters of the generalized equivalent electrical model of non-contact biopotential electrodes and their interface with the body. The resulting framework analytically corroborates the common practice, leading to the proposal for guidelines for designing non-contact electrode–body interfaces. This first objective results in four contributions, which are listed and visualized later in this section.
• This study also serves as a review paper, covering the trends that have been followed since some of the most recent general overviews on non-contact and capacitive biopotential electrodes [90,93] have been published. Hence, when possible and convenient, papers dating from 2016 onward are chosen for modern applications and system examples.
• By surveying discussions provided in the cited papers and summarizing their outcomes through a bottom-up analytical approach, up-to-date feedback to other research groups in the area of non-contact and capacitive biopotential measurement is given. The aim of such a report is to identify major challenges and speed up future research, steering it towards open questions that have to be addressed in order for non-contact and capacitive biopotential electrodes to live up to their commercialization and become standard practice in long-term biomonitoring and wearable biomedical devices. The adopted systematic approach and narrative style have allowed authors to carefully choose the most relevant papers and thoroughly tailor the order of their introduction, creating an “all-in-one-place” digital library that garners the fundamentals of non-contact and capacitive biopotential sensing. The established corpus can later be used to gain quick access to important historical milestones, as well as to instructive theoretical discussions and practical design examples. The build-up of a modern, up-to-date non-contact biopotential electrode, with all of its intricacies, is supported by book chapters, theses, papers, reports, lecture notes, handouts, application notes, and datasheets to accompany the calculations and simulations with practice, thereby verifying the theoretical background scattered across papers published over the last several decades. To fully grasp the context behind common practice and important tradeoffs in the design, no steps in the analysis are skipped, even in the case of applying elementary mathematical derivations and fundamental concepts of electronics or signals and systems analysis, which are elsewhere in the respective literature often implied or considered to be self-explanatory. The ultimate goal of these guiding principles is to make this study as suitable as possible, even for readers without a priori knowledge of biopotential electrodes, providing them with an educative example of using mathematical apparatus and engineering fundamentals in practice and encouraging them to join the deep dive into this sometimes-daunting research niche. Strategies for improving readability will be further described in Section 2 on Materials and Methods.

Based on the first objective of this study, the provided synthesis of the literature yields four main contributions as follows:

• Contribution 1: Proposal for a test for assessing the predominant mechanism of coupling achieved over an insulating layer (Section 3.1);
• Contribution 2: Proposal for a generalized equivalent electrical model of the interface between the body and a non-contact biopotential electrode (Contribution 2.1, Section 3.1), that is further developed into an equivalent electrical model of a buffer active non-contact electrode (Contribution 2.2, Section 4.1);
• Contribution 3: Proposal for revised classification of biopotential electrodes with respect to inaccuracies in the used terminology (Section 3.1);
• Contribution 4: Proposal for guidelines for designing non-contact electrode–body interfaces (Section 4.2), along with practical aspects of their implementation (Section 4.2 and Section 4.3).

These four contributions are ordered chronologically and visualized in the following flow diagram (Figure 1), along with related sections and appendixes. More information on the paper structure will be given in Section 1.3.

1.3. Paper Structure

In terms of linear narrative, the paper is structured as follows (Figure 1).

Section 2 expounds on the research methodology and tools used, reporting on the genesis of this study, strategies for improving readability, and the rationale for using four appendixes.

In Section 3.1, non-contact and capacitive biopotential electrodes are introduced. Before a detailed examination of the subject, Appendix A, Appendix B and Appendix C are cross-referenced to conveniently provide the reader with the knowledge that is a prerequisite for understanding non-contact and capacitive biopotential electrodes (Prerequisite 1).

Firstly, Appendix A gives a brief introduction into the origin of biopotential signals and the role of electrodes in their acquisition. After cross-referencing Appendix A, with respect to the observed biopotential signals, Section 3.1 establishes the frequency range of interest for the rest of the paper.

Next, Section 3.1 cross-references Appendix B, where fundamentals of electrochemistry and a comprehensive comparison of commonly used wet and dry surface-contact biopotential electrodes are given, along with a detailed breakdown of their structure design and equivalent electrical models of their interfaces with the body. Through listing advantages and disadvantages of various surface-contact biopotential electrodes, motivation for using non-contact and capacitive biopotential electrodes as an alternative is further elaborated.

Thereafter, Section 3.1 continues by presenting the concept of non-contact and capacitive biopotential electrodes and highlighting the key differences with respect to surface-contact biopotential electrodes, described in Appendix B. After a brief introduction into capacitive coupling, Appendix C is cross-referenced. Therein, fundamentals of dielectrics, capacitors, phasors, and impedance are recalled.

After cross-referencing three appendixes and establishing the frequency range of interest for the rest of the paper, Section 3.1 proceeds with a discussion on the appropriate model that will be used for describing the interface between the body and non-contact and capacitive biopotential electrodes. Based on practical aspects of capacitive coupling and caveats behind electrical properties of real-world dielectrics and insulating layers, a test is proposed for assessing the predominant mechanism of coupling (capacitive or resistive) achieved over an insulating layer (Contribution 1). A model of a single insulating layer is further expanded into a generalized equivalent electrical model of the interface between the body and a non-contact biopotential electrode (Contribution 2.1). As a result of discussions on misterming and methodological inconsistencies present in the existing literature, the initially used term “insulating layer” and group term “non-contact and capacitive electrodes” (or occasionally just “non-contact electrodes” for simplicity’s sake) are refined and corrected. Accordingly, the end of Section 3.1 brings a proposal for the classification of biopotential electrodes based on their distance from the body, type of contact, and invasiveness (Contribution 3).

Afterward, in Section 3.2, the difference between single-ended and differential signals is explained, various amplifier configurations are compared, and benefits of using an active electrode design approach over a passive design are discussed. Based on the comparison between two main types of active electrodes, the buffer active electrode is highlighted, and the rest of the prerequisites and assumptions for further investigation of its interface with the body are established.

Next, in Section 4, the developed model of a non-contact electrode–body interface is expanded into a model of a single buffer active non-contact biopotential electrode (Contribution 2.2). The entire system (interface–electrode–preamplifier) is defined as a cascade of two subsystems: the input voltage divider and the buffer preamplifier.

Section 4.1 begins by cross-referencing Appendix D. Therein, the buffer preamplifier subsystem is analyzed, starting from the fundamentals of operational amplifiers, negative feedback, Bode plot, and amplifier stability (Prerequisite 2).

Thereafter, Section 4.1 continues by identifying equivalent impedance blocks and deriving the transfer function of the entire system, observed as a two-port network. Based on the fundamentals of system analysis, the properties and stability of the entire system are inspected from its poles and zeros. Accordingly, the impulse response of the system and its frequency characteristic (frequency response) are derived as well.

In Section 4.2, the influence of the second subsystem, the input voltage divider, is further examined in depth. Therein, investigation of the influence of equivalent electrical parameters of the electrode–body interface is divided into three steps: influence on the area of predominantly capacitive coupling (step 1), influence on voltage attenuation and the magnitude response (step 2), and influence on the phase response and signal morphology, which is demonstrated on ECG signals (step 3). In each of the three steps, calculations are corroborated by simulation results, and each step ends with a summarizing paragraph. Following the analytical discussion and simulation results within these three steps, guidelines for designing the investigated electrode–body interfaces are given, along with considerations on practical aspects of their implementation (Contribution 4).

Lastly, in Section 4.3, investigation of the frequency response of the input voltage divider subsystem from Section 4.2 is extended to include the buffer preamplifier subsystem from Appendix D. This way, results of the analysis are further confirmed and abstracted by observing the total frequency response in accordance with the total transfer function and impulse response from Section 4.1. The analysis ends with a discussion on intricacies of designing systems for experimental validation of non-contact electrode–body interfaces, and the generalized model is further evaluated in the context of physical measurements that can be found in the literature.

Finally, Section 5 provides conclusions, laid out as another layer of abstraction. This section highlights the most important findings, equations, figures, and proposals, providing further motivation for future research.

2. Materials and Methods

In the previous section, the motivation, main objectives, contributions, and structure of this study were elaborated. This section encompasses the means and methods of the performed meta-analysis, carried out in accordance with the updated Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) 2020 statement [100,101]. In the Supplementary Materials, the corresponding PRISMA 2020 for Abstracts Checklist is given in Table S1, the PRISMA 2020 Checklist in Table S2, and the PRISMA 2020 Flow Diagram in Figure S1. For details on the availability of this material, readers can refer to the paragraph “Supplementary Materials” at the end of this paper. A detailed and complete report that complies with PRISMA 2020 statement guidance is available in this section. Additionally, sources of financial support for this research are listed in the paragraph “Funding” at the end of this paper.

This study stemmed from state-of-the-art research carried out during the authors’ Master’s studies and ongoing PhD studies. Rather than being a routine update report, it is an unforeseen by-product of a slowly but steadily increasing learning curve, boosted through the authors’ self-practice, as well as experience built during their several-years-long joint research activities. In that sense, the initial search strategy was driven by the need for extending personal knowledge and clarifying the unclear concepts within the research group. In the beginning (year 2022), each of the four authors independently provided the starting syllabus with a list of articles that they considered to be representative and essential for familiarizing them with the research area of non-contact and capacitive biopotential monitoring. The collected papers were stratified into several levels:

• Biomedical monitoring;
• Biopotential monitoring;
• Non-invasive biopotential monitoring;
• Non-contact and capacitive non-invasive biopotential monitoring.

Additionally, the papers were given various category tags for future use, depending on the topics they cover:

• Motivation for biomedical and biopotential measurements;
• Type of biopotential electrodes discussed based on their distance from the body, type of contact, and invasiveness;
• Motivation for using non-contact and capacitive biopotential electrodes specifically;
• Type of biomedical and/or biopotential application of non-contact and capacitive electrodes;
• Original and unique applications of non-contact and capacitive biopotential electrodes;
• Important milestones in the historical development of non-contact and capacitive biopotential electrodes;
• Elaborate equivalent electrical models;
• Elaborate numerical models;
• Elaborate analytical models;
• Elaborate calculation examples;
• Elaborate methods of procedure, protocols, and evaluation metrics;
• Elaborate procedures for a specific electrical measurement;
• Illustrative measurement results;
• Discussions on electrical modeling of electrode–body interfaces;
• Discussions on insulating materials and their fabrication;
• Discussions on electrode size, shape, material, and fabrication;
• Discussions on the number of electrodes used and their configuration;
• Discussions on feasibility and limitations in activity monitoring;
• Discussions on feasibility and limitations in long-term monitoring;
• Discussions on feasibility and limitations in clinical-grade applications;
• Subcategories on analysis and illustrative examples of remaining challenges (parasitic elements and stability analysis, intrinsic noise, crosstalk, motion artifacts, extrinsic noise);
• Subcategories on addressing these remaining challenges (electrode–body interface design, discrete and integrated hardware solutions, digital signal processing, machine learning approach and deep learning methods).

The literature was exchanged among the authors, thoroughly examined, and further inspected in the context of fundamental concepts of electronics and system analysis. As the knowledge was expanded, a deeper insight obtained through discussions, thought experiments, and physical measurements provided a clearer understanding of various phenomena behind non-contact biopotential monitoring and intricacies of capacitive coupling. This allowed authors to delve into more complex matter, reexamine the papers from a new perspective, and eventually create a densely connected network of abstract relations between distinct studies. This way, the reference list was iteratively expanded and reshaped.

Soon, the examined literature revealed room for a bottom-up analysis with a scope stretching from early work to the newest advanced solutions. The turning point for the decision to compose a detailed report on the basis of joint research efforts was the discovery of the aforementioned letters and comments on earlier papers and reviews [96,97,98,99], which suggested the presence of erroneous assumptions and methodological issues in common practice that remain to be remedied. Afterward, further investigation was accompanied by a systematic electronic search, performed in the pursuit of works similar to the ones already explored, as well as new findings that had emerged in the meantime.

An electronic search was performed iteratively throughout the years 2023, 2024, and 2025. Eligible studies were obtained by exploring the Scopus database, IEEE Xplore digital library, and the Google Scholar search engine. The coverage was from 1965 to the present date in 2025. Papers were explored in the case of both surface-contact biopotential measurement and non-contact and capacitive biopotential measurement to allow for the drawing of further parallels between various types of biopotential electrodes.

With regard to biopotential electrodes, an individual search of the following terms was undertaken:

• biomedical/bio-medical;
• biopotential/bio-potential;
• unobtrusive/non-obtrusive/nonobtrusive/non-intrusive/nonintrusive;
• non-invasive/noninvasive;
• surface;
• contact;
• wet;
• gel;
• electrolyte;
• paste;
• dry/dry-contact/dry contact;
• gelless/gel-less;
• pasteless/paste-less;
• active;
• passive;
• on-body;
• off-body;
• non-contact/noncontact;
• indirect;
• contactless;
• insulating/insulated;
• capacitive.

Considering electrode material and fabrication, search queries were as follows:

• rigid;
• multi-layer;
• substrate;
• soft;
• flex;
• stretch;
• nano;
• fabric;
• textile;
• patch.

Considering biopotential signal measurements and typical applications, search queries were as follows:

• ECG;
• cECG;
• heart;
• heart rate;
• EMG;
• muscle;
• EEG;
• brain;
• evoked;
• EOG;
• seat;
• chair;
• bed;
• mattress;
• cushion;
• drive/driving;
• ambulatory;
• body area/body-area/BAN;
• long-term;
• home.

Finally, in terms of analysis of electrode–body interfaces, search queries were as follows:

• skin;
• impedance;
• sweat;
• perspiration;
• humid;
• moist;
• motion;
• movement;
• artifact/artefact;
• air gap;
• triboelectric;
• microphony/microphonic;
• noise;
• interference;
• powerline/power-line/power line;
• right leg.

To further boost the applicability of discussions, aside from conference and journal papers and several aforementioned letters and comments [96,97,98,99] that allowed us to minimize the risk of bias and limitations in the searching process, book chapters and guides were also included, along with lecture notes and handouts (Massachusetts Institute of Technology and University of Edinburgh), as well as several contributive dissertations and theses [102,103,104]. Also, application notes, reports and bulletins, and technical datasheets from various companies such as Texas Instruments (Dallas, TX, USA), along with Burr-Brown (Tucson, AZ, USA), Analog Devices (Wilmington, MA, USA), Renesas Electronics (Tokyo, Japan), Microchip Technology (Chandler, AZ, USA), Murata Manufacturing (Kyoto, Japan), and AIC tech (Moka, Japan), were added in pursuit of analogous reports at the level of fundamental knowledge. Several topics of the technical literature were explored, with an emphasis on fundamentals of:

• Action potentials and biopotential signals;
• ECG biopotential signal and ECG measurement methods;
• Electrochemistry;
• Electrical modeling of real-world electric components and electrode–body interfaces;
• Capacitors and dielectrics;
• Electromagnetism and Maxwell’s equations, fringing fields, and finite models;
• Operational amplifiers and negative feedback;
• Stability analysis;
• Time-domain and frequency-domain system analysis;
• Intrinsic noise analysis;
• Triboelectricity;
• Microphony;
• Interference analysis, electric and magnetic coupling, cabling, and shielding;
• Fabrication of e-textiles and smart fabrics.

An overview of the technical literature will be given on several occasions throughout the rest of the paper.

Aside from the electronic search, further studies were also searched by screening the reference list in every relevant article and report examined. In addition, authors and research groups were identified. For instance, for many prominent authors, the rest of their publications were explored, such as in the case of the [78,83,87,103,105,106,107,108,109] collection. In such cases, both published and unpublished papers were considered. However, to enhance the accessibility of the study, analogous works published in English were preferred. Additionally, in accordance with the second objective (Section 1.2), when possible, papers dating from 2016 onward were chosen for modern applications and system examples.

Finally, amongst the extensive collection of examined papers, the most representative ones were chosen based on the following questionnaire:

• Is the work subject to the issues addressed in the reference letters and comments [96,97,98,99]?
• What is the level of complexity and which level of prior knowledge is required for understanding the work?
• Does the work have an educational value?
• Is the work significant in the context of historical development?
• Is there an informative literature review present?
• Is there an instructive research methodology or measurement protocol provided?
• Are there any analytical or equivalent electrical models, illustrative calculation examples, or illustrative measurement results provided?
• Are the phenomena described in the work clearly distinguishable and separately addressed?
• Are the challenges presented in the work clearly distinguishable and separately addressed?
• Is the presented solution clearly elaborated and differentiated from the existing solutions?

Based on these inclusion criteria, when possible, studies with a clear and intuitive structure, illustrative examples, and detailed analysis, that could be of use both at a beginner and at an intermediate level of knowledge, were preferred. For instance, paper [90] is included as an important recent review, yet commentary on this paper [98] is additionally referenced when appropriate. The chosen papers are then analyzed once again from the context of each category tag they had been assigned. For every cited source, full citation, along with the Digital Object Identifier (DOI), web address, and date of last access, are provided in the References section. Numbers of identified reports, screened reports, reports assessed for eligibility, and included reports are given in the PRISMA 2020 Flow Diagram in Figure S1 in the Supplementary Materials.

Specifically, the relevant review literature that will be cited throughout this study can be classified into the following categories:

• Dry surface-contact electrodes with a focus on noise analysis [110], as well as on material and structural design [111,112,113,114,115,116,117,118,119];
• Comparison of various types of non-invasive electrodes—in the context of EEG [120,121,122], BCIs [54,78,123], and human body communication [74];
• Non-contact and capacitive electrodes: general overview of capacitive sensing applications [12], overview specifically focused on biochemical applications [20], review from the perspective of electrosurgery [55], integrated unobtrusive biomedical sensing solutions [43], and specifically electric field and biopotential sensing [86,88,89,90]; reviews on biopotential measurement with the addition of noise analysis [87]; reviews focused on ECG [93] and EMG [91] applications, dielectrics [92], and structural design for wearability [21];
• Wearable design: modalities and prospects [124], advancements in devices for arrhythmia detection [125], hydrogel-based devices [126], textile-based electrodes [127,128,129,130], printed wearable electronics [131], biocompatibility [132] and biodegradability [133], wearable antennae [134,135,136], energy harvesting [137], plant monitoring [17], BANs [10,79,80,81], exoskeletons [84], and specifically smart-textile exosuits [85];
• Integrated design [138,139,140,141,142];
• Problem-oriented review papers: interference in ECG recordings [143] and motion artifacts removal techniques for wearable EEG [144,145] and specifically for non-contact and capacitive ECG measurements [94,95].

The resulting study is script-like, written on the basis of an elaborate storyline that was revised multiple times as earlier findings would stumble upon a contradiction or an exception, which allowed for further generalization of the analysis, as well as a finer reassessment of the authors’ perception. Based on the available reviews and discussions, it brings an integration of the gathered findings into a comprehensive step-by-step creation of the most frequently used up-to-date equivalent electrical model of non-contact and capacitive biopotential electrodes and their interface with the body. To corroborate the outcomes of the analysis, mathematical apparatus and engineering fundamentals are extensively used and followed by simulations in the MathWorks^® MATLAB R2025b environment [146] and LTspice^® 24.1.10 simulation software [147], illustrative examples, and supporting information obtained from the datasheets of off-the-shelf components that have been recently used in the 2010s and 2020s in similar applications. Since ECG is the most popular application commonly chosen by many papers delving into non-contact and capacitive biopotential sensing, most of the examples will be demonstrated on ECG measurement systems. For this purpose, two reference datasets will be used:

• PhysioNet service from the MIT-BIH Arrhythmia database for standard clinical vital signs measurements [148];
• UnoViS database for unobtrusive and non-contact medical monitoring in various scenarios ranging from a clinical study to measurements obtained while lying in bed and driving a car [149].

Incidentally, datasets for other methods of contactless sensing are reviewed in [11].

To further improve readability, several additional strategies are employed:

• Beginnings of discussions inside each section are highlighted with a brief heading indicating the subject of the discussion;
• Assumptions and simplifications that are used throughout this study are enumerated and highlighted within a dedicated paragraph titled “Assumption X.” This way, each assumption represents a stepping stone in the bottom-up analysis, summarizing the findings built upon the previous assumptions, as well as offering an abridged version of the content for quick access. Throughout the paper, nine assumptions are established in total.
• Each topic discussed contains a brief literature overview, which serves as a collecting point for all the cited literature on the respective topic. The provided references offer a progressive learning path, as they span a broad spectrum of complexity, ranging from foundational concepts for novices to advanced, expert-level content.
• Finally, to reduce the main body of the text, elementary mathematical derivations and prior knowledge (Prerequisites 1 and 2 in Figure 1) are strategically relocated to four appendixes, which are appropriately cross-referenced at the beginning of Section 3.1 and Section 4.1. Their content is described in Figure 1 and Section 1.3:
  • Appendix A. Origin and Acquisition of Biopotential Signals;
  • Appendix B. Surface-Contact Electrodes;
  • Appendix C. Fundamentals of Capacitors and Phasor Algebra;
  • Appendix D. First Subsystem: Operational Amplifier.

To preserve the linearity of the narrative, cross-reference of an appendix is treated as the point at which the reader should switch to the cross-referenced appendix and read its content before moving forward with the main body of the text. Appendix A, Appendix B and Appendix C are cross-referenced at the beginning of Section 3.1, whereas Appendix D is cross-referenced at the beginning of Section 4.1. Figures and equations in appendixes are prefixed with “A”.

3. Electrodes

3.1. Non-Contact Electrodes

Origin and acquisition of biopotential signals. In Appendix A, the origin of biopotential signals, such as ECG, EMG, and EEG, is briefly explained, as well as the role of electrodes in their acquisition. Also, references [150,151,152,153,154,155,156,157,158] are introduced, along with Figure A1. The comparison between invasive and non-invasive biopotential electrodes given therein further explains why the rest of the paper will be focused on non-invasive and surface-contact electrodes and their interfaces with the body. With respect to the biopotential signals that will be observed in this paper, the first assumption out of nine in total can be expressed as follows.

Assumption 1.

Biopotential amplitudes and frequencies of interest. The vast majority of biopotential signals lie in the frequency range from sub-Hz frequencies (0.05 Hz or even 0.02 Hz in the case of clinical-grade ECG [154]) up to several kHz (20 kHz in the case of electrocochleography (EcochG) [104]), with most of the power concentrated at frequencies of up to 500 Hz. Therefore, the frequency range 0–20 kHz will from now on be referred to as the frequencies of interest, frequency range of interest, bandwidth of interest, frequency band of interest, or area of interest. On the other hand, typical amplitudes of biopotential signals reach the order of 1 mV, and the order of 10 mV in the case of EMG signals specifically. Amplitudes and bandwidths for different biopotentials are given in Table 1 in [92] and Table 1 in [95] (ECG), page 11 in [157] (EMG), Table 1 in [144] and Section 2.1 in [145] (EEG), and Table 5.1 in [104] (EOG and EcochG). These properties of biopotential signals will be recalled later and used to draw eight additional assumptions and simplifications. Most of the examples will be demonstrated on ECG measurements and compared with clinical ECG features (Figure A1b). More on noise sources in ECG signals, as well as on ECG signal frequency spectrum and clinical features, can be found in [152,153,154]. A brief history of ECG recording is given in [125].

Surface-contact electrodes. In Appendix B, surface-contact biopotential electrodes are investigated and described. Appropriately, references [159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240] and Figure A2 and Figure A3 are introduced. Furthermore, various types of surface-contact electrodes (wet, semi-dry, and dry) are compared, and in addition, their structural design is commented (rigid, flexible, and textile). Although their utilization has long been a standard clinical practice, understanding their theory of operation is a prerequisite for delving into non-contact and capacitive biopotential electrodes. The electrical model of the interface between a surface-contact biopotential electrode and the skin, developed and described therein (Figure A2), is the basis for the model of non-contact electrode–body interfaces, which will be developed and generalized in the following pages of this section. Therefore, the next paragraph, which introduces non-contact and capacitive biopotential electrodes, is written as a continuation of the discussion provided in Appendix B.

Capacitive coupling, polarization, and displacement currents. Dry surface-contact electrodes, which concluded Appendix B, are non-insulated and still intended for direct contact with skin, so therein mentioned problems of half-cell and skin potential variation, as well as significantly higher coupling impedance, must be addressed with careful material selection and microstructures, which often implies a sacrifice of comfort and non-invasiveness, impeding their commercialization and rendering the manufacturing process complex and expensive. Following this reasoning, an alternative approach has been drawing more and more attention: the idea of designing non-invasive electrodes that would not require contact with skin at all, and which could instead be applied over clothing or embedded into material, allowing even measurements from a distance and without the subject’s awareness [241]. Dry and non-invasive by definition, such electrodes are called non-contact, contactless, or indirect-contact electrodes. They were first introduced by Richardson in the second half of the 1960s, with papers such as [242], followed by other examples in the early 1970s [243,244,245], later undergoing a renaissance through a series of fundamental quantum physics studies on electric potential sensing (EPS), where non-contact electrodes were used as a more user-friendly alternative to SQUID magnetometers [88,246,247,248,249,250,251,252,253,254]. Contrary to previously described electrodes, which rely on the resistive (ohmic, conductive) method of coupling, the coupling should now be primarily capacitive: the electrode sensing surface and the skin form two plates of a capacitor, whereas the material that separates the electrode from the skin becomes the capacitor dielectric. Dielectric, as a polarizable insulating material [255,256,257], impedes the flow of free charge, and in addition becomes polarized under an externally applied electric field (Figure 2). Hence, the translation of ionic currents in the body into electron currents in the electrical circuit is now based on displacement currents instead [258,259,260,261]. Unlike conduction currents in resistive coupling, which are proportional to the electric field and explained by the physical motion of electrons under electric potential difference, displacement currents are proportional to the rate of change of the electric field in time and represent Maxwell’s correction of Ampère’s law [262,263]. They describe an apparent flow of current through a capacitor, which is in fact a macroscopic manifestation of the displacement of charges stored on the plates: for each electron drawn to one plate of the capacitor, one electron becomes pulled away from the other plate. This way, capacitors resist sudden changes in voltage, counteracting them by creating a flow of charge. As a result, direct current (DC current) flow is blocked, and displacement current i_dis(t) exists only as long as either the electric potential difference v_coupling(t) across capacitor C_coupling, or C_coupling itself, changes in time. If a linear time-varying model of a capacitor is assumed [264], this behavior can be expressed mathematically as follows (1):

$$i_{\mathrm{d}\mathrm{i}\mathrm{s}}(t)={\displaystyle \frac{dq\left(t\right)}{dt}}=C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left(t\right){\displaystyle \frac{d{v}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left(t\right)}{dt}}+v_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left(t\right){\displaystyle \frac{d{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left(t\right)}{dt}} ,$$

(1)

where dq(t)/dt is the rate of apparent charge flow over a given short time interval dt. For a C_coupling assumed to be constant, Assumption 2 can be expressed as follows:

Assumption 2.

Coupling capacitance. Unless stated otherwise, C_coupling is assumed to be linear and constant with time. In addition, a stable environment is assumed without variations in any condition that would change C_coupling. More on capacitor modeling and its fundamental properties can be found in [264].

Figure 2. Theoretical principle of an ideal dry non-contact and capacitive electrode created with respect to Figure A2. In the process of dielectric polarization, electric dipoles are displaced under the influence of the applied external electric field. This in turn creates an internal electric field of opposite direction [256,257,258]. Due to safety regulations, the human body is ungrounded, so its floating body potential is denoted by a bar symbol under the v_source voltage.

Figure 2. Theoretical principle of an ideal dry non-contact and capacitive electrode created with respect to Figure A2. In the process of dielectric polarization, electric dipoles are displaced under the influence of the applied external electric field. This in turn creates an internal electric field of opposite direction [256,257,258]. Due to safety regulations, the human body is ungrounded, so its floating body potential is denoted by a bar symbol under the v_source voltage.

Under Assumption 2, (1) can be simplified to (2):

$$i_{\mathrm{d}\mathrm{i}\mathrm{s}}(t)=C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{\displaystyle \frac{d{v}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left(t\right)}{dt}}$$

$$\leftrightarrow v_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left(t\right)={\displaystyle \frac{1}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{\int }_{-\infty }^t{i}_{\mathrm{d}\mathrm{i}\mathrm{s}}\left(\tau \right)d\tau =v_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left(t_0\right)+{\displaystyle \frac{1}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{\int }_{t_0}^t{i}_{\mathrm{d}\mathrm{i}\mathrm{s}}\left(\tau \right)d\tau ,t\ge t_0 .$$

(2)

In other words, the existence of the displacement current calls for alternating phases of capacitor charging and discharging. On the other hand, integral equation for voltage across the capacitor, v_coupling(t), shows that unlike a resistor, a capacitor has memory. This memory property is contained in the initial condition v_coupling(t₀) that represents the voltage present in the chosen initial moment t₀ [264].

Fundamentals of capacitors and phasor algebra. Fundamental considerations on capacitors, as well as fundamentals of phasor algebra and the concept of impedance, are briefly recalled in Appendix C. Also, references [265,266,267,268,269], Equations (A1)–(A5), and Figure A4 are introduced. Therein described properties will be extensively used throughout this study and, in the following paragraphs, employed in the context of coupling with a non-contact biopotential electrode.

Parallel plate capacitor approximation. The actual capacitance C_coupling between two conductive points separated in space is determined by the geometry and electrical properties of the material in between, as well as the material in the surroundings. In the case of non-contact biopotential electrodes, C_coupling is the equivalent capacitance between the electrode sensing surface and the surface of the skin. The resulting capacitive coupling mechanism is most often approximated with the behavior of a parallel plate capacitor [256,258], as indicated earlier in Figure 2. This approximation assumes two flat plates and negligible fringing fields around their perimeter (3):

$$C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}={\epsilon }_0{\epsilon }_{\mathrm{r}}{\displaystyle \frac{A}{d}} ,$$

(3)

where C_coupling is the capacitance of the parallel plate capacitor formed by the body and the electrode sensing surface with an area of A. On the other hand, d represents the thickness of the material between the plates, i.e., the distance between the electrode and the body. Finally, the extent to which the external electric field is suppressed by the internal electric field, created in the opposite direction by induced charges in the dielectric, is described with permittivity: ε₀ represents the permittivity of free space (vacuum) (8.854 × 10⁻¹² F/m), ε_r represents the relative permittivity of the material that separates the electrode from the skin, and their product ε₀ε_r represents absolute permittivity. Often, dielectric constant ϰ is used (4) as a synonym for ε_r [270], and the same will be applied throughout this study:

$$\kappa ={\displaystyle \frac{{{\epsilon }_0\epsilon }_{\mathrm{r}}}{{\epsilon }_0}}={\epsilon }_{\mathrm{r}} .$$

(4)

Other useful approximations. Other approximations for modeling non-contact and capacitive electrode–body interfaces are also feasible, such as cylindrical capacitor approximation [258,271], but the parallel plate capacitor approximation prevails as by far the most often used method. In fact, approximations for the capacitance of practical geometric configurations and printed circuit board (PCB) structures often stem from the parallel plate capacitor model. Many of these approximations will be used later in this study to model parasitic and stray capacitances. For instance, expressions for capacitance in a coaxial cable, capacitance between two parallel rods, and capacitance of a rod over a conductive plane, can be found in [259]. Further, in [272], capacitances found in several cable configurations are compared. At the PCB design level, additional useful expressions for the capacitance of adjacent PCB copper traces, capacitance between parallel copper planes, and capacitance between a copper trace and an adjacent copper plane can be found in [273]. Finally, calculations for the capacitance of a PCB transmission line in microstrip and stripline configuration can be found in [259,273,274]. The rest of the analysis will use the parallel plate approximation (3), and the employed simplifications will be summarized in the assumption at the end of this section.

Model accuracy, fringing fields, and finite models. Equation (3) revealed that the parallel plate capacitor approximation can be used as a tool for qualitatively relating the electrode–body capacitance C_coupling to electrode sensing area A, as well as to thickness d and dielectric constant ε_r of the material inserted between the body and the electrode. In reality, direct proportionality of capacitance to the parallel plate area is an oversimplification because of real-world dielectric material properties and additional layout parasitic capacitance, which will be further discussed in Section 4.1. As a rule of thumb, the greater the ratio $\sqrt{A}$/d (i.e., the wider the plates and the smaller the distance between the plates), the more accurate the chosen parallel plate approximation [102,259]. Ideally, the electric field that arises within the equivalent parallel plate capacitor as a result of charge accumulated on the surface of its plates is predominantly uniform and perpendicular to the electrode–body interface. This is in accordance with Gauss’s law for electric fields (first Maxwell’s equation) [258,262]. However, in reality, fringing electric fields that spread into the surroundings from the edges of the electrode–body interface will contribute to additional capacitance that is not accounted for in the parallel plate capacitor model. The contribution of these fringing fields becomes especially evident as d is increased [259]. For this reason, when it comes to using electrodes that are physically separated from the body via an air gap (ε_r,air ≈ 1), validity of the parallel plate approximation is limited to very small d up to about the order of 1 mm [9]. Later in this section, this will prove to be an important limiting factor in the employment of off-body biopotential electrodes. More on electric field calculation can be found in [255,258,275,276], and accompanying analysis of finite parallel plate capacitor models that account for edge effects of electric field distribution can be found in [15,277,278,279,280,281]. Likewise, if the space around the parallel plate capacitor were not uniformly filled with the material of same properties, as assumed here, accurate modeling would require accounting for partial dielectric filling and using the effective dielectric constant for describing the dielectric properties of inhomogeneous material (see [259] and subchapter 7.3 in [257]).

Dielectric constant and dielectric loss. The dielectric constant of the dielectric between the plates of the capacitor is an intrinsic macroscopic property of the material. It is roughly related to the size and number of electric dipoles and describes the magnitude of the polarization effect, as well as the extent to which the light is slowed down in the dielectric [259]. Just like electrical properties, such as conductivity σ and its reciprocal, resistivity $\rho$, the dielectric constant also depends on various parameters—from ambient-induced conditions (e.g., temperature and moisture content) to manufacturing process and material structure (surface roughness, material purity and homogeneity). Furthermore, it may also depend on the strength of the electric field and the applied voltage, which is an issue known as the DC bias effect, typical for high-κ class 2 and class 3 ceramic capacitors, with a dielectric constant that can reach the order of 1000 (see subchapter 1x.3 in [282]). Additionally, the dielectric constant may also depend on the direction of the electric field, which would require treating it as a tensor [257,276]. Finally, when discussing dielectric properties, there is also the aspect of alternating voltage. Namely, unlike conductors, in which the power dissipation is proportional to the square of the applied voltage, electrical energy transformation into heat in dielectrics displays additional dependence on the frequency. Among all mechanisms of these dielectric losses, the simplest and most often considered in electrical circuits is the dielectric loss conditioned by conductivity (see subchapter 7.8 in [257], and more specifically, [283]). Higher frequencies increase the motion of the dipoles, thereby increasing the conductivity that arises from collisions and scattering of charge carriers, as well as from the heat generated by the friction of increased dipole motion. To model these dielectric losses, complex number formalism is used: imaginary part ε_r″ is added, and the so far observed dielectric constant ε_r becomes the real part ε_r′. The total permittivity can therefore be observed as a complex number ε* $=$ ε′ − jε″ $=$ ε₀ε_r′ − jε₀ε_r″ $=$ ε₀ε_r − jε₀ε_r″. To describe how lossy the dielectric material is, the tangent of the dielectric loss angle tanδ is used. It is also called the loss tangent or dissipation factor, and it is roughly related to the number of dipoles and their freedom of movement in the applied alternating current (AC) electric field [283]. Also, its reciprocal, the quality factor Q of the dielectric material, can be obtained [256,270]. Specifically, dielectric loss conditioned by conductivity manifests itself in an additional conduction or active current i_cond(t) that is in phase with the applied sinusoidal voltage v_coupling(t) (Figure 3). In that sense, tanδ indicates the extent to which the total current phasor $\dot{I_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}$ deviates from the desired, purely reactive displacement current phasor, $\dot{I_{\mathrm{d}\mathrm{i}\mathrm{s}}}$, and so, lower tanδ means lower dielectric loss. $\dot{I_{\mathrm{d}\mathrm{i}\mathrm{s}}}$ can be observed through the capacitive reactance X_Ccoupling (A4), whereas the additional conduction current phasor $\dot{I_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}}$ can be perceived through an equivalent resistive leakage component R_coupling. Applying alternating sinusoidal electric field and parallel plate approximation as in (3) yields (5):

$$\mathrm{t}\mathrm{a}\mathrm{n}\delta ={\displaystyle \frac{\dot{I_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}}}{\dot{I_{\mathrm{d}\mathrm{i}\mathrm{s}}}}}={\displaystyle \frac{\frac{\dot{V_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{\frac{\dot{V_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{X_{\mathrm{Ccoupling}}\left(\omega \right)}}}={\displaystyle \frac{X_{\mathrm{Ccoupling}}\left(\omega \right)}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}={\displaystyle \frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}·\omega C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$$

$$={\displaystyle \frac{1}{\frac{1}{\sigma }\frac{d}{A}·\omega {\epsilon }_0{\epsilon }_{\mathrm{r}}\frac{A}{d}}}={\displaystyle \frac{\frac{\sigma }{\omega }}{{\epsilon }_0{\epsilon }_{\mathrm{r}}}}={\displaystyle \frac{{\epsilon }^{″}}{{\epsilon }_0{\epsilon }_{\mathrm{r}}}}={\displaystyle \frac{{\epsilon }_0{{\epsilon }_{\mathrm{r}}}^{″}}{{\epsilon }_0{{\epsilon }_{\mathrm{r}}}^{\prime }}}={\displaystyle \frac{\mathrm{I}\mathrm{m}\left\{{\epsilon }^{*}\right\}}{\mathrm{R}\mathrm{e}\left\{{\epsilon }^{*}\right\}}}\approx {\displaystyle \frac{1}{Q}}={\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{p}\mathrm{e}\mathrm{r} \mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}{\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{d} \mathrm{p}\mathrm{e}\mathrm{r} \mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}},$$

(5)

where σ is the bulk or volume conductivity of the dielectric material. As a rule of thumb for polymers, a tighter confinement of dipoles inside the structure will result in a smaller real part of relative permittivity ε_r′ and lower tanδ [283]. The existence of dielectric losses explains why additional precaution must be taken at higher frequencies, e.g., when wearable antennae design is considered [134,135,136,229]. Aside from introducing the dielectric loss, the frequency of the applied alternating electric field will also contribute to characterizing the mechanisms of dielectric polarization, as described in [256,257,260], and specifically in [284], concerning triboelectric effect. The existence of various polarization mechanisms is tightly related to frequency-dependent changes in ε_r* called dielectric dispersion and described with dielectric spectroscopy. Similar observations are applicable to dielectric properties of biological tissues as well [56,75]. Lastly, depending on the physical mechanisms of dielectric loss under given conditions, frequency-dependent variations may also be visible in tanδ itself due to, e.g., dipole inertia. For more details on all these phenomena, refer to subchapters 7.1, 7.3, and 7.8 in [257], and subchapter 9.5 in [283]. Assuming that tanδ $=$ ${\displaystyle \frac{\sigma }{{\epsilon }_0{\epsilon }_{\mathrm{r}}\omega }}$ is fairly constant means that conductivity σ is strongly frequency-dependent. After summarizing the observations in the following Assumption 3, this will be further investigated in the context of the insulating layer between the body and a non-contact electrode. Afterward, the analysis will be extended to the generalized electrical model of the electrode–body interface.

Assumption 3.

Insulating layers and dielectric properties. In this paper, the terms “dielectric”, “dielectric layer”, and “insulating layer” are used interchangeably. The real part ε_r′ of the complex relative permittivity, ε_r* = ε_r′ − jε_r″, will be considered equal to the dielectric constant, and denoted simply by ε_r′ = ε_r, whereas the imaginary part will be denoted by ε_r″. For the purpose of accompanying the model with calculations that can allow for a qualitative analysis at frequencies of interest (refer to Assumption 1 for details), linear, isotropic, and homogeneous dielectrics [255,257,285] with uniform cross-sectional areas will be assumed. In other words, it is assumed that insulating layers have a uniform structure with properties equal in all directions and that the superposition principle is applicable in the case of multiple sources of electromagnetic fields. Therefore, ε_r′ = ε_r will be treated as a scalar. In accordance with Assumption 2, the influence of ambient conditions on ε_r* and tanδ will not be considered. Similarly, practical aspects of manufacturing processes, structural imperfections, and aging effect in the material will be neglected. Finally, assuming that dielectric loss conditioned by conductivity is the predominant mechanism of dielectric losses at frequencies of interest (refer to Assumption 1), any additional possible variations in ε_r* and tanδ with frequency are neglected. Thus, at frequencies of interest, ε_r′ = ε_r is approximated with the quasi-static dielectric constant that would be measured in near-DC or constant electric fields. Based on these assumptions, the following analysis regarding the leakage resistance as a function of frequency is performed in accordance with [283].

Dielectric leakage resistance and insulator breakdown. In the case of wet and non-insulated dry surface-contact electrodes described in Appendix B, imperfections of resistive coupling were modeled with parallel capacitances that represented the unwanted dielectric properties in each conductive layer (Figure A2). Now, from the perspective of non-contact and capacitive electrodes, the imperfections of capacitive coupling achieved via an insulating layer are modeled with a leakage or insulation resistance R_coupling that shunts the desired coupling capacitance C_coupling, giving rise to resistive coupling (Figure 3). In other words, the equivalent impedance of a dielectric can be modeled as Z_coupling $=$ (−jX_Ccoupling)||R_coupling (A4). On the one hand, this parallel leakage component R_coupling accounts for the finite intrinsic conduction of the insulating material due to impurities and defects in its structure, as well as for the current from the motion of mobile charge carriers such as ions if present (see chapter 3 in [260] and subchapter 9.4 in [283]). This leakage usually increases with the applied voltage until the dielectric strength or the breakdown field is reached (see [255,256,275,286] and subchapter 7.10 in [257]). However, aside from this finite DC leakage at 0 Hz, R_coupling,DC, there is also an additional frequency-dependent leakage arising from the dielectric loss and the finite conduction current $\dot{I_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}}$. In accordance with the parallel plate capacitor approximation (3), this AC-leakage resistance R_coupling,AC(ω) can be expressed as a function of geometric configuration and frequency (6) [283,287,288]:

$$R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{A}\mathrm{C}}\left(\mathsf{\omega }\right)={\displaystyle \frac{\dot{V_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{\dot{I_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}}}}={\displaystyle \frac{\dot{V_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{\frac{\dot{V_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{\frac{1}{j\omega {\epsilon }_0\left(-j{{\epsilon }_{\mathrm{r}}}^{″}\right)\frac{A}{d}}}}}={\displaystyle \frac{1}{\omega {{{\epsilon }_0\epsilon }_{\mathrm{r}}}^{″}}}{\displaystyle \frac{d}{A}}={\displaystyle \frac{1}{\omega {{\epsilon }_0\epsilon }_{\mathrm{r}}\mathrm{t}\mathrm{a}\mathrm{n}\delta }}{\displaystyle \frac{d}{A}}={\displaystyle \frac{1}{{\sigma }_{\mathrm{A}\mathrm{C}}\left(\omega \right)}}{\displaystyle \frac{d}{A}} ,$$

(6)

where σ_AC(ω) is the bulk or volume AC conductivity of the dielectric material. In other words, the dielectric constant ε_r as the real part of ε_r*, along with $\dot{I_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}}}$, determines the reactive, capacitive portion X_Ccoupling(ω) (3,A4) of the total Z_coupling, whereas the imaginary part of ε_r*, along with $\dot{I_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}}$, determines the resistive portion R_coupling,AC(ω) (6) of the total Z_coupling. Both this capacitive and this resistive component are frequency-dependent. Following Assumption 3 and the analysis from subchapters 9.4–9.6 in [283], the R_coupling,AC(ω) component does not emerge abruptly above 0 Hz, but rather appears above a certain ω_δ angular frequency. Thus, the total shunt leakage resistance R_coupling(ω) can be treated as a DC leakage resistance R_coupling,DC (7):

$$R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}={\displaystyle \frac{1}{{\sigma }_{\mathrm{D}\mathrm{C}}}}{\displaystyle \frac{d}{A}}={\rho }_{\mathrm{D}\mathrm{C}}{\displaystyle \frac{d}{A}} \mathrm{f}\mathrm{o}\mathrm{r} \omega <{\omega }_{\mathsf{\delta }} ,$$

(7)

until reaching ω_δ, above which it decreases with frequency (8):

$$R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{A}\mathrm{C}}\left(\omega \right)={\displaystyle \frac{1}{{\sigma }_{\mathrm{D}\mathrm{C}}+{\sigma }_{\mathrm{A}\mathrm{C}}\left(\omega \right)}}{\displaystyle \frac{d}{A}}={\displaystyle \frac{1}{\frac{1}{{\rho }_{\mathrm{D}\mathrm{C}}}+(\omega -{\omega }_{\mathsf{\delta }}){{\epsilon }_0\epsilon }_{\mathrm{r}}\mathrm{t}\mathrm{a}\mathrm{n}\delta }}{\displaystyle \frac{d}{A}}$$

$$={\displaystyle \frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}}{1+{{\epsilon }_0\epsilon }_{\mathrm{r}}{\rho }_{\mathrm{D}\mathrm{C}}\mathrm{t}\mathrm{a}\mathrm{n}\delta (\omega -{\omega }_{\mathsf{\delta }})}} \mathrm{f}\mathrm{o}\mathrm{r} \omega \ge {\omega }_{\mathsf{\delta }},$$

(8)

where $\rho$_DC is the bulk DC resistivity of the dielectric material. Now, the previous statement on the frequency-dependence of dielectric losses becomes clearer: the increase in frequency of the applied voltage and the generated electric field translates into a faster back-and-forth rotation of electric dipoles, which increases the bulk conductivity and decreases the bulk resistivity. The larger the tanδ, the lossier the dielectric and the larger the steepness of that decrease [283]. Substituting further the parallel plate approximations (3) and (7) into (8) finally yields the piecewise first-order equation for the total shunt leakage resistance of the dielectric, R_coupling(ω) (9):

$$R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}(\omega )=\left\{\begin{array}{cc}\hspace{1em}\hspace{1em}\hspace{1em} R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}={\rho }_{\mathrm{D}\mathrm{C}}{\displaystyle \frac{d}{A}},& \omega <{\omega }_{\mathsf{\delta }}\hfill \\ {\displaystyle \frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}}{1+R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\mathrm{t}\mathrm{a}\mathrm{n}\delta (\omega -{\omega }_{\mathsf{\delta }})}},& \omega \ge {\omega }_{\mathsf{\delta }}\hfill \end{array}\right..$$

(9)

The letter R for leakage resistance is consistently used, despite its frequency-dependence, to stress the distinction between the leakage resistance R_coupling(ω) on the one side and the corresponding Z_Ccoupling $=$ −jX_Ccoupling, along with the total Z_coupling $=$ (−jX_Ccoupling)||R_coupling(ω), on the other side.

Assessment of the predominant coupling mechanism. In the preceding discussion, the dielectric between the capacitor plates (surface of the skin on the one side, and electrode sensing surface on the other side) was modeled as an R_coupling(ω)C_coupling-parallel based on parallel plate approximations (3,9). Therein, the existence of the finite leakage component R_coupling(ω) revealed that the coupling over a dielectric layer is seldom solely capacitive, and even if it is predominantly capacitive, it is not necessarily so across the entire frequency range. Therefore, Test 1 for evaluating the predominant coupling mechanism over a dielectric can be devised:

Test 1.

For a particular insulating layer, if across a certain frequency range:

$$R_{\mathrm{coupling}}\left(\omega \right)\gg X_{\mathrm{Ccoupling}}\left(\omega \right)\to R_{\mathrm{coupling}}\left(\omega \right)\gg {\displaystyle \frac{1}{\omega C_{\mathrm{coupling}}}} ,$$

i.e.,

$$R_{\mathrm{coupling}}\left(\omega \right)\ge {\displaystyle \frac{10}{\omega C_{\mathrm{coupling}}}} or \omega \ge {\displaystyle \frac{10}{R_{\mathrm{coupling}}{\left(\omega \right)C}_{\mathrm{coupling}}}} ,$$

then the predominant coupling mechanism at these frequencies is capacitive.

Conversely, if across a certain frequency range:

$$R_{\mathrm{coupling}}\left(\omega \right)\ll X_{\mathrm{Ccoupling}}\left(\omega \right) \to R_{\mathrm{coupling}}\left(\omega \right)\ll {\displaystyle \frac{1}{\omega C_{\mathrm{coupling}}}} ,$$

i.e.,

$$R_{\mathrm{coupling}}\left(\omega \right)\le {\displaystyle \frac{1}{10}}{\displaystyle \frac{1}{\omega C_{\mathrm{coupling}}}} or \omega \le {\displaystyle \frac{1}{10}}{\displaystyle \frac{1}{R_{\mathrm{coupling}}{\left(\omega \right)C}_{\mathrm{coupling}}}} ,$$

then the predominant coupling mechanism at these frequencies is resistive (ohmic, conductive).

The corresponding frequency f is further calculated from ω with the use of relation (A2).

Two important consequences can be noted. Firstly, if a wet surface-contact electrode with its typical analog of R_coupling up to the order of 10 kΩ and a typical analog of C_coupling in the order of 10 nF is considered, Test 1 corroborates that the achieved coupling is predominantly resistive, at least up to frequencies in the order of 1 kHz. Interestingly, it can be noted that their C_coupling in the order of 10 nF is in fact a value higher than most C_coupling values achieved with predominantly capacitive dry electrodes. In other words, rather than the absolute value of C_coupling, it is the relative value of X_Ccoupling(ω) with respect to R_coupling(ω) that matters. This can be extended further to the second important consequence: whenever a dielectric exhibits a finite amount of leakage (R_coupling < $\mathrm{\infty }$), there exists a frequency up to which the resistive coupling predominates. Accordingly, a frequency at which the coupling mechanism transitions from predominantly resistive to predominantly capacitive can be defined. Specifically, since R_coupling(ω) $=$ R_coupling,DC at ω ≤ ω_δ, this angular frequency of transition is equal to ${\displaystyle \frac{10}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ whenever ω_δ ≥ ${\displaystyle \frac{10}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$. This indicates that the bandwidth, band, frequency range, or area of predominantly capacitive coupling can be further explored based on the position of ω_δ with respect to ${\displaystyle \frac{10}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$. To generalize the investigation further and explore the influence of the difference between ω_δ and ${\displaystyle \frac{10}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$, ω_δ can be expressed as a portion (or a multiple) of ${\displaystyle \frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ with the use of a positive real factor F. Accordingly, notation ω_δ $=$ ${\displaystyle \frac{F}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ can be used. Moreover, the factor of 10 that substitutes the “>>” and “<<” conditions of “at least an order of difference” can also be generalized to a positive real factor of T ≥ 10 to investigate the influence of the stringency of Test 1. This way, the area of predominantly capacitive coupling would be defined by the condition $R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left(\omega \right)\ge {T·X}_{\mathrm{C}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left(\omega \right)\ge {10X}_{\mathrm{C}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left(\omega \right)$. Based on this criterion, the area of predominantly capacitive coupling can be assessed by observing the R_coupling(ω) and X_Ccoupling(ω) curves. This finally leads to results in

Table 1. Two cases (A and B) based on the position of ω_δ $=$ ${\displaystyle \frac{F}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ with respect to angular frequency ${\displaystyle \frac{T}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$. Factors F and T are positive real numbers. For each of the four subcases, the resulting bandwidths, frequency bands, frequency ranges, or areas of predominantly capacitive coupling are given based on the criterion $R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left(\omega \right)\ge {T·X}_{\mathrm{C}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}$ for T ≥ 10. For R_coupling(ω), a first-order model that is monotonically decreasing with frequency as in (9) is assumed.

Case A	Condition: ${\omega }_{\mathsf{\delta }}\ge {\displaystyle \frac{T}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ $, T =$ 10 in this paper
	${\omega }_{\mathsf{\delta }}={\displaystyle \frac{F}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}, F\ge T\ge 10, {\displaystyle \frac{1}{10}}\ge {\displaystyle \frac{1}{T}}\ge {\displaystyle \frac{1}{F}}$
	Area of predominantly capacitive coupling	Subcase A.1 $\omega \ge {\displaystyle \frac{T}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{t}\mathrm{a}\mathrm{n}\delta \le {\displaystyle \frac{1}{T}}$ $T =$ 10 in this paper
		Subcase A.2 ${\displaystyle \frac{T}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}\le \omega \le {\omega }_{\mathsf{\delta }}{\displaystyle \frac{\mathrm{t}\mathrm{a}\mathrm{n}\delta -\frac{1}{F}}{\mathrm{t}\mathrm{a}\mathrm{n}\delta -\frac{1}{T}}} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{t}\mathrm{a}\mathrm{n}\delta >{\displaystyle \frac{1}{T}}$ $T =$ 10 in this paper
Case B	Condition: ${\omega }_{\mathsf{\delta }}<{\displaystyle \frac{T}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$$, T =$ 10 in this paper
	${\omega }_{\mathsf{\delta }}={\displaystyle \frac{F}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}, F<T, T\ge 10$
	Area of predominantly capacitive coupling	Subcase B.1 $\omega \ge {\omega }_{\mathsf{\delta }}{\displaystyle \frac{\frac{1}{F}-\mathrm{t}\mathrm{a}\mathrm{n}\delta }{\frac{1}{T}-\mathrm{t}\mathrm{a}\mathrm{n}\delta }} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{t}\mathrm{a}\mathrm{n}\delta <{\displaystyle \frac{1}{T}}$ $T =$ 10 in this paper
		Subcase B.2 $\omega \in \varnothing \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{t}\mathrm{a}\mathrm{n}\delta \ge {\displaystyle \frac{1}{T}}$ $T =$ 10 in this paper

, which are accompanied by visual examples in Figure 4.

The results are as expected: the angular frequency at which the coupling becomes predominantly capacitive is equal to ${\displaystyle \frac{T}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ whenever ω_δ ≥ ${\displaystyle \frac{T}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$, and it can only be higher than ω_δ whenever ω_δ < ${\displaystyle \frac{T}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$. If the upper angular frequency limit to the area of predominantly capacitive coupling exists, it is situated at ω ≥ ω_δ. To increase the upper frequency limit and reduce the lower frequency limit, i.e., broaden the area of predominantly capacitive coupling, several measures can be taken:

• Increasing C_coupling (reactance reduction (A4));
• Increasing R_coupling,DC (leakage reduction (9));
• Reducing tanδ (slowing down the leakage increase (9));
• Increasing factor F and consequently ω_δ (delaying the leakage increase (9));
• Decreasing T (relaxing the criterion for assessing the frequency at which the coupling becomes predominantly capacitive (Table 1)).

Naturally, the presented cases are more of a generalized mathematical construct rather than physically meaningful possibilities. In practice, tanδ of wearable fabrics is largely below 0.1 [134,135,229], and the dielectric loss of used material is not of concern before reaching the frequencies in the order of 1 MHz or higher, which are used, e.g., for capacitive power transfer [76,77,279,280,289], intrabody communication [75], and wearable antennae design [134,135,136,229]. These practical aspects narrow the cases from Table 1 down to subcase A.1, with ω_δ outside of the frequency range of interest (Assumption 1). Therefore, in the rest of this paper, only the R_coupling,DC component of the leakage resistance R_coupling(ω) (9) will be considered. In other words, parallel plate approximations (3,7) will be used. Accordingly, simplified notation R_coupling,DC $=$ R_coupling and ρ_DC $=$ ρ will be used. Also, test factor T $=$ 10 will continue to be used for “>>” and “<<” conditions. This will be summarized in Assumption 4 at the end of this section.

Multiple insulating or coupling layers, equivalent coupling capacitance, and equivalent coupling resistance. So far, a single insulating layer has been analyzed. In reality, there could be more than just one insulating layer: the non-contact interface between the body and the electrode can be achieved by means of fabric layers that cover the skin, an air gap that separates the electrode from the body, and insulating layers coated on the conductive sensing surface of the electrode. With respect to their equivalent capacitive and resistive components, each of these layers can be more generally referred to as a coupling layer. Such a generalized case of a non-contact and insulated electrode–body interface is given in Figure 5. Therein, “sk,index” is the “layer” notation used for referring to one of the n coupling layers placed on the surface of the skin (for each “index” from 1 to n), whereas “el,index” is the “layer” notation used for referring to one of the k coupling layers coated on the electrode sensing surface (for each “index” from 1 to k). Following this naming convention, the bottommost coupling layer placed directly on the surface of the skin is layer sk,1 with a thickness d_sk,1, dielectric constant ε_r,sk,1, and bulk DC resistivity ρ_DC,sk,1 $=$ ρ_sk,1, whereas the topmost coupling layer coated directly on the electrode sensing surface is layer el,1 with a thickness d_el,1, dielectric constant ε_r,el,1, and bulk DC resistivity ρ_DC,el,1 $=$ ρ_el,1. The air gap is then denoted by thickness h and dielectric constant ε_r,air. In accordance with the parallel plate approximations for subcase A.1 from Table 1 (3,7), these multiple coupling layers stacked on top of each other can be treated as a series of capacitors C_{coupling,layer} and their shunt leakage resistances R_{coupling,DC,layer} $=$ R_{coupling,layer}. Herein, the notation “layer” in subscripts can be replaced by “sk,index”, “air”, and “el,index”. The equivalent C_coupling of the entire stackup can then be calculated as follows (10) [258]:

$${\displaystyle \frac{1}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}=\sum {\displaystyle \frac{1}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}}}}$$

$$={\displaystyle \frac{1}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{s}\mathrm{k},1}}}+\cdots +{\displaystyle \frac{1}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{s}\mathrm{k},n}}}+{\displaystyle \frac{1}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{a}\mathrm{i}\mathrm{r}}}}+{\displaystyle \frac{1}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{e}\mathrm{l},k}}}+\cdots +{\displaystyle \frac{1}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{e}\mathrm{l},1}}}$$

$$={\displaystyle \frac{1}{{\epsilon }_0{\epsilon }_{\mathrm{r},\mathrm{s}\mathrm{k},1}\frac{A}{d_{\mathrm{s}\mathrm{k},1}}}}+\cdots +{\displaystyle \frac{1}{{\epsilon }_0{\epsilon }_{\mathrm{r},\mathrm{s}\mathrm{k},n}\frac{A}{d_{\mathrm{s}\mathrm{k},n}}}}+{\displaystyle \frac{1}{{\epsilon }_0{\epsilon }_{\mathrm{r},\mathrm{a}\mathrm{i}\mathrm{r}}\frac{A}{h}}}+{\displaystyle \frac{1}{{\epsilon }_0{\epsilon }_{\mathrm{r},\mathrm{e}\mathrm{l},k}\frac{A}{d_{\mathrm{e}\mathrm{l},k}}}}+\cdots +{\displaystyle \frac{1}{{\epsilon }_0{\epsilon }_{\mathrm{r},\mathrm{e}\mathrm{l},1}\frac{A}{d_{\mathrm{e}\mathrm{l},1}}}}$$

$$={\displaystyle \frac{\sum _{i=1}^n\frac{d_{\mathrm{s}\mathrm{k},i}}{{\epsilon }_{\mathrm{r},\mathrm{s}\mathrm{k},i}}+\frac{h}{{\epsilon }_{\mathrm{r},\mathrm{a}\mathrm{i}\mathrm{r}}}+\sum _{j=0}^{k-1}\frac{d_{\mathrm{e}\mathrm{l},k-j}}{{\epsilon }_{\mathrm{r},\mathrm{e}\mathrm{l},k-j}}}{{\epsilon }_{0}A}} .$$

(10)

Conveniently, the equivalent dielectric constant ε_r of the equivalent series capacitance C_coupling can be calculated; ε_r (11) follows directly from the equivalence based on (10):

$${\displaystyle \frac{1}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}={\displaystyle \frac{\sum _{i=1}^n\frac{d_{\mathrm{s}\mathrm{k},i}}{{\epsilon }_{\mathrm{r},\mathrm{s}\mathrm{k},i}}+\frac{h}{{\epsilon }_{\mathrm{r},\mathrm{a}\mathrm{i}\mathrm{r}}}+\sum _{j=0}^{k-1}\frac{d_{\mathrm{e}\mathrm{l},k-j}}{{\epsilon }_{\mathrm{r},\mathrm{e}\mathrm{l},k-j}}}{{\epsilon }_{0}A}}={\displaystyle \frac{1}{{\epsilon }_0{\epsilon }_{\mathrm{r}}\frac{A}{\sum _{i=1}^n{d}_{\mathrm{s}\mathrm{k},i}+h+\sum _{j=0}^{k-1}{d}_{\mathrm{e}\mathrm{l},k-j}}}}$$

$$\to {\epsilon }_{\mathrm{r}}={\displaystyle \frac{\sum _{i=1}^n{d}_{\mathrm{s}\mathrm{k},i}+h+\sum _{j=0}^{k-1}{d}_{\mathrm{e}\mathrm{l},k-j}}{{\sum }_{i=1}^n\frac{d_{\mathrm{s}\mathrm{k},i}}{{\epsilon }_{\mathrm{r},\mathrm{s}\mathrm{k},i}}+\frac{h}{{\epsilon }_{\mathrm{r},\mathrm{a}\mathrm{i}\mathrm{r}}}+{\sum }_{j=0}^{k-1}\frac{d_{\mathrm{e}\mathrm{l},k-j}}{{\epsilon }_{\mathrm{r},\mathrm{e}\mathrm{l},k-j}}}} .$$

(11)

Other methods for estimating the equivalent dielectric constant ε_r can be found in the literature, such as the weighted average often used in PCB manufacturing [290].

Similarly, equivalent series resistance R_coupling (12), along with the equivalent bulk DC resistivity $\rho$ of the entire stackup (13), can be calculated based on (7). In accordance with the results from Table 1, only R_{coupling,DC,layer} components are considered (subcase A.1), and, for simplicity’s sake, they are denoted by R_{coupling,layer}. Similarly, $\rho$_DC,layer components are denoted by $\rho$_layer:

$$R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}=\sum R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}}\left(\omega \right)=\sum R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{D}\mathrm{C},\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}}=\sum R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}}$$

$$=R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{s}\mathrm{k},1}+\cdots +R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{s}\mathrm{k},n}+R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{a}\mathrm{i}\mathrm{r}}+R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{e}\mathrm{l},k}+\cdots +R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{e}\mathrm{l},1}$$

$$={\rho }_{\mathrm{s}\mathrm{k},1}{\displaystyle \frac{d_{\mathrm{s}\mathrm{k},1}}{A}}+\cdots +{\rho }_{\mathrm{s}\mathrm{k},n}{\displaystyle \frac{d_{\mathrm{s}\mathrm{k},n}}{A}}+R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{a}\mathrm{i}\mathrm{r}}+{\rho }_{\mathrm{e}\mathrm{l},k}{\displaystyle \frac{d_{\mathrm{e}\mathrm{l},k}}{A}}+\cdots +{\rho }_{\mathrm{e}\mathrm{l},1}{\displaystyle \frac{d_{\mathrm{e}\mathrm{l},1}}{A}}$$

$$={\displaystyle \frac{\sum _{i=1}^n{\rho }_{\mathrm{s}\mathrm{k},i}{d}_{\mathrm{s}\mathrm{k},i}+R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{a}\mathrm{i}\mathrm{r}}+\sum _{j=0}^{k-1}{\rho }_{\mathrm{e}\mathrm{l},k-j}{d}_{\mathrm{e}\mathrm{l},k-j}}{A}} ,$$

(12)

$${\displaystyle \frac{\sum _{i=1}^n{\rho }_{\mathrm{s}\mathrm{k},i}{d}_{\mathrm{s}\mathrm{k},i}+R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{a}\mathrm{i}\mathrm{r}}+\sum _{j=0}^{k-1}{\rho }_{\mathrm{e}\mathrm{l},k-j}{d}_{\mathrm{e}\mathrm{l},k-j}}{A}}=\rho {\displaystyle \frac{{\sum }_{i=1}^n{d}_{\mathrm{s}\mathrm{k},i}+h+{\sum }_{j=0}^{k-1}{d}_{\mathrm{e}\mathrm{l},k-j}}{A}}$$

$$\to \rho ={\displaystyle \frac{\sum _{i=1}^n{\rho }_{\mathrm{s}\mathrm{k},i}{d}_{\mathrm{s}\mathrm{k},i}+R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g},\mathrm{a}\mathrm{i}\mathrm{r}}+\sum _{j=0}^{k-1}{\rho }_{\mathrm{e}\mathrm{l},k-j}{d}_{\mathrm{e}\mathrm{l},k-j}}{{\sum }_{i=1}^n{d}_{\mathrm{s}\mathrm{k},i}+h+{\sum }_{j=0}^{k-1}{d}_{\mathrm{e}\mathrm{l},k-j}}} .$$

(13)

Naturally, very high leakage resistance of the air gap layer, R_coupling,air, cannot be accurately described with the parallel plate approximation (7); also, transfer of electrons through the air can occur in the event of electrical discharge (refer to Figure 6 in [275], and to [284,291]). Hence, equations for equivalent R_coupling (12) and ρ (13) can be used whenever h $=$ 0.

Non-contact versus capacitive. Now that the electrical model of a single insulating (coupling) layer has been defined (Figure 3) and expanded into the stackup model of the electrode–body interface (Figure 5), the abstracted interface can be further adapted for various types of non-contact and capacitive biopotential electrodes. In that process, care must be taken: not all non-contact electrodes are necessarily capacitive, and not all capacitive electrodes are necessarily non-contact. This is often a source of inconsistencies and misunderstandings in the literature, which presented the motivation for this paper in the first place [96,97,98,99]. Three different cases can be observed: insulated, off-body, and on-body electrodes.

Insulated electrodes. Especially in the early days of non-contact electrodes [242,243,244,245], high coupling capacitance was achieved by coating the sensing surface of a non-insulated dry surface-contact electrode with a few mils thin layer of high dielectric constant (for instance, metal oxides or barium-titanate ceramics). That way, C_{coupling,el,1} in the order of 1 nF and a very weak resistive contact R_{coupling,el,1} higher than 1 TΩ was achieved in series with the stratum corneum impedance [98]. With these values of C_coupling $\approx$ C_{coupling,el,1} and R_coupling $\approx$ R_{coupling,el,1}, predominantly capacitive coupling could have been achieved already at frequencies in the order of 1 mHz (${\displaystyle \frac{10}{{2\pi R}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$, refer to Test 1). Concomitantly, given the high values of the leakage resistance, the coupling layer indeed served as an electrically insulating layer, and the resistive component R_{coupling,el,1} was often omitted in the literature (e.g., Figure 1 in [87]). Since this type of dry electrode exhibits both a high coupling capacitance and a high leakage resistance with the use of an insulating layer coated on the sensing surface, it is more specifically called an insulated or insulating electrode, and often regarded as capacitive even when it is surface-contact, i.e., placed directly on the skin. Moreover, there are even rare examples of immersing insulated electrodes in the electrolytic paste [245]. However, it is safer to treat such cases rather as poor wet surface-contact interfaces. In fact, this is advisable whenever there is a possibility of electrochemical reaction between the electrode sensing surface material and an electrolyte solution such as sweat, as turned out to be the case when anodic aluminum oxide coating was used [242,243,244,245]. To conclude, the case of an insulated electrode, either surface-contact or non-contact, can be considered whenever at least one “el,index” coupling layer is present in Figure 5.

Off-body electrodes. Aside from being insulated or non-insulated, electrodes could also be physically separated and isolated from the body, thereby completely removing both galvanic and mechanical contact [292]. Unlike the on-body electrodes described so far, these electrodes are off-body. In that case, naturally, electrodes are non-contact even if they are non-insulated, and even if the skin is exposed and uncovered. Therefore, this case of off-body electrodes is considered whenever an air gap exists in Figure 5. Detaching the electrode from the body and placing it to a height h > 0 adds the parallel plate capacitor C_coupling,air, which represents the created air gap. Since R_coupling in (12) is now theoretically approaching infinity due to the very high R_coupling,air, off-body electrodes draw virtually no current from the body and, therefore, achieve the purest capacitive coupling that truly operates on small displacement currents only. This entirely capacitive sensing mechanism is based on electric potential sensing (EPS) [9], which expands the utilization of non-contact biopotential electrodes to a more generalized concept of electric potential probes, displacement current sensing, and electric field detectors [88,248,253,254,293]. With recently developed off-body electrodes, researchers carried out biopotential measurements even at a distance of several meters [294]. However, although creation of an air gap may ensure pure capacitive coupling, the dielectric constant of the air layer, ε_r,air, is only about 1, and hence, inclusion of a low capacitance C_coupling,air in series drastically reduces the total equivalent capacitance C_coupling (10). For instance, using a 1 cm² sensing area electrode at a distance of only 1 mm from the body reduces C_coupling to the values of about 1 pF at best (3). Therefore, as concluded in [96,98], firm mechanical contact between the electrode and the body, established without an air gap, is preferred—i.e., on-body electrodes. The same was discussed earlier in the paragraph titled “Model accuracy, fringing fields, and finite models”, considering the validity of parallel plate capacitor approximation.

Non-contact on-body electrodes. Finally, the third option is a compromise: insulated or not, dry electrodes could be placed on the skin indirectly—most commonly over a layer of fabric or cloth (e.g., over a shirt or under the bed sheets). Therefore, in accordance with Figure 5, this case can be considered whenever there is no air gap (h $=$ 0, on-body electrodes) and whenever there is at least one “sk,index” layer present (insulated or non-insulated, non-contact electrodes). However, these conditions do not necessarily make the electrodes capacitive. Unlike the case of off-body electrodes (h > 0), in most of practical cases of on-body biopotential measurements, the concept of the capacitive electrode is an oversimplification, especially when the electrode is non-insulated (no “el,index” layers present). Namely, as discussed earlier in this section (Figure 3), there is no perfect insulator; hence, some finite leakage current always flows through the interface, creating a conductive path [103,104,249,260]. This is especially the case when a layer of clothing material is used (wearable non-contact on-body electrodes), since such wearable coupling layers often comprise a fairly low resistive component R_{coupling,sk,index} in the order of 100 MΩ or even lower [92,98], allowing a significant faradaic reaction at the interface. Moreover, if the fabric is hygroscopic (moisture absorbent), the corresponding coupling impedance Z_{coupling,sk,index} could be further reduced in the presence of moisture content. For instance, the cotton sample provided in [92], as the only layer covering the skin, exhibited a dielectric constant ε_r,sk,1 of about 3 and R_{coupling,sk,1} higher than 40 MΩ. The electrode used was non-insulated; hence, R_{coupling,sk,1} $\approx$ R_coupling, X_{Ccoupling,sk,1} $\approx$ X_Ccoupling, ε_r,sk,1 $\approx$ ε_r, and d_sk,1 $\approx$ d. In accordance with (3) and Test 1, for the used 510 mm² of non-insulated electrode sensing area A, even if the dried-up cotton sample provided 10 GΩ of R_coupling, it should still not be thicker than about 4 μm (14) for the condition R_coupling >> X_Ccoupling to be satisfied across the entire clinical ECG frequency range (lowest frequency not higher than 0.05 Hz in accordance with Assumption 1):

$$R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\gg X_{\mathrm{C}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\to R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\ge 10{\displaystyle \frac{1}{\omega C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$$

$$\to 10 \mathrm{G}\mathsf{\Omega }\ge {\displaystyle \frac{10d}{2\pi ·0.05 \mathrm{H}\mathrm{z}·{{\epsilon }_0\epsilon }_{\mathrm{r}}A}} \to d\le 4 \mathsf{\mu }\mathrm{m}.$$

(14)

Therefore, in practice, the resulting coupling in the case of on-body electrodes is seldom predominantly capacitive across the entire frequency range of the measured biopotential signals. Nevertheless, coupling capacitances C_{coupling,sk,index} achieved with wearable fabrics (ε_r,sk,index usually in the range of 1–4 depending on the material and manufacturing process [92,135]) are still higher than the values of total coupling capacitances C_coupling achieved with off-body electrodes (order of 10 pF compared to the order of 100 fF or 1 pF at best). For a given fabric material, C_{coupling,sk,index} can be further increased if a thinner layer is used (3,14).

Important conclusions and misterming issues as a motivation for the classification of biopotential electrodes. In the existing literature, all the discussed cases can be found under the terms non-contact, contactless, indirect-contact, and capacitive electrodes. These electrodes are also referred to as non-intrusive, non-obtrusive, or unobtrusive. However, in the previous paragraphs it has been shown that all these terms are not always interchangeable. For instance, the last three terms regarding intrusiveness are broader and describe any sort of unobtrusively applied electrodes, which might consider not only off-body, non-contact on-body, and insulated dry surface-contact electrodes, but also, e.g., non-insulated dry surface-contact electrodes embedded in clothing. Furthermore, in terms of the coupling mechanism, wet surface-contact electrodes described in Appendix B could in fact achieve a notably higher C_double-layer (order of 10 nFcm⁻²) than the C_coupling achievable with common dry electrodes intended to be used capacitively, but that still does not make their coupling capacitive. Conversely, while off-body electrodes achieve the purest capacitive coupling, their C_coupling is reduced to the limits of usability for biopotential measurements [96,98]. On the other hand, whenever a mechanical contact without an air gap is established with the body (on-body electrodes), some amount of leakage current will always flow through the electrode–body interface—especially in the case of using clothing wearable layers. In other words, coating the sensing surface of a dry on-body electrode or separating it from the skin with an on-skin insulating layer does not guarantee that there will be no resistive coupling across the inspected frequency range. For instance, with the use of silicon dioxide coating in [295], C_coupling in the order of 1 nF and higher was achieved, whilst R_coupling might have been in the order of 100 MΩ. Yet, in accordance with Test 1 and (14), even for such a low R_coupling, coupling would become predominantly capacitive already at frequencies in the order of 1 Hz. Moreover, additional effects, such as the aforementioned moisturization and humidification, can easily make an on-skin insulating layer conductive. In that case, a layer would remain insulating only in terms of physically separating the on-body electrode from the skin, rather than providing electrical insulation. Similarly, while insulated surface-contact electrodes could be treated as both electrically insulating and capacitive as long as there is no electrochemical reaction at the interface, their coupling conditions might easily worsen once they are applied over a layer of fabric or separated via an air gap. Namely, while they could still be treated as insulated due to preserved high R_coupling (12), the total C_coupling would become dominated by the lowest of all individual layer capacitances, C_{coupling,layer} (10). Due to all these considerations, the initially used term “insulating layer” has been broadened and replaced with the term “coupling layer”. Just as a combination of faradaic and non-faradaic processes is always present at the interface with a non-insulated surface-contact electrode (Figure A2), so can the coupling with a non-contact on-body electrode always be described as a combination of capacitive and resistive coupling, rather than as pure capacitive coupling. Accordingly, a more reliable representation of the impedance of each on-skin (“sk,index”) and on-electrode (“el,index”) coupling layer would be the coupling capacitance C_{coupling,layer} (e.g., C_{coupling,sk,1} or C_{coupling,el,1}) shunted by the insulation or leakage resistance R_{coupling,layer} (e.g., R_{coupling,sk,1} or R_{coupling,el,1}), as in Figure 5. Additionally, and especially in the case of fabric layers, some papers [66,296] add a separate resistance R_{contact,sk,index} in series with the R_{coupling,sk,index}C_{coupling,sk,index}-parallel as a reminder of non-zero conductivity at the boundaries with adjacent conductive surfaces. To conclude, instead of treating electrodes as pure resistive or pure capacitive, they should rather be treated as predominantly resistive or predominantly capacitive across a certain frequency range. Moreover, contact with the surface of the skin does not guarantee predominantly resistive coupling, nor does placing the electrode over a layer of fabric or coating its sensing surface guarantee predominantly capacitive coupling. In fact, it is almost always more appropriate to treat non-contact on-body electrodes as dry surface-contact electrodes with a higher leakage resistance. Similarly, it is almost never advisable to treat non-contact biopotential measurements as capacitive (e.g., it is more appropriate to use the term non-contact ECG rather than capacitive ECG). In accordance with these issues, it becomes clear that using the group term “non-contact and capacitive electrodes” to describe electrodes alternative to conventional wet and dry surface-contact electrodes is not entirely appropriate. This will be further discussed in the next paragraph.

Proposal for the classification of biopotential electrodes. To avoid misterming in the future, a categorization of biopotential electrodes is proposed in Figure 6 based on their distance from the body, type of contact, and invasiveness, along with an assessment of their predominant coupling mechanism. In the bottom row of the classification tree, red, sharp diamond blocks represent invasive electrodes, whereas the rest of the electrodes are non-invasive, meaning that they do not require bypassing the stratum corneum or skin protrusion of any kind. Among them, blue octagons represent non-insulated surface-contact electrodes, which are described in Appendix B and which rely mostly on the faradaic process and resistive coupling with the skin. On the other hand, green ellipsoidal blocks represent various types of dry electrodes, in which the influence of resistive and capacitive coupling depends on the electrical and dielectric properties of the electrode–body interface. All five proposed green ellipsoidal cases can be gathered under the term “non-contact and insulated electrodes” as a more appropriate replacement for the previous group term “non-contact and capacitive electrodes”. To emphasize the strength of capacitive coupling in comparison with resistive coupling for each of these cases, stronger capacitive coupling is depicted by a darker shade of green. In that sense, cases 4 and 5 (true non-contact, off-body electrodes) are the ones with the most capacitive coupling and the least resistive coupling. With respect to Figure 5, these cases would correspond to the presence of an air gap (h > 0). On the other side, cases 1, 2, and 3 correspond to on-body electrodes (no air gap, h $=$ 0). The presence of at least one layer on the surface of the skin (sk,1) is needed to consider the electrode as non-contact, whereas the presence of at least one layer coated on the sensing surface (el,1) is needed to consider the electrode as insulated. However, as mentioned in the paragraph titled “Insulated electrodes”, care must be taken: since wet electrodes are not characterized by the presence of the electrolytic solution itself, but rather by the electrochemical reaction between the electrode material and the solution, an electrochemical reaction with solutions such as sweat could effectively turn any non-insulated or even insulated dry on-body electrode into a poor surface-contact electrode. In any case, based on (3), the total coupling capacitance C_coupling can be approximated with a series of parallel plate capacitors, as in (10,11). Similarly, based on (7), the total resistance R_coupling can be approximated with (12,13). The terminology presented in Figure 6 will be used throughout the rest of this paper for identifying and comparing different categories of electrodes: wet surface-contact and non-insulated dry surface-contact electrodes described in Appendix B on the one side, and off-body, non-contact on-body, and insulated dry surface-contact electrodes on the other side. This latter group of electrodes will be referred to as non-contact and insulated electrodes. In the context of non-contact and insulated electrodes, the total coupling impedance Z_coupling of the coupling path, i.e., the impedance between the electrode sensing surface and the surface of the skin, is often called the electrode–body impedance (EBI). Finally, Assumption 4 can be expressed.

Assumption 4.

Electrode–body interface, coupling layers, and coupling impedance. In accordance with the discussion laid out in this section and the proposal for revised classification given in Figure 6, “non-contact and capacitive electrodes” (or occasionally just “non-contact electrodes” for simplicity’s sake) will from now on be more precisely referred to as “non-contact and insulated electrodes”. This term will consider off-body, non-contact on-body, and insulated dry surface-contact electrodes, with a caveat that on-body electrodes have proved to be preferred. The term “coupling impedance” will be used for describing the total equivalent impedance of the coupling path between the electrode sensing surface and the surface of the skin, Z_coupling (Figure 5). In other words, internal body and tissue impedance, as well as impedance of the stratum corneum, are excluded from the coupling impedance and will be denoted by a separate equivalent skin–tissue impedance Z_{body & skin}. This will allow the employment of the voltage measured at the surface of the skin, v_surface(t), to be the starting point for circuit analysis. Specifically, in the case of wet and non-insulated dry surface-contact electrodes (Appendix B), the term “coupling impedance” would correspond to the total impedance of the electrode–hair–electrolyte–skin path (electrode–skin path for short), comprising both the electrode or electrode–electrolyte impedance and the contact or electrolyte impedance. To make a distinction between wet and non-insulated dry surface-contact electrodes on the one side (Figure A2), and non-contact and insulated electrodes on the other side (Figure 5), and allow easier comparison with the existing papers, the “electrode–skin” term will be primarily used for addressing the coupling with the former, whereas a more general term “electrode–body” will be primarily used to stress the coupling with the latter. Similarly, terms such as “contact surface” and “contact area” will be primarily used to address the sensing surface of the former, whereas terms such as “coupling surface” and “coupling area” will be primarily used to address the sensing surface of the latter. Therefore, the coupling impedance Z_coupling of wet and non-insulated dry surface-contact electrodes will also be called the electrode–skin impedance (ESI), whereas Z_coupling of non-contact and insulated electrodes will also be called the electrode–body impedance (EBI). Accordingly, the electrode–tissue impedance (ETI) [176] would denote the equivalent impedance that accounts for both the skin–tissue impedance Z_{body & skin} and the coupling impedance Z_coupling. Further, insulation, leakage, or coupling resistance as the real (resistive) part of Z_coupling shall be denoted by R_coupling, whereas the coupling capacitance and the imaginary (reactive) part of Z_coupling shall be denoted by C_coupling and X_Ccoupling, respectively. The relation between C_coupling and X_Ccoupling is expressed in (A4). In total, Z_coupling = Z_Ccoupling||R_coupling = (−jX_Ccoupling)||R_coupling. Accordingly, each layer that builds the electrode–body interface represents one coupling layer. The term “coupling layer” is preferred to the term “insulating layer” to remind that coupling layers are not necessarily insulating from the standpoint of electrical conductivity. For each of the coupling layers, Assumptions 2 and 3 are applicable, and the corresponding coupling layer impedance is Z_{coupling,layer} = Z_{Ccoupling,layer}||R_{coupling,layer} = (−jX_{Ccoupling,layer})||R_{coupling,layer}, where the subscript “layer” can stand for “sk,index”, “air”, and “el,index”. In the case of air gap, only the parameter C_coupling,air is considered. This notation is described in more detail in paragraph “Multiple insulating or coupling layers, equivalent coupling capacitance, and equivalent coupling resistance.” Based on parallel plate approximations for each C_{coupling,layer}, Equations (3), (10) and (11) will be used to describe the equivalent C_coupling and ε_r of the entire stackup. On the other hand, based on the analysis given in Table 1, only the DC component of the leakage resistance is considered in (9) (subcase A.1). Therefore, R_{coupling,DC,layer} = R_{coupling,layer}, ρ_DC,layer = ρ_layer, and so Equations (7), (12) and (13) will be used to describe the equivalent R_coupling and ρ of the stackup. Finally, the predominant coupling mechanism will be determined in accordance with Test 1. Test factor T = 10 will continue to be used, as well as “>>” and “<<” conditions that indicate “at least an order of difference”. Practical examples are given in (14) and the paragraph titled “Important conclusions and misterming issues as a motivation for the classification of biopotential electrodes.”

Examples of measurements and motivation for further research on non-contact and insulated biopotential electrodes. Through contactless and remote sensing properties independent of the skin, non-contact and insulated electrodes offer maximization of user safety, at the same time minimizing preparation, intrusiveness, and subjects’ anxiety, as well as the possibility of irritation and allergic reaction. All these advantages show that with the use of a non-contact and insulated measurement approach, we are one step closer to out-of-hospital unobtrusive biomonitoring in a day-to-day environment. Nevertheless, the challenges of low coupling capacitances and absence of direct contact with skin remain to be tackled. The resulting differences between signals measured with non-contact and insulated electrodes on the one side, and surface-contact electrodes on the other side, can be significant (Figure 7) [32,297]. Other interesting examples of comparison in performance and recorded biopotential signals are available in the literature—e.g., comparison between wet, dry, and insulated surface-contact electrodes [197], comparison between all types of non-invasive electrodes [121,122,123], comparison between non-contact on-body electrodes placed directly on the skin and over fabric (Figure 3.5 in [103]), comparison between wet surface-contact electrodes and non-contact on-body electrodes placed over various fabric material [32,297,298], and comparison between surface-contact electrodes and off-body electrodes at various distances [9]. In this study, the phenomena behind these results will be addressed through a thorough system analysis of non-contact and insulated electrode–body interfaces, as well as by demystifying the key requirements of their design. As a common application example, surface-contact ECG and non-contact ECG (sometimes ill-advisedly referred to as capacitive ECG or cECG) measurements will be used. More on noise sources in ECG signals, as well as on the properties and frequency spectrum of ECG signals, can be found in [152,153,154].

Fundamental literature overview. Throughout the following sections, fundamentals of passive components and capacitor working principle will be extensively used. The literature on this fundamental knowledge can be categorized as follows:

• Fundamentals of dielectrics, polarization, and capacitors: insulators and dielectrics on a quantum level in [256], and polarization effect in [255,256,257,258,260,284]; extensive discussion on electrical, mechanical, and thermal properties of dielectrics is given in [257]; physical basis of capacitance is given in [256,258,259,263,285] and further extended to linear and non-linear capacitor models and their properties in [264], and applications of capacitor circuits in [261,264]; more on properties and manufacturing of real-world capacitors is given in [260,261] and subchapter 1x.3 in [282]; an overview of historical development of capacitive sensing is given in [10].
• Considerations for higher frequencies, such as transmission line effects, controlled impedance, and dielectric loss are given in [257,283,289,299,300,301]; more on dielectric loss specifically can be found in [256,260,270,283,289] and subchapters 7.1 and 7.8 in [257].
• Considerations for higher voltages, electrical discharge, electrical breakdown, and dielectric failure mechanisms can be found in [255,256,275,286] and subchapter 7.10 in [257].
• Physical basis of resistance [287] and inductance [302]; therein presented fundamentals, along with references on physical basis of capacitance listed in the previous bullets, can be extended to discussions on modeling PCB parasitic elements ([274] and subchapter 1x.1 in [282]), package lead parasitics [303], and on-chip interconnects parasitics [304], as well as equivalent circuits of capacitors, resistors, inductors, and diodes [260,305,306,307,308]. More on manufacturing of real-world passive components and diodes along with their non-idealities can be found in [282,305,306,307], with a focus on capacitors in [260] and subchapter 1x.3 in [282], and with a focus on resistors in [288] and subchapter 1x.2 in [282].
• Useful approximations for capacitance, resistance, and inductance of various practical geometric configurations can be found in [258,259,272,274,287,302,306]. Incidentally, an example of theoretical modeling of wire inductance and capacitance between the wire and the ground can be found in [280];
• Comparison between working principles of capacitors and inductors [264,265,309] that will provide a further insight into analogies between capacitive and inductive coupling mechanisms.
• Basics of phasor analysis [260,265,266,267,268].
• Guide to understanding Maxwell’s equations [262,263,285,289,291]; more on electric field calculation and numerical simulations can be found in [255,258,275,276,289,291].
• Finite parallel plate capacitor models [15,277,278,279,280,281].
• Various important rules and useful formulae for board-level and system-level design are highlighted in [273] and will be occasionally referenced. Various PCB design issues are covered in [274,306], chapter 3 in [104], and subchapter 1x.1 in [282]; specifically, more on power supply noise and decoupling is given in [310].
• With the addition of the literature on operational amplifiers and systems theory that will be listed throughout Section 3.2 and Section 4.1, all these references will serve as the basis for the rest of the assumptions.

3.2. Passive and Active Design

Single-ended and differential measurement. Throughout the following analysis of a single non-contact and/or insulated biopotential electrode, signals will be considered as single-ended and referenced to the circuit common. However, in reality, one must be aware that at least two electrodes are required for an ECG measurement [98]. The obtained ECG signal can be either single-ended (unipolar, monopolar), meaning that one electrode measures the ECG signal, whereas the other serves as a reference node with assumed zero electric potential (e.g., circuit common), or it can be differential (also known as single-differential or bipolar), meaning that both electrodes measure the ECG signal. Also, hybrid configurations are possible, such as the pseudo-differential configuration [102]. In any case, the obtained electric potential difference between the two electrodes represents one lead or channel. More on unipolar and bipolar leads can be found in [152,311].

Passive design and the main amplifier. In a classic two-electrode biopotential measurement (one-channel ECG), the input signal is differential, and the most convenient approach to achieve signal acquisition and convert that differential signal into a single-ended output signal is to use either a differential-input amplifier or an instrumentation amplifier (Figure 8) that offers improvements in terms of higher input impedance, lower sensitivity to component tolerances, and easier gain adjustment [102]. The input non-inverting amplifiers in the first stage of the instrumentation amplifier are usually coupled or, less frequently, non-coupled. Alternatively, if a differential output is desired instead of a single-ended output, fully differential amplifiers with a differential input and a differential output can be used [312,313,314]. More on various voltage amplifier configurations is given in [315]. Because there are no active electronics mounted on the electrode and applied to the signal path between the electrode sensing surface and the input of the main amplifier, such a design is referred to as passive.

Buffer and amplifying active electrodes. Passive design is prevalent in the case of wet surface-contact electrodes, but rather rare in the realm of non-contact and insulated electrodes (some examples of passive non-contact design are given in [316,317]). Namely, input of a high-coupling impedance electrode, just like any high-impedance node, is susceptible to electric coupling with the environment, and thus behaves like a charge-sensitive electrometer due to a lack of low-impedance path to a reference potential (refer to Figures 2–9 in [318], and to [110,274,314]). Therefore, the high coupling impedance of dry electrodes leaves the signal path more vulnerable to environmental noise. This results in attenuation of the signal and in a significantly lower signal-to-noise ratio (SNR). To counteract this issue, the first amplifying stage can be implemented directly on the electrode—most commonly as an additional voltage buffer. While a single-transistor source follower was typical in the early days [242,243,244,245,319], such buffer electrodes are today mostly realized with an operational amplifier in a voltage follower configuration, comprising a negative voltage feedback and a unity closed-loop gain. Since this amplifier precedes the main amplifier (differential or instrumentation, as described in the last paragraph), it is also called a preamplifier. One of the first notable papers describing the importance of such a buffer preamplifier implementation was [319]. Serving as impedance converters, these preamplifiers convert the high input impedance into low output impedance, reducing the noise pickup and signal loss. This allows for a longer wired signal transmission path toward the data acquisition (DAQ) system and its main amplifier (Figure 8). Alternatively, it also enables the option of wireless transmission in the case of a more complex design with a dedicated microprocessor that enables wireless connectivity. However, this comes at the cost of additional wires required for powering the preamplifiers, which is why this design is called active (Figure 9). Usually, the sensing surface is implemented on the bottom layer of the active electrode PCB and connected through vias with the top layer of the PCB, where the analog front-end with the preamplifier is implemented. Such an active approach to combating the high coupling impedance is not limited to non-contact and insulated electrodes—on the contrary, there are many examples of using it for non-insulated dry surface-contact electrodes [218,228,320], and even examples of using it for wet surface-contact electrodes [321]. Aside from the described buffer active electrodes, which use unity voltage gain preamplifiers, active electrodes can also be amplifying or preamplified. In this case, the electrode preamplifiers employ voltage gains higher than unity and, therefore, entirely replace the first stage of the instrumentation amplifier, allowing for a reduction in power consumption and the number of necessary operational amplifiers [102,203,210,322,323]. However, this increases sensitivity to component tolerances, making the possible gain mismatch between the electrodes more problematic. In addition, accounting for the possibility of a DC electrode offset at the input may demand considerable gain limitation. In spite of gain adaptation and DC offset compensation solutions that address these challenges of amplifying active electrodes [322,323], buffer active electrodes with unity-gain voltage preamplifiers still prevail as the most widely used method of signal acquisition in non-contact and insulated biopotential measurements. Further developed from papers such as [319] through fundamental electric potential sensing (EPS) research of the Sussex research group [88,246,247,248,249,250,251,252,253,254], such an approach has remained a standard to this day. Therefore, buffer active electrodes will be further investigated in the rest of the paper. More on the performance and architecture of various active electrodes can be found in [139]. These solutions will be revisited at the end of Section 4.2.

Transimpedance and charge amplifiers as an alternative. Specifically, for cases of low-capacitance coupling (C_coupling in the order of 1 pF or lower) or underwater environments, where impedance of the electrode–body interface Z_coupling is shunted by the low impedance of the water medium [41,230,289], trans-impedance (current-to-voltage) and charge (charge-to-voltage) amplifiers are a promising alternative [9,293,294,324,325,326,327,328].

Active versus passive with respect to the proposed categorization. In conclusion, non-contact and insulated biopotential electrodes are commonly active and, more specifically, designed in a buffer configuration, whereas non-insulated surface-contact biopotential electrodes (especially the wet ones) are commonly passive. However, previous paragraphs have shown that active design does not imply a contactless biopotential measurement method, nor does passive design imply a surface-contact biopotential measurement method. Examples of measurement and comparison between active and passive approaches can be found in [110,329]; also, refer to Figure 1 in [197] and Figures 6 and 7 in [228].

Assumption 5.

Buffer active non-contact and/or insulated electrode. In Section 3.1, the equivalent electrical model of the interface between the body and a non-contact and insulated biopotential electrode was developed (Figure 5 and Assumption 4). In Section 4, it will be combined with the model of a buffer preamplifier from the appendix, cross-referenced at the beginning of Section 4.1, to analyze a single buffer active non-contact and/or insulated electrode and its interface with the body. Throughout the paper, the terms “voltage follower” and “buffer” will be used interchangeably. The entire system (interface–electrode–preamplifier) will be denoted by the subscript “el” and separated into two subsystems: the input voltage divider (subscript “div”) and the buffer preamplifier (subscript “preamp”), with individual equivalent electrical parameters defined in the first pages of Section 4.1. Details on variable naming is given in Assumption 6. The analysis is applicable both to non-contact and to insulated active electrodes. For this reason, the “and/or” conjunction is used to emphasize that the generalized non-contact and insulated electrode–body interface can be non-contact only, insulated only, and both non-contact and insulated. In addition, depending on the validity of assumptions on the operational amplifier, which will be summarized in Section 4.1, and depending on the interpretation of the term “coupling impedance” with respect to the type of the electrode (Assumption 4), the analysis is further applicable to any given non-invasive active biopotential electrode. The electrode will be employed for ECG measurements, with its input denoted by v_in and output denoted by v_out. In accordance with Assumption 4, voltage v_surface, that would have been measured with an ideal equivalent passive non-insulated dry surface-contact electrode at the surface of the skin, will be the starting point for circuit analysis. For the purpose of achieving a gradually complexified analysis throughout the study, electric potential difference between the floating body potential (bar symbol) and the circuit common (triangle symbol) will be neglected for the rest of this paper. Therefore, voltages v_surface, v_in, and v_out will be referenced to the circuit common. Similarly, zero initial conditions will be considered.

Assumption 6.

Variable naming. In general, the following convention will be used for naming currents and voltages: lowercase letters will indicate time-domain variables with the argument (t). On the other hand, uppercase letters will be used to denote frequency-domain variables with either argument (s) as the complex frequency s = σ + jω that will indicate the Laplace transform, or argument (jω) or (jf) in accordance with (A2). Likewise, unless used in the Laplace s-domain, impedances Z are assumed to be functions of (jω) or (jf). Accordingly, reactances X are functions of (ω) or (f) (A4). Specifically, $\dot{V}{e}^{j\omega t}$ and $\dot{I}{e}^{j\omega t}$ represent rotating phasors as analytic representations of voltage and current with their static parts or phasors $\dot{V}$ and $\dot{I}$, respectively (A1). In addition, uppercase letters will also be used to denote any type of quantity that is assumed to be constant or fixed within the observed time frame, such as power supply voltages and mean average values.

4. System Analysis

4.1. System Transfer Function and Impulse Response

Introduction to Section 4 and the first subsystem: operational amplifier. In accordance with Assumption 5, in Section 4, a buffer active non-contact and/or insulated biopotential electrode and its interface with the body will be modeled and analyzed. The model is based on two subsystems: the generalized model of non-contact and insulated electrode–body interfaces, presented in Section 3.1, and a buffer preamplifier, presented in Appendix D. Therein, fundamentals of operational amplifiers, negative feedback, Bode plot, and amplifier stability are recalled, with a focus on the voltage follower (buffer) configuration. By beginning this section with Appendix D, analysis of the system conveniently starts from the voltage-feedback operational amplifier, which is then configured as a voltage follower and mounted as a buffer preamplifier on a passive non-contact and insulated biopotential electrode presented in Figure 5. This way, a buffer active non-contact and/or insulated electrode is created. Analysis in Appendix D is based on several book chapters and application reports, discussing fundamentals of operational amplifiers and their specifications [330,331,332,333,334], as well as on the previously listed fundamental literature (paragraph “Fundamental literature overview”). Useful calculations are summarized in [273]. In addition, detailed gain analysis of various practical voltage amplifiers is given in [315]. Also, steps will occasionally be supported by a compendium of interesting blog topics on operational amplifier design given in [335]. The analysis will also be corroborated by several examples of amplifiers commonly used as preamplifiers for buffer active electrodes: LMP7721 [336] and OPA129 [337] (Texas Instruments, Inc., Dallas, TX, USA), and AD8641 [338] (Analog Devices, Inc., Wilmington, MA, USA). After introducing references [339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373], Equations (A6)–(A21), and Figure A5, Figure A6, Figure A7, Figure A8, Figure A9 and Figure A10, the provided recollection of the working principles of operational amplifiers and, specifically, buffer amplifiers, leads from Appendix D to Assumptions 7 and 8. Therein, the key assumptions on the operational amplifier, used as a buffer preamplifier, and the preamplifier analog front-end are highlighted, along with PCB design considerations. Afterward, the derived first-order (single-pole) model of a buffer preamplifier A_diff(f) (A17)–(A19) from Appendix D will be combined with the generalized model of non-contact and insulated electrode–body interfaces from Section 3.1. The resulting system (interface–electrode–preamplifier) will represent the model of a buffer active non-contact and/or insulated electrode and its interface with the body. Thereafter, the last assumption, Assumption 9, will be expressed, and the rest of this section will analyze the transfer function and impulse response of the entire system. Along the way, additional useful literature on theory of systems and their stability will be provided. Specifically, stability of operational amplifiers, discussed in Appendix D, will be revisited later in Section 4.3. The next paragraph is written as a continuation of Appendix D.

In summary, in Appendix D, the departure from the ideal operational amplifier model was achieved in several aspects: finite input impedance, non-zero output impedance, and finite bandwidth (for more on the ideal operational amplifier model, refer to [330,331,332]). These small-signal dynamic limitations of operational amplifiers, along with large-signal dynamic limitations, such as a finite slew rate [331,332,335,344], will be surveyed in the following Assumption 7. Additionally, static limitations, representing various input-referred DC errors, rejection capabilities, and operating limits, will also be addressed, along with the departure from the rest of the ideal characteristics—zero bias currents, zero input offset voltage and zero input offset current, and absence of noise. More on static limitations and error modeling can be found in [315,330,332,333,335,374,375]. The following assumption on the buffer preamplifier, along with assumptions on parasitic elements expressed in Assumption 8, will be used in the rest of this paper.

Assumption 7.

Operational amplifier employed as a buffer preamplifier.

• (I) For the given amplitudes of interest (Assumption 1), the operational amplifier is considered to operate in the linear, small-signal regime. In other words, analysis considers a small-amplitude, time-varying AC biopotential signal superimposed to any DC voltage present at the interface.
• (II) Open-loop common-mode voltage gain a_com is neglected in comparison with the open-loop differential-mode voltage gain a_diff, i.e., a ≈ a_diff and v_{out,open-loop} ≈ a_diffv_diff (A8). Also, a_diff is sufficiently high at frequencies of interest (at least 90 dB [336,337,338]) that the virtual short concept can be adopted and that the resulting closed-loop differential-mode voltage gain A_diff at frequencies of interest depends only on the gain of the external feedback path β. In other words, Equations (A17) and (A18) can be used. The operational amplifier is assumed to be internally compensated and modeled with a first-order system: open-loop gain a_diff(f) (A14) and closed-loop gain A_diff(f) (A19) with a unity closed-loop gain A₀ = 1 V/V = 1 (0 dB) for a voltage follower (buffer) configuration. Given the typical gain-bandwidth product (GBP) of 1 MHz [336,337,338], the influence of capacitance at the inverting input and at the output will be neglected (A21) and revisited in Section 4.3. On the other hand, the capacitance at the non-inverting input will be introduced in the subsequent pages.
• (III) Given that the implementation of a series–parallel negative feedback increases Z_diff, the relation Z_diff >> Z_com−,Z_com+ will be assumed. Similarly, Z_out is decreased by the series–parallel negative feedback and will be neglected [331,333,334,340,342]. Further, Z_com− and Z_com+ are considered to be approximately equal (matched, balanced). Since the negative feedback path is short-circuited (R_f = 0 Ω) and since the signals source is present at the non-inverting input, for practical considerations of the equivalent electrical circuit at the buffer preamplifier input, only Z_com+ will be of interest and, from now on, named Z_in [332].
• (IV) Furthermore, internal errors of the operational amplifier and effects of its native non-linearities, significantly reduced by the presence of negative feedback, will not be considered.
• (V) Similarly, the influence of power supply fluctuations on output voltage and their finite rejection, expressed by the power supply rejection ratio (PSRR) [315,330,332,375], are neglected. In general, the implementation and intricacies of amplifier power supply are not considered [333,376], and in addition, proper power supply decoupling, along with proper biasing of internal circuitry, is assumed (refer to Section 27 in [335], and to [310]).
• (VI) Also, manufacturer tolerances and mismatch in components and input stages, as well as asymmetry in the signal path, are neglected. Therefore, output offset voltage or zero error, as the output voltage present when input terminals are short-circuited and grounded, and input offset current, as the difference between bias currents I₊ and I₋ (Figure A8) [315,330,332,333,335,374,375], are neglected. Just like DC and slow-fluctuating errors arising from the electrode–body interface, such as the half-cell potential, these DC limitations and offset errors are in practice mitigated through various procedures of correction, filtering, cancelation, and calibration, more on which can be found in [332,333,335,374,375]. For instance, instead of an input series blocking capacitor, DC rejection can be achieved with the use of a DC servo loop that allows for automatic control of amplifier output offset [314,377,378]. Specifically, the influence of non-zero bias currents I₊ and I₋ will be addressed in the subsequent pages in the context of DC biasing circuitry.
• (VII) Next, constant ambient conditions and constant ambient temperature are assumed (Assumption 2). Hence, output voltage drift due to the temperature dependence of reverse-biased leakage of p-n junctions, as well as due to the temperature dependence of amplifier input offset voltage and input offset current, is not considered. Similarly, drift with time due to aging is not considered.
• (VIII) Finally, AC and large-signal limitations can also be considered—more specifically, slew rate and swing limitations. Although the addition of a capacitance for internal compensation severely impacted the slew rate, and although lower amplifier consumption usually draws lower slew rates, commonly used amplifiers achieve slew rates of at least 2 V/μs [336,337,338]. With these values of slew rate, even a 10 V amplitude sine wave would yield a full power bandwidth of at least 31.8 kHz (${\displaystyle \frac{2 V/\mu s}{2\pi ·10 V}}$in accordance with the magnitude of the time derivative of a sinusoid; refer to subchapter 2.2 in [331], and [273,332,335,344,379]). Therefore, slew rates of off-the-shelf amplifiers commonly employed for biopotential monitoring are more than sufficient for most biopotential signals. In other words, rate of change in output voltage is fast enough to track the input voltage. Hence, the slew rate-induced distortion is not considered, and the small-signal bandwidth, described in Appendix D, is rather limited by the input and output swing limitations [330,332,375]. Given the amplitudes of biopotential signals (Assumption 1), clipping distortion (i.e., truncation of signal peaks) as a result of exceeding the swing specifications will not be considered for the system analysis.
• (IX) In conclusion, the established assumptions are in accordance with similar analyses, such as [21,32,38,95,380].

Assumption 8.

Parasitic elements.

• (I) Given the frequencies of interest (Assumption 1), transmission line effects will be neglected.
• (II) Furthermore, PCB parasitics can be considered. Firstly, for a typical FR-4 PCB at room temperature, given that a 1 oz (35 microns) thick copper trace exhibits about 0.5 mOhm per square area, unit-length resistance of a PCB copper trace achieves orders of 10–100 mOhms/cm [273,287]. Secondly, PCB trace inductance per unit length is in the order of 10 nH/cm, whereas via inductance achieves values of about 1–2 nH [273,302], which is negligible at frequencies of interest (Assumption 1). Thirdly, unit-length stray capacitance between adjacent copper traces on the same layer can reach the order of 100 fF/cm for very small distances of 0.1 mm, which is negligible even for fairly long coupling lengths [259,273]. Therefore, PCB trace resistance and inductance, as well as stray capacitance between adjacent traces on the same layer, will be neglected and considered minimized with proper PCB stack-up and PCB layout (refer to paragraph “Fundamental literature overview” at the end of Section 3.1 for details). Likewise, in the case of using e-textile devices, the parasitic capacitance of conductive yarns that is usually in the order of 1 fF/cm would be neglected at frequencies of interest [381].
• (III) Aside from PCB and e-textile parasitics, parasitic elements of the operational amplifier (buffer preamplifier) can also be considered. Firstly, in accordance with estimation (A21), capacitance present at the inverting input terminal and output capacitance will not be considered. Their influence, elaborated in Appendix D, will be appropriately revisited in Section 4.3. On the other hand, the paragraph “Parasitic input capacitance” in Section 4.1. will introduce the parasitic capacitance present at the non-inverting input terminal. Secondly, considering the frequencies of interest (Assumption 1), amplifier input inductance [352], as well as parasitic resistance and inductance of package leads [303] and on-chip interconnects [304], will be neglected.
• (IV) Lastly, non-idealities and parasitic elements of any additional discrete component are neglected at frequencies of interest (Assumption 1); hence, equivalent electrical models for components such as resistors, inductors, ceramic capacitors, electrolytic capacitors, and diodes are not considered. More on equivalent electrical circuits and non-idealities of various components can be found in the paragraph “Fundamental literature overview” in Section 3.1.

Model of a buffer active non-contact and/or insulated electrode and its interface with the body with three equivalent impedances. In Section 3.1, an equivalent electrical model of the interface between the body and a non-contact and insulated biopotential electrode was built and generalized (Figure 5) in accordance with Assumption 4. In Section 3.2, a preamplifier was added to the passive electrode, and the electrode became active (Figure 9). In Appendix D, the buffer preamplifier was modeled (Figure A8 and Figure A10), which led to Assumption 7. Now, the model of the electrode–body interface can be extended toward the input of the buffer preamplifier in accordance with Assumptions 5 and 8. This yields the model of a buffer active non-contact and/or insulated electrode and its interface with the body that will be analyzed in the rest of this paper. The generalized model, presented in Figure 10, reveals the existence of three equivalent impedance blocks. The first one, Z_{body & skin}, represents the equivalent skin–tissue impedance of the layered RC-parallels describing electrical properties of the subcutaneous layer, dermis, viable epidermis, and stratum corneum. The second one, Z_coupling, represents the impedance of the electrode–body path or the coupling impedance, i.e., the equivalent impedance of all the coupling layers between the electrode and the surface of the skin, described with the equivalent R_couplingC_coupling-parallel (Assumption 4). Altogether, these two equivalent impedances, Z_{body & skin} and Z_coupling, would correspond to the electrode–tissue impedance (ETI) [176]. Finally, the third equivalent impedance block, Z_in, represents the preamplifier common-mode input impedance Z_com+ (Assumption 7). In accordance with the general definition [330], just like Z_coupling and Z_diff, Z_in can be modeled with an RC-parallel as well: the common-mode input resistance R_com+ $=$ R_in representing Z_in at 0 Hz, and the common-mode input capacitance C_com+ $=$ C_in, which models the decrease in the R_in with increasing frequency. This third impedance block will be further investigated in the next two paragraphs, leading to Assumption 9.

Parasitic input capacitance. While the capacitances present at the preamplifier inverting input pin and output were discussed in Appendix D, now, capacitance present at the non-inverting input pin will be introduced. This parasitic capacitance can be modeled with two parallel capacitances referenced to the preamplifier circuit common: the aforementioned common-mode input capacitance C_com+ $=$ C_in and the additional capacitance C_par. Unlike C_in, which is an internal capacitance inherent to the semiconductor junctions of the preamplifier input stage transistors and electrostatic discharge (ESD) input protection structures, C_par is similar to C₋ described in Appendix D. It represents the external pin, pad, and layout capacitance: pin and pad capacitance, via capacitance, and PCB trace-to-reference-plane capacitance. Each of these components are far from negligible, as they could easily add a few pF of capacitance. In fact, for a thin multi-layer stackup, a capacitance of a copper surface to a copper surface on an adjacent layer (in this case, the circuit common plane) can reach the order of 10 pF/cm² (per unit length and width) [99,273,274]. Since the copper PCB path at the non-inverting input pin often includes not only the PCB trace, but also the sensing surface itself (as in the herein analyzed model), it is not surprising that C_par could total over 120 pF, as simulated in [380]. In addition, if a cable exists at the input of the preamplifier, such as in the case when the preamplifier analog front-end is not implemented directly on the electrode but rather connected via wires, C_par also comprises cable capacitance similar to C_cable in Appendix D, indicating the capacitance between the inner conductor and cable shield. Even for low-capacitance cables, this additional capacitance exhibits about 1 pF/cm [259,272,356].

Bias currents and DC biasing. As explained in Appendix D with the concept of negative feedback, an operational amplifier will ideally force v₋ to track v₊ by outputting the required voltage and current, but without the need for drawing any current at input terminals. However, in reality, since the common-mode input impedance is finite, DC bias currents I₊ and I₋ exist in the input stage and ESD input protection structures (Figure A8). In fact, their presence at the electrode–body interface is one of the reasons why proper modeling of the electrode–body coupling is essential for fully grasping the phenomena of non-contact and insulated biopotential measurements. Similarly to v_diff (A6) and v_com (A7), specifications of operational amplifiers usually define the input bias current as the average ${\displaystyle \frac{I_{+}-I_{-}}{2}}$ and the input offset current as the difference between the bias currents, I₊ − I₋ [330]. Specifically, the contribution of I₋ can be neglected in comparison with the contribution of I₊, since the inverting input terminal is connected to the preamplifier low output impedance. On the other hand, as described in [382] for an AC-coupled amplifier, if the bias current I₊ is not shunted to circuit common, it charges up the C_coupling and becomes integrated into voltage (2), which is further amplified by the non-inverting DC closed-loop gain of the preamplifier (A18). Depending on the direction of I₊, this process drives the non-inverting input toward one of the supply rails until eventually the common-mode input voltage range or the output voltage swing is exceeded and the preamplifier is driven out of its range of linear operation and into saturation. The speed of the charging process will be defined by the capacitance and the resistance of the charging path, the product of which describes the time constant. The smaller bias current will take longer to achieve the same effect. Therefore, one way to mitigate this effect is to reduce the bias currents with the choice of the preamplifier. Aside from using amplifiers with lower bandwidth, amplifiers with bipolar junction transistor (BJT) input stages, commonly exhibiting bias currents of up to the order of 1 μA, could be replaced with amplifiers with field-effect transistor (FET) input stages. This way, bias currents are reduced to the leakage currents in the order of 1 pA or even 1 fA, as in the case of the chosen preamplifiers [336,337,338]. However, this comes with an increase in the common-mode input resistance R_in—from typical values up to the order of 1 MΩ in the case of BJT-input stages to the order of 1 TΩ and higher in the case of FET-input stages. Furthermore, the choice of input stage technology comes with additional tradeoffs. For example, although bias currents of the FET-input stages are several orders smaller, they exhibit a severe increase with temperature because they arise from the leakage currents of reverse-biased p-n junctions. Also, while using BJT-input stages typically results in larger bias currents and higher current noise, using FET-input stages may result in larger input offset voltage and higher voltage noise. More on these tradeoffs between input stage technologies can be found in [332,333,383], and subchapters 4x.5 and 4x.10 in [355]. Finally, even if bias currents are minimized, biasing will still remain an issue to be resolved for long-term measurements. To define the DC operating point and provide an alternative pathway for discharging C_coupling, thereby maximizing the output signal range and preventing the input voltage from drifting outside of the common-mode input voltage range, an external DC biasing net is added to the preamplifier input. Most simply, it can be realized with an external input resistor, R_bias (later, at the end of Section 4.2, more complex biasing circuits will be mentioned). However, additional bias pathways could be inadvertently created in the system as well. Namely, the previously described devastating effect of I₊ considered the electrode–body interface as an AC coupler (infinite R_coupling). In reality, the effect will depend not only on the DC biasing circuit, but also on R_coupling. For instance, as mentioned in paragraph “Non-contact on-body electrodes” in Section 3.1, using hygroscopic fabric as a coupling layer would allow the bias currents to flow through the coupling material toward the skin. With increased perspiration, this would increase the faradaic reaction at the interface and eventually render the DC biasing path ineffective. While this would be an effective way to resolve the issue of bias currents, it would also effectively turn a non-contact electrode into a surface-contact electrode. Therefore, from the methodological point of view, the hygroscopic fabric would begin to serve as merely a spacer that physically separates the electrode from the skin, rather than as an electrically insulating layer with predominantly capacitive coupling at the frequencies of interest [98]. This example stresses the importance of preserving the dedicated DC biasing path and taking into account the finite conductivity of the used coupling layers.

Now, the assumption on the total equivalent input impedance can be expressed.

Assumption 9.

The third impedance block—the total equivalent input impedance. The first equivalent impedance block in Figure 10, Z_{body & skin}, was investigated in Appendix B. The second equivalent impedance block, Z_coupling, was investigated in Section 3.1 and defined in Assumption 4. Now, the third equivalent impedance block, describing the total equivalent input impedance Z_IN, can be defined. To summarize, similarly to Z_coupling, the total input impedance Z_IN can be expressed as the R_INC_IN-parallel. Therein, R_IN represents the parallel combination of the equivalent DC biasing resistor R_bias and the preamplifier common-mode input resistance R_com+ = R_in (i.e., R_IN = R_bias||R_in). On the other hand, C_IN represents the total capacitance seen between the preamplifier non-inverting input pin (sensing surface) and the circuit common, expressed as a parallel combination of the pin, pad, layout, and cable parasitic input capacitance C_par and the preamplifier common-mode input capacitance C_com+ = C_in (i.e., C_IN = C_par||C_in). In conclusion, subscripts “in” with lowercase letters represent equivalent elements inherent to the preamplifier (R_in, C_in, Z_in = (−jX_Cin)||R_in), whereas subscripts “IN” with uppercase letters represent equivalent elements of the total input impedance (R_IN, C_IN, Z_IN = (−jX_CIN)||R_IN). Following the idea behind modeling the layers and stages with equivalent RC-parallels, some papers [384] add a parasitic resistance component, expressed as R_par, in parallel with C_par to formally create a distinction with respect to the R_bias component, especially for the case of more complex models that consider shielding enclosures and additional methods of input capacitance compensation, which will be mentioned later at the end of Section 4.2. However, for the most basic conceptual model generalized in this paper, this separate R_par component will not be considered.

Unique solvability, linearity, and time invariance. In accordance with Assumption 5, the system interface–electrode–preamplifier comprises two subsystems: the electrode–body interface, modeled as an input voltage divider, and the buffer preamplifier, modeled as a first-order system (A14,A19). It can be analyzed as a two-port network with a common ground (Figure 11). Since the purpose of this system is to allow electric potential sensing without the loading effect and at the same time preserve the shape of the measured signal without introducing any distortion, and since only lumped-element passive electrical components and an operational amplifier serving as a voltage follower for a small AC signal (Assumption 7) are used in the model, the properties of homogeneity and additivity are satisfied. In addition, the network contains only independent sources with the assumption of constant behavior over the time of analysis (i.e., response changes only when the input changes). Hence, the system can be treated as a continuous-time, single-input single-output (SISO), uniquely solvable [385], linear time-invariant (LTI) system. In other words, across the entire frequency range of interest (Assumption 1), the output contains only the frequency components that are already present at the input. The caveat behind these simplifications will be discussed later in Section 4.2, when phase response and group delay of the input voltage divider subsystem will be investigated. Treating the system as LTI also means that the principle of superposition is applicable: every voltage and every current in the network can be expressed as the sum of two terms—zero-state response (response to external inputs as independent sources of perturbation, with initial conditions set to zero) and zero-input response (response to initial conditions with external inputs set to zero). Whereas zero-state response is a result of externally applied independent voltage and current sources, zero-input response is a result of energy previously stored in inductors and capacitors. More on LTI systems and their fundamental properties that will be extensively used in the following pages can be found in [386,387].

Laplace transform and transfer function. Under assumed zero initial conditions (Assumption 5), the Laplace transform [387,388] of the impulse response h_el(t) (zero-state response to a unit impulse) gives the transfer function or network function H_el(s) of this uniquely solvable LTI system. The letter s, s $=$ σ + jω, denotes the complex frequency, which comprises the real part σ, and the imaginary part, angular frequency ω. In accordance with Assumption 5, the voltage sensed with the analyzed active non-contact and/or insulated electrode can be calculated with respect to the voltage V_surface(s) as follows (15):

$$V_{\mathrm{o}\mathrm{u}\mathrm{t}}\left(s\right)=V_{\mathrm{i}\mathrm{n}}·A\left(s\right)\approx V_{\mathrm{i}\mathrm{n}}·A_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}\left(s\right)=V_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}\left(s\right)·{\displaystyle \frac{Z_{\mathrm{I}\mathrm{N}}\left(s\right)}{Z_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left(s\right)+Z_{\mathrm{I}\mathrm{N}}\left(s\right)}}· {\displaystyle \frac{A_0}{1+\frac{s}{{\omega }_{\mathrm{A}}}}} ,$$

(15)

where ω_A is the angular frequency corresponding to the preamplifier closed-loop bandwidth f_A (A2), which is, in the case of the first-order model of a buffer preamplifier (A₀ $=$ 1 V/V $=$ 1), equal to its unity-gain closed-loop bandwidth f₁ (A17)–(A20). It can be observed that the presence of finite Z_IN reduces the open-circuit voltage, resulting in circuit loading and voltage divider effect. As mentioned in Assumption 9, Z_IN can be expressed as the R_INC_IN-parallel, where R_IN is equal to R_bias||R_in, and C_IN is equal to C_par||C_in; hence (16)

$$V_{\mathrm{o}\mathrm{u}\mathrm{t}}\left(s\right)=V_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}\left(s\right)·{\displaystyle \frac{\frac{R_{\mathrm{I}\mathrm{N}}}{1+s{R}_{\mathrm{I}\mathrm{N}}{C}_{\mathrm{I}\mathrm{N}}}}{\frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{1+s{R}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}+\frac{R_{\mathrm{I}\mathrm{N}}}{1+s{R}_{\mathrm{I}\mathrm{N}}{C}_{\mathrm{I}\mathrm{N}}}}}·{\displaystyle \frac{A_0}{1+\frac{s}{{\omega }_{\mathrm{A}}}}} .$$

(16)

Using Laplace transform directly on the input–output form as a more efficient alternative to the state-space approach for SISO systems [388,389] gives the transfer function H_el(s) of the analyzed system interface–electrode–preamplifier. As announced in Assumption 5, H_el(s) can be simplified into a cascade of the input voltage divider and preamplifier transfer functions, H_div(s) and H_preamp(s), respectively (17):

$$H_{\mathrm{e}\mathrm{l}}\left(s\right)={\displaystyle \frac{V_{\mathrm{o}\mathrm{u}\mathrm{t}}\left(s\right)}{V_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}\left(s\right)}}={\displaystyle \frac{V_{\mathrm{i}\mathrm{n}}\left(s\right)}{V_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}\left(s\right)}}·{\displaystyle \frac{V_{\mathrm{o}\mathrm{u}\mathrm{t}}\left(s\right)}{V_{\mathrm{i}\mathrm{n}}\left(s\right)}}=H_{\mathrm{d}\mathrm{i}\mathrm{v}}\left(s\right)· H_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{m}\mathrm{p}}\left(s\right)$$

$$={\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{1+\frac{s}{\frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}}{1+\frac{s}{\frac{1}{{{(R}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}||R}_{\mathrm{I}\mathrm{N}})(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}||C_{\mathrm{I}\mathrm{N}})}}}}·{\displaystyle \frac{A_0}{1+\frac{s}{{\omega }_{\mathrm{A}}}}}$$

$$={\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{s+\frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{s+\frac{1}{{{(R}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}||R}_{\mathrm{I}\mathrm{N}})(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}||C_{\mathrm{I}\mathrm{N}})}}}·{\displaystyle \frac{A_0{\omega }_{\mathrm{A}}}{s+{\omega }_{\mathrm{A}}}} .$$

(17)

Poles and zeros. Since the analyzed system is a uniquely solvable LTI network, the resulting transfer function H_el(s) (17) is a ratio of two polynomials with real coefficients. Also, the null points (roots) of these polynomials are generally either real numbers or they occur in complex conjugate pairs. The null points of the denominator polynomial are called poles and represent complex frequencies at which the transfer function approaches infinity, whereas null points of the numerator polynomial are called zeros and represent complex frequencies at which the transfer function approaches zero. More on the transfer function and geometrical interpretation of poles and zeros can be found at the beginning of [387], and in [390,391]. The degree of the denominator polynomial is 2. Since there are no additional poles canceled by pole–zero cancelation (see [387], subchapter 4.9 in [333], and [346]), both H_el(s) and the uniquely solvable LTI system itself are second-order, with two single real poles s_p,1 and s_p,2 (18) (Figure 12):

$$\left[s_{\mathrm{p},1}+{\displaystyle \frac{1}{{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})}}\right]·\left(s_{\mathrm{p},2}+{\omega }_A\right)=0$$

$$\to s_{\mathrm{p},1}=-{\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}=-\left|s_{\mathrm{p},1}\right| , {\mathrm{s}}_{\mathrm{p},2}=-{\omega }_{\mathrm{A}}=-\left|s_{\mathrm{p},2}\right| .$$

(18)

It can be noticed that |s_p,2| $=$ ω_A $=$ 2πf_A is the angular frequency that corresponds to the preamplifier closed-loop bandwidth f_A (A19,A20). This explains why the first-order model of the preamplifier subsystem (A14,A19), established in Appendix D, is also referred to as single-pole. In Bode approximation of a first-order (single-pole) system (Figure A9) [273,341,348,349], f_A is also the break frequency of the closed-loop frequency response—|A_diff(f)|_dB and $\measuredangle$A_diff(f).

Similarly, the degree of the numerator polynomial is 1; hence, the analyzed system contains a single non-trivial zero s_z,1 and a single trivial zero s_z,2 at infinity (19), corroborating that the transfer function approaches zero as the frequency increases:

$$s_{\mathrm{z}}+{\displaystyle \frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}=0 \to s_{\mathrm{z},1}=-{\displaystyle \frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}=-\left|s_{\mathrm{z},1}\right| , {\mathrm{s}}_{\mathrm{z},2}=\infty .$$

(19)

Finite bandwidth and causality. As a result of an excess of non-trivial poles over the number of non-trivial zeros, the numerator polynomial is of smaller degree than the denominator polynomial. Consequently, H_el(s) is strictly proper, and the system, just like the preamplifier itself, has a finite bandwidth. Also, the system is strictly causal, meaning that the output depends only on the previous values of the input—it can neither instantaneously nor anticipatorily respond to changes. Given that a finite amount of propagation delay always exists in physical systems, causality is a commonly satisfied property in practical applications. As a result of causality, the region of convergence (ROC) of Laplace transform H_el(s) extends to the right of the rightmost pole [392] (Figure 12).

Stability. Furthermore, stability analysis can be revisited. In Appendix D, stability analysis has already been carried out for the single-pole model of the preamplifier, A_diff(f) (A19). For this purpose, the simplified Nyquist criterion on the basis of the Bode plot was used [350,351]. Now, system stability analysis [392,393,394] can be carried out at the level of the entire system H_el(s) described with its poles and zeros. In accordance with the superposition principle, stability can be observed on the zero-input response from the perspective of internal stability, and on the zero-state response from the perspective of input–output stability. From the perspective of internal stability, since the system is causal uniquely solvable LTI, since both poles have negative real parts (18), and since there are no additional canceled poles, the ROC of Laplace transform H_el(s) contains the imaginary axis (Figure 12) and the system is asymptotically stable. In other words, the system zero-input response is bounded in magnitude and decays to equilibrium over time for every finite initial condition. Also, asymptotic stability automatically draws bounded-input bounded-output (BIBO) stability as input–output stability: the system zero-state response to a bounded input remains bounded.

Impulse response. For the described causal, continuous-time, uniquely solvable LTI system, having all poles in the open left-half complex s-plane guarantees its stability. However, it does not guarantee a response without oscillatory decay. Namely, in addition to stability itself, the phase margin must be observed to assess how close the stable system is to becoming unstable. This issue was mentioned in Appendix D, where it was explained that the loop gain determines not only the case of marginal stability and whether the system is stable, but also the amount of gain peaking in the closed-loop frequency response of a stable system, along with the amount of overshoot, ringing, and settling time in its time-domain transient response. Now, that discussion can be related to the location of poles in the complex s-plane and their impact on the impulse response h_el. Often described as a fingerprint of an LTI system, the impulse response represents its reaction to a sudden, brief external input—specifically, the zero-state response to a unit impulse—offering further insight into the natural dynamics, transient behavior, and stability of the system “at rest” (i.e., for zero initial conditions). For the analyzed uniquely solvable LTI system, impulse response h_el can be obtained as the time derivative of the zero-state step response [387], or from the inverse Laplace transform (20,21) after partial fraction expansion of H_el(s) (17) [387,388]:

$$H_{\mathrm{e}\mathrm{l}}\left(s\right)={\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}{A}_0{\omega }_{\mathrm{A}} {\displaystyle \frac{1}{{\omega }_{\mathrm{A}}-\frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}}$$

$$·\left({\displaystyle \frac{\frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}-\frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}{s+\frac{1}{{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})}}}+{\displaystyle \frac{{\omega }_{\mathrm{A}}-\frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{s+{\omega }_{\mathrm{A}}}}\right) ,$$

(20)

$$h_{\mathrm{e}\mathrm{l}}\left(t\right)={\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}{A}_0{\omega }_{\mathrm{A}} ·\left({\displaystyle \frac{\left|s_{\mathrm{z},1}\right|-\left|s_{\mathrm{p},1}\right|}{{\omega }_{\mathrm{A}}-\left|s_{\mathrm{p},1}\right|}}{e}^{-\left|s_{\mathrm{p},1}\right|t}+{\displaystyle \frac{{\omega }_{\mathrm{A}}-\left|s_{\mathrm{z},1}\right|}{{\omega }_{\mathrm{A}}-\left|s_{\mathrm{p},1}\right|}}{e}^{-{\omega }_{\mathrm{A}}t}\right)·\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}\left(t\right).$$

(21)

Firstly, since the system is real and comprises zeros and poles that are either real-valued or they occur in complex conjugate pairs, the resulting impulse response h_el(t) is a real function. Secondly, as a consequence of causality, h_el(t) is zero before the unit impulse is applied at t $=$ 0 s, hence h_el(t) is right-sided (Figure 13). This is indicated in (21) with multiplication by the Heaviside unit step function step(t). Thirdly, it has already been noted that the analyzed uniquely solvable LTI system is asymptotically stable and BIBO stable as a result of being causal and having all poles in the open left-half complex s-plane (Figure 12). In addition, h_el(t) is absolutely integrable, i.e., the area under the curve |h_el(t)| in time (t) is finite. Namely, requirement ω_A > |s_p,1| yields (22)

$${\int }_0^{\infty }\left|h_{\mathrm{e}\mathrm{l}}\left(t\right)\right|dt=$$

$${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}{\displaystyle \frac{A_0{\omega }_{\mathrm{A}}}{{\omega }_{\mathrm{A}}-\left|s_{\mathrm{p},1}\right|}} ·{\int }_0^{\infty }\left|\left(\left|s_{\mathrm{z},1}\right|-\left|s_{\mathrm{p},1}\right|\right)e^{-\left|s_{\mathrm{p},1}\right|t}+\left({\omega }_{\mathrm{A}}-\left|s_{\mathrm{z},1}\right|\right)e^{-{\omega }_{\mathrm{A}}t}\right|dt<\mathrm{\infty } .$$

(22)

Exponential internal stability and the generalized contribution of poles. Finally, asymptotic internal stability of an LTI system also draws the more stringent property of exponential internal stability: for all initial conditions and zero input, all the circuit variables return to their equilibrium at an exponentially decaying rate. In fact, the prerequisite for using the simplified Nyquist stability criterion with the use of Bode plots in Appendix D [350,351] was to assume that the open-loop gain a_diff(f) is exponentially stable, and the stability of the closed-loop response, assessed in Figure A9, was in fact the exponential stability. The same is indicated by the location of poles in the complex s-plane (Figure 12): each pole s_p $=$ σ + jω in the Laplace s-domain (transfer function) contributes to a summation term in the time domain (impulse response). Let the analyzed real system be generalized into a rational transfer function H(s), expressed as a ratio of polynomials with real coefficients. Then, as explained in subchapter 3 in [387], after partial fraction expansion and inverse Laplace transform, the time-domain modes from each of n poles with multiplicity m can be generalized as follows (23):

$$h\left(t\right)={\mathcal{L}}^{-1}\left\{s_{\mathrm{p},1} \mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m} \mathrm{i}\mathrm{n} H\left(s\right)\right\}+\cdots +{\mathcal{L}}^{-1}\left\{s_{\mathrm{p},n} \mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m} \mathrm{i}\mathrm{n} H\left(s\right)\right\}$$

$$=\left\{\sum _{i=0}^{m_1-1}{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}_{1,i}{·t}^i{e}^{({\sigma }_1+j{\omega }_1)t}+\cdots +\sum _{i=0}^{m_n-1}{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}_{n,i}{·t}^i{e}^{\left({\sigma }_n+j{\omega }_n\right)t}\right\}·\mathrm{step}(t) .$$

(23)

Accordingly, as described in [390,391], the imaginary parts of poles in e^jωt terms determine the frequencies of oscillations in the impulse response h(t), whereas the real parts of poles in e^σt terms control the rates of their exponential change (decay or growth) (Figure 14). In the case of H_el(s), both poles (18) are real-valued (ω₁ $=$ 0 and ω₂ $=$ 0), negative (σ₁ < 0 and σ₂ < 0), and single (m₁ $=$ 1 and m₂ $=$ 1); thus, the resulting behavior of h_el(t) is overdamped, as obtained in (21) and depicted in Figure 13. To give a physical interpretation from an electrical engineering perspective, this behavior can be explained via an RLC circuit: first, there is no inductor to counteract the discharging and charging of the capacitor by allowing for the back-and-forth energy transfer, and second, the resistive component of the circuit dissipates the energy available for storage in the electric field of the capacitor. This translates into an exponential decay without oscillation.

Convolution and multiplication. Given how tangled the solution for h_el(t) might be in comparison with H_el(s), the power of the transfer function as a tool for evaluating LTI systems “at rest” (i.e., for zero initial conditions) now becomes evident. Namely, instead of using the convolution operation with the impulse response to obtain a zero-state response in the time domain, Laplace transform can be used to transform this operation into multiplication with the transfer function in the s-domain [387] (24):

$$V_{\mathrm{o}\mathrm{u}\mathrm{t}}\left(s\right)=\mathcal{L}\left\{v_{\mathrm{o}\mathrm{u}\mathrm{t}}\left(t\right)\right\}=\mathcal{L}\left\{\left(v_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}\ast h_{\mathrm{e}\mathrm{l}}\right)\left(t\right)\right\}=\mathcal{L}\left\{{\int }_0^{t}h\left(t-\tau \right)v_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}\left(\tau \right)d\tau \right\}=$$

$$=V_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}\left(s\right)·H_{\mathrm{e}\mathrm{l}}\left(s\right) .$$

(24)

Fourier transform and frequency characteristic. If the excitation applied to this uniquely solvable LTI system were a complex exponential Ae^jωt with a constant magnitude A instead of an exponentially modulated complex exponential Ae^(σ+jω)t, the steady-state response of all voltages and currents would be obtained [268,386,387], in which the contribution of the transient response vanishes. In that case, s is substituted with jω, and Laplace transform is evaluated along the imaginary axis and reduced to Continuous-Time Fourier Transform (CTFT), which exists since h_el(t) satisfies Dirichlet conditions [392]. Lastly, as depicted in Figure 15, the transfer function H_el(s) (17) turns into frequency characteristic or frequency response H_el(jω) (25), and consequently, circuit variables can be described with phasors (A1), whereas the relation between the voltage and current phasors can be expressed with impedances (A3,A4) [268]:

$$H_{\mathrm{e}\mathrm{l}}\left(j\omega \right)={\displaystyle \frac{\dot{V_{\mathrm{o}\mathrm{u}\mathrm{t}}\left(j\omega \right)}}{\dot{V_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}\left(j\omega \right)}}}={\displaystyle \frac{\dot{V_{\mathrm{i}\mathrm{n}}\left(j\omega \right)}}{\dot{V_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}\left(j\omega \right)}}}·{\displaystyle \frac{\dot{V_{\mathrm{o}\mathrm{u}\mathrm{t}}\left(j\omega \right)}}{\dot{V_{\mathrm{i}\mathrm{n}}\left(j\omega \right)}}}=H_{\mathrm{d}\mathrm{i}\mathrm{v}}\left(j\omega \right)· H_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{m}\mathrm{p}}\left(j\omega \right)$$

$$={\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{1+\frac{j\omega }{\frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}}{1+\frac{j\omega }{\frac{1}{{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})}}}}·{\displaystyle \frac{A_0}{1+j\frac{\omega }{{\omega }_{\mathrm{A}}}}}$$

$$={\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{j\omega +\frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{j\omega +\frac{1}{{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})}}}·{\displaystyle \frac{A_0{\omega }_{\mathrm{A}}}{j\omega +{\omega }_{\mathrm{A}}}} .$$

(25)

The frequency response H_el(jω) can be visualized by depicting its frequency-dependent magnitude characteristic (magnitude response) |H_el(jω)| and phase characteristic (phase response) $\measuredangle$H_el(jω). In accordance with convolution (24) and the superposition principle, the magnitude response of this LTI system acts as amplitude scaling, whereas the phase response acts as a phase shift. In other words, in steady state, each frequency component ω_k $=$ 2πf_k of the input signal v_surface will be scaled by |H_el(jω_k)| and shifted in phase by $\measuredangle$H_el(jω_k) [268]. This will be further explored in the next two sections. Since the observed system is real, with real-valued inputs and outputs, the frequency response is conjugate symmetric: H_el(jω) $=$ H_el^*(−jω). As a result, magnitude response |H_el(jω)| is even (symmetric about the ordinate) and phase response $\measuredangle$H_el(jω) is odd (antisymmetric about the ordinate), so it is sufficient to observe the response only at positive frequencies [268,386]. The same is valid for both subsystems, H_div(jω) and H_preamp(jω). By plotting the frequency response curves (or their linear piecewise approximation—Bode plot [273,341,348,349]), the plot of the three-dimensional (3D) transfer function is replaced with its two-dimensional (2D) cross-section along the plane σ $=$ 0 (Figure 15). This way, poles, as complex frequencies in the s-plane at which the transfer function goes to infinity, are mapped to break frequencies in the Bode plot. Interesting graphical relations on this subject can be found in [396]. In that sense, just like the transfer function and impulse response, the frequency response is also shaped by the location of poles and zeros, as described in [390,391]. With regard to the frequency domain, in the vicinity of poles close to the imaginary jω-axis, magnitude response will be raised toward a local maximum (peak), whereas in the vicinity of zeros close to the imaginary jω-axis, magnitude response will be lowered toward a local minimum (dip or notch), with a rapidly varying phase in both cases. Conversely, in terms of the time-domain transient response, the distance of the poles from the real axis will define the frequencies of oscillations, whereas the distance from the imaginary axis will define the rate of their exponential change. Specifically, a complex conjugate pair of non-repeated poles, that lies directly on the imaginary axis, contains only imaginary parts. Therefore, these poles are entirely imprinted in the 2D frequency response, rendering the system marginally stable with sustained, undamped oscillations at the angular frequency that corresponds to the absolute value of their imaginary parts.

Figure 15. (a) 3D transfer function H_el(s) in decibels (dB); locations of poles and zeros in the complex s-plane correspond to the pole–zero plot depicted in Figure 12; (b) The same H_el(s) in dB with cross-sections along the planes σ $=$ 0 and ω $=$ 0; (c) translation of H_el(s) into 2D magnitude response in dB, |H_el(jω)|_dB; (d) once again, |H_el(jω)|_dB is plotted, but now, angular frequency ω $=$ 2πf on abscissa axis is replaced with frequency f in logarithmic scale. This representation will be used in the rest of the paper and it can be further approximated with the Bode linear piecewise approximation as presented in Figure A9. Parameter values correspond to the ones chosen for impulse response h_el(t) in Figure 13. Plots are obtained in MathWorks^® MATLAB R2025b environment [146].

Figure 15. (a) 3D transfer function H_el(s) in decibels (dB); locations of poles and zeros in the complex s-plane correspond to the pole–zero plot depicted in Figure 12; (b) The same H_el(s) in dB with cross-sections along the planes σ $=$ 0 and ω $=$ 0; (c) translation of H_el(s) into 2D magnitude response in dB, |H_el(jω)|_dB; (d) once again, |H_el(jω)|_dB is plotted, but now, angular frequency ω $=$ 2πf on abscissa axis is replaced with frequency f in logarithmic scale. This representation will be used in the rest of the paper and it can be further approximated with the Bode linear piecewise approximation as presented in Figure A9. Parameter values correspond to the ones chosen for impulse response h_el(t) in Figure 13. Plots are obtained in MathWorks^® MATLAB R2025b environment [146].

The concept of frequency characteristic has already been introduced in Appendix D with the single-pole model of an operational amplifier (preamplifier), a_diff(f) (A14) and A_diff(f) (A19). The only difference is that now, a more formal expression, with the use of jω or jf argument rather than f, is employed in accordance with Assumption 6. In other words, the model A_diff(f) analyzed in Appendix D corresponds to the preamplifier subsystem H_preamp(jω) and, specifically, to the buffer preamplifier subsystem for A₀ $=$ 1 and ω_A $=$ ω₁ $=$ 2πf₁ in (25). Therefore, the rest of the system analysis will focus on the second subsystem, input voltage divider H_div(jω) (Section 4.2). Subsequently, the findings will be employed at the level of the entire system, H_el(jω) (Section 4.3). More on the frequency characteristic and its construction can be found in [268,347,348,386,390,391,396].

4.2. Second Subsystem: Input Voltage Divider

Input voltage divider effect. The existence of a voltage divider at the preamplifier input has already revealed the number one reason for the difference between the biopotential that is being sensed by the system, v_surface, and the signal that is acquired at the preamplifier non-inverting input, v_in. This voltage divider effect depends not only on the parameters of the electrode–body interface, but also on frequency. Hence, the frequency characteristic (frequency response) of the input voltage divider subsystem, H_div(jω), can be observed in more detail (26):

$$H_{\mathrm{d}\mathrm{i}\mathrm{v}}\left(j\omega \right)={\displaystyle \frac{\dot{V_{\mathrm{i}\mathrm{n}}\left(j\omega \right)}}{\dot{V_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}\left(j\omega \right)}}}={\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{1+\frac{j\omega }{\frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}}{1+\frac{j\omega }{\frac{1}{{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})}}}}$$

$$={\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{j\omega +\frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{j\omega +\frac{1}{{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})}}}=K·{\displaystyle \frac{j\omega -s_{\mathrm{z},1}}{j\omega -s_{\mathrm{p},1}}} .$$

(26)

As seen from (26), H_div(jω) is first-order and bilinear, with one real pole s_p,1 (18) and one real zero s_z,1 (19) [397]. If the polar form [268,269,390,397] is used (27), magnitude response in dB, |H_div(jω)|_dB (28), and phase response in degrees, $\measuredangle$H_div(jω) (29), can be further computed from (26)

$$H_{\mathrm{d}\mathrm{i}\mathrm{v}}\left(j\omega \right)=\left|H_{\mathrm{d}\mathrm{i}\mathrm{v}}\left(j\omega \right)\right|e^{j\measuredangle H_{\mathrm{d}\mathrm{i}\mathrm{v}}\left(j\omega \right)} ,$$

(27)

$${\left|H_{\mathrm{d}\mathrm{i}\mathrm{v}}\left(j\omega \right)\right|}_{\mathrm{d}\mathrm{B}}=20{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left[{\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{\sqrt{1+({\omega R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}})}^2}}{\sqrt{1+{\left[\omega {\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})\right]}^2}}}\right]$$

$$= 20{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left[{\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{\sqrt{{\omega }^2+{\left(\frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}\right)}^2}}{\sqrt{{\omega }^2+{\left(\frac{1}{{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})}\right)}^2}}}\right] ,$$

(28)

$$\measuredangle H_{\mathrm{d}\mathrm{i}\mathrm{v}}\left(j\omega \right)=\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}2\left(\omega R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right)-\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}2\left[\omega \left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)\right] ,$$

(29)

where atan2 is the two-argument arctangent function that accounts for the quadrant of the result based on the signs of both arguments, wrapping the results in a [−180°, 180°] or [−π, π] rad interval in accordance with (30) [398]:

$$\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}2\left({\displaystyle \frac{y}{x}}\right)=\left\{\begin{array}{cc}\hfill \arctan \left({\displaystyle \frac{y}{x}}\right),& x>0\hfill \\ \hfill \arctan \left({\displaystyle \frac{y}{x}}\right)+180°,& x<0 \mathrm{a}\mathrm{n}\mathrm{d} y\ge 0\hfill \\ \hfill \arctan \left({\displaystyle \frac{y}{x}}\right)-180°,& x<0 \mathrm{a}\mathrm{n}\mathrm{d} y<0\hfill \\ \hfill \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left(y\right)·90°,& x=0 \mathrm{a}\mathrm{n}\mathrm{d} y\ne 0\hfill \\ \hfill \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d},& x=0 \mathrm{a}\mathrm{n}\mathrm{d} y=0\hfill \end{array}\right. .$$

(30)

In accordance with convolution (24) and the superposition principle, in steady state, H_div(jω) will affect each frequency component V_kcos(ω_kt + φ_k) of the input signal v_surface in such a way that it will scale its amplitude into V_k$·$|H_div(jω_k)| and shift its phase by $\measuredangle$H_div(jω_k) [268]. In this section, |H_div(jω)|_dB will be preferred for denoting the magnitude response.

Four subcases. In Section 3.1, it was discussed that the term capacitive electrode is an oversimplification for most cases of practical biopotential measurement. As an alternative, Test 1 was established as a tool for evaluating the predominant coupling mechanism in each of the coupling layers. Now, the predominant coupling mechanism can be investigated in the context of the generalized buffer active non-contact and/or insulated electrode and its interface with the body, presented in Figure 10. In that sense, Test 1 can be applied to the total equivalent coupling impedance Z_coupling, which comprises the equivalent R_coupling and C_coupling in accordance with Assumption 4. This equivalent Z_coupling is loaded by the equivalent input impedance Z_IN, which comprises R_IN and C_IN in accordance with Assumption 9. The resulting H_div(jω) (26) has one pole, s_p,1 $=$ $-{\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$ (18), and one zero, s_z,1 $=$ $-{\displaystyle \frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ (19). Hence, two cases can be discerned based on the position of the pole s_p,1 with respect to the position of the zero s_z,1 in the pole–zero plot (Figure 12). For this purpose, absolute value of the pole, |s_p,1| $=$ ${\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$, and absolute value of the zero, |s_z,1| $=$ ${\displaystyle \frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$, can be compared. Condition |s_p,1| ≤ |s_z,1| yields the first case (case I), whereas |s_p,1| > |s_z,1| yields the second case (case II). The resulting two cases are depicted in Figure 16.

As recalled from (18) and (19), the zero s_z,1 $=$ $-{\displaystyle \frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ is defined with the combination of values R_coupling and C_coupling, whereas the pole s_p,1 $=$ $-{\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$ is defined with the combination of values R_coupling, R_IN, C_coupling, and C_IN. This means that the position of the pole with respect to the position of the zero is determined by the value of the R_coupling/R_IN ratio with respect to the C_IN/C_coupling ratio, as shown in (31):

$$\left|s_{\mathrm{p},1}\right|={\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}={\displaystyle \frac{1}{\frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{R}_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{+R}_{\mathrm{I}\mathrm{N}}}\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}\right)}}=$$

$$={\displaystyle \frac{1}{\frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{\frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{R_{\mathrm{I}\mathrm{N}}}+1}\left(1+\frac{C_{\mathrm{I}\mathrm{N}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}\right)}}=\left|s_{\mathrm{z},1}\right|{\displaystyle \frac{1+\frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{R_{\mathrm{I}\mathrm{N}}}}{1+\frac{C_{\mathrm{I}\mathrm{N}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}} .$$

(31)

The ratio R_coupling/R_IN can be treated as the attenuation of the resistive voltage divider: the higher the ratio R_coupling/R_IN, the more attenuation is caused by the resistive voltage divider. Similarly, the ratio C_IN/C_coupling can be treated as the attenuation of the capacitive voltage divider: the higher the ratio C_IN/C_coupling, the more attenuation is caused by the capacitive voltage divider. If |s_p,1| ≤ |s_z,1|, then C_IN/C_coupling ≥ R_coupling/R_IN (case I in Figure 16). Otherwise, if |s_p,1| > |s_z,1|, then C_IN/C_coupling < R_coupling/R_IN (case II in Figure 16).

Furthermore, two additional conditions are possible: R_IN < R_coupling (subcases 1) and R_IN ≥ R_coupling (subcases 2). These two conditions will determine the extent to which the attenuations of the resistive and capacitive voltage dividers can be manipulated.

In total, combination of the aforementioned conditions gives four subcases (I.1, I.2, II.1, and II.2), which are listed in

Table 2. Four subcases (I.1, I.2, II.1, and II.2) based on the position of the pole s_p,1 (18) with respect to the zero s_z,1 (19) (case I and case II) and based on the R_coupling/R_IN ratio (subcases 1 and subcases 2). First, bandwidth, frequency band, frequency range, or area of predominantly capacitive coupling is given in accordance with Test 1; it will be further explored in step 1. Second, for cases I and II, cutoff frequencies are given, along with areas of predominant capacitive divider, which will be derived in step 2. Third, area is given in which the influence of phase shifts from the pole and the zero is minimized. This topic will be covered in step 3. Lastly, for all four subcases, the feasibility of achieving minimized C_IN (C_coupling >> C_IN, i.e., C_coupling ≥ 10C_IN) is investigated. To generalize the results, a positive real factor T ≥ 10 can be used instead of number 10. This will be further explored in step 2.

zero:  ${\mathit{s}}_{\mathbf{z},\mathbf{1}}=-{\displaystyle \frac{\mathbf{1}}{{\mathit{R}}_{\mathbf{c}\mathbf{o}\mathbf{u}\mathbf{p}\mathbf{l}\mathbf{i}\mathbf{n}\mathbf{g}}{\mathit{C}}_{\mathbf{c}\mathbf{o}\mathbf{u}\mathbf{p}\mathbf{l}\mathbf{i}\mathbf{n}\mathbf{g}}}}$				$\left|{\mathit{s}}_{\mathbf{p},\mathbf{1}}\right|={\displaystyle \frac{\mathbf{1}+\frac{{\mathit{R}}_{\mathbf{c}\mathbf{o}\mathbf{u}\mathbf{p}\mathbf{l}\mathbf{i}\mathbf{n}\mathbf{g}}}{{\mathit{R}}_{\mathbf{I}\mathbf{N}}}}{\mathbf{1}+\frac{{\mathit{C}}_{\mathbf{I}\mathbf{N}}}{{\mathit{C}}_{\mathbf{c}\mathbf{o}\mathbf{u}\mathbf{p}\mathbf{l}\mathbf{i}\mathbf{n}\mathbf{g}}}}}·\left|{\mathit{s}}_{\mathbf{z},\mathbf{1}}\right|$
pole: $s_{\mathrm{p},1}=-{\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$
Area of predominantly capacitive coupling: $\omega \ge 10\left|s_{\mathrm{z},1}\right|={\displaystyle \frac{10}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ no high-side frequency limit (only subcase A.1 from Table 1 is considered for each of the coupling layers)
		Case I		Case II
Condition		$\left|s_{\mathrm{p},1}\right|\le \left|s_{\mathrm{z},1}\right|$ i.e., ${\displaystyle \frac{C_{\mathrm{I}\mathrm{N}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}\ge {\displaystyle \frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{R_{\mathrm{I}\mathrm{N}}}}$		$\left|s_{\mathrm{p},1}\right|>\left|s_{\mathrm{z},1}\right|$ i.e., ${\displaystyle \frac{C_{\mathrm{I}\mathrm{N}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}<{\displaystyle \frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{R_{\mathrm{I}\mathrm{N}}}}$
Cutoff frequencies		${\omega }_{\mathrm{R},\mathrm{I}}={\displaystyle \frac{1}{\sqrt{ {\left(\frac{1}{s_{\mathrm{p},1}}\right)}^2-2{\left(\frac{1}{s_{\mathrm{z},1}}\right)}^2}}}$ ${\omega }_{\mathrm{C},\mathrm{I}}=\sqrt{{s_{\mathrm{z},1}}^2-2{s_{\mathrm{p},1}}^2}$		${\omega }_{\mathrm{R},\mathrm{I}\mathrm{I}}={\displaystyle \frac{1}{\sqrt{ {\left(\frac{1}{s_{\mathrm{z},1}}\right)}^2-2{\left(\frac{1}{s_{\mathrm{p},1}}\right)}^2}}}$ ${\omega }_{\mathrm{C},\mathrm{I}\mathrm{I}}=\sqrt{{s_{\mathrm{p},1}}^2-2{s_{\mathrm{z},1}}^2}$
		They exist if ${\displaystyle \frac{\left|s_{\mathrm{z},1}\right|}{\left|s_{\mathrm{p},1}\right|}}={\displaystyle \frac{1+\frac{C_{\mathrm{I}\mathrm{N}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{1+\frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{R_{\mathrm{I}\mathrm{N}}}}}\ge \sqrt{3}$		They exist if ${\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{\left|s_{\mathrm{z},1}\right|}}={\displaystyle \frac{1+\frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{R_{\mathrm{I}\mathrm{N}}}}{1+\frac{C_{\mathrm{I}\mathrm{N}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}}\ge \sqrt{3}$
		Area of predominant capacitive divider$: \omega \ge {\omega }_{\mathrm{C},\mathrm{I}}$$\mathrm{for} \mathrm{case} \mathrm{I} \mathrm{and} \omega \ge {\omega }_{\mathrm{C},\mathrm{I}\mathrm{I}}$ for case II
Area of minimized phase shifts		$\omega \ge 10\left|s_{\mathrm{z},1}\right|={\displaystyle \frac{10}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$		$\omega \ge 10|s_{\mathrm{p},1}|=10\left|s_{\mathrm{z},1}\right|{\displaystyle \frac{1+\frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{R_{\mathrm{I}\mathrm{N}}}}{1+\frac{C_{\mathrm{I}\mathrm{N}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}}$
Subcases		I.1	I.2	II.1	II.2
Condition		$R_{\mathrm{I}\mathrm{N}}<R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}$	$R_{\mathrm{I}\mathrm{N}}\ge R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}$	$R_{\mathrm{I}\mathrm{N}}<R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}$	$R_{\mathrm{I}\mathrm{N}}\ge R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}$
${\mathit{C}}_{\mathbf{c}\mathbf{o}\mathbf{u}\mathbf{p}\mathbf{l}\mathbf{i}\mathbf{n}\mathbf{g}}\ge \mathbf{10}{\mathit{C}}_{\mathbf{I}\mathbf{N}}$ feasible?		Never because $C_{\mathrm{I}\mathrm{N}}>C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}$	$C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}=10C_{\mathrm{I}\mathrm{N}}$ if $R_{\mathrm{I}\mathrm{N}}\ge {10R}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}$	Always	$C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}=10C_{\mathrm{I}\mathrm{N}}$ if $R_{\mathrm{I}\mathrm{N}}<{10R}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}$
Area of predominantly capacitive coupling + area of predominant capacitive divider + area of minimized phase shifts: $\omega \ge 10\left|s_{\mathrm{z},1}\right|$$\mathrm{for} \mathrm{case} \mathrm{I} \mathrm{and} \omega \ge 10\left|s_{\mathrm{p},1}\right|$ for case II

. The four parameters of the interface, R_coupling, R_IN, C_coupling, and C_IN, will define the attenuation of the input voltage divider, as well as the position of the pole and its |s_p,1| absolute value. In addition, R_coupling and C_coupling will define the position of the zero and its |s_z,1| value. As explained earlier in Figure 12, these absolute values |s_z,1| and |s_p,1| correspond to the distance of the negative, real-valued zero s_z,1 and the negative, real-valued pole s_p,1 from the origin of the complex s-plane, respectively. Additionally, |s_z,1| and |s_p,1| also correspond to the zero and the pole angular break frequency in the Bode plot of H_div(jω), respectively. Similarly, ${\displaystyle \frac{\left|s_{\mathrm{z},1}\right|}{2\pi }}$ and ${\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$ denote the corresponding break frequencies in accordance with the relation ω $=$ 2πf (A2). Shifting the angular break frequencies |s_z,1| and |s_p,1| toward lower frequencies would be translated into moving the zero s_z,1 and the pole s_p,1 toward the origin of the complex s-plane. In accordance with Bode approximations [273,341,348,349,397] and the example provided of analysis with Bode plots (Figure A9), |s_z,1| in the term $(1+{\displaystyle \frac{s}{\left|s_{\mathrm{z},1}\right|}})$ is the zero angular break frequency above which the slope of the magnitude response |H_div(jω)|_dB increases for 20 dB/decade. Accordingly, |s_z,1| introduces leading (positive) phase shifts into $\measuredangle$H_div(jω), with a positive slope of +45°/decade that begins approximately at angular frequency 0.1|s_z,1| and ends approximately at angular frequency 10|s_z,1|, achieving a total of +90° across the two decades. Similarly, |s_p,1| in the term $1/(1+{\displaystyle \frac{s}{\left|s_{\mathrm{p},1}\right|}})$ is the pole angular break frequency above which the slope of the magnitude response |H_div(jω)|_dB decreases for 20 dB/decade. Accordingly, |s_p,1| introduces lagging (negative, delaying) phase shifts into $\measuredangle$H_div(jω), with a negative slope of −45°/decade that begins approximately at angular frequency 0.1|s_p,1| and ends approximately at angular frequency 10|s_p,1|, achieving a total of −90° across the two decades. For these reasons, case I (|s_p,1| ≤ |s_z,1|) yields a magnitude response |H_div(jω)|_dB that can contain a portion descending with increasing frequency (roll-off), whereas case II (|s_p,1| > |s_z,1|) yields a magnitude response |H_div(jω)|_dB that will always contain a portion ascending with increasing frequency (roll-on). The concept of lagging and leading is illustrated in [397], along with frequency responses of various bilinear transfer functions. In the following pages, all these effects will be inspected in the context of the input voltage divider subsystem of the generalized buffer active non-contact and/or insulated electrode and its interface with the body. Analysis will be further divided into three steps. Analytical discussions will be accompanied by simulation results in MathWorks^® MATLAB R2025b environment [146]. To visually separate the discussions in each of the three steps, each step ends with a summarizing paragraph, and the respective figures are given afterward. Lastly, summaries from each of the three steps will be assembled to define guidelines for designing non-contact and insulated electrode–body interfaces.

STEP 1: Extending the area (frequency band) of predominantly capacitive coupling. The first interesting observation for each of the four subcases (Figure 16 and Table 2) is the following: since |s_z,1| is equal to ${\displaystyle \frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ (19), |s_z,1| could serve as a rough transition point from predominantly resistive to predominantly capacitive coupling. Namely, in accordance with Test 1, the coupling mechanism is predominantly resistive up to the angular frequency of 0.1|s_z,1|. Concomitantly, the magnitude at ω << min{|s_z,1|, |s_p,1|} (ω at least a decade lower than the lowest value in the braces) flattens and the frequency response is predominantly defined by the resistive divider, ${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$or 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ in decibels (32), which provides a conductive path at DC:

$$\underset{\omega \to 0}{\lim }{H}_{\mathrm{d}\mathrm{i}\mathrm{v}}\left(j\omega \right)=\underset{\omega \to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}{\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{1+\frac{j\omega }{\frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}}{1+\frac{j\omega }{\frac{1}{{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})}}}}$$

$$={\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}} ,$$

(32)

Conversely, above the angular frequency of 10|s_z,1|, the coupling mechanism is predominantly capacitive. Concomitantly, the magnitude at ω >> max{|s_z,1|, |s_p,1|} (ω at least a decade higher than the highest value in the braces) flattens once again and the frequency response is predominantly defined by the capacitive divider, ${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$or 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$ in decibels (33):

$$\underset{\omega \to \infty }{\lim }{H}_{\mathrm{d}\mathrm{i}\mathrm{v}}\left(j\omega \right)=\underset{\omega \to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{j\omega +\frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{j\omega +\frac{1}{{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})}}}$$

$$={\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}} ,$$

(33)

The influence of C_coupling and R_coupling on the position of the zero can be observed in Figure 17 and Figure 18, respectively.

Aside from the low-side frequency limit, the high-side frequency limit to predominantly capacitive coupling can also be noted. Since only subcase A.1 from Table 1 is considered for any coupling layer given (Assumption 4), once the angular frequency 10|s_z,1| is reached, coupling stays predominantly capacitive up to the highest frequency of interest (Assumption 1). Thus, for the purpose of biopotential measurement, no high-side frequency limit to the predominantly capacitive coupling area will be considered.

Capacitive electrode. The existence of a frequency below which the coupling is not predominantly capacitive is the reason why it is often stated that non-contact and insulated electrodes possess an inherent high-pass frequency characteristic, which would lead to the conclusion they could not be used down to DC (0 Hz). This would be true for pure capacitive electrodes (R_coupling virtually infinite). Indeed, their input stage would be AC-coupled by nature due to the pronounced capacitive character of the coupling impedance and an extremely attenuating resistive voltage divider. Therefore, their H_div(jω) (26) would come down to (34) as a special case of subcase II.1 (|s_p,1| > |s_z,1| and R_IN << R_coupling, Figure 19):

$$H_{\mathrm{d}\mathrm{i}\mathrm{v}}(j\omega ){|}_{R_{\mathrm{coupling}}\to \infty }={\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{j\omega }{j\omega +\frac{1}{R_{\mathrm{I}\mathrm{N}}\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}} .$$

(34)

However, in most practical cases of on-body measurements, which are preferred for biopotential monitoring, R_coupling is finite. Therefore, the influence of finite leakage through the electrode–body interface should be taken into account, as discussed in Section 3.1. Also, an increase in R_coupling is in practice limited by the interface material and ambient conditions. In that sense, the existence of a frequency below which the coupling mechanism is not predominantly capacitive does not necessarily impose a pronounced high-pass behavior, nor does it preclude the usage of non-contact and insulated electrodes below that frequency. Rather, it stresses the methodological issue that has already been commented on in paragraphs “Non-contact on-body electrodes” and “Bias currents and DC biasing” in previous sections: if, e.g., a non-insulated non-contact on-body electrode were used over a moisture-absorbent fabric, R_coupling could decrease to the order of 100 MΩ or even lower with time, which could cause |s_z,1| $=$ ${\displaystyle \frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ to increase. This would in turn expand the predominantly resistive coupling area toward higher frequencies. Thus, it would no longer be correct to treat the electrode as predominantly capacitive across the initial frequency range.

STEP 1: SUMMARY. To sum up, in accordance with Test 1 and Assumption 4, the area of predominantly capacitive coupling spreads above the angular frequency 10|s_z,1|. Thus, the first step in using active electrodes with a predominantly capacitive coupling is to make R_coupling and C_coupling interface parameters sufficiently high so that 10|s_z,1| $=$ ${\displaystyle \frac{10}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ is decreased below the lowest angular frequency of interest (ω_min). This way, the bandwidth of the measured signal will be entirely situated in the area of predominantly capacitive coupling. On the one hand, a higher R_coupling will reduce the leakage through the interface. On the other hand, the higher the C_coupling, the closer to short it will appear at frequencies of the measured biopotential signal (A4), which again enhances the capacitive coupling mechanism. In conclusion, in step 1, parameters R_coupling, C_coupling, and the zero s_z,1 are chosen and fixed.

Figure 17. Impact of increased C_coupling on the magnitude response |H_div(jω)|_dB for case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN) and case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN). Vertical lines denote the ${\displaystyle \frac{10\left|s_{\mathrm{z},1}\right|}{2\pi }}$frequencies, along with the shaded areas of predominantly capacitive coupling. It can be noted that the magnitude response can contain a roll-off in case I due to |s_p,1| ≤ |s_z,1| and that it contains a roll-on in case II due to |s_p,1| > |s_z,1|.

Figure 17. Impact of increased C_coupling on the magnitude response |H_div(jω)|_dB for case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN) and case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN). Vertical lines denote the ${\displaystyle \frac{10\left|s_{\mathrm{z},1}\right|}{2\pi }}$frequencies, along with the shaded areas of predominantly capacitive coupling. It can be noted that the magnitude response can contain a roll-off in case I due to |s_p,1| ≤ |s_z,1| and that it contains a roll-on in case II due to |s_p,1| > |s_z,1|.

Figure 18. Impact of increased R_coupling on the magnitude response |H_div(jω)|_dB for case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN) and case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN). Vertical lines denote the ${\displaystyle \frac{10\left|s_{\mathrm{z},1}\right|}{2\pi }}$frequencies along with the shaded areas of predominantly capacitive coupling. It can be noted that the magnitude response can contain a roll-off in case I due to |s_p,1| ≤ |s_z,1| and that it contains a roll-on in case II due to |s_p,1| > |s_z,1|.

Figure 18. Impact of increased R_coupling on the magnitude response |H_div(jω)|_dB for case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN) and case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN). Vertical lines denote the ${\displaystyle \frac{10\left|s_{\mathrm{z},1}\right|}{2\pi }}$frequencies along with the shaded areas of predominantly capacitive coupling. It can be noted that the magnitude response can contain a roll-off in case I due to |s_p,1| ≤ |s_z,1| and that it contains a roll-on in case II due to |s_p,1| > |s_z,1|.

Figure 19. Impact on the magnitude response |H_div(jω)|_dB when R_coupling goes to infinity. Vertical lines denote the ${\displaystyle \frac{10\left|s_{\mathrm{z},1}\right|}{2\pi }}$frequencies, along with the shaded areas of predominantly capacitive coupling. By increasing R_coupling for all other parameters fixed, the electrode approaches to the concept of a true pure capacitive electrode with a pronounced high-pass frequency response—the special case of subcase II.1 (|s_p,1| > |s_z,1|, i.e., C_IN/C_coupling < R_coupling/R_IN, and R_IN << R_coupling). This will be further investigated in step 2.

Figure 19. Impact on the magnitude response |H_div(jω)|_dB when R_coupling goes to infinity. Vertical lines denote the ${\displaystyle \frac{10\left|s_{\mathrm{z},1}\right|}{2\pi }}$frequencies, along with the shaded areas of predominantly capacitive coupling. By increasing R_coupling for all other parameters fixed, the electrode approaches to the concept of a true pure capacitive electrode with a pronounced high-pass frequency response—the special case of subcase II.1 (|s_p,1| > |s_z,1|, i.e., C_IN/C_coupling < R_coupling/R_IN, and R_IN << R_coupling). This will be further investigated in step 2.

STEP 2: Manipulating the attenuation of the input voltage divider. In the previous step, the equivalent coupling impedance Z_coupling, along with its parameters, R_coupling and C_coupling, was chosen and fixed. Consequently, the zero s_z,1 $=$ $-{\displaystyle \frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ (19) was chosen and fixed, as well as its angular break frequency |s_z,1| and the area of predominantly capacitive coupling, with its low-side limit at an angular frequency of 10|s_z,1|. Aside from the area of predominantly capacitive coupling, the second important aspect is the magnitude response |H_div(jω)|_dB and its attenuation, which arises from the resistive and capacitive voltage dividers. To observe their influence, the other two parameters of the interface, C_IN and R_IN, should be accounted for, along with their effect on the pole s_p,1 $=$ $-{\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$ (18) and its angular break frequency |s_p,1|. First, the influence of C_IN and R_IN on the voltage divider attenuation and the pole position will be inspected. Next, cutoff frequencies will be defined to emphasize the possible distinction between the area of predominantly capacitive coupling and the area of predominant capacitive divider. Lastly, the preferred position of |s_p,1| with respect to the fixed position of |s_z,1| will be observed based on possible configurations of the electrode–body interface: high-pass, all-pass, and low-pass. The summary of the analysis is available at the end of step 2.

Influence of C_IN. Firstly, the capacitive voltage divider effect can be tackled. It dominates the magnitude response at ω >> max{|s_z,1|, |s_p,1|}, simplifying it to ${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$ (33). As seen from the shape of the capacitive divider, since C_coupling is fixed in step 1, this magnitude level comes down to dependence on C_IN. To minimize the capacitive divider attenuation and raise its magnitude toward the maximum value of 1 (or 0 dB), C_IN should be 0, i.e., as low as possible. To achieve this, requirement C_coupling >> C_IN (i.e., C_coupling ≥ 10C_IN) should be satisfied. This will reduce the attenuation to 1/11 in the case when C_coupling $=$ 10C_IN, or to a lower value when C_coupling > 10C_IN. Therefore, the rest of the analysis in the paper will be carried out with the aim of fulfilling the condition C_coupling >> C_IN. However, as seen from |s_p,1| $=$ ${\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$, reducing C_IN increases |s_p,1| (Figure 20). Ultimately, minimizing C_IN to the ideal value of 0 would maximize the |s_p,1| value for all other parameters fixed. This and similar interdependencies between the pole and the interface parameters will prove to be of use later in this step.

Influence of R_IN. On the other side of the frequency response, at the lowest frequencies, the resistive voltage divider effect can be investigated. It dominates the magnitude response at ω << min{|s_z,1|, |s_p,1|}, simplifying it to ${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ (32). As seen from the shape of the resistive divider, since R_coupling is fixed in step 1, this magnitude level comes down to dependence on the fourth variable of the interface, R_IN. The importance of R_IN has already been noticeable in Figure 19, where the creation of a pure capacitive electrode was depicted through an increase in R_coupling (for all other parameters fixed). Therein, once R_coupling reached the value of 10 GΩ, which was chosen to be 10R_IN, further increase in R_coupling allowed for a further increase in attenuation of the resistive divider and further extension of the predominantly capacitive coupling area toward lower frequencies, but the angular frequency |s_p,1| $=$ ${\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$ remained unchanged because it became dominated by R_IN << R_coupling. This influence of R_IN on the pole s_p,1 can be generalized: for a given R_coupling and C_coupling (step 1) and for the chosen C_IN, lower R_IN would decrease the R_coupling||R_IN parallel, increasing the value of |s_p,1|. This means that the highest possible |s_p,1| for a given R_coupling and C_coupling would be achieved when both C_IN and R_IN are minimized. In other words, the by-product of achieving the least attenuating capacitive voltage divider and the most attenuating resistive voltage divider would be maximization of |s_p,1|. Conversely, a higher R_IN increases the R_coupling||R_IN parallel, lowering the value of |s_p,1| $=$ ${\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$ for the chosen C_IN (Figure 21).

To further analyze the influence of R_IN, subcases 1 and 2 determined by the R_coupling/R_IN ratio can be included in the analysis. As shown earlier in Figure 16, case I (|s_p,1| ≤ |s_z,1|) results in a magnitude response that can contain a roll-off portion, whereas case II (|s_p,1| > |s_z,1|) results in a magnitude response that certainly contains a roll-on portion. Now going a step further, as shown in Figure 22 for subcase I.1 and I.2, and in Figure 23 for subcase II.1 and II.2, the achieved magnitude levels 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ in subcases 1 (R_IN < R_coupling) are of values below 20log₁₀0.5 $\approx$ −6 dB (50% attenuation). On the other hand, in subcases 2 (R_IN ≥ R_coupling), the achieved magnitude levels 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ are greater than or equal to about −6 dB. In that sense, subcases 1 (R_IN < R_coupling) represent the cases of aggressive resistive voltage divider (50% or more attenuation) and allow a maximization of resistive divider attenuation. On the contrary, subcases 2 (R_IN ≥ R_coupling) allow a minimization of the resistive voltage divider attenuation (50% or less attenuation). The rough boundary between subcases 1 and subcases 2 is the −6 dB magnitude level (50% attenuation) at the low-frequency end.

Limitations on minimizing C_IN. In previous paragraphs, influence of C_IN and R_IN on the voltage attenuation and the pole position were addressed separately. Now, the combined effect of the capacitive and the resistive voltage divider can be examined to inspect how the achievable values of the C_IN/C_coupling ratio are affected by the R_coupling/R_IN ratio. For each of the four subcases and for a fixed R_coupling, C_coupling, and |s_z,1|, a change in R_IN and C_IN corresponds to two effects: first, a vertical shift in the magnitude of the resistive and capacitive voltage dividers, 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ and 20log₁₀${\displaystyle \frac{C_{\mathrm{coupling}}}{C_{\mathrm{coupling}}+C_{\mathrm{IN}}}}$ respectively, and second, a horizontal shift in the |s_p,1| angular break frequency. However, the range of achievable values for C_IN and R_IN might not be unconstrained: only in subcase II.1 (C_IN/C_coupling < R_coupling/R_IN and R_IN < R_coupling) can C_IN be chosen to be as low as possible for any given value of R_coupling, R_IN, and C_coupling for which the subcase is applicable (subcase II.1 in Figure 23). In the rest of the subcases, a decrease in the C_IN/C_coupling ratio is limited by the R_coupling/R_IN ratio. Specifically, prerequisites for subcase I.1, C_IN/C_coupling ≥ R_coupling/R_IN and R_IN < R_coupling, lead to the requirement C_IN > C_coupling. This means that in subcase I.1, both the capacitive voltage divider 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$ and the resistive voltage divider 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ attenuate at least 50% of the input signal. As a result, the magnitude response |H_div(jω)|_dB in subcase I.1 is entirely situated below the −6 dB magnitude level. Therefore, subcase I.1 is not a practical solution for biopotential measurement (subcase I.1 in Figure 22). On the other hand, in subcase I.2 (C_IN/C_coupling ≥ R_coupling/R_IN and R_IN ≥ R_coupling), condition C_coupling $=$ 10C_IN requires R_IN ≥ 10R_coupling. To generalize this with the use of a positive real factor T ≥ 10, condition C_coupling $=$ T$·$C_IN requires R_IN ≥ T$·$R_coupling. In other words, in subcase I.2, a less attenuating resistive divider (lower R_coupling/R_IN ratio) allows for the realization of a less attenuating capacitive divider (lower C_IN/C_coupling ratio). Specifically, since R_coupling and C_coupling are fixed in step 1, this means that a higher R_IN allows us to achieve a lower C_IN (subcase I.2 in Figure 22). Lastly, in subcase II.2 (C_IN/C_coupling < R_coupling/R_IN and R_IN ≥ R_coupling), condition C_coupling $=$ 10C_IN is feasible when R_IN < 10R_coupling. To generalize this once again with the use of a positive real factor T ≥ 10, condition C_coupling $=$ T$·$C_IN is feasible whenever R_IN < T$·$R_coupling. In other words, minimization of the capacitive voltage divider attenuation imposes a limit on the minimization of the resistive voltage divider attenuation (subcase II.2 in Figure 23).

Cutoff frequencies and the area of predominant divider. Aside from observing the magnitude levels of the resistive and capacitive voltage dividers, it is also interesting to observe the frequencies at which these magnitude levels are reached with respect to the areas of predominant coupling mechanisms. In particular, the area in which the magnitude response |H_div(jω)|_dB simplifies to the capacitive divider 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$ (the area of predominant capacitive divider) (33) can be observed with respect to the area of predominantly capacitive coupling. Similarly, the area in which the magnitude response |H_div(jω)|_dB simplifies to the resistive divider 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ (the area of predominant resistive divider) (32) can be observed with respect to the area of predominantly resistive coupling. So far, it has been stated that the area of predominant resistive divider corresponds to angular frequencies ω << min{|s_z,1|, |s_p,1|}, and that the area of predominant capacitive divider corresponds to angular frequencies ω >> max{|s_z,1|, |s_p,1|}. Instead of using vague mathematical definitions, these frequency limits can be estimated with the use of conventional cutoff frequencies, as introduced in Appendix D and [347]. For the case when |s_p,1| ≤ |s_z,1| (case I, Figure 16), the magnitude response |H_div(jω)|_dB can contain a roll-off; hence, ω_R,I at low-end frequencies can be defined as the angular frequency at which the descending |H_div(jω)|_dB falls to a value that is 3 dB below 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ (35,36):

$$\mathrm{C}\mathrm{A}\mathrm{S}\mathrm{E} \mathrm{I}:{\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{\sqrt{1+{\left({\omega }_{\mathrm{R},\mathrm{I}}{R}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right)}^2}}{\sqrt{1+{\left[{\omega }_{\mathrm{R},\mathrm{I}}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})\right]}^2}}}$$

$$\approx {\displaystyle \frac{1}{\sqrt{2}}}·{\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}},$$

(35)

$${\omega }_{\mathrm{R},\mathrm{I}}={\displaystyle \frac{1}{\sqrt{{\left[{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})\right]}^2-2{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}·C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right)}^2}}}$$

$$={\displaystyle \frac{1}{\sqrt{{\left(\frac{1}{s_{\mathrm{p},1}}\right)}^2-2{\left(\frac{1}{s_{\mathrm{z},1}}\right)}^2}}}.$$

(36)

At the high-end side, ω_C,I in case I can be defined as the angular frequency at which the descending |H_div(jω)|_dB falls to a value that is 3 dB above 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$ (37,38):

$$\mathrm{C}\mathrm{A}\mathrm{S}\mathrm{E} \mathrm{I}:{\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{\sqrt{{{\omega }^2}_{\mathrm{C},\mathrm{I}}+{\left(\frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}\right)}^2}}{\sqrt{{{\omega }^2}_{\mathrm{C},\mathrm{I}}+{\left(\frac{1}{{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})}\right)}^2}}}$$

$$\approx \sqrt{2}·{\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}} ,$$

(37)

$${\omega }_{\mathrm{C},\mathrm{I}}=\sqrt{{\left[{\displaystyle \frac{1}{{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})}}\right]}^2-2{\left({\displaystyle \frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}\right)}^2 }$$

$$=\sqrt{{s_{\mathrm{z},1}}^2-2{s_{\mathrm{p},1}}^2} .$$

(38)

This pair of cutoff frequencies ω_R,I and ω_C,I for case I exists whenever (39) is satisfied:

$$\mathrm{C}\mathrm{A}\mathrm{S}\mathrm{E} \mathrm{I}:{{|s}_{\mathrm{p},1}|\le \omega }_{\mathrm{R},\mathrm{I}},{\omega }_{\mathrm{C},\mathrm{I}}\le \left|s_{\mathrm{z},1}\right|\to {\displaystyle \frac{\left|s_{\mathrm{z},1}\right|}{\left|s_{\mathrm{p},1}\right|}}={\displaystyle \frac{1+\frac{C_{\mathrm{I}\mathrm{N}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}{1+\frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{R_{\mathrm{I}\mathrm{N}}}}}\ge \sqrt{3} .$$

(39)

The resulting calculations are summarized in Table 2 and visualized in Figure 24 (case I).

Similarly, in the case when |s_p,1| > |s_z,1| (case II, Figure 16), the magnitude response |H_div(jω)|_dB contains a roll-on; hence, ω_R,II at low-end frequencies can be defined as the angular frequency at which the ascending |H_div(jω)|_dB increases to a value that is 3 dB above 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ (40,41):

$$\mathrm{C}\mathrm{A}\mathrm{S}\mathrm{E} \mathrm{I}\mathrm{I}:{\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{\sqrt{1+{\left({\omega }_{\mathrm{R},\mathrm{I}\mathrm{I}}{R}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right)}^2}}{\sqrt{1+{\left[{\omega }_{\mathrm{R},\mathrm{I}\mathrm{I}}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})\right]}^2}}}$$

$$\approx \sqrt{2}·{\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}},$$

(40)

$${\omega }_{\mathrm{R},\mathrm{I}\mathrm{I}}={\displaystyle \frac{1}{\sqrt{{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}·C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right)}^2-2{\left[{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})\right]}^2}}}$$

$$={\displaystyle \frac{1}{\sqrt{{\left(\frac{1}{s_{\mathrm{z},1}}\right)}^2-2{\left(\frac{1}{s_{\mathrm{p},1}}\right)}^2}}}.$$

(41)

At the high-end side, ω_C,II in case II can be defined as the angular frequency at which the ascending |H_div(jω)|_dB increases to a value that is 3 dB below $20{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}{\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$ (42,43):

$$\mathrm{C}\mathrm{A}\mathrm{S}\mathrm{E} \mathrm{I}\mathrm{I}:{\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}·{\displaystyle \frac{\sqrt{{{\omega }^2}_{\mathrm{C},\mathrm{I}\mathrm{I}}+{\left(\frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}\right)}^2}}{\sqrt{{{\omega }^2}_{\mathrm{C},\mathrm{I}\mathrm{I}}+{\left(\frac{1}{{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})}\right)}^2}}}$$

$$\approx {\displaystyle \frac{1}{\sqrt{2}}}·{\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}} ,$$

(42)

$${\omega }_{\mathrm{C},\mathrm{I}\mathrm{I}}=\sqrt{{\left[{\displaystyle \frac{1}{{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right||R}_{\mathrm{I}\mathrm{N}}\left)\right(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\left|\right|C_{\mathrm{I}\mathrm{N}})}}\right]}^2-2{\left({\displaystyle \frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}\right)}^2 }$$

$$=\sqrt{{s_{\mathrm{p},1}}^2-2{s_{\mathrm{z},1}}^2} .$$

(43)

This pair of cutoff frequencies ω_R,II and ω_C,II for case II exists whenever (44) is satisfied:

$$\mathrm{C}\mathrm{A}\mathrm{S}\mathrm{E} \mathrm{I}\mathrm{I}: |s_{\mathrm{z},1}|{\le \omega }_{\mathrm{R},\mathrm{I}\mathrm{I}},{\omega }_{\mathrm{C},\mathrm{II}}\le {|s}_{\mathrm{p},1}|\to {\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{\left|s_{\mathrm{z},1}\right|}}={\displaystyle \frac{1+\frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{R_{\mathrm{I}\mathrm{N}}}}{1+\frac{C_{\mathrm{I}\mathrm{N}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}}\ge \sqrt{3} .$$

(44)

The resulting calculations are again summarized in Table 2 and visualized in Figure 24 (case II).

Area of predominantly capacitive coupling with respect to the area of predominant capacitive divider. Unlike the 10|s_z,1| value, which determines the low-side frequency of the predominantly capacitive coupling area (step 1), conventional angular cutoff frequencies ω_C,I and ω_C,II determine the low-side frequency of the predominant capacitive divider area with a 3 dB tolerance. In case I, the condition |s_p,1| ≤ |s_z,1| is valid, hence |s_z,1| ≥ ω_C,I. Also, 10|s_z,1| > ω_C,I. Thus, in case I, the area of predominantly capacitive coupling implies the area of predominant capacitive divider. Conversely, in case II, the condition |s_p,1| > |s_z,1| is valid. Therefore, ω_C,II might be higher than 10|s_z,1| and, consequently, the area of predominantly capacitive coupling would begin at lower frequencies than the area of predominant capacitive divider. Specifically, for this condition ω_C,II > 10|s_z,1| to occur, requirement |s_p,1| > $\sqrt{102}$|s_z,1| should be fulfilled (43), which can be achieved only in subcase II.1 (C_IN/C_coupling < R_coupling/R_IN and R_IN < R_coupling) (44). The existence of this special case reveals why it is important to keep track of both |s_z,1| and |s_p,1| values and also track both the area of predominantly capacitive coupling and the area of predominant capacitive divider with respect to the bandwidth of the measured signal. This will be further discussed in step 3.

Interdependence of the difference between the pole and the zero and the difference between the magnitude levels of the resistive and capacitive voltage dividers. The use of the derived pairs of angular cutoff frequencies, ω_R,I and ω_C,I for case I, and ω_R,II and ω_C,II for case II, is three-fold (Figure 24). First, ω_R,I and ω_R,II estimate the high-side frequency of the predominant resistive divider area, whereas ω_C,I and ω_C,II estimate the low-side frequency of the predominant capacitive divider area, as explained in previous paragraphs. Second, the difference between the cutoff frequencies in a pair indicates the extent of the difference between the magnitude levels 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ and 20log₁₀${\displaystyle \frac{C_{\mathrm{coupling}}}{C_{\mathrm{coupling}}+C_{\mathrm{IN}}}}$, as well as the extent of their vertical distance in the magnitude response. Third, the difference between the cutoff frequencies in a pair also indicates the extent of the difference between the zero s_z,1 and the pole s_p,1 in the complex s-plane, as well as the extent of the difference between their corresponding angular break frequencies |s_z,1| and |s_p,1|. Accordingly, whenever a pair of these angular cutoff frequencies exists, which is determined by (39) and (44) respectively, the difference between the magnitude levels of the resistive and capacitive voltage dividers will be at least 3 dB, and one of the values |s_z,1| and |s_p,1| will be at least $\sqrt{3}$ greater than the other. Naturally, if the difference between the respective magnitude levels is at least 3 dB, but less than 6 dB, cutoff frequencies will overlap (ω_R,I > ω_C,I and ω_R,II > ω_C,II). Once the difference becomes exactly 6 dB, cutoff frequencies in a pair will be equal (ω_R,I $=$ ω_C,I and ω_R,II $=$ ω_C,II), as in dashed red examples in Figure 24. As the difference between the magnitude levels is further increased, conditions ω_C,I > ω_R,I and ω_C,II > ω_R,II are valid. Moreover, an interdependency exists: the further apart the pole and the zero are set from one another, the greater the difference between the magnitude levels of the dividers. This is in accordance with the Bode approximation of contribution of s_z,1 and s_p,1 to the magnitude response, depicted in Figure A9: above angular break frequency |s_z,1|, the slope of |H_div(jω)|_dB increases for 20 dB/decade, whereas above |s_p,1|, it decreases for 20 dB/decade. Thus, the greater the difference between s_z,1 and s_p,1, the greater the achieved shift in magnitude between |s_z,1| and |s_p,1|. This can be observed separately for case I, which is based on the |s_z,1|/|s_p,1| ratio (39), and for case II, which is based on the |s_p,1|/|s_z,1| ratio (44). To inspect the interdependence of the increasing difference between angular break frequencies |s_z,1| and |s_p,1|, and the increasing difference between the magnitude levels of the dividers, the magnitude response can be observed when the ratios in (39) and (44) are increased. Specifically, in case I (|s_p,1| ≤ |s_z,1|), the ratio is |s_z,1|/|s_p,1|. To increase it for a fixed value of |s_z,1|, s_p,1 should be moved away from s_z,1 toward the origin of the complex s-plane, i.e., |s_p,1| should be reduced. As seen in (39), the ratio |s_z,1|/|s_p,1| can be increased by decreasing R_coupling/R_IN and by increasing C_IN/C_coupling. For a fixed R_coupling and C_coupling, this means that both R_IN and C_IN should be increased, which is in accordance with the minimization of |s_p,1| $=$ ${\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$. Graphically, an increase in R_IN corresponds to an upward shift of the resistive divider level 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$, whereas an increase in C_IN corresponds to a downward shift of the capacitive divider level 20log₁₀${\displaystyle \frac{C_{\mathrm{coupling}}}{C_{\mathrm{coupling}}+C_{\mathrm{IN}}}}$ (case I in Figure 24). In total, with an increase in R_IN and C_IN, |s_p,1| decreases, |s_p,1| and |s_z,1| are further separated, and the realized magnitude response will have a more pronounced roll-off (low-pass characteristic). On the other hand, in case II (|s_p,1| > |s_z,1|), the ratio is |s_p,1|/|s_z,1|. To increase it for a fixed value of |s_z,1|, s_p,1 should be once again moved away from s_z,1, but this time away from the origin of the complex s-plane, so its value |s_p,1| should now be increased. As seen in (44), the ratio |s_p,1|/|s_z,1| can be increased by increasing R_coupling/R_IN and by decreasing C_IN/C_coupling. For a fixed R_coupling and C_coupling, this means that both R_IN and C_IN should be decreased, which is in accordance with the maximization of |s_p,1| $=$ ${\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$. Graphically, a decrease in R_IN corresponds to a downward shift of the resistive divider level 20log₁₀${\displaystyle \frac{C_{\mathrm{coupling}}}{C_{\mathrm{coupling}}+C_{\mathrm{IN}}}}$, whereas a decrease in C_IN corresponds to an upward shift of the capacitive divider level 20log₁₀${\displaystyle \frac{C_{\mathrm{coupling}}}{C_{\mathrm{coupling}}+C_{\mathrm{IN}}}}$ (case II in Figure 24). In total, with a decrease in R_IN and C_IN, |s_p,1| increases, |s_z,1| and |s_p,1| are further separated, and the realized magnitude response will have a more pronounced roll-on (high-pass characteristic). These effects will be applied in the following paragraphs to explore the feasibility of practical configurations of the interface between the body and a non-contact and/or insulated biopotential electrode.

High-pass configuration. Finally, the previous findings can be used to further inspect the practicality and usability of the specific subcases. Intuitively, the design of non-contact and insulated electrodes and their interface with the body could strive for the concept of a purely capacitive electrode (34), as depicted earlier in Figure 19. In that case, the electrode would be treated as a high-pass, AC-coupling filter with the predominantly resistive coupling area as the stopband and the predominantly capacitive coupling area as the passband. This means that the attenuation of the capacitive voltage divider should be minimized (minimum C_IN/C_coupling), whereas the attenuation of the resistive voltage divider should be maximized (maximum R_coupling/R_IN). For a fixed R_coupling and C_coupling, this requires a minimization of both C_IN and R_IN. To allow a roll-on in the magnitude response, case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN, Figure 16 and Figure 23) is desirable. Appropriately, ω_C,II (43) can be used as the low-side cutoff frequency of the system. Accordingly, when ω_C,II > ω_R,II, i.e., when $20{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}{\displaystyle \frac{C_{\mathrm{coupling}}}{C_{\mathrm{coupling}}+C_{\mathrm{IN}}}}-20{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}{\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ is greater than 6 dB, the frequency band between ω_R,II and ω_C,II can be treated as the transition band. In addition, separating the |s_p,1| value from the |s_z,1| value is desirable to achieve a greater distinction in magnitude levels between the passband and the stopband region. More specifically, since |s_p,1| > |s_z,1| in case II, |s_p,1| $=$ ${\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$ should be increased, which is in accordance with the reduction in R_IN and C_IN. For this reason, subcase II.1 would be preferred (R_IN < R_coupling, Figure 23), since it allows for fulfilling the requirement C_coupling >> C_IN and at the same time maximizing the ratio R_coupling/R_IN without any limitations. An example of a high-pass configuration design can be found in [380]. However, a caveat arising from the discussions in the previous paragraphs should be kept in mind. Namely, for subcase II.1, it can be recalled that if |s_p,1| < $\sqrt{102}$|s_z,1|, the condition 10|s_z,1| > ω_C,II is valid. Under such circumstances, the area of predominant capacitive divider as the passband is not entirely situated in the predominantly capacitive coupling area, so it would be particularly erroneous to treat the electrode as capacitive. In other words, |s_p,1| in the high-pass configuration for approximating the pure capacitive electrode should be at least $\sqrt{102}$ times higher than |s_z,1|. In Figure 19, this is achieved in the dotted blue example. However, an increase in |s_p,1| increases ω_C,II (43) and decreases ω_R,II (41). Thus, the higher the |s_p,1| value with respect to |s_z,1|, the closer the electrode to a pure capacitive electrode, but the wider the transition band in its frequency response. Because of this, such tendencies to create a sharp distinction between magnitude levels in the area of predominantly resistive coupling and the area of predominantly capacitive coupling could limit the use of such pronouncedly capacitive biopotential electrodes in clinical-grade applications, where sensing the sub-Hz frequencies could be important for accurate diagnostics. This will be further discussed in step 3 in terms of phase response.

All-pass configuration. Aside from the high-pass configuration, another option is to treat the electrode and the electrode–body interface as an all-pass system, but again with the objective of maximizing the area of predominantly capacitive coupling and the area of predominant capacitive divider. For this purpose, keeping the pole s_p,1 as close as possible to the zero s_z,1 (i.e., bringing the |s_p,1| value closer to the |s_z,1| value) would be desirable to reduce the difference between magnitude levels of the resistive and capacitive voltage dividers. Ideally, if the pole s_p,1 were canceled with the zero s_z,1, the levels of attenuation from the resistive and capacitive voltage dividers would be equal and there would be no cutoff frequencies. The frequency response H_div(jω) would come down to a constant level of attenuation as in (45)

$$H_{\mathrm{d}\mathrm{i}\mathrm{v}}\left(j\omega \right){|}_{s_{\mathrm{p},1}=s_{\mathrm{z},1}}={\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}={\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}} .$$

(45)

For this condition s_z,1 $=$ s_p,1 to happen, ratios C_IN/C_coupling and R_coupling/R_IN should be equal (46). This is not surprising, since this derived relation between |s_p,1| and |s_z,1| (46) has shown that the input voltage divider effect is the reason why a pole exists in the first place. Specifically, if the ratios C_IN/C_coupling and R_coupling/R_IN were minimized and equal to zero, the achieved gain of the magnitude response would be raised toward its maximum value of unity (0 dB). This shows that for an all-pass configuration of the interface, attenuation of both the capacitive and the resistive voltage divider should be minimized. For a fixed R_coupling and C_coupling, this means that minimization of C_IN and maximization of R_IN is beneficial. To inspect which subcases can be used for the all-pass configuration of the electrode–body interface, each of the four subcases can be further observed in terms of the preferable pole position and in terms of fulfilling the requirement C_coupling >> C_IN. For this purpose, the result from (31) can be recalled (46) and rewritten:

$$\left|s_{\mathrm{p},1}\right|=\left|s_{\mathrm{z},1}\right|{\displaystyle \frac{1+\frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{R_{\mathrm{I}\mathrm{N}}}}{1+\frac{C_{\mathrm{I}\mathrm{N}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}}=\left|s_{\mathrm{z},1}\right|{\displaystyle \frac{\frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}{\frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{I}\mathrm{N}}+R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}}=\left|s_{\mathrm{z},1}\right|{\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}+R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{R_{\mathrm{I}\mathrm{N}}}}{\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}} .$$

(46)

To use case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN, Figure 16 and Figure 22) for the all-pass configuration, |s_p,1| value should be maximized. Specifically, only subcase I.2 (R_IN ≥ R_coupling, Figure 22) enables the desired requirement C_coupling >> C_IN to be fulfilled. In fact, this is the only subcase that allows combinations C_IN/C_coupling $=$ R_coupling/R_IN, which would truly yield |s_p,1| $=$ |s_z,1| and an all-pass configuration with a constant level of attenuation. To further reduce the attenuation and raise the flattened magnitude response, C_IN/C_coupling and R_coupling/R_IN can be equally decreased. For a fixed R_coupling and C_coupling, the requirement C_coupling $=$ 10C_IN would call for R_IN ≥ 10R_coupling. Generally, for a positive real factor T ≥ 10, requirement C_coupling $=$ T$·$C_IN would call for R_IN ≥ T$·$R_coupling. In the ideal case of C_IN $=$ 0 and infinite R_IN, unity gain would be achieved. In other words, although maximizing |s_p,1| $=$ ${\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$ is desirable in subcase I.2, maximizing R_IN is still beneficial because it allows for further decrease in C_IN (subcase I.2 in Figure 25).

On the other hand, case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN, Figure 16 and Figure 23) can be considered for the all-pass configuration. The difference with respect to case I is that the |s_p,1| value should now be minimized. Minimizing C_IN increases |s_p,1| $=$ ${\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$, but maximizing R_IN compensates for this increase. Therefore, minimum C_IN and maximum R_IN can still be the preferred choice of parameters. Specifically, in subcase II.1 (R_IN < R_coupling, Figure 23), requirement C_coupling >> C_IN can be fulfilled without any limitations. The lowest C_IN is 0, and the highest achievable R_IN is slightly lower than R_coupling. In that case, the achieved |s_p,1| is slightly higher than 2|s_z,1| (46), and this is the closest that |s_p,1| can be to |s_z,1| in subcase II.1. For C_IN $=$ 0, value |s_p,1| would further increase toward ${\displaystyle \frac{1}{R_{\mathrm{I}\mathrm{N}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ as R_IN is decreased to 0 (46) (subcase II.1 in Figure 25). This shows once again how s_p,1 becomes dominated by R_IN when R_coupling >> R_IN and C_coupling >> C_IN. Lastly, in subcase II.2 (R_IN ≥ R_coupling, Figure 23), minimization of C_IN limits the maximization of R_IN. In particular, requirement C_coupling $=$ 10C_IN calls for R_IN < 10R_coupling. Generally, for a positive real factor T ≥ 10, requirement C_coupling $=$ T$·$C_IN calls for R_IN < T$·$R_coupling. However, this does not pose a severe issue on the large scale: the lowest achievable C_IN is 0, and to achieve it, maximum R_IN can be any finite number higher than R_coupling. Therefore, |s_p,1| could theoretically be decreased to a value that is infinitesimally larger than |s_z,1|. For C_IN $=$ 0, as R_IN is decreased to its minimum (R_IN $=$ R_coupling), |s_p,1| would increase toward 2|s_z,1| (46) (subcase II.2 in Figure 25).

Now, the three subcases I.2, II.1, and II.2 can be compared to determine which one is preferred for the all-pass configuration. In case II, |s_p,1| > |s_z,1|, so |s_p,1| can never be equal to |s_z,1|, and as a result, a roll-on will always be present in the magnitude response. On the other hand, in subcase I.2, |s_p,1| ≤ |s_z,1|, so |s_p,1| can theoretically be equal to |s_z,1|. However, if it is not, then a roll-off is present in the magnitude response, which would reduce the gain in the area of predominantly capacitive coupling. Also, minimization of C_IN down to 0 is feasible, but it must be accompanied by maximization of R_IN toward infinity. In particular, if a positive real factor T ≥ 10 is used once again, requirement C_coupling $=$ T$·$C_IN calls for R_IN ≥ T$·$R_coupling, and whenever these two conditions are fulfilled, |s_z,1|/|s_p,1| ratio is lower than $\sqrt{3}$ (39). Consequently, cutoff frequencies do not exist, and the difference between the magnitude levels of the resistive and capacitive voltage dividers is less than 3 dB. On the contrary, in subcase II.1, minimization of C_IN is unconstrained. However, because R_IN < R_coupling, attenuation of the resistive voltage divider is at least 50% (roughly −6 dB), so whenever C_coupling ≥ 10C_IN is fulfilled, which corresponds to 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$ ≥ −0.8 dB, cutoff frequencies exist. Namely, in accordance with (44), cutoff frequencies in subcase II.1 do not exist when (47) is satisfied:

$$R_{\mathrm{I}\mathrm{N}}>{\displaystyle \frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{\sqrt{3}(1+\frac{C_{\mathrm{I}\mathrm{N}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}})-1}} .$$

(47)

Under requirement C_coupling ≥ 10C_IN, this would require R_IN > R_coupling, which is in contradiction with the prerequisite for subcase II.1 (R_IN < R_coupling). In the ideal case when C_IN would be minimized to 0, the smallest achievable difference between magnitude levels of the resistive and capacitive voltage dividers would still be roughly 6 dB. Therefore, although minimization of C_IN in subcase II.1 is unconstrained, it will always yield the existence of cutoff frequencies. Lastly, in subcase II.2, for a positive real factor T ≥ 10, requirement C_coupling $=$ T$·$C_IN calls for R_IN < T$·$R_coupling, and whenever these two conditions are fulfilled, the |s_p,1|/|s_z,1| ratio is lower than $\sqrt{3}$, cutoff frequencies do not exist, and the difference between the magnitude levels of the resistive and capacitive voltage dividers is less than 3 dB (44). Compared to subcase I.2, the constraint on minimizing C_IN and maximizing R_IN is now relaxed, because minimization of C_IN down to 0 can be achieved for any finite R_IN. In conclusion, among the three subcases feasible for the all-pass configuration (subcase I.2, II.1, II.2), subcase II.2 is the preferred one, since it allows us to bring the pole in the vicinity of the zero without the possibility for creating a roll-off, whilst relaxing the constraints on the minimization of C_IN and maximization of R_IN. Once again, since |s_p,1| > |s_z,1| in case II, both the area of predominantly capacitive coupling and the area of predominant capacitive divider should be checked and ensured.

Low-pass configuration. Naturally, the third option—a low-pass configuration—can be discussed. This configuration is a conceptual inverse of the high-pass configuration: it would use the area of predominantly resistive coupling as the passband and the area of predominantly capacitive coupling as the stopband. In that sense, maximization of both R_IN and C_IN would be preferred for a fixed R_coupling and C_coupling. Using subcase I.2 (|s_p,1| ≤ |s_z,1| and R_IN ≥ R_coupling, Figure 22) would be desirable to allow for a roll-off in the magnitude response without imposing an upper limit to the increase in R_IN. Appropriately, ω_R,I (36) would be used as the high-side cutoff frequency of the system, and the frequency band between ω_R,I and ω_C,I would be treated as the transition band whenever ω_C,I > ω_R,I. However, it can be seen that this configuration would favor the resistive character of the coupling impedance Z_coupling and that it would require measures opposite to the ones that have been preferred so far. In particular, it requires the extension of the predominantly resistive coupling area toward higher frequencies, the maximization of R_IN and C_IN, and the minimization of R_coupling and C_coupling. Therefore, it contradicts the rationale behind using the non-contact and insulated electrodes, and it would be rather the preferred choice for non-insulated surface-contact electrodes. Nevertheless, as discussed in paragraphs “Non-contact on-body electrodes” and “Bias currents and DC biasing” and step 1, the electrode–body interface for non-contact and insulated electrodes could be inadvertently driven into this configuration in the events of decreased R_coupling or C_coupling.

STEP 2: SUMMARY. By choosing R_coupling and C_coupling in step 1, the equivalent coupling impedance Z_coupling was fixed, as well as the zero s_z,1 and its angular break frequency |s_z,1|. Now, in step 2, the other two parameters of the interface were investigated: C_IN and R_IN, which define the equivalent input impedance Z_IN. The derived relation between |s_p,1| and |s_z,1| (31,46) revealed that the existence of the pole s_p,1 is a direct consequence of the input voltage divider effect. Therefore, parameters C_IN and R_IN have been shown to affect not only the attenuation of the resistive and capacitive voltage dividers (20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ and 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$), but also the pole s_p,1 and its angular break frequency |s_p,1|. Based on Bode approximation of contribution of s_z,1 and s_p,1 to the magnitude response, an interdependence was discovered: greater difference between |s_z,1| and |s_p,1| draws a greater difference between the magnitude levels of the resistive and capacitive voltage dividers, and vice versa. This was quantified by pairs of cutoff frequencies: ω_R,I and ω_C,I for case I (36,38,39), and ω_R,II and ω_C,II for case II (41,43,44). In general, while 10|s_z,1| defined the low-side frequency of the predominantly capacitive coupling area (step 1), ω_C,I and ω_C,II define the low-side frequency of the predominant capacitive divider area. Non-contact and insulated biopotential electrodes rely on predominantly capacitive coupling, hence they should be used in both specified areas. To determine the frequency range across which these two areas overlap, each of the four established subcases was investigated separately. For this purpose, two essential differences between the subcases were identified and visualized on curves of the magnitude response |H_div(jω)|_dB. The first difference can be used to distinguish case I from case II: in case I, |H_div(jω)|_dB can contain a roll-off due to |s_p,1| ≤ |s_z,1|, whereas in case II, it contains a roll-on due to |s_p,1| >| s_z,1|. The second difference can be used to distinguish subcases 1 from subcases 2: in subcases 1, the achieved magnitude levels of the resistive voltage divider, 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$, are of values below 20log₁₀0.5 $\approx$ −6 dB (50% or more attenuation) due to R_IN < R_coupling, whereas in subcases 2, they are of values above −6 dB (50% or less attenuation) due to R_IN ≥ R_coupling. These characteristics were further observed in the context of two specified areas for each of the four subcases. Specifically, subcase I.1 (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN, and R_IN < R_coupling) is not a practical solution for biopotential measurement, since both the resistive and capacitive voltage dividers attenuate at least 50% of the input signal. Therefore, the analysis continued for subcases I.2, II.1, and II.2. In case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN) and subcase II.2 (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN, and R_IN ≥ R_coupling), the range ω ≥ 10|s_z,1| ensures that both the predominantly capacitive coupling area and the predominant capacitive divider area are used. However, in subcase II.1 (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN, and R_IN < R_coupling), the area of predominant capacitive coupling does not imply the area of predominant capacitive divider if |s_p,1| > $\sqrt{102}$|s_z,1|. In that case, the range ω ≥ ω_C,II ensures that both specified areas are used. This special case shows why it is important to check the values of both |s_z,1| and |s_p,1| and also track both the predominantly capacitive coupling area and the predominant capacitive divider area with respect to the bandwidth of the measured signal. Finally, based on the favorable position of s_p,1 with respect to the fixed s_z,1, possible configurations of the electrode–body interface were investigated. As a tool for relocating the pole s_p,1, values of ratios R_coupling/R_IN and C_IN/C_coupling were manipulated. The first analyzed configuration—high-pass configuration, relies on the concept of a purely capacitive electrode (34). This is typical for low-leakage and high-R_coupling interfaces, such as those achieved with insulated electrodes, which allow very high R_coupling in the order of 1 TΩ and C_coupling in the order of 1 nF (refer to paragraph “Insulated electrodes”). Hence, this configuration strives to maximize the resistive divider attenuation (R_coupling/R_IN) and minimize the capacitive divider attenuation (C_IN/C_coupling). For this purpose, achieving a roll-on in the magnitude response and moving s_p,1 away from s_z,1 is beneficial. Thus, subcase II.1 (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN, and R_IN < R_coupling) is preferred. The value |s_p,1| $=$ ${\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$ should be increased, and hence, for a fixed R_coupling and C_coupling, both R_IN and C_IN should be minimized. In doing so, whenever R_coupling >> R_IN and C_coupling >> C_IN is achieved, |s_p,1| simplifies to ${\displaystyle \frac{1}{R_{\mathrm{I}\mathrm{N}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$. In general, the further apart |s_p,1| is set from |s_z,1|, the greater the difference between the magnitude levels of the resistive and capacitive voltage dividers, and the closer the electrode to the concept of a pure capacitive electrode (Figure 19). However, this comes at the cost of an expanded transition band (decreased ω_R,II and increased ω_C,II). As a result, issues of achieving accurate sensing at sub-Hz frequencies arise, which will be further corroborated in step 3. On the other hand, the low-pass configuration poses opposite requirements, and so subcase I.2 (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN, and R_IN ≥ R_coupling) is preferred. Accordingly, this configuration favors the resistive coupling, so it is desirable for non-insulated surface-contact electrodes. Nevertheless, in the context of non-contact and insulated electrodes, it stresses an important methodological trap of treating non-contact and insulated electrodes as predominantly capacitive, when in fact the realized coupling mechanism at the frequencies of interest is predominantly resistive due to an insufficiently low value of 10|s_z,1| $=$ ${\displaystyle \frac{10}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ (refer to paragraphs “Non-contact on-body electrodes” and “Bias currents and DC biasing”). Finally, the all-pass configuration is an alternative to the high-pass configuration that takes advantage of the finite leakage present at the electrode–body interface. Hence, it strives to flatten and maximize the magnitude response, as illustrated in Figure 25 (subcase I.2). In terms of flattening, bringing s_p,1 close to s_z,1 is beneficial. This comes down to reducing the difference between the ratios R_coupling/R_IN and C_IN/C_coupling (46). In terms of maximizing the magnitude response and raising it toward 0 dB, these two ratios should be minimized. In other words, minimization of both the resistive and the capacitive voltage attenuation should be achieved. For a fixed R_coupling and C_coupling, this means that R_IN should be maximized and C_IN minimized. Subcase II.2 (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN, and R_IN ≥ R_coupling) is preferred, because it allows for relaxed constraints on the simultaneous minimization of C_IN and maximization of R_IN. Although this configuration strives to maximize R_IN for a fixed R_coupling, that does not necessarily preclude it from being used with high values of R_coupling in the order of 100 GΩ and 1 TΩ. Rather, the extent of its utilization for interfaces with high values of R_coupling is limited by the extent to which R_IN can be increased with respect to R_coupling. On the other hand, all-pass configuration is particularly interesting in the context of wearable non-contact on-body electrodes, with lower values of R_coupling in the order of 1 GΩ, 100 MΩ, or even lower, and lower values of C_coupling in the order of 10 pF and 100 pF. High-pass and all-pass configuration will be further compared considering phase response in step 3.

Figure 20. Impact of decreased C_IN on the magnitude response |H_div(jω)|_dB for case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN) and case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN). Vertical lines denote the ${\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$pole break frequencies, along with the shaded areas of predominantly capacitive coupling.

Figure 20. Impact of decreased C_IN on the magnitude response |H_div(jω)|_dB for case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN) and case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN). Vertical lines denote the ${\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$pole break frequencies, along with the shaded areas of predominantly capacitive coupling.

Figure 21. Impact of increased R_IN on the magnitude response |H_div(jω)|_dB for case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN) and case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN). Vertical lines denote the ${\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$ pole break frequencies, along with the shaded areas of predominantly capacitive coupling. Note that the upper figure represents case II, whereas the lower figure represents case I.

Figure 21. Impact of increased R_IN on the magnitude response |H_div(jω)|_dB for case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN) and case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN). Vertical lines denote the ${\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$ pole break frequencies, along with the shaded areas of predominantly capacitive coupling. Note that the upper figure represents case II, whereas the lower figure represents case I.

Figure 22. Mutual impact of R_IN and C_IN on the magnitude response |H_div(jω)|_dB for case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN): subcase I.1 (R_IN < R_coupling) and subcase I.2 (R_IN ≥ R_coupling). Vertical lines denote the ${\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$pole break frequencies, along with the shaded areas of predominantly capacitive coupling. It can be noted that the magnitude response can contain a roll-off due to |s_p,1| ≤ |s_z,1|. Also, the achieved magnitude levels of the resistive voltage divider, 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$, are below 20log₁₀0.5 $\approx$ −6 dB (50% or more attenuation) in subcase I.1 (R_IN < R_coupling), and above −6 dB (50% or less attenuation) in subcase I.2 (R_IN ≥ R_coupling).

Figure 22. Mutual impact of R_IN and C_IN on the magnitude response |H_div(jω)|_dB for case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN): subcase I.1 (R_IN < R_coupling) and subcase I.2 (R_IN ≥ R_coupling). Vertical lines denote the ${\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$pole break frequencies, along with the shaded areas of predominantly capacitive coupling. It can be noted that the magnitude response can contain a roll-off due to |s_p,1| ≤ |s_z,1|. Also, the achieved magnitude levels of the resistive voltage divider, 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$, are below 20log₁₀0.5 $\approx$ −6 dB (50% or more attenuation) in subcase I.1 (R_IN < R_coupling), and above −6 dB (50% or less attenuation) in subcase I.2 (R_IN ≥ R_coupling).

Figure 23. Mutual impact of R_IN and C_IN on the magnitude response |H_div(jω)|_dB for case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN): subcase II.1 (R_IN < R_coupling) and subcase II.2 (R_IN ≥ R_coupling). Vertical lines denote the ${\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$pole break frequencies, along with the shaded areas of predominantly capacitive coupling. It can be noted that the magnitude response contains a roll-on due to |s_p,1| > |s_z,1|. Also, the achieved magnitude levels of the resistive voltage divider, 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$, are below 20log₁₀0.5 $\approx$ −6 dB (50% or more attenuation) in subcase II.1 (R_IN < R_coupling), and above −6 dB (50% or less attenuation) in subcase II.2 (R_IN ≥ R_coupling).

Figure 23. Mutual impact of R_IN and C_IN on the magnitude response |H_div(jω)|_dB for case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN): subcase II.1 (R_IN < R_coupling) and subcase II.2 (R_IN ≥ R_coupling). Vertical lines denote the ${\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$pole break frequencies, along with the shaded areas of predominantly capacitive coupling. It can be noted that the magnitude response contains a roll-on due to |s_p,1| > |s_z,1|. Also, the achieved magnitude levels of the resistive voltage divider, 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$, are below 20log₁₀0.5 $\approx$ −6 dB (50% or more attenuation) in subcase II.1 (R_IN < R_coupling), and above −6 dB (50% or less attenuation) in subcase II.2 (R_IN ≥ R_coupling).

Figure 24. Magnitude response |H_div(jω)|_dB along with the pairs of cutoff frequencies (vertical lines): ω_R,I and ω_C,I in case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN), and ω_R,II and ω_C,II in case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN). In these graphs, the existence of cutoff frequencies and their relative position with respect to C_IN/C_coupling and R_coupling/R_IN ratios is inspected. The first example in each of the two cases (solid black line) depicts a scenario in which there are no cutoff frequencies. Accordingly, the difference between magnitude levels of the resistive and capacitive voltage dividers is less than 3 dB. Also, the ratio |s_z,1|/|s_p,1| in case I or, equivalently, the ratio |s_p,1|/|s_z,1| in case II, is lower than $\sqrt{3}$. The other three examples in each of the two cases are arbitrarily chosen to illustrate the interdependence between the relative position of voltage divider magnitude levels, 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ and 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$, and the relative position of angular break frequencies |s_z,1| and |s_p,1| and cutoff frequencies in a pair. Starting from the scenario in which the difference between magnitude levels of the resistive and capacitive voltage dividers is 6 dB and the cutoff frequencies are equal (dashed red line examples), greater difference between the cutoff frequencies sets the magnitude levels of the voltage dividers, as well as the angular break frequencies, further apart.

Figure 24. Magnitude response |H_div(jω)|_dB along with the pairs of cutoff frequencies (vertical lines): ω_R,I and ω_C,I in case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN), and ω_R,II and ω_C,II in case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN). In these graphs, the existence of cutoff frequencies and their relative position with respect to C_IN/C_coupling and R_coupling/R_IN ratios is inspected. The first example in each of the two cases (solid black line) depicts a scenario in which there are no cutoff frequencies. Accordingly, the difference between magnitude levels of the resistive and capacitive voltage dividers is less than 3 dB. Also, the ratio |s_z,1|/|s_p,1| in case I or, equivalently, the ratio |s_p,1|/|s_z,1| in case II, is lower than $\sqrt{3}$. The other three examples in each of the two cases are arbitrarily chosen to illustrate the interdependence between the relative position of voltage divider magnitude levels, 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ and 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$, and the relative position of angular break frequencies |s_z,1| and |s_p,1| and cutoff frequencies in a pair. Starting from the scenario in which the difference between magnitude levels of the resistive and capacitive voltage dividers is 6 dB and the cutoff frequencies are equal (dashed red line examples), greater difference between the cutoff frequencies sets the magnitude levels of the voltage dividers, as well as the angular break frequencies, further apart.

Figure 25. Positions of maximum |s_p,1|, denoted by vertical dashed lines, with respect to the minimum C_IN and R_IN achievable in each of the three practical subcases: subcase I.2 (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN, and R_IN ≥ R_coupling), subcase II.1 (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN, and R_IN < R_coupling), and subcase II.2 (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN, and R_IN ≥ R_coupling).

Figure 25. Positions of maximum |s_p,1|, denoted by vertical dashed lines, with respect to the minimum C_IN and R_IN achievable in each of the three practical subcases: subcase I.2 (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN, and R_IN ≥ R_coupling), subcase II.1 (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN, and R_IN < R_coupling), and subcase II.2 (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN, and R_IN ≥ R_coupling).

STEP 3: Mitigating phase shifts and morphological changes. In step 1, the area of predominantly capacitive coupling was inspected, and in step 2, it was accompanied by the area of predominant capacitive divider. Finally, aside from ensuring predominantly capacitive coupling in step 1 and improving the magnitude response by manipulating the influence of input voltage divider in step 2, the last important aspect is the phase response $\measuredangle$H_div(jω). Once again, analysis can start from engineering fundamentals. The summary of the analysis is available at the end of step 3.

Linearity of the phase response. As is known from the technical literature [399,400,401], to minimize time-domain distortion, the phase response of the system should be linear. Namely, a linear phase response will ensure that individual frequency components are delayed by the same amount of time, and hence that the signal shape and its temporal characteristics are preserved at the output. From the phasor perspective, ω is the rate of change in phase Δφ over a time interval Δt. This means that in steady state, phase shift Δφ_k between a frequency component V_kcos(ω_kt + φ_k) at the input of the subsystem (v_surface) and the corresponding frequency component V_k$·$|H_div(jω_k)|cos[ω_kt + φ_k + $\measuredangle$H_div(jω_k)] at the output of the subsystem (v_in) can be translated into time shift Δt_k as follows (48):

$${\omega }_{\mathrm{k}}=-{\displaystyle \frac{∆{\phi }_{\mathrm{k}}}{∆t_{\mathrm{k}}}}=-{\displaystyle \frac{[{\phi }_{\mathrm{k}}+\measuredangle H_{\mathrm{div}}\left(j{\omega }_{\mathrm{k}}\right)]-{\phi }_{\mathrm{k}}}{∆t_{\mathrm{k}}}} \to ∆t_{\mathrm{k}}=-{\displaystyle \frac{\measuredangle H_{\mathrm{div}}\left(j{\omega }_{\mathrm{k}}\right)}{{\omega }_{\mathrm{k}}}} .$$

(48)

The negative sign indicates that a lagging phase shift, $\measuredangle$H_div(jω_k) < 0, will result in a time delay, Δt_k > 0. Now, it can be shown that if the phase response $\measuredangle$H_div(jω) were linear with a slope const, each input frequency component propagating through the subsystem H_div(jω) would be delayed at its output for the same value const (49):

$$∆t_{\mathrm{k}}=-{\displaystyle \frac{\measuredangle H_{\mathrm{div}}\left(j{\omega }_{\mathrm{k}}\right)}{{\omega }_{\mathrm{k}}}}=-{\displaystyle \frac{-const·{\omega }_{\mathrm{k}}}{{\omega }_{\mathrm{k}}}}=const .$$

(49)

Group delay. The previously analyzed Δt_k, also called phase delay, represents the time shift of each frequency component separately. Additionally, group delay gd can be calculated to analyze the time shift of the signal envelope. Group delay is equal to the negative gradient of phase response. Consequently, a linear phase response would draw a constant group delay, whereas a constant phase response would draw zero group delay. In that sense, frequency bands or areas of constant and zero group delay could be used to investigate at which frequencies a system can reliably transfer, amplify or filter the input signal. Specifically, group delay gd_div of the observed subsystem H_div(jω) can be calculated from the phase response (29) as follows (50):

$${gd}_{\mathrm{d}\mathrm{i}\mathrm{v}}(\omega )=-{\displaystyle \frac{d\left[\measuredangle H_{\mathrm{d}\mathrm{i}\mathrm{v}}\left(j\omega \right)\right]}{d\omega }}=$$

$$={\displaystyle \frac{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}{1+{\left[\omega \left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)\right]}^2}}-{\displaystyle \frac{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{1+{\left(\omega R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right)}^2}}$$

$$={\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{{s_{\mathrm{p},1}}^2+{\omega }^2}}-{\displaystyle \frac{\left|s_{\mathrm{z},1}\right|}{{s_{\mathrm{z},1}}^2+{\omega }^2}} .$$

(50)

Signal morphology issues. From Section 4.1, it can be recalled that the output of a stable and causal, uniquely solvable LTI system in steady state is a scaled and delayed version of the input. However, phase response $\measuredangle$H_div(jω) (29) is not truly linear, and therefore, group delay gd_div (50) deviates from a constant value. This indicates that the waveform of the signal v_surface may not necessarily be preserved after being sensed on the electrode sensing surface and translated to the input of the preamplifier. In other words, the subsystem H_div(jω) can indeed only be approximated with an LTI system at narrow bands of frequencies where gd_div is approximately flat and constant. In fact, the same is true for any analog system, given that the Bode plot is, after all, a linear approximation with a finite error [273,349]. Therefore, the wider the bandwidth of the input signal, the more distortion it could experience. Focusing again on the biopotential measurement, this is especially important to take into account when the feasibility of electrodes in clinical-grade applications is considered. Namely, a non-linear phase response of the measurement system could alter the morphology of the measured biopotential signal and thereby preclude reliable diagnostics. For instance, in the case of clinical ECG measurements, a commonly used low-side frequency limit is 0.05 Hz or even lower, and the influence of the measurement system on near-DC and sub-Hz ECG content, especially ST-segments and T-waves (ST–T changes), but also P-waves, is a well-known issue (refer to [402] and Figure 7). An example can be observed: for this purpose, ECG signals obtained at the input of the simulated subsystem (v_surface at the surface of the skin) and at the output of the simulated subsystem (v_in at the preamplifier non-inverting input) can be compared in the time domain. In addition, their power spectra are compared in the frequency domain. Calculations are performed in MathWorks^® MATLAB R2025b environment [146] with the use of one-channel ECG signals recorded at a sampling frequency of 360 Hz and obtained from the PhysioNet service from the MIT-BIH Arrhythmia database [148]. Specifically, the results are observed for the case of premature ventricular contraction (Figure 26). Further discussions on this subject in the context of non-contact and insulated electrodes can be found in [403,404,405,406]. More on the morphology of ECG signal and its frequency spectrum can be found in [152,153,154]. Various ECG abnormalities and methods of automatic heart anomaly detection are surveyed in [153,407]. The issue of morphological changes in clinical ECG will be revisited later in paragraph “Practical aspects of DC biasing, feasibility of clinical-grade diagnostics, surface leakage, and dielectric absorption.”

Zero and pole phase shifts. Now that the influence of phase response has been demonstrated, phase response $\measuredangle$H_div(jω) can be inspected more thoroughly. Specifically, entire phase shift created between the skin surface and the input of the preamplifier arises from the pole s_p,1 and the zero s_z,1. As described in Figure A9 with the Bode plot [273,341,348,349,397], the left-half plane pole s_p,1 introduces lagging phase shifts with a negative slope of −45°/decade that begins approximately at angular frequency 0.1|s_p,1| and ends approximately at angular frequency 10|s_p,1|. Similarly, the left-half plane zero s_z,1 introduces leading phase shifts with a positive slope of +45°/decade that begins approximately at angular frequency 0.1|s_z,1| and ends approximately at angular frequency 10|s_z,1|. This means that the phase response $\measuredangle$H_div(jω) will be fairly constant at ω ≤ min{0.1|s_z,1|, 0.1|s_p,1|} and at ω ≥ max{10|s_z,1|, 10|s_p,1|}. However, across the range in between, min{0.1|s_z,1|, 0.1|s_p,1|} < ω < max{10|s_z,1|, 10|s_p,1|}, $\measuredangle$H_div(jω) will form a dip if |s_p,1| < |s_z,1| (case I) or a bell-like curve if |s_p,1| > |s_z,1| (case II). This is depicted in Figure 27 and Figure 28, and the same can be further inspected for other types of bilinear transfer functions, as given in [397]. In other words, linearity of $\measuredangle$H_div(jω) cannot be achieved across the entire frequency range of interest (Assumption 1). Rather, it is tied to frequencies that are a decade or less apart from |s_z,1| and |s_p,1| values. Because it is localized, area of linear phase shifts in $\measuredangle$H_div(jω) turns out to be as troublesome as any area of non-linear phase shifts. Therefore, instead of searching for areas of linear phase shifts in $\measuredangle$H_div(jω) and constant group delay gd_div, it is more useful to investigate the areas of approximately constant $\measuredangle$H_div(jω) and near-zero gd_div.

Influence of equating and decreasing |s_z,1| and |s_p,1|. The following effect can be observed: as |s_z,1| and |s_p,1| are brought closer to one another, the phase shifts introduced by s_z,1 and s_p,1 (i.e., the dip or the bell) are smaller. In fact, if s_z,1 and s_p,1 canceled out, which would be achieved when R_coupling/R_IN $=$ C_IN/C_coupling (46), phase response would flatten out to a value of 0° regardless of the attenuation level in the magnitude response (Figure 29). In other words, bringing s_p,1 closer to s_z,1 flattens not only the magnitude response, but also the phase response, raising it toward 0°. In addition to constant and zero phase response $\measuredangle$H_div(jω), if these two ratios were minimized (R_coupling/R_IN $=$ C_IN/C_coupling $=$ 0), a unity gain (0 dB) magnitude response |H_div(jω)|_dB would be achieved, as described in step 2, and the subsystem H_div(jω) would turn into the ideal voltage follower. This speaks in favor of bringing |s_p,1| closer to |s_p,1| and minimizing R_coupling/R_IN and C_IN/C_coupling ratios, corroborating once again that the all-pass configuration from step 2 has a greater potential for accurate sensing of sub-Hz frequency components than the high-pass configuration. Namely, in the case of a high-pass configuration, a greater roll-on in magnitude response would cause more devastating phase shifts (Figure 26 and Figure 30). Aside from reducing the difference between |s_p,1| and the fixed |s_z,1| by manipulating the ratios R_coupling/R_IN and C_IN/C_coupling for a fixed R_coupling and C_coupling, both |s_z,1| and |s_p,1| can be further decreased by increasing R_coupling and C_coupling. This way, the phase shifts in $\measuredangle$H_div(jω) will be moved toward lower frequencies and affect a narrower band of frequencies. Now, if equating and decreasing |s_z,1| and |s_p,1| is observed at the level of individual parameters of the interface, conclusions again show that increase in R_coupling and C_coupling from step 1, followed by minimization of C_IN and maximization of R_IN from step 2, is beneficial. Accordingly, subcase I.2 (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN) and subcase II.2 (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN) with their prerequisite R_IN ≥ R_coupling are desirable, since they allow for the implementation of the least attenuating resistive divider without upper limits on R_IN. Among them, subcase II.2 is preferred, since it allows for relaxed constraints on simultaneous minimization of C_IN and maximization of R_IN (step 2).

Area of minimized phase shifts. In the last paragraph, we described how to narrow the area of phase shifts of $\measuredangle$H_div(jω) and bring it to lower frequencies. Since these phase shifts are largely concentrated at min{0.1|s_z,1|, 0.1|s_p,1|} < ω < max{10|s_z,1|, 10|s_p,1|}, the usable area of minimized phase shifts can be determined accordingly. Interestingly, step 1 has shown that predominantly resistive coupling is achieved below 0.1|s_z,1|, which in fact corresponds to the angular frequency at which the zero s_z,1 introduces its phase shifts in accordance with Bode approximation [273,341,348,349,397]. On the other hand, predominantly capacitive coupling is achieved above 10|s_z,1|, which, in accordance with Bode approximation, corresponds to the angular frequency at which the zero s_z,1 no longer contributes to the phase shifts of the system. In other words, the point of transition into the area of predominantly capacitive coupling, 10|s_z,1| (step 1), corresponds to the frequency of approximate end of phase shifts introduced by s_z,1. Further, in step 2, the area of predominant capacitive divider was presented and distinguished from the area of predominantly capacitive coupling. Therein, it was explained that non-contact and insulated biopotential electrodes, which rely on predominantly capacitive coupling, should be used in both the area of predominantly capacitive coupling and the area of predominant capacitive divider. For this purpose, it was shown that tracking both |s_z,1| and |s_p,1| values is important, since |s_p,1| might be significantly higher than |s_z,1|. Now, in step 3, this can be stressed even further in terms of the phase response: to mitigate the impact of phase shifts present in $\measuredangle$H_div(jω), measurements should be carried out at frequencies at which the phase shifts of both s_z,1 and s_p,1 are minimized. These ranges, ω ≤ min{0.1|s_z,1|, 0.1|s_p,1|} and ω ≥ max{10|s_z,1|, 10|s_p,1|}, correspond to the desired area within which gd_div has fallen to values in the vicinity of zero. Given that the designed electrode–body interface is non-contact and/or insulated, only the latter frequency range is of interest. From now on, this area ω ≥ max{10|s_z,1|, 10|s_p,1|} will be referred to as the area of minimized phase shifts or area of near-zero group delay. Therein, using the term “minimized” rather than “vanished” and stressing the “near-zero” part is important. Namely, it can be recalled that this entire calculation is based on Bode approximation, which yields a phase error of ±5.7° at 0.1|s_z,1|, 10|s_z,1|, 0.1|s_p,1|, and 10|s_p,1| angular frequencies [273,349]. This means that once the angular frequency max{10|s_z,1|, 10|s_p,1|} is reached and crossed, there is still a small residue of phase shifts, due to which $\measuredangle$H_div(jω) continues to gradually increase (case I) or decline (step II) toward 0° as the frequency is further increased.

Relation between the three established areas. Finally, the requirements for measuring in the predominantly capacitive coupling area (step 1) and the predominant capacitive divider area (step 2) can be combined with the requirement for measuring in the area of minimized phase shifts. Accordingly, the target low-side angular frequency limit for measurements should be ω_min ≥ max{10|s_p,1|, ω_C,I, 10|s_z,1|} $=$ 10|s_z,1| in case I and ω_min ≥ max{10|s_z,1|, ω_C,II, 10|s_p,1|} $=$ 10|s_p,1| in case II (Table 2). This again shows that aside from tracking and bringing the values |s_z,1| and |s_p,1| closer to one another, decreasing them to lower values is also important to further extend the predominantly capacitive coupling area, predominant capacitive divider area, and area of minimized phase shifts toward lower frequencies. Given the fact that area of minimized phase shifts is only an approximation, ω_min should in practice be as high as possible compared to the respective low-side angular frequency.

STEP 3: SUMMARY. After considering the magnitude response |H_div(jω)|_dB and voltage divider attenuation in step 2, the analysis in step 3 is extended to the phase response $\measuredangle$H_div(jω). First, phase delay as the time delay of individual frequency components, Δt_k, and group delay as the time delay of the signal envelope, gd_div, are derived. Second, the issue behind their non-constant values, arising from non-linearities in the phase response, is observed on clinical-grade ECG signals as a motivational example. Third, based on Bode approximation of phase shifts introduced by the left-half plane zero s_z,1 and the left-half plane pole s_p,1, dips and bell-like shapes are observed in the phase response at min{0.1|s_z,1|, 0.1|s_p,1|} < ω < max{10|s_z,1|, 10|s_p,1|}. Suppressing these phase shifts to lower values close to 0° and limiting their influence to a narrower frequency range will draw a smaller deviation between the individual frequency components of the input signal v_surface, hence mitigating signal distortion and morphological changes. For this purpose, three operations prove to be beneficial: first, the values |s_z,1| and |s_p,1| should be brought closer to one another by equating the ratios R_coupling/R_IN and C_IN/C_coupling (46). The closer the |s_p,1| value to the |s_z,1| value, the flatter the magnitude response |H_div(jω)|_dB and the phase response $\measuredangle$H_div(jω), and the closer the phase shifts in $\measuredangle$H_div(jω) to 0°. Second, the smaller the ratios R_coupling/R_IN and C_IN/C_coupling, the closer the magnitude response |H_div(jω)|_dB to 0 dB (Figure 29). Third, decreasing |s_z,1| and |s_p,1| will further narrow the area of phase shifts present in $\measuredangle$H_div(jω) and move it toward lower frequencies. Accordingly, an increase in R_coupling and C_coupling (step 1), followed by minimization of C_IN and maximization of R_IN (step 2), is beneficial. Finally, after introducing the area of predominantly capacitive coupling in step 1 and the area of predominant capacitive divider in step 2, step 3 introduced the area of minimized phase shifts. To ensure that the frequency range of interest is situated in all three areas, the following should be ensured for the lowest angular frequency of interest, ω_min: ω_min ≥ 10|s_z,1| in case I (|s_p,1| ≤ |s_z,1|), and ω_min ≥ 10|s_p,1| in case II (|s_p,1| > |s_z,1|), with a caveat that ω_min should be as high as possible compared to these low-side limits to account for the error of Bode approximation. This shows once again that aside from equating|s_z,1| and |s_p,1| values, decreasing them is also important to extend each of the three respective areas toward lower frequencies. All these solutions favor the all-pass configuration from step 2 (Figure 26 and Figure 30). Unlike the high-pass configuration, its focus is not on AC-coupling of the electrode–body interface, but rather on taking the advantage of finite leakage and enhancing the reduction of phase shifts and signal distortion.

Figure 26. Magnitude response |H_div(jω)|_dB and phase response $\measuredangle$H_div(jω) of the input voltage divider subsystem, along with an ECG signal at its input and the corresponding response at its output, obtained in MathWorks^® MATLAB R2025b environment [146]. Input and output one-channel ECG is given in both the frequency domain (power spectra) and the time domain. Recordings are obtained from the PhysioNet service from the MIT-BIH Arrhythmia database [148]. Signals are recorded at a sampling frequency of 360 Hz and they represent the cardiac disorder of premature ventricular contraction. Results reveal different degrees of changes in signal morphology depending on |s_z,1| and |s_p,1| values defined by the four interface parameters. Three cases are observed: (upper) neither C_coupling >> C_IN nor R_IN >> R_coupling condition is satisfied (high-pass configuration with R_coupling >> R_IN and 16%-attenuation of the capacitive voltage divider); (middle) high-pass configuration with R_coupling >> R_IN and C_coupling >> C_IN; (lower) all-pass configuration with R_IN >> R_coupling and C_coupling >> C_IN.

Figure 26. Magnitude response |H_div(jω)|_dB and phase response $\measuredangle$H_div(jω) of the input voltage divider subsystem, along with an ECG signal at its input and the corresponding response at its output, obtained in MathWorks^® MATLAB R2025b environment [146]. Input and output one-channel ECG is given in both the frequency domain (power spectra) and the time domain. Recordings are obtained from the PhysioNet service from the MIT-BIH Arrhythmia database [148]. Signals are recorded at a sampling frequency of 360 Hz and they represent the cardiac disorder of premature ventricular contraction. Results reveal different degrees of changes in signal morphology depending on |s_z,1| and |s_p,1| values defined by the four interface parameters. Three cases are observed: (upper) neither C_coupling >> C_IN nor R_IN >> R_coupling condition is satisfied (high-pass configuration with R_coupling >> R_IN and 16%-attenuation of the capacitive voltage divider); (middle) high-pass configuration with R_coupling >> R_IN and C_coupling >> C_IN; (lower) all-pass configuration with R_IN >> R_coupling and C_coupling >> C_IN.

Figure 27. Impact of increased R_coupling on the magnitude response |H_div(jω)|_dB and the phase response $\measuredangle$H_div(jω). In the upper figure, vertical lines denote the $10{\displaystyle \frac{\left|s_{\mathrm{z},1}\right|}{2\pi }}$zero frequencies as the low-side limit of the area of minimized phase shifts for case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN). In the lower figure, vertical lines denote the $10{\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$pole frequencies as the low-side limit of the area of minimized phase shifts for case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN).

Figure 27. Impact of increased R_coupling on the magnitude response |H_div(jω)|_dB and the phase response $\measuredangle$H_div(jω). In the upper figure, vertical lines denote the $10{\displaystyle \frac{\left|s_{\mathrm{z},1}\right|}{2\pi }}$zero frequencies as the low-side limit of the area of minimized phase shifts for case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN). In the lower figure, vertical lines denote the $10{\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$pole frequencies as the low-side limit of the area of minimized phase shifts for case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN).

Figure 28. Impact of increased C_coupling on the magnitude response |H_div(jω)|_dB and the phase response $\measuredangle$H_div(jω). In the upper figure, vertical lines denote the $10{\displaystyle \frac{\left|s_{\mathrm{z},1}\right|}{2\pi }}$zero frequencies as the low-side limit of the area of minimized phase shifts for case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN). In the lower figure, vertical lines denote the $10{\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$pole frequencies as the low-side limit of the area of minimized phase shifts for case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN).

Figure 28. Impact of increased C_coupling on the magnitude response |H_div(jω)|_dB and the phase response $\measuredangle$H_div(jω). In the upper figure, vertical lines denote the $10{\displaystyle \frac{\left|s_{\mathrm{z},1}\right|}{2\pi }}$zero frequencies as the low-side limit of the area of minimized phase shifts for case I (|s_p,1| ≤ |s_z,1|, C_IN/C_coupling ≥ R_coupling/R_IN). In the lower figure, vertical lines denote the $10{\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$pole frequencies as the low-side limit of the area of minimized phase shifts for case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN).

Figure 29. Impact of pole–zero cancelation on the magnitude response |H_div(jω)|_dB and the phase response $\measuredangle$H_div(jω). When R_coupling/R_IN $=$ C_IN/C_coupling, |s_p,1| is brought closer to |s_z,1|, and the frequency response flattens out. In addition, as R_coupling/R_IN and C_IN/C_coupling ratios decrease to 0, |H_div(jω)|_dB is raised toward 0 dB.

Figure 29. Impact of pole–zero cancelation on the magnitude response |H_div(jω)|_dB and the phase response $\measuredangle$H_div(jω). When R_coupling/R_IN $=$ C_IN/C_coupling, |s_p,1| is brought closer to |s_z,1|, and the frequency response flattens out. In addition, as R_coupling/R_IN and C_IN/C_coupling ratios decrease to 0, |H_div(jω)|_dB is raised toward 0 dB.

Figure 30. Impact of simultaneous increase in R_IN and decrease in C_IN on the magnitude response |H_div(jω)|_dB and the phase response $\measuredangle$H_div(jω). For R_coupling and C_coupling fixed in step 1, decrease in R_coupling/R_IN and C_IN/C_coupling ratios can be achieved by minimization of C_IN and maximization of R_IN. In addition to minimizing phase shifts, this also maximizes the magnitude gain. Examples are tailored in a way to show how a high-pass configuration (solid black example) transforms into a close-to-ideal all-pass configuration (dotted blue example). Vertical lines denote the $10{\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$pole frequencies as the low-side limit of the area of minimized phase shifts for the observed case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN).

Figure 30. Impact of simultaneous increase in R_IN and decrease in C_IN on the magnitude response |H_div(jω)|_dB and the phase response $\measuredangle$H_div(jω). For R_coupling and C_coupling fixed in step 1, decrease in R_coupling/R_IN and C_IN/C_coupling ratios can be achieved by minimization of C_IN and maximization of R_IN. In addition to minimizing phase shifts, this also maximizes the magnitude gain. Examples are tailored in a way to show how a high-pass configuration (solid black example) transforms into a close-to-ideal all-pass configuration (dotted blue example). Vertical lines denote the $10{\displaystyle \frac{\left|s_{\mathrm{p},1}\right|}{2\pi }}$pole frequencies as the low-side limit of the area of minimized phase shifts for the observed case II (|s_p,1| > |s_z,1|, C_IN/C_coupling < R_coupling/R_IN).

Guidelines for designing non-contact and insulated electrode–body interfaces. Now, based on the results in Table 2, the laid out analysis can be summarized into four guidelines. To ensure predominantly capacitive coupling in accordance with Test 1, with minimized attenuation and minimized morphological changes at ω_min ≤ ω ≤ ω_max (refer to Assumption 1), the following steps should be taken:

• Decrease |s_z,1| $=$ ${\displaystyle \frac{1}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ below ω_min/10 by increasing R_coupling and C_coupling of the interface.
• Ensure C_coupling >> C_IN to minimize attenuation from the capacitive voltage divider;
• For a high-pass configuration, ensure R_coupling >> R_IN to achieve a roll-on in magnitude response (subcase II.1) and move |s_p,1| $\approx$ ${\displaystyle \frac{1}{R_{\mathrm{I}\mathrm{N}}{C}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}}$ away from |s_z,1| toward higher frequencies. For an all-pass configuration, ensure R_IN >> R_coupling and keep |s_p,1| $=$ ${\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$in proximity of |s_z,1| to flatten the frequency response, minimize phase shifts in the phase response, and maximize the magnitude response (subcase I.2 and subcase II.2). These requirements are relaxed in subcase II.2 compared to subcase I.2.
• Finally, constrain the measurements to ω ≥ max{10|s_z,1|, 10|s_p,1|} to mitigate phase shifts and degradation of signal morphology. In addition, bear in mind that ω_min should be as high as possible compared to this low-side limit to account for the error of Bode approximation.

Guideline number 1 arises from Test 1 applied to the total equivalent Z_coupling. On the other hand, guidelines number 2 and 3 can be understood intuitively as a result of the aim to maximize Z_IN and thereby minimize the influence of the impedance voltage divider ${\displaystyle \frac{Z_{\mathrm{IN}}}{Z_{\mathrm{coupling}}+Z_{\mathrm{IN}}}}$, separated into the resistive and the capacitive component. The difference is that for an all-pass configuration, the design strives to maximize Z_IN and reduce input voltage divider effect across the entire frequency range, whereas the high-pass design focuses on achieving this only in the area of predominantly capacitive coupling and the area of predominantly capacitive divider. Finally, guideline number 4 stresses the importance of satisfying the first three guidelines from the perspective of preserving the morphology of the acquired signal. The positive influence of high Z_IN on the mitigation of signal distortion has already been highlighted in papers that investigated the input impedance requirements for dry surface-contact ECG recording and its specifications with respect to IEC 60601 performance standard [408]. Overall, in accordance with Assumption 5, the designing principles behind the proposed guidelines are applicable to any given non-invasive active biopotential electrode. They will be further inspected from the perspective of the frequency response of the entire system H_el(jω), accounting for the buffer preamplifier, in the next section.

Practical aspects of achieving a high input impedance of the preamplifier. Lastly, practical aspects of realizing the expressed guidelines from the perspective of the preamplifier can be discussed. As shown in Figure 10 and in accordance with Assumption 9, R_IN can be modeled as R_bias||R_in, whereas C_IN can be modeled as C_par||C_in. This means that achieving as high Z_IN as possible requires both high R_bias and R_in, as well as low C_par and C_in. To accomplish this, various analog techniques can be employed, which will be summarized in the following paragraphs. More details can be found in [86,94,104].

Practical aspects of DC biasing, feasibility of clinical-grade diagnostics, surface leakage, and dielectric absorption. As shown in step 2, increasing R_IN reduces the |s_p,1| value and the attenuation of the resistive voltage divider. From this point of view, R_IN should be maximized. On the other hand, the paragraph “Bias currents and DC biasing” has shown that high input resistance R_in makes shunting the bias current I₊ to the circuit common more problematic. Bias currents can be reduced with the use of FET-input stage amplifiers, but this in turn increases R_in even further (refer to [332,333,383], and subchapters 4x.5 and 4x.10 in [355]). Also, no matter how small they are, they may still limit the long-term performance of the measurement system. Therefore, using active high-input-impedance electrodes necessitates external means of DC biasing, which was most simply modeled as R_bias. To serve its purpose, R_bias should be at least an order lower than R_IN. Therefore, whenever it is present, it can replace R_IN in all equations discussed throughout this paper. On the other hand, R_bias should still be as high as possible in order to avoid degradation of the input impedance. This shows that R_bias is the limiting factor in the process of increasing R_IN. Furthermore, it leads to the issue of the measured bandwidth and low-side frequency limitations. Capturing the entire bandwidth of ECG signal to allow for clinical diagnostics comparable to hospital-grade systems requires the low-side measurement frequency to be of value 0.05 Hz, or even 0.02 Hz and lower [154] (Assumption 1). According to step 3, this means that |s_p,1| $=$ ${\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$ $\approx$ ${\displaystyle \frac{1}{R_{\mathrm{I}\mathrm{N}}\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$ (18) should not be higher than 2π$·2$ mHz. For common C_coupling||C_IN values of tens of pF, this calls for R_bias values in the order of 500 GΩ or even higher. Such resistors are expensive, contain a wide tolerance range, and impose a problem of sensitivity to temperature and humidity. This issue of impact on signal morphology at frequencies in the order of 1 Hz and lower is in fact the reason why the majority of predominantly capacitive non-contact and insulated ECG applications have been focused on ambulatory ECG and smart gadget applications such as heart rate measurement, where portability, compactness, and simplicity are of primary concern. Alternatively to discrete resistors with values of 500 GΩ and higher, insulated cable leakage and PCB leakage can be taken advantage of with the use of gimmick resistors from a pair of twisted insulated wires connected to the non-inverting input pin at one end and unconnected on the other end, or with the use of additional PCB pads and a grounded PCB trace placed close to the trace leading to the non-inverting pin. Nevertheless, these are rather prototype solutions, and also, they could increase C_IN by extra capacitance in the order of 1 pF (see [86] and subchapter 1x.3 in [282]). Even if the design allows for an increase in R_bias, R_bias cannot be increased indefinitely. Rather, its value is limited by the preamplifier common-mode input voltage range and the output voltage swing with respect to the bias current I₊ [86]. In addition, extremely high values of R_bias could make the voltage drop introduced by I₊ across R_bias significant, resulting in an increase in input offset voltage. Also, the higher the R_bias, the more important it becomes to minimize surface leakage between circuit nodes that could occur due to various accumulated contaminants and debris, from flux residue and dust to skin sebum and other products of exocrine glands. For instance, for only 1 V present between two nodes, an additional leakage path between them with resistance as high as 1 TΩ would create a leakage current of 1 V/1 TΩ $=$ 1 pA, which can easily be of the same order as bias currents, or even higher. To avoid such additional spurious bias paths and minimize surface leakage, the guarding technique can be employed, which extends the idea of a grounded PCB trace in close proximity to the non-inverting input. Namely, it surrounds the sensitive, high-input-impedance non-inverting node with a conductor of the same electric potential, thereby effectively minimizing the electric potential difference needed for leakage current flow. Aside from a passive connection of the guard trace to the desired electric potential, this can also be achieved by active driving, similarly to shield drivers [367], to mitigate the influence of additional parasitic capacitance. Aside from externally implemented active guard drivers, active guarding can also be achieved via specialized on-chip guard terminals, as in [249]. Additional leakage might be caused by the dielectric absorption of the dielectric material itself [260,282,305]. Along with intrinsic frequency-dependent variation in dielectric constant, this effect, also called dielectric soakage and dielectric memory, is a second important example of frequency-dependence in the concept of capacitance. It describes a phenomenon of spontaneous recovery of charge and a consequential residual voltage build-up after discharging. This finite error of residual polarization can be accounted for in the model by additional RC-series in parallel with the capacitance C_coupling (i.e., a capacitance in series with the leakage resistance R_coupling). Lastly, finite leakage exists in PCB solder mask as well, hence, solder mask can be removed, and the non-inverting input pin can be bent upwards from the PCB to employ the technique of point-to-point air-wiring or sky-wiring. More on all these techniques can be found in [249,274,336,380,409,410]. Finaly, in any of the aforementioned high-value R_bias solutions, the final two issues still remain to be solved: first, the issue of long-term stability, and second, the issue of recovery time after a voltage transient. Namely, as seen from the pole angular break frequency |s_p,1| $=$ ${\displaystyle \frac{1}{\left(R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|R_{\mathrm{I}\mathrm{N}}\right)\left(C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\right|\left|C_{\mathrm{I}\mathrm{N}}\right)}}$ (18) and h_el(t) (21), the capacitive and resistive elements of the electrode–body interface define the settling time after a transient change, which can reach orders of 10 s when R_IN in the order of 1 TΩ is achieved [86,99]. This requires additional fast reset circuitries [411] to recover the baseline in reasonable time and prevent high-amplitude artifacts from driving the analog front-end into saturation and obstructing the measurements. For this purpose, voltage-dependent switch-and-transistor-based [411], varistor-based [412], and diode-based circuits [83,104,105,248,411] can be employed. Finally, input impedance can be effectively increased at the frequencies of interest with the use of the bootstrapping method, while at the same time securing a low-impedance bias path for a stable baseline [369,370,410]. Alternative input impedance-boosting methods such as, e.g., conductance neutralization or using a negative impedance converter, can be employed. They are especially suitable for integrated design [140,413].

Practical aspects of decreasing the input capacitance. Secondly, reduction in C_IN can be considered in practice. In the early works on non-contact and insulated electrodes in the 1960s and 1970s [242,243,244,245], C_coupling in the order of nF was achieved by coating the electrode sensing surface with insulating materials of high dielectric constant. The resulting capacitive voltage divider endured higher input capacitances. With the rapid advancements in electronics that took place in the following decades, C_in was reduced from the order of 100 pF to the order of 1 pF [414], which, conversely, relaxed the C_coupling requirement down to the order of 10 pF and 100 pF. This allowed for the sensing surface to be moved away from the immediate vicinity of the body and the utilization of wearable fabrics (which, due to porosity and air cavities, generally achieve lower dielectric constants usually in the range of 1–4 [92,135]). Also, it removed the need for using special coating materials that exhibited unwanted piezoelectric effect under mechanical stress, such as barium-titanate ceramics used in high-κ class 2 and class 3 ceramic capacitors (refer to [245], subchapter 1x.3 in [282], and [98]). Nevertheless, as shown in the paragraph “Parasitic input capacitance”, the resulting pin, pad, layout, and cable parasitic capacitance, C_par, is at least in the order of 1 pF, which is still in the order of C_in. Moreover, C_in could be further increased to the order of 10 pF if large-area input-stage transistors are used [86]. As explained in [97], C_IN of only 1 pF exhibits impedance of about 16 GΩ at the frequency of 10 Hz (A4), where the maximum power spectral density of ECG is. This way, Z_IN is degraded even for very low input capacitances. Therefore, unlike R_bias which is, compared to R_in, a limiting factor in achieving an increase in R_IN, in the case of C_IN, achieving effective decrease in C_in and C_par is of equal importance. On the one hand, the aforementioned passive and active guarding schemes [249,274,336,380,409,410] can be employed to effectively reduce spurious conductive paths and minimize surface leakage whilst preserving R_bias. On the other hand, effective reduction of “grounded” capacitances, C_par and C_in, can be achieved with the use of capacitance neutralization [86,94,104]. In addition, compensating for the internal amplifier capacitance C_in can also be achieved with power supply bootstrapping [371,372,415].

Alternative topologies and integrated design. However, if C_coupling drops to the order of 1 pF or lower, achieving C_IN << C_coupling becomes particularly challenging. In that case, trans-impedance and charge amplifiers can be employed [9,293,294,324,325,326,327,328]. Moreover, various biasing, input impedance boosting, and input capacitance reduction techniques can be employed inside the integrated circuit itself. Such techniques lead to complexified modern designs [104,371,413,416], as well as integrated bioamplifiers [103,109,138,139,140,141,142,176,314,413]. Minimizing the number of external discrete components and using custom integrated solutions instead of off-the-shelf chips allows for greater flexibility in the design: parameters, such as cutoff frequencies, are now designer-defined functions of the designed on-chip elements [417]. This way, the amplifier can be optimized for the specific purpose, and hence, the reduction in both board-level and on-chip parasitic elements can be enhanced. For instance, input capacitances can be further reduced down to the order of 10 fF and 100 fF, as in [109]. Aside from achieving the desired specifications, the choice and design of a biopotential amplifier should also include complying with the requirements of the specific measurement—for instance, performance requirements for ECG measurements can be found in [138,311,408]. Finally, unlike the classic ultra-high-input-impedance active electrodes investigated in this study, two-wired topologies can be used [102]. They can be implemented with the use of a current source bias and either a buffer active electrode, realized with a single transistor or with an operational amplifier [418], or an amplifying active electrode [323]. Also, a combination of both worlds—a high-input-impedance solution and two-wired electrodes—is feasible [415].

4.3. System Frequency Response

Appendix D and Section 4.2 brought the analysis of two single-pole subsystems, H_preamp and H_div, respectively. Now, these two subsystems can be cascaded into the frequency characteristic (frequency response) of the entire system, H_el(jω) (25), with magnitude response |H_el(jω)|_dB and phase response $\measuredangle$H_el(jω) given in (51) [268,269,390,397]. The corresponding transfer function H_el(s) was derived in (17), and impulse response h_el(t) in (21) in Section 4.1:

$$H_{\mathrm{e}\mathrm{l}}\left(j\omega \right)=H_{\mathrm{d}\mathrm{i}\mathrm{v}}\left(j\omega \right)· H_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{m}\mathrm{p}}\left(j\omega \right)=\left|H_{\mathrm{el}}\left(j\omega \right)\right|e^{j\measuredangle H_{\mathrm{el}}\left(j\omega \right)},$$

$${\left|H_{\mathrm{el}}\left(j\omega \right)\right|}_{\mathrm{dB}}=20{\log }_{10}\left[{\displaystyle \frac{C_{\mathrm{coupling}}}{C_{\mathrm{coupling}}+C_{\mathrm{IN}}}}·{\displaystyle \frac{\sqrt{{\omega }^2+{\left(\frac{1}{R_{\mathrm{coupling}}{C}_{\mathrm{coupling}}}\right)}^2}}{\sqrt{{\omega }^2+{\left(\frac{1}{{\left(R_{\mathrm{coupling}}\right||R}_{\mathrm{IN}}\left)\right(C_{\mathrm{coupling}}\left|\right|C_{\mathrm{IN}})}\right)}^2}}}·{\displaystyle \frac{A_0{\omega }_{\mathrm{A}}}{\sqrt{{\omega }^2+{{\omega }_{\mathrm{A}}}^2}}}\right],$$

$$\measuredangle H_{\mathrm{el}}\left(j\omega \right)=\mathrm{atan}2\left(\omega R_{\mathrm{coupling}}{C}_{\mathrm{coupling}}\right)-\mathrm{atan}2\left[\omega \left(R_{\mathrm{coupling}}\right|\left|R_{\mathrm{IN}}\right)\left(C_{\mathrm{coupling}}\right|\left|C_{\mathrm{IN}}\right)\right]-\mathrm{atan}2\left({\displaystyle \frac{\omega }{{\omega }_{\mathrm{A}}}}\right),$$

(51)

where atan2 is the four-quadrant arctangent function defined in (30).

Summary of upcoming figures. As inspected in Section 4.1, the entire system H_el is second-order with two non-trivial, single real poles s_p,1 and s_p,2 (18), and one non-trivial, single real zero s_z,1 (19). To investigate its magnitude response |H_el(jω)|_dB and phase response $\measuredangle$H_el(jω) (51), simulation is performed in LTspice^® 24.1.10 simulation software [147] with the use of SPICE models of two referenced operational amplifiers: LMP7721 [336,419] and OPA129 [337,420]. Firstly, with the use of a model based on LMP7721 (Figure 31), a parametric sweep of four electrode–body interface parameters was performed: R_coupling (Figure 32) and C_coupling (Figure 33), which form Z_coupling, and R_bias (Figure 34) and C_par (Figure 35) as external parts of Z_IN. These parameters were carefully modeled with values that appropriately reflect the previously investigated effects. Next, the influence of external parasitic capacitance C₋, present at the inverting input terminal and introduced in Appendix D in the discussion titled “Generalized analysis: frequency-dependent feedback network and higher-order models”, is observed in Figure 36. Likewise, the influence of the total effective capacitive load, C_out, representing the parallel combination of C_cable and C_load (Figure A10), is investigated in Figure 37. In addition, the influence of the preamplifier and its GBP (A20) is inspected in Figure 38, where frequency response is simulated for both operational amplifiers. Lastly, Figure 39 shows the frequency response for three types of electrodes analyzed in Section 3.1: insulated electrode, off-body electrode, and a wearable non-contact on-body electrode. In this last figure, parameters were modeled with values commonly found in practice to allow for easier comparison with the relevant papers. Results are summarized in the following paragraphs.

Low-end frequencies and influence of interface parameters. The cutoff frequencies of the predominant capacitive divider area, ω_C,I and ω_C,II, derived in step 2 in the previous section (38,43), represent the low-side cutoff frequency of the system H_el. Similarly, high-pass and low-pass configurations, described in step 2 of the previous section, are now observable at the low end of the frequency spectrum. In that area, frequency response H_el(jω) is predominantly shaped by the input voltage divider subsystem H_div and its four interface parameters, explored in the previous section. Resulting curves once again indicate the importance of achieving C_coupling >> C_IN (Figure 33 and Figure 35), using on-body electrodes (Figure 39), and taking into account the value of R_IN with respect to R_coupling (Figure 32 and Figure 34). In particular, for R_coupling << R_bias (black and red curves in Figure 32), resistance R_bias has no influence on the frequency response, and the area of predominantly resistive coupling stretches up to the frequencies in the order of 1 kHz. This was explained earlier in paragraph “Bias currents and DC biasing” considering the bias current path provided by low R_coupling, and now, it can also be understood in terms of the resistive input voltage divider, as explained in step 2 in the previous section. Furthermore, an increase in R_coupling from very low values (R_coupling << R_bias) to very high values (R_coupling >> R_bias) turns the configuration of the interface at low-end frequencies from all-pass to high-pass, shifting |s_z,1| to lower frequencies. An increase in R_coupling without a corresponding increase in R_bias increases the difference between the magnitude levels of the resistive and capacitive voltage dividers, 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ and 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$, but it also increases the phase shifts present in the phase response $\measuredangle$H_el(jω) between angular frequencies 0.1|s_z,1| and 10|s_p,1|, as explained in step 3 in the previous section. Moreover, R_bias is only an external part of R_IN, which is formed by the parallel combination of R_bias and R_in introduced by the preamplifier. Therefore, for R_bias << R_in, R_IN comes down to R_bias (R_IN $\approx$ R_bias), whereas for R_bias >> R_in, R_IN comes down to R_in (R_IN $\approx$ R_in). In other words, for a specified R_bias, the value of R_in can be estimated by observing the magnitude of the resistive voltage divider in the magnitude response |H_el(jω)|_dB (Figure 32 and Figure 34). The resistance R_in turns out to be about 195 GΩ. This explains why 66 GΩ of R_coupling gives 6 dB (50%) attenuation of the resistive input voltage divider in Figure 32. Similarly, C_par is only an external part of C_IN, which is formed by the parallel combination of C_par and C_in. In other words, for a chosen C_par, the value of C_in can be estimated by observing the magnitude of the capacitive voltage divider in |H_el(jω)|_dB (Figure 33 and Figure 35). The capacitance C_in turns out to be fairly high—15 pF, as estimated in the LMP7721 datasheet [336]. This explains why 35 pF of C_par for 50 pF of C_coupling is sufficient to yield 6 dB (50%) attenuation of the capacitive input voltage divider 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$. On the other hand, Figure 38 reveals that the estimated value of C_in in the case of OPA129 is about 2–3 pF, as stated in the OPA129 datasheet [337].

Mid-band and high-end frequencies. At the high-end frequencies, the high-side cutoff frequency of the system H_el is defined by the closed-loop bandwidth ω_A. For the observed single-pole model of a buffer preamplifier A_diff(f) (A17)–(A19) and the ideal unity closed-loop gain A₀ $=$ 1, ω_A is equal to the open-loop unity-gain crossover angular frequency ω₁, and hence the high-side cutoff frequency is determined by the GBP of the preamplifier (A20) (Figure 38). The area between the low-side and the high-side cutoff frequency corresponds to the mid-band gain, where the magnitude response |H_el(jω)|_dB is dominated by the magnitude level of the capacitive voltage divider. In accordance with the guidelines given in Section 4.2, the mid-band should be used between angular frequencies max{10|s_z,1|, 10|s_p,1|} and ω₁/10.

Higher-order models and stability issues revisited. Finally, effects at high-end frequencies can be observed more carefully. For this purpose, the influence of additional poles and parasitic capacitances C₋ and C_out that were described in the discussion titled “Generalized analysis: frequency-dependent feedback network and higher-order models” in Appendix D can be recalled. Treating these capacitances as being parallel to one another is now corroborated by their almost identical separate influence depicted in Figure 36 and Figure 37. With respect to the adopted conservative estimation of their influence (A21), the typical GBP of LMP7721 is about ten times higher—15 MHz [336], and its closed-loop output impedance is reduced to very low values in the order of 1 Ω. Additionally, gain peaking is observed at a frequency of about the GBP even when C₋ and C_out are 0 pF. Adding 35 pF of external C₋||C_out capacitance reduces the phase margin to 45°, whereas 330 pF reduces it to 0°. Similar simulations for various operational amplifiers and their configurations are available in [355]. As discussed in Appendix D, this shows why modeling at higher frequencies demands higher-order models of operational amplifiers, as well as performing thorough stability analysis with respect to the influence of reactive components in the system, especially the ones in the amplifier feedback, at its inverting input, and at its output. To quantify gain peaking and damping of the response, damping ratio and quality factor [421] can be used as indicators of system behavior under disturbance. More on peaking and phase margin can be found in [273,335,353]. More on damping ratio and quality factor in the context of second-order systems can be found in [341,345,354,390], with details behind various practical engineering examples in [395,422,423]. Examples of second-order and third-order modeling of operational amplifiers [341,346,356] can be found in [367] for the case of shield-driving circuits, in [369,370] for the case of bootstrapping networks, in [371] for a combination of a shield-driving circuit, bootstrapping network, and power supply bootstrapping network, and in [313] for fully differential biopotential amplifiers. Considerations on dry electrode transfer function and amplifier specifications with respect to standards for ECG recording can be found in [311,408]. Additional useful references can be found in discussions titled “Generalized analysis: frequency-dependent feedback network and higher-order models” and “Stability in practical biopotential measurements” in Appendix D.

Examples of physical measurements and concluding remarks. To carry out the corresponding physical measurements of electrode–body coupling impedance Z_coupling and system frequency response H_el(jω), it is necessary to come up with solutions that will allow for a reliable operation of non-contact and insulated electrodes in spite of all the adverse phenomena present in real-life experiments. These phenomena are usually neglected in the system analysis due to their high and difficult-to-control dependence on measurement setup, ambient conditions, and environmental noise. Firstly, aside from a complexified design of electrodes and bioamplifiers based on methods described at the end of Section 4.2 [104,371,413,416], such measurement systems also require designing a dedicated DAQ unit, along with a firmware solution for communication protocol, data transmission, and storage, as well as for rejection of power line interference and mitigation of motion artifacts [94,95], which must be implemented in such a way that it does not affect the real-time performance or signal quality. Secondly, such systems call for carefully planned mechanical layouts that will enable stable contact conditions. Lastly, they demand sophisticated methods of monitoring Z_coupling, which would ensure the predominantly contactless coupling principle. This is achieved by implementing strict galvanic isolation or (preferably) battery-powered instrumentation, as well as by managing the transient behavior and the so-called hidden parameters, such as perspiration, temperature, and pressure, which endanger the long-term temporal stability [424]. For these reasons, a considerable number of papers in the field of non-contact and insulated biopotential monitoring have been dedicated solely to a certain aspect of these demanding practical challenges [102,176,177,226,371,380,424,425,426,427,428,429]. However, each of these ingeniously devised measurement protocols and systems is commonly based on a use-case-specific analog front-end and DAQ solution. Also, they are evaluated under significantly different ambient conditions, which are accompanied by sources of electromagnetic interference that are uniquely tied to the measurement environment. When the issue of inherently problematic repeatability of non-contact measurements is added, along with the fact that the dielectric behavior and electrical performance of real-world dielectrics depend on their manufacturing process, structural properties, and ambient conditions, it becomes clear why the obtained measurements are in most cases only qualitatively comparable. Nevertheless, this does not lessen their use in further validation of the model presented in this paper. Namely, in contrast to the aforementioned papers dedicated to solutions for experimental validation, the goal of this paper was to consolidate theoretical foundations with existing practical design implications, and address long-standing terminological inconsistencies in the field of non-contact and insulated biopotential monitoring. To achieve this, an electrical model of the non-contact and insulated electrode–body interface, based on physical data and developed over the last several decades, is generalized and abstracted into a simplified, yet handy engineering tool that allows for the qualitative assessment of systems for non-contact and insulated biopotential measurements. In addition, it also provides a theoretical framework for understanding the phenomena behind real-life experiments. Ultimately, analytical discussions in Section 4.1 and Section 4.2 helped to identify the four interface parameters. Understanding their physical origin is the first step toward embracing the practical limitations of capacitive coupling and using that knowledge to qualitatively anticipate the behavior of real-world non-contact and insulated biopotential monitoring systems. In that sense, the developed model can indeed support examples of physical measurements available in the literature. For instance, in [380], simulation and physical measurements of magnitude response are provided. Simulation is based on OPA124 (Texas Instruments, Inc., Dallas, TX, USA), with parameters R_coupling $=$ 1 TΩ, C_coupling $=$ 150 pF, R_bias in the order of 1 GΩ, R_in $=$ 100 TΩ, and C_IN $=$ 3 pF. Since R_bias << R_in, R_IN comes down to R_bias. Therefore, C_IN/C_coupling < R_coupling/R_IN, which belongs to case II (|s_p,1| > |s_z,1|). As a result, low-side frequency of the predominant capacitive divider area, ω_C,II, is defined as in (43). When the effect of an increase in R_bias is observed for all other parameters fixed (Figure 4 in [380]), since R_IN $\approx$ R_bias for all considered values of R_bias, the corresponding f_C,II is shifted toward lower frequencies. On the other hand, when the effect of increase in C_IN is observed for all other parameters fixed and R_bias $=$ 3 GΩ (Figure 5 in [380]), magnitude level 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$ is decreased due to increased attenuation of the capacitive voltage divider. Furthermore, physical measurements of magnitude response are provided for a system based on AD8642 [338] with active guarding circuitry. |H_preamp(jω)|_dB is simulated and measured separately in Figure 15 in [380], yielding the high-side cutoff frequency f_A in accordance with the typical GBP of about 3 MHz, chosen closed-loop gain A₀ $=$ 40 dB $=$ 10^40/20 $=$ 100 V/V, and the fact that two operational amplifiers were used. Lastly, magnitude response is plotted for three different wearable fabrics: 100% cotton of thickness 0.45 mm, 100% polyester of thickness 0.26 mm, and 100% wool of thickness 0.91 mm. For the predefined coupling (sensing) area A $=$ 16 cm², parallel plate capacitor formula (3) could be used to calculate C_coupling in each of the three cases. However, no data is available for the values of corresponding dielectric constants ε_r and bulk resistivities ρ. As noted earlier in this paragraph, these values depend on the moisture content and structural properties, such as volume density (packing factor) and, consequently, applied pressure [430]. Based on the expected values for ε_r (1–3 [92,135]), expected C_coupling is in the range of 30–100 pF (3). Thus, for the simulated C_in of 150 pF, C_coupling will be of the same order as C_in, which yields the magnitude level 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}\approx$ −8 dB or lower. Similar qualitative comparison is available with other papers: in [384], magnitude response is obtained for three values of sensing (coupling) area A; in [296], surface-contact and over-cloth magnitude response are compared; and finally, in [49], magnitude response is obtained for three values of dielectric thickness, d. In all three cases, parallel plate capacitor approximation (3) can be used to assess the influence of the varying parameters: larger coupling area A increases C_coupling, thereby increasing the corresponding magnitude level 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$. On the contrary, greater thickness d decreases C_coupling, thereby decreasing the corresponding magnitude level 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$.

Figure 31. Simulation model of the analyzed system H_el based on LMP7721 operational amplifier [336,419] and created in LTspice^® 24.1.10 simulation software [147] in accordance with Figure 10. Default values of parameters are as follows: R_coupling $=$ 100 GΩ, C_coupling $=$ 50 pF, R_bias $=$ 100 GΩ, C_par $=$ 10 pF, C₋ $=$ 0 pF, and C_out $=$ 0 pF.

Figure 31. Simulation model of the analyzed system H_el based on LMP7721 operational amplifier [336,419] and created in LTspice^® 24.1.10 simulation software [147] in accordance with Figure 10. Default values of parameters are as follows: R_coupling $=$ 100 GΩ, C_coupling $=$ 50 pF, R_bias $=$ 100 GΩ, C_par $=$ 10 pF, C₋ $=$ 0 pF, and C_out $=$ 0 pF.

Figure 32. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for four values of R_coupling (100 kΩ, 100 MΩ, 66 GΩ, and 1 TΩ). The rest of the parameters are fixed: C_coupling $=$ 50 pF, R_bias $=$ 100 GΩ, C_par $=$ 10 pF, C₋ $=$ 0 pF, and C_out $=$ 0 pF.

Figure 32. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for four values of R_coupling (100 kΩ, 100 MΩ, 66 GΩ, and 1 TΩ). The rest of the parameters are fixed: C_coupling $=$ 50 pF, R_bias $=$ 100 GΩ, C_par $=$ 10 pF, C₋ $=$ 0 pF, and C_out $=$ 0 pF.

Figure 33. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for four values of C_coupling (100 fF, 5 pF, 25 pF, and 1 nF). The rest of the parameters are fixed: R_coupling $=$ 100 GΩ, R_bias $=$ 100 GΩ, C_par $=$ 10 pF, C₋ $=$ 0 pF, and C_out $=$ 0 pF.

Figure 33. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for four values of C_coupling (100 fF, 5 pF, 25 pF, and 1 nF). The rest of the parameters are fixed: R_coupling $=$ 100 GΩ, R_bias $=$ 100 GΩ, C_par $=$ 10 pF, C₋ $=$ 0 pF, and C_out $=$ 0 pF.

Figure 34. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for four values of R_bias (1 GΩ, 10 GΩ, 1 TΩ, and 10 TΩ). The rest of the parameters are fixed: R_coupling $=$ 100 GΩ, C_coupling $=$ 50 pF, C_par $=$ 10 pF, C₋ $=$ 0 pF, and C_out $=$ 0 pF.

Figure 34. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for four values of R_bias (1 GΩ, 10 GΩ, 1 TΩ, and 10 TΩ). The rest of the parameters are fixed: R_coupling $=$ 100 GΩ, C_coupling $=$ 50 pF, C_par $=$ 10 pF, C₋ $=$ 0 pF, and C_out $=$ 0 pF.

Figure 35. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for four values of C_par (0 pF, 35 pF, 50 pF, and 100 pF). The rest of the parameters are fixed: R_coupling $=$ 100 GΩ, C_coupling $=$ 50 pF, R_bias $=$ 100 GΩ, C₋ $=$ 0 pF, and C_out $=$ 0 pF.

Figure 35. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for four values of C_par (0 pF, 35 pF, 50 pF, and 100 pF). The rest of the parameters are fixed: R_coupling $=$ 100 GΩ, C_coupling $=$ 50 pF, R_bias $=$ 100 GΩ, C₋ $=$ 0 pF, and C_out $=$ 0 pF.

Figure 36. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for four values of C₋, denoted by C_inv (0 pF, 35 pF, 330 pF, and 500 pF). In addition, frequency response is zoomed at frequencies between 3 MHz and 30 MHz. The rest of the parameters are fixed: R_coupling $=$ 100 GΩ, C_coupling $=$ 50 pF, R_bias $=$ 100 GΩ, C_par $=$ 10 pF, and C_out $=$ 0 pF.

Figure 36. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for four values of C₋, denoted by C_inv (0 pF, 35 pF, 330 pF, and 500 pF). In addition, frequency response is zoomed at frequencies between 3 MHz and 30 MHz. The rest of the parameters are fixed: R_coupling $=$ 100 GΩ, C_coupling $=$ 50 pF, R_bias $=$ 100 GΩ, C_par $=$ 10 pF, and C_out $=$ 0 pF.

Figure 37. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for four values of C_out (0 pF, 35 pF, 330 pF, and 500 pF). In addition, frequency response is zoomed at frequencies between 3 MHz and 30 MHz. The rest of the parameters are fixed: R_coupling $=$ 100 GΩ, C_coupling $=$ 50 pF, R_bias $=$ 100 GΩ, C_par $=$ 10 pF, and C₋ $=$ 0 pF.

Figure 37. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for four values of C_out (0 pF, 35 pF, 330 pF, and 500 pF). In addition, frequency response is zoomed at frequencies between 3 MHz and 30 MHz. The rest of the parameters are fixed: R_coupling $=$ 100 GΩ, C_coupling $=$ 50 pF, R_bias $=$ 100 GΩ, C_par $=$ 10 pF, and C₋ $=$ 0 pF.

Figure 38. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for two operational amplifiers: (black) LMP7721 [336,419] and (red) OPA129 [337,420]. The rest of the parameters are fixed: R_coupling $=$ 100 GΩ, C_coupling $=$ 50 pF, R_bias $=$ 100 GΩ, C_par $=$ 10 pF, C₋ $=$ 0 pF, and C_out $=$ 0 pF.

Figure 38. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for two operational amplifiers: (black) LMP7721 [336,419] and (red) OPA129 [337,420]. The rest of the parameters are fixed: R_coupling $=$ 100 GΩ, C_coupling $=$ 50 pF, R_bias $=$ 100 GΩ, C_par $=$ 10 pF, C₋ $=$ 0 pF, and C_out $=$ 0 pF.

Figure 39. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for three different types of electrodes analyzed in Section 3.1: (upper) insulated electrode, (middle) off-body electrode, and (lower) wearable non-contact on-body electrode.

Figure 39. (Solid lines) magnitude response |H_el(jω)|_dB and (dotted lines) phase response $\measuredangle$H_el(jω) of the entire system based on simulation model in Figure 31 for three different types of electrodes analyzed in Section 3.1: (upper) insulated electrode, (middle) off-body electrode, and (lower) wearable non-contact on-body electrode.

5. Conclusions

In Section 1.1, various examples of biomedical and non-biomedical measurements demonstrated versatility of the capacitive sensing strategy, as well as its promising role in electrophysiological and biopotential sensing. Relying on capacitive coupling over insulating layers, the non-contact biopotential measurement principle offers zero preparation and seamless integration of electrodes into everyday garments and wearable body area networks, paving the way for unobtrusive, long-term, remote health monitoring to become standard practice. However, there are challenges left to tackle before unlocking their full potential and widespread use in applications for reliable early diagnostics.

In Section 3.1 and Appendix B, after establishing the frequency range of interest in Assumption 1, motivation for further research on non-contact and capacitive biopotential monitoring was found through a comparison of various types of biopotential electrodes according to their working principle and characteristics: wet and non-insulated dry surface-contact electrodes on the one side (Appendix B) and dry non-contact and capacitive electrodes on the other side (Section 3.1). The analysis of their equivalent electrical models, as well as electrochemical processes at the interface (Figure A2 and Figure A3) and finite leakage resistances of insulating layers (Figure 3), showed that coupling with biopotential electrodes comes down to a combination of resistive and capacitive elements. Discussion in Section 3.1, starting from the coupling over a single insulating layer, resulted in the creation of a generalized model of the interface between the body and a non-contact biopotential electrode (Figure 5, Contribution 2.1). The model is based on parallel plate approximations described with Equations (3), (7), and (9)–(13). These equations revealed that the equivalent electrode–body coupling impedance Z_coupling can be manipulated with a careful choice of electrode sensing (coupling) area, as well as the coupling material and its thickness. By accounting for a finite leakage resistance of insulating layers, the distinction was emphasized between idealized capacitive electrodes on the one side, and non-contact and insulated electrodes with predominantly capacitive coupling on the other side. In brief, contactless biomonitoring does not guarantee predominantly capacitive coupling, nor does surface-contact biomonitoring guarantee predominantly resistive coupling. At DC (f $=$ 0 Hz), equivalent leakage resistance of the electrode–body interface, R_coupling, defines the bulk DC conductivity of the interface, as well as its electrical insulation performance. On the other hand, at a certain frequency f > 0 Hz, the value of frequency-dependent R_coupling(ω) (9) relative to the capacitive reactance X_Ccoupling(ω) (A4) determines the predominant coupling mechanism (resistive or capacitive). Therefore, Test 1 was proposed as a tool for assessing the predominant coupling mechanism achieved over an insulating layer at a certain frequency (Contribution 1). Results are given in Table 1, visualized in Figure 4, and applied to three specific cases of the generalized electrode–body interface model in Figure 5: insulated electrodes, off-body electrodes, and non-contact on-body electrodes. Due to the devastating effect of an air gap (off-body case), on-body electrodes were highlighted as preferred for biopotential measurement. Important findings and examples, summarized in the paragraph titled “Important conclusions and misterming issues as a motivation for the classification of biopotential electrodes”, led to Assumption 4 and replacement of the initially used term “insulating layer” with a more general term “coupling layer”. Likewise, the initially used group term “non-contact and capacitive biopotential electrodes” was replaced with a more appropriate group term “non-contact and insulated biopotential electrodes”. Finally, refined classification of biopotential electrodes was proposed to clearly emphasize that the terms “non-invasive”, “on-body”, “off-body”, “surface-contact”, “non-contact”, “wet”, “dry”, “insulated”, and “capacitive” are not synonyms (Figure 6, Contribution 3).

After explaining the benefits of active electrode design over passive design in Section 3.2, the generalized model of non-contact and insulated electrode–body interfaces was expanded into a model of a single buffer active non-contact and/or insulated biopotential electrode and its interface with the body (Contribution 2.2). The model is based on Assumption 5, introduced in paragraph “Introduction to Section 4 and the first subsystem: operational amplifier”, and depicted in Figure 10. Therein, two equivalent impedance blocks were highlighted for further analysis: the total equivalent electrode–body coupling impedance Z_coupling, defined in Assumption 4, and the total equivalent input impedance Z_IN, defined in Assumption 9. These two equivalent impedances comprise a resistive and a capacitive component, described by four interface parameters: R_coupling, C_coupling, R_IN $=$ R_bias||R_in, and C_IN $=$ C_par||C_in. The parameters R_in and C_in of the preamplifier common-mode input impedance were introduced in paragraph “Model of a buffer active non-contact and/or insulated electrode and its interface with the body with three equivalent impedances.” On the other hand, the parameter C_par, as the pin, pad, layout, and cable capacitance between the non-inverting input pin (sensing surface) and the preamplifier circuit common, was introduced in paragraph “Parasitic input capacitance”, whereas the parameter R_bias, as the equivalent resistance of the external DC biasing network, was introduced in paragraph “Bias currents and DC biasing.” Influence of the rest of the parameters—additional capacitances C₋, C_com−, C_diff, and C_out $=$ C_cable||C_load seen at the preamplifier inverting input and output—was analyzed and neglected in Appendix D in the discussion titled “Generalized analysis: frequency-dependent feedback network and higher-order models”, and revisited in Section 4.3.

The model presented in Figure 10 was further employed throughout Section 4 to carry out the bottom-up analysis of the entire system interface–electrode–preamplifier, which was defined as a cascade of two first-order (single-pole) subsystems: the input voltage divider and the buffer preamplifier. In accordance with Assumption 5, the provided analysis is also applicable to other non-invasive active biopotential electrodes.

The first subsystem—buffer preamplifier—was built in Appendix D. Therein, the analysis started from the working principle of a voltage-feedback operational amplifier, as well as considerations on its stability and voltage follower (buffer) configuration. As a result, the single-pole model of a buffer preamplifier, A_diff(f) (A17)–(A19), was established and adopted for the rest of the paper as the subsystem H_preamp. Assumptions on the operational amplifier and the preamplifier analog front-end, as well as on the PCB design considerations, are listed in Assumptions 7 and 8.

In Section 4.1, the transfer function of the entire system interface–electrode–preamplifier, H_el(s) (17), and its impulse response, h_el(t) (21), were derived. The pole–zero plot was observed (Figure 12), and an example of impulse response and the corresponding transfer function were visualized in Figure 13 and Figure 15, respectively. Specifically, Figure 14 revealed how the location of poles in the complex s-plane shapes the impulse response and the time-domain transient response. The transfer function H_el(s) confirmed the key influence of the input voltage divider effect: voltage sensing depends on the frequency-dependent voltage divider ${\displaystyle \frac{Z_{\mathrm{IN}}}{Z_{\mathrm{coupling}}+Z_{\mathrm{IN}}}}$, created by the equivalent electrode–body coupling impedance Z_coupling and the finite equivalent input impedance Z_IN. Lastly, for the steady-state analysis, Laplace transform and transfer function were reduced to Fourier transform and frequency characteristic (frequency response) H_el(jω) (25) with its magnitude and phase response (51). In accordance with Bode approximations presented in Figure A9, an example in Figure 15 showed how poles and zeros of the transfer function were translated into break frequencies of the corresponding magnitude response.

The second subsystem—input voltage divider and its frequency response H_div(jω) (26)—were further inspected in Section 4.2. Therein, it has been shown how voltage attenuation and additional phase shifts, introduced by the zero s_z,1 (19) and the pole s_p,1 (18), degrade the frequency response and adversely affect ECG signal morphology. This was demonstrated in Figure 26. Under the assumption of a negligible contribution of the AC conductivity at the frequencies of interest (subcase A.1 in Table 1), two main cases (case I and case II) were identified and illustrated in Figure 16 based on the position of s_p,1 with respect to s_z,1. Additionally, subcases 1 and 2 were observed based on the value of R_IN with respect to R_coupling. The results are listed in Table 2 for each of the four subcases. Applying Test 1 and Bode approximations to Z_coupling and subsystem H_div(jω) in three steps revealed how the position of the zero s_z,1 and the pole s_p,1 determine each of the three specific areas: area of predominantly capacitive coupling (step 1), area of predominant capacitive divider (step 2), and area of minimized phase shifts and near-zero group delay (step 3). Accordingly, step 2 analyzed the magnitude response |H_div(jω)|_dB and magnitude levels of the resistive and capacitive voltage dividers, 20log₁₀${\displaystyle \frac{R_{\mathrm{I}\mathrm{N}}}{R_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+R_{\mathrm{I}\mathrm{N}}}}$ and 20log₁₀${\displaystyle \frac{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}}{C_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+C_{\mathrm{I}\mathrm{N}}}}$, respectively. On the other hand, step 3 analyzed the phase response $\measuredangle$H_div(jω) and area of minimized phase shifts. Discussion in each step was corroborated by graphical examples and concluded with a summarizing paragraph.

The derived relation between zero and pole angular break frequencies, |s_z,1| and |s_p,1| (31,46), showed that the existence of the pole is a direct consequence of the input voltage divider effect. Further analysis in step 2 revealed an important interdependence: greater difference between |s_z,1| and |s_p,1| draws a greater difference between the magnitude levels of the resistive and capacitive voltage dividers, and vice versa. In that sense, the shape of the frequency response H_div(jω) can be modeled by relocating s_p,1 with respect to s_z,1. This comes down to manipulating the four interface parameters and their ratios R_coupling/R_IN and C_IN/C_coupling, depending on the target configuration of the interface. On the one hand, high-pass configuration achieves a roll-on in the magnitude response. Hence, moving s_p,1 further away from s_z,1, increasing R_coupling/R_IN, and decreasing C_IN/C_coupling is beneficial. On the other hand, all-pass configuration achieves a flattened magnitude response. Hence, keeping s_p,1 in the vicinity of s_z,1, as well as equating and decreasing the ratios R_coupling/R_IN and C_IN/C_coupling, is beneficial. For both configurations, the goal is to minimize the attenuation of the capacitive divider and achieve C_coupling >> C_IN, i.e., C_coupling ≥ 10C_IN. However, high-pass configuration strives for the concept of a pure capacitive electrode with a focus on AC-coupling and electrical insulation of the electrode–body interface; thus, it favors increasing R_coupling. On the other hand, all-pass configuration takes advantage of the finite R_coupling (i.e., finite interface leakage) to focus on extending the measurement range further toward lower frequencies, offering the additional benefit of flattened phase response. As a result, all-pass configuration is less prone to distortion of the signal.

Three-step analysis in Section 4.2 resulted in guidelines for designing non-contact and insulated electrode–body interfaces, proposed in the paragraph “Guidelines for designing non-contact and insulated electrode–body interfaces” (Contribution 4). The aim of these guidelines is to extend the predominantly capacitive coupling area, predominant capacitive divider area, and area of minimized phase shifts toward lower frequencies. Ultimately, flattening the frequency response of the input voltage divider H_div(jω) whilst reducing attenuation of its magnitude response and minimizing phase shifts in its phase response is beneficial. This comes down to an increase in R_coupling and C_coupling, followed by maximization of R_IN and minimization of C_IN.

These results were further confirmed in Section 4.3 with the use of SPICE simulation of the entire system. Therein, both subsystems, H_div analyzed in Section 4.2 and H_preamp analyzed in Appendix D, were combined to observe the total frequency response H_el(jω). The analysis has shown that the input voltage divider and electrode–body interface shape the frequency response at low-end frequencies, whereas the preamplifier with its closed-loop bandwidth shapes the frequency response at high-end frequencies.

In practice, maximization of R_IN is limited by the DC biasing network and its equivalent resistance R_bias. On the other hand, minimization of C_IN is limited by both the preamplifier common-mode input capacitance C_in and the pin, pad, layout, and cable parasitic capacitance C_par. The presence of these elements adversely affects the low-side frequency limit of the system mid-band gain, and ignoring them could impede the use of active non-contact and insulated biopotential electrodes in clinical-grade applications. Therefore, their design calls for reconciling the goal of achieving a high input impedance on the one side, which comes with the issue of vulnerability to electric field interference and sensitivity to surface leakage, with the goal of achieving proper DC biasing and fast recovery from large input transients on the other side. For this purpose, various advanced techniques for biasing, input impedance boosting, and input capacitance reduction can be employed, taking into account the balance between cost, size, consumption, and performance. Moreover, specialized integrated bioamplifiers can be employed, as well as solutions other than the analyzed classic high-input-impedance models, such as two-wired topologies, transimpedance amplifiers, and charge amplifiers. These practical aspects of the proposed guidelines were surveyed at the end of Section 4.2.

As discussed in Section 4.3, the goal of this paper was to bring an analytical interpretation of commonly obtained measurement results, as well as explain the caveats behind modeling of biopotential electrodes and their interface with the body. Since the purpose of the laid out system analysis was to provide a deeper insight into the influence of individual parameters that build the equivalent electrical model of non-contact and insulated electrode–body interfaces, simulations were the preferred way of communicating the results of analytical calculation. Accordingly, the intent of this paper was not to introduce novel experiments, but rather to synthesize, reconcile, and systematize the extensive and sometimes inconsistent body of literature in accordance with three objectives and four contributions that were listed and visualized in Section 1.2. Based on other review papers in this research area, categorized in Section 2, this review serves the purpose of enhancing the transparency and unification of earlier findings, translating them from between-the-lines outcomes of a specific research approach into an elaborate list of abstracted conclusions that streamline the access to existing work and allow for easier dissemination of knowledge among research groups, with the hope of preventing the pinpointed conceptual errors and terminological inconsistencies in the future. Accordingly, instead of skipping elementary derivations and common engineering knowledge, fundamentals of the following were thoroughly revisited: biopotential signals and their acquisition (Appendix A), electrochemistry (Appendix B), dielectrics, dielectric loss, capacitors, phasors and impedance (Appendix C and Section 3.1), operational amplifiers, negative feedback, Bode plot and amplifier stability (Appendix D and Section 3.2), transfer function, impulse response, frequency characteristic and system stability (Section 4.1), and phase delay and group delay (Section 4.2). The analysis was built upon each of the nine established assumptions, which offer an abridged version of the content for quick access. The results were additionally corroborated by examples of real-life parameter values, practical aspects of real-world electronic components, and several examples of commonly used operational amplifiers.

Finally, to further place the analysis in the context of real-life measurements available in the existing literature, the last paragraph in Section 4.3 titled “Examples of physical measurements and concluding remarks” expounds on how the existing experimental data relate to the presented theoretical framework. While the developed and generalized model is a cornerstone for modern non-contact and insulated biopotential electrodes, papers dedicated to the challenges of real-world measurements reveal the presence of additional phenomena, which arise from electromagnetic interference with the measurement environment and from variations in interface parameters as a result of motion and non-constant ambient conditions. Nevertheless, the provided model can be used to qualitatively assess non-contact and insulated electrode–body interfaces with respect to the clarified terminology and resolved misconceptions. Further accounting for the adverse effects of environment, ambient conditions, and real-world electronic components would require the addition of interference and noise analysis to the presented system analysis.