Low-Frequency Measurements in Electrochemical Impedance Spectroscopy: A Brief Review

Abstract

Some properties of electrochemical systems can be properly characterized by low- and extremely low-frequency impedance measurements. The frequency range down to 1 mHz requires peculiar attention in terms of accuracy, measurement speed, portability and costs, and commercial systems are often unable to guarantee good performances without expensive or bulky solutions. Here, we first review some of the applications of electrochemical impedance spectroscopy and their requirements and then report on recent works proposing circuits and systems focused on optimizing performances, reducing costs, and increasing versatility. Circuits for generating controllable current sources are described, optimizing output impedance and measurable impedance range. Then, integrated systems are taken into account, aiming at realizing portable and versatile instruments.

1. Introduction

Electrochemical Impedance Spectroscopy (EIS) is a widely adopted, non-invasive analytical technique for investigating the dynamic behavior of electrochemical systems [1]. Its versatility and sensitivity make it a valuable tool across a broad range of applications, including energy storage devices [2], corrosion science [3], and electrocatalysis [4]. An electrochemical system inherently presents multiple barriers to current flow, including ionic transport resistance in the electrolyte, diffusion limitations of reactive species approaching the electrode surface, capacitive impedance associated with the electrical double layer, and activation overpotentials linked to electron transfer reactions. Each of these steps can be represented by an electrical element or a combination of elements in an equivalent circuit model [5]. EIS is commonly utilized to derive the parameters required for equivalent circuit model development [6,7]. Through impedance analysis, it is thus possible to extract quantitative and phenomenological insights into each of these processes. However, a key challenge lies in deconvoluting the overall impedance response and correctly attributing each spectral feature to its corresponding physical process. This requires the development and validation of circuit models that accurately reflect the electrochemical system under investigation and closely reproduce the experimental impedance spectra with high accuracy. The main advantage of EIS lies in its frequency-resolved nature: except for purely resistive elements, most impedance contributions—such as capacitive and diffusional reactance—strongly depend on frequency. This allows for the isolation and analysis of individual electrochemical phenomena, which is essential for both fundamental understanding and the optimization of functional materials and devices.

EIS involves the measurement of small-amplitude sinusoidal voltage and current signals over a broad frequency range, typically from 10⁻³ Hz to 10⁶ Hz. In the frequency (f) domain, the impedance can be written as:

$$Z\left(\omega \right)={\displaystyle \frac{V\left(\omega \right)}{I\left(\omega \right)}}=Z^{\prime }\left(\omega \right)+jZ″\left(\omega \right)=|Z(\omega )|e^{j\varphi \left(\omega \right)}$$

(1)

where ω = 2πf is the angular frequency; Z′(ω) and Z″(ω) are the real and imaginary components of the impedance, respectively; $\left|Z\right(\omega \left)\right|$ is the impedance modulus and ϕ(ω) the phase shift between the applied (measured) voltage V(ω) and the measured (applied) current I(ω). In particular, Z(ω) encodes information about the physicochemical processes occurring at the electrode/electrolyte interface and within the bulk phases.

The ability of EIS to resolve different electrochemical phenomena relies on the fact that various processes exhibit characteristic time constants and thus dominate in different regions of the frequency spectrum. At high frequencies, the impedance is typically governed by the ohmic resistance of the electrolyte and contact resistances. Intermediate frequencies often reflect charge transfer kinetics and double-layer capacitance, while low-frequency measurements reveal slower processes such as ionic diffusion, mass transport limitations, capacitive storage, and interfacial polarization effects [8]. The investigation of system behavior at low frequencies is particularly important. Indeed, measurements performed in the low-frequency domain—typically below 1 Hz—enable the analysis of phenomena governed by large time constants, such as ionic diffusion within porous electrodes, mass transport limitations across membranes or viscous electrolytes, and capacitive charge accumulation in energy storage devices [2]. These contributions—not readily observable at higher frequencies and only properly resolved through low-frequency analysis—are often represented by Warburg impedance [9] or constant phase elements [10], which are estimated by specifically developed models [11,12].

The ability to probe these processes is essential for the deconvolution of the various physical mechanisms contributing to the overall electrochemical response. Specifically, low-frequency measurements enhance the resolution of slow kinetic phenomena, allowing for more accurate parameter extraction in equivalent circuit models, including charge transfer resistance, double-layer capacitance, and diffusion-related impedance components. For instance, this level of detail is critical in the study of batteries, supercapacitors, and fuel cells, where long-term performance, charge storage mechanisms, and degradation pathways are intrinsically linked to such slow dynamic processes [13]. Moreover, EIS in the low-frequency regime provides diagnostic insight into systems exhibiting relaxation behavior, polarization build-up, or interfacial instability [14].

Dedicated systems operating at specific frequency bands must be designed and implemented to address the challenges of low-frequency measurements, including the need for high accuracy, extended acquisition time, and sensitivity to non-stationary conditions.

Commercial impedance meters are usually employed for broadband applications but are generally limited to frequencies in the kHz range, unless one resorts to expensive or bulky equipment. On the other hand, most of the applications of low-frequency EIS often require low-cost dedicated solutions, in situ measurement, and high portability, without sacrificing the high performances within the frequency band of interest. Improvements in impedance modeling and measurement methodologies can be helpful, but they must be supported by the development of custom-designed circuits. Therefore, hardware for EIS systems has to be optimized using new circuital architectures, embedded solutions, and integrated systems, trying to balance between performance, costs, and versatility.

This work offers a brief review of recently proposed circuits and systems designed to meet the aforementioned requirements of low-frequency EIS. The paper is organized as follows: Section 2 is an overview of the most common applications of EIS with the aim to summarize the main findings and challenges; Section 3 analyzes the required characteristics for the electronics of EIS systems and goes through some circuital hardware solutions adopted to meet these requirements; Section 4 provides some general considerations and discussion points on the reported circuits in the framework of future trends.

2. A Brief Summary of EIS Applications

In some specific applications of EIS, very tight constraints must be met in terms of current levels and frequency bandwidth. Several studies have reported impedance measurement in the very low-frequency range. In this context, several applications of low-frequency spectroscopy are further examined below, including battery diagnostics, early tumor detection, tissue and cellular composition analysis, sensor characterization, perovskite solar cell studies, and electrode degradation monitoring.

2.1. Energy Storage Systems

EIS is extensively employed to characterize the dynamic behavior of batteries and electrochemical energy conversion systems [15]. Its ability to provide insights into the internal state of batteries makes it a valuable tool for real-time condition monitoring [16]. EIS involves applying a low-amplitude perturbation to the battery and measuring the output response, typically using single-frequency sinusoidal signals [17,18,19,20]. As a result, the impedance is calculated at discrete frequencies. In the sub-millihertz range, the Nyquist plot reveals crucial information about the battery’s state of charge (SoC) and state of health (SoH) [21,22,23], highlighting the importance of accurate low-frequency measurements.

Olderburger et al. [21] investigate the impact of time-domain (TD) measurement parameters on the low-frequency impedance of Li-ion cells, focusing on excitation current amplitude, pulse duration, and pulse direction (charge or discharge of the cell). Experiments conducted at 25 °C and various states of battery charge (30%, 50%, and 70%) reveal that all these factors significantly affect the Warburg impedance below 2 mHz, with deviations reaching up to 60% depending on excitation history. The observed discrepancies are attributed to open circuit voltage (OCV) hysteresis, as confirmed by simulations using a one-dimensional transmission line model with integrated hysteresis. Classical frequency-domain EIS fails to capture these effects unless an OCV loop correction is applied. The study highlights that accurate low-frequency impedance measurements require careful selection of excitation parameters. To avoid errors and reduce measurement time, the authors recommend using TD methods with short, high-current pulses for frequencies below the mHz range.

McCarthy et al. [24] introduce a robust approach to monitor the temperature of the battery core in real time using impedance data obtained during operation. In the first phase, they systematically analyze how temperature, SoC, and SoH affect battery impedance at various frequencies. They identify 200 Hz as a specific frequency at which the impedance signal is highly sensitive to temperature variations, while remaining relatively insensitive to changes in SoC and SoH, making it an optimal feature for temperature estimation. Based on this insight, the authors develop an estimation model relying on single-point impedance measurements at 200 Hz, performed online under load conditions. The method is validated with data from nine independent commercial Li-ion cells, each cycled more than 100 times, achieving an average root mean square error of 1.41 °C across all datasets. This performance underscores the feasibility of using EIS for accurate, sensorless internal temperature monitoring without interrupting normal battery operation. The approach is likely to be implemented in Battery Management Systems aimed at improving safety and efficiency.

In ref. [25], EIS was used to estimate the charge-transfer resistance and double-layer capacitance of low-capacity Ni-Cd cells. A two-term impedance calibration process was applied, using a short and a known shunt-resistor standard, to obtain error coefficients and correct the system response. The calibration had a notable effect at frequencies above 200 Hz, with average changes of approximately 8% in the real part and 6% in the imaginary part of the impedance. Finite element method simulations were conducted to analyze the impact of cable inductance and geometry. Calibrated EIS data were collected at various SoC from single-cell and two-cell blocks. Equivalent circuit models were fitted to the data to extract electrochemical parameters such as ohmic resistance, charge transfer resistance, and double-layer capacitance.

Du et al. [26] propose a perturbation signal processing framework for real-time measurement of Differential Impedance Spectroscopy (DIS) to assess battery charging and overcharging dynamics. Unlike conventional methods based on voltage and temperature, the proposed approach uses DIS to extract overcharge-related parameters by fitting the frequency response to an equivalent circuit model. Experimental validation, including Kramers–Kronig tests with residuals below 0.5%, confirms the accuracy of the DIS measurements. The disjoint coefficient D² reaches 100% under varied conditions, indicating reliable detection of overcharging based on changes in charge-transfer resistance. Overcharge warnings are triggered at approximately 98% SoC. Although the method is validated on three cells, the authors point out that further investigation is needed to generalize the methodology across different battery types, temperatures, aging levels, and pack configurations, as well as to evaluate its long-term effects on battery health and thermal stability.

2.2. Biomedical Applications

Impedance measurement techniques are useful for evaluating the electrical characteristics of biological tissues [27], offering important information on body composition and overall tissue condition [28,29]. These methods have also been employed to monitor cardiovascular indicators such as heart rate variability and cardiac output [30]. In the context of respiratory assessment, an impedance measurement of lung provides critical data regarding pulmonary function and health status [31]. These applications typically involve impedance measurements conducted within the frequency range of 5 kHz to 250 kHz [32].

EIS at low frequencies is particularly useful for investigating the electrical behavior of biological tissues at the cellular and interfacial level. In this frequency range (typically from 0 Hz to a few hundred Hertz), EIS is sensitive to the α-dispersion phenomenon, which is associated with the relaxation of ions near charged cell membranes [33,34].

Additionally, low-frequency impedance-based approaches have demonstrated potential in cancer diagnostics by identifying abnormal tissue properties and monitoring tumor development [35] and chemoresistance [36,37]. For accurate discrimination among different cancer cell types, it is essential to perform EIS over a broad frequency range, beginning at low frequencies (as low as 100 Hz), where critical information related to cell membrane properties and interfacial polarization is most prominent, and extending to the MHz range to capture intracellular characteristics [38].

In ref. [39], a novel bioimpedance device was developed for early breast cancer detection, aiming for a non-invasive wearable device. The system employs low-frequency (<5 kHz) alternating current, which primarily flows through the extracellular space [40], as the cell membrane acts as a barrier at these frequencies. Since malignant breast tissue exhibits lower extracellular resistance compared to healthy tissue, the system detects abnormalities by measuring the impedance difference between the left and right breasts, where a resistance difference exceeding 50 Ω indicates the need for further clinical evaluation. The device was validated through both in vitro and in vivo testing. In vitro results showed a strong correlation between NaCl concentration and resistance (r = 0.97), with stable readings achieved after five minutes. In the same context, Huerta-Nunez et al. [41] present an EIS-based biosensor capable of detecting low quantities of breast cancer cells. Impedance measurements were performed over a frequency range from 100 Hz to 1 MHz to capture both α and β dielectric dispersion regions, which are associated with surface heterogeneity and membrane polarization, respectively. The low-frequency range (starting at 100 Hz) was particularly relevant for detecting extracellular and membrane-related electrical properties. The biosensor demonstrated high sensitivity in distinguishing cancerous from non-cancerous cells, indicating its potential for early-stage breast cancer diagnostics through label-free, non-invasive electrical measurements.

In ref. [42], Jeong et al. introduce a micro-dimensional EIS platform combined with machine learning for classifying normal and cancerous human urothelial cells. The system utilizes low-frequency impedance measurements, enhanced by a flow cytometry design that ensures close contact between cells and electrodes. This configuration allows the electric field to penetrate the cell membrane and cytoplasm, enabling accurate detection of intracellular electrical properties even in the low-frequency range.

2.3. Other Applications

EIS is widely used to investigate fouling phenomena in microbial fuel cells [43,44]. In ref. [45], EIS was employed in the range 100 mHz–1000 kHz to investigate the charge transfer resistance (R_ct) at the electrode/electrolyte interface within the microbial fuel cell-ceramic membrane bioreactor systems. EIS enabled the assessment of how extracellular polymeric substance composition affects the electrochemical behavior of the membrane and contributes to membrane fouling. By monitoring changes in R_ct, the technique provided insight into the fouling progression and its impact on electron transfer processes. This allows for a better understanding of the relationship between fouling development and the system’s electrical performance.

Achtenberg et al. [46] employed EIS to study the frequency-dependent impedance behavior of the HgCdTe heterostructure, particularly in the negative differential resistance (NDR) region. The technique allowed for a detailed analysis of the device’s dynamic electrical properties as a function of DC bias, temperature, and frequency. EIS revealed unusual characteristics in the NDR region, such as two local maxima in complex impedance, drastic variations in parallel capacitance and resistance (including sign reversal), and strong frequency dependence.

In ref. [47], EIS was used in the frequency range 1 Hz–1 MHz to probe slow carrier dynamics and interfacial phenomena in triple cation perovskite solar cells. In this frequency range, impedance measurements are sensitive to long-timescale processes such as ion migration, charge accumulation at interfaces, and trap-assisted recombination. These effects play a crucial role in device performance, particularly in relation to hysteresis and operational stability. The low-frequency response revealed signatures of Maxwell–Wagner polarization and allowed the separation of ionic and electronic contributions. By combining impedance and modulus spectroscopy, the study provided insight into localized carrier relaxation and charge transport mechanisms that cannot be resolved at higher frequencies. Additionally, slow ionic processes and interfacial phenomena in perovskite solar cells were investigated by means of EIS from 1 MHz down to 10 mHz [48]. The study introduced an active circuit model with a gyrator element to replicate unusual impedance features, such as inductive loops and negative capacitance, observed only at low frequencies, highlighting strong ion–electron coupling effects.

Impedance measurements, over a frequency of 100 kHz down to a few mHz, allow for the investigation of low corrosion processes of magnesium, such as film formation and breakdown, adsorption of intermediates, and mass transport through corrosion layers [49]. These phenomena occur over long timescales and produce capacitive and inductive responses detectable only at low frequencies. This allowed detailed insight into the dynamic evolution of corrosion mechanisms and changes in charge-transfer resistance.

EIS was also used to characterize boron-doped diamond thin films grown by Chemical Vapor Deposition, intended for use in multi-electrode array biosensors [50]. Measurements were performed in a three-electrode cell over a frequency range from 1 Hz to 1 MHz. The impedance spectra, represented as Nyquist plots, allowed for the extraction of charge transfer resistance values, which varied depending on the boron doping level, surface morphology, and grain size of the diamond films. Results demonstrated that optimized films exhibited low impedance and favorable electrochemical behavior, indicating their suitability for integration into stable and sensitive amperometric biosensing systems.

EIS was applied in ref. [51] to examine the low-frequency electrochemical characteristics of polyaniline (PANI). The objective was to analyze the resistive and capacitive contributions of PANI in acidic solution, distinguishing charge transfer processes, double-layer capacitance, and ionic diffusion. EIS measurements were performed using a three-electrode setup over a frequency range from 10 mHz to 10 kHz. The results demonstrate that low-frequency EIS provides critical insights into the electrochemical properties of PANI, supporting its potential application in electronic devices and chemical sensors.

The work by Placidi et al. [52] presents the development of a low-cost, low-frequency impedance meter for measuring soil water content, aimed at precision agriculture applications. EIS was employed as the soil’s electrical impedance significantly varies with moisture content, affecting both resistive and capacitive components. The device operates within a frequency range of 10–100 kHz, where sensitivity to water content changes is maximized. Low-frequency measurements enable accurate assessment of soil moisture by differentiating resistive and capacitive contributions. Due to its simplicity and affordability, the system is suitable for continuous and widespread monitoring, which is essential for optimizing irrigation management and resource use in precision farming scenarios.

2.4. Challenges in Experimental Environments

Despite being a highly versatile and powerful technique, EIS applications across different domains may reveal several recurring limitations. In energy storage systems, EIS measurements are often affected by the non-linear and non-stationary behavior of batteries under dynamic operating conditions. This limits the applicability of traditional linear equivalent circuit models, increasing the risk of data misinterpretation and requiring more complex non-linear models [53].

In perovskite solar cells, time-dependent phenomena such as ion migration and current–voltage hysteresis introduce memory effects that complicate the attribution of impedance features to specific physical processes. Although EIS is more suitable for quantifying hysteresis than conventional I–V sweeps, features such as inductive loops and low-frequency negative capacitance still require further experimental validation [54].

Biomedical and biosensing applications are subject to distinct yet equally critical limitations, including signal instability resulting from electrode fouling, inherent variability in biological samples, and the difficulty of discriminating specific binding events from non-specific interactions —all of which compromise reproducibility and lower the signal-to-noise ratio [55].

Specialized applications—such as microbial fuel cells, corrosion monitoring, and sensors based on conductive polymers or diamond-based biosensors—often involve systems that are highly dynamic, structurally heterogeneous, and not adequately captured by conventional impedance models [56,57]. In these contexts, overlapping time constants, interfacial evolution, and environmental variability introduce substantial obstacles to achieving accurate and reproducible EIS characterization.

The aforementioned limitations underscore the need for context-specific experimental design, the integration of complementary analytical techniques, and the advancement of more sophisticated modeling strategies—including time-varying and parameter-varying impedance frameworks—aimed at enhancing both the applicability and reliability of EIS in complex electrochemical systems [58].

3. Electronic Circuits and Solutions for EIS

Operatively, common EIS measurements involve the acquisition of the voltage drop across a sample and the current flowing through it for a driving harmonic component, the extraction of the impedance parameters, i.e., module and phase, or real and imaginary part, and the repetition of the procedure for the whole spectrum range. Figure 1 sketches the block diagram for a general system for the construction of the impedance spectrum: X is a source signal, used as a stimulus, while Y is the measured one.

A first classification can be made between the so-called potentiostatic and galvanostatic systems. In the former, X = V_S, i.e., the stimulus voltage, whereas Y = I_m is the measured current. In the latter, X = I_S is the applied current and Y = V_m is the measured voltage across the Device Under Test (DUT). A further way to distinguish between different systems is by the measurement method, where the electrodes used to acquire X and Y signals can be superposed (in the case of a 2-wire measurement) or separated (in the case of a 4-wire measurement). The choice between the two methods is made depending on the physical characteristics of the DUT [59].

In any case, an EIS system includes a signal generator, an instrument for the measured signal, a processing block where the information about the DUT impedance is extracted from the ratio of the amplitudes of the two signals and their relative phase, and a control system for the automatic frequency scanning. Each block can be realized using either analog or digital technology and can be implemented through hardware or software.

Regardless of the details about the architectural solution, an EIS system has to guarantee some properties, which need to be balanced for a specific application:

−

Accuracy: the impedance measurement has to be reliable, even if the performance parameters to be evaluated are highly dependent on the specific application. Along with the applied method, the measurement accuracy is critically affected by the stimulus signal, which needs to be stable and highly controllable. For this reason, special care is usually dedicated to the waveform generator design.

−

Portability: most of the applications of EIS can involve in-situ measurement, requiring the system to be embedded or having a certain degree of integration. This is specific, for instance, to a bio-impedance sensor with wearability requirements, but can be preferable for material or system characterization as well.

−

Measurement time: as said, the impedance spectrum requires multiple acquisitions over the entire frequency range of operation; moreover, the investigation at extremely low frequencies usually implies time-consuming measurements even for the single harmonic. When possible, dedicated strategies have to be put in place to reduce, manage, or control the measurement time.

−

Versatility: as already mentioned, different frequency ranges evaluate different properties; hence, broadband solutions ensure a robust system, albeit at the expense of architectural complexity. At the same time, a wide measurable impedance range could allow the system to adapt to more than one application.

Each designed circuital solution tries to optimize at least one of these properties, acting on one or more functional blocks, allowing a differentiation of the systems according to the specific investigated system. Moreover, a number of EIS applications demand low current levels to satisfy safety requirements (for instance, bio-impedance measurements on living tissues have to fulfill the EN-60601-1 standard [60]). As a consequence, to directly control the current injection into the sample, galvanostatic methods are likely to be preferred.

In the following, we overlook a couple of EIS system classes with larger impact on the overall performances: one looking at the improvement or optimization of the signal (current) generator, thus focusing on the front-end analog circuit, the other focusing on component integration to increase the degree of portability.

3.1. Solutions Based on Howland’s Circuit

The main constraint when performing an impedance measurement through a galvanostatic method deals with having a stable and controllable supply current of the DUT. Hence, such systems must encompass a high-performance current generator, mainly at low and extremely low frequencies. Howland’s circuit [61] represents the most reliable solution for this scope, allowing us to obtain a customizable Voltage Controlled Current Source (VCCS).

Figure 2 shows a couple of main configurations for Howland’s Circuit: the basic configuration (Figure 2a) includes both positive and negative feedback used in a difference amplifier based on one operational amplifier (op-amp). To simplify the analysis, and without loss of generality, let us assume that $V_{in-}$is grounded and $V_s=V_{in+}$. Under these conditions, the output current I_L in the load (Z_L) assumes the value:

$$I_L={\displaystyle \frac{V_S}{R_3}}-{\displaystyle \frac{R_1{R}_4-R_2{R}_3}{R_1{R}_3}}{\displaystyle \frac{V_L}{R_3}}$$

(2)

Under the balanced-bridge condition $R_4/R_3=R_2{/R}_1$, any dependence on the load voltage V_L is eliminated (i.e., Z_L sees an infinite resistance) and the current assumes the value:

$$I_L={\displaystyle \frac{V_S}{R_3}}$$

(3)

meaning that the circuit behaves as a VCCS, where 1/R₃ is the transconductance parameter defining the voltage-to-current conversion coefficient. The circuit illustrated in Figure 2a operates as a VCCS over the entire output voltage range, defined as V_O-MAX across the Z_L load, provided the op-amp remains in its linear region. By circuit inspection, V_O-MAX is given by:

$$V_{O-MAX}={\displaystyle \frac{R_1}{R_1{+R}_2}}{V}_{sat}$$

(4)

where V_sat is the saturation voltage of the op-amp output.

The improved version of the Howland’s (Figure 2b) splits the resistor in the positive feedback path into two component R_A and R_B in order to increase the transconductance coefficient compared to the basic configuration shown in Figure 2a. For the improved Howland’s circuit, with the new balanced-bridge condition defined by (R_A + R_B)/R₃ = R₂/R₁, the current in the load is expressed as:

$$I_L=V_S{\displaystyle \frac{1}{R_3}}{\displaystyle \frac{R_A{+R}_B}{R_B}}=V_S{\displaystyle \frac{1}{R_B}}{\displaystyle \frac{R_2}{R_1}}$$

(5)

thus allowing the transconductance coefficient to be predominantly determined by the R_B component. Additionally, the R₃−R_B voltage divider allows the current source to maintain a constant current over a wider range of load voltages. Anyway, in both its basic and improved versions [62], under perfect resistors matching condition, the circuit behaves like a current pump, with current values independent of the voltage across the load and controlled by the input voltage signal.

Hence, in the framework of impedance spectroscopy, the Howland’s current pump can be a practical and reliable solution to generate a controllable current to supply the DUT. When employed in such an application, there are some issues that need to be carefully addressed for a Howland’s circuit:

−

Resistors matching: the VCCS behavior is possible only if the bridge condition is exactly fulfilled. Indeed, only in this case the output impedance of the circuit is infinite, while a small deviation from the perfect matching condition leads to a finite output impedance, or in other words, a current dependent on the load value. The acceptable tolerance for the resistors depends on the accuracy needed for the measurements, although it affects mostly the high-frequency measurements [63]. Remarkably, resistor matching certainly represents a constraint for all read-out circuits employed in low-frequency impedance spectroscopy as well.

−

Output voltage swing: the input-voltage-dependent current is provided as long as the op-amp output voltage is below the saturation region, directly affecting the maximum range of measurable impedance. In this case both the maximum current and the transconductance coefficient can be optimized to adapt the device to different impedance values.

As the ideal behavior depends on the exact matching condition on resistors, it is immediate to observe that the employment of all circuits for impedance measurement relies on extremely precise resistors with high accuracy along the whole frequency range of operation. Hence, in all the circuits reported in this section, it is necessary to mount high-quality resistors in order to have a proper VCCS. On the other hand, in low and extremely low frequency impedance application, circuit modification, architecture design, and measurement methods can contribute to optimizing the performance in terms of voltage swing or bandwidth.

When specifically referred to bio-impedance measurement, the double layer capacitance at the electrode–electrolyte interface can invalidate a two-wire measurement [64]. The solution can be the implementation of a four-wire connection to independently deal with current injection and voltage measurement. In ref. [65], the authors propose the realization of an integrated circuit for impedance measurement, comparing the results from 4- and 2-wire configurations.

For the employment of a versatile device for current injection, a modified version of the Howland’s circuit is designed, as sketched in Figure 3a, while a discrete instrumentation amplifier is employed to acquire the voltage values. The measurement setup is completed with a commercial signal generator and a high-bandwidth oscilloscope to acquire the signals. The aim of the authors is to optimize the front-end device parameters to find a solution to mediate between high accuracy measurement, bandwidth and portability.

The critical device is thus the current generator, implemented by a modified version of the Howland’s pump using a fully differential AD8132 Op-Amp [66] and implementing double feedback in both positive and negative paths to double the output dynamics: with reference to Figure 3a, the circuit analysis shows that when the bridge matching condition R_f = R_O + R_p is fulfilled, the voltage-to-current conversion parameter is R_f/2R_gR_O, thus depending only on R_O under the choice R_g = R_f.

The circuit input and output voltage swing are studied, choosing a specific set of values for the components, therefore obtaining a transconductance parameter of 5 × 10⁻⁴ A/V, while the op-amp operates within its linear region for a load resistance up to 25 kΩ. The measured bandwidth of 41 MHz can be extended up to 100 MHz by adding further poles and zeroes through the feedback networks to reduce the overshooting and linearize the gain, respectively.

Once the accuracy at high frequency has been optimized through circuit design, the voltage measurement is performed by an instrumentation amplifier realized with a low-noise OPA659 amplifier to minimize the 1/f noise, so that accurate measurements can be performed at low and very low (<1 Hz) frequencies as well. The system is able to measure impedances on a MΩ range, considering open- and short- load measurements, as shown in Figure 3b, and it has been tested on biological samples over a broad bandwidth from 1 mHz to 25 MHz, showing good matching with commercial instruments with an accuracy of <5%.

The authors underline critical aspects that can be improved both to optimize the circuital performances and favor the system portability, including lower current levels, power consumption, relatively high common mode voltage and necessity to integrate other devices for automatic signal acquisition, processing and storing of data.

Pettinato et al. [67] proposed a solution to overcome the problem of resistor matching when meeting the bridge condition, providing a device for the measurement of bio-impedance at extremely low frequencies. Based on the idea developed in ref. [68,69], the authors use a Howland’s pump exploiting the superior performance of single-chip gain selectable amplifiers, with the measurement based on homodyne detection of small amplitude signals by a commercial Lock-In Amplifier (LIA).

The architectural solution provides an accurate current signal employing a commercial selectable-gain amplifier to implement a Howland’s current pump by properly connecting the IC internal thin-film resistor, avoiding the need to add external components [70]. The employed LT199x series features excellent matching (error within 0.05%) and extremely low temperature coefficient (3 ppm/°C) of the internal resistors [71], making them ideal for meeting the requirement of a high output impedance of the voltage-to-current converter [70].

As the IC is not designed to be a current pump, the issue is to maximize the output voltage swing: with reference to Figure 4a, this is obtained by choosing the lowest resistors available as input resistors and using parallel connection for the feedback resistors to lower the gain. For the LT1995, the voltage to current conversion coefficient is relatively low (2.5 × 10⁻⁴ S), thus allowing for the generation of extremely low currents: this means that the circuit can properly operate with a large range of impedance values, making the circuit ideal for measuring bio-impedance or for sensor conditioning. Experimental results demonstrate that the proposed architectural solution features a single dominant pole response with a bandwidth of 30 MHz for DUT resistance up to 10 kΩ, reducing to 50 kHz for DUT values of the order of 1 MΩ.

The main drawback of this solution is the voltage drop across the internal resistor in series with the load, which compromises the maximum output voltage swing, especially for low-resistance DUTs.

To increase the output voltage range, a precision current divider—based on the LT1996 selectable-gain amplifier—can be selected, reducing the overall transconductance parameter down to one order of magnitude. Finally, a third stage composed of a decoupling feedback buffer (by OPA277) can further decrease the V/I conversion coefficient by another factor of 10. The final configuration is completed by two switches to exclude the current divider and/or the output-buffer stages from the measurement, allowing for the maximum versatility of this solution for a different range of values and different kinds of DUTs. In fact, exploiting the three different configurations, the voltage to current parameter can vary over three orders of magnitude from 0.2 to 250 μA/V. The measurements show excellent performances when measuring both resistive loads over six orders of magnitude up to 100 MΩ, for frequencies down to 1 mHz (Figure 4b,c), and capacitive from 10 nF to 1 µF (Figure 4d). Very remarkably, in the latter case, a further feedback stage has been employed to avoid voltage drift for purely capacitive loads [67].

The whole circuit has been tested to measure equivalent bio-sensor impedances, providing good agreement with the expected values [50] in the whole frequency range (Figure 4e). This circuit represents an extremely versatile low-cost solution, which does not compromise the low-frequency performances when measuring low-level signals and can be thought of as a device by itself or a practical add-on for LIAs.

As said, one issue when dealing with extremely low frequencies is the time required to obtain a complete dataset, considering both the single measurement and the frequency scanning. One solution involves replacing the single harmonic signal (either current or voltage) with a multi-frequency signal, thereby probing the complete spectrum at once [72,73], albeit at the expense of signal accuracy [74]. In ref. [75], the authors propose using a random signal generated by a microcontroller, based on the so-called Chaotic Pulse Width Modulation (CPWM) method. Without going into the details of such a technique [76], the microcontroller is programmed to solve a chaotic oscillator model whose output is used to generate the PWM signal: the resulting output waveform is a voltage signal with a random broad spectrum, which can be used to feed the impedance meter.

The authors demonstrate that the multi-frequency signal from CPWM can substantially reduce the time-consuming measurements at low frequency, but it requires an accurate measurement method to keep a high value for accuracy.

In this framework, one of the methods used to test the system is a modified version of Howland’s current pump previously introduced in [77], as sketched in Figure 5. This enhanced version of the circuit employs a second operational amplifier in buffer configuration to increase the output impedance in non-ideal conditions: in fact, when considering the finite open-loop gain, the output impedance results are proportional to the gain itself: while for the standard improved Howland’s solution (Figure 2b) the expression of the impedance is Z_out = 0.5A_OL × R_2B where A_OL is the open-loop gain of op-amps, for the modified circuit, assuming the same A_OL value for the three op-amps, Z_out = A_OL × R_2B, hence doubling the output range.

This enhanced Howland’s pump is implemented by 65 nm CMOS technology to obtain an integrated circuit, using a microcontroller’s internal operational amplifier in a buffered feedback configuration to drive the load. The employed components resulted in an open-loop gain of 7 × 10⁷ (97 dB) and an output impedance Z_out of 2 × 10¹² Ω, with a voltage-to-current conversion coefficient of 33.3 μA/V.

The experimental measurements were carried out by generating 21-bit data for a CPWM signal of 1.25 V on RC loads and biological samples, comparing the results with two potentiostatic methods and a commercial impedance meter over a frequency range of 10 mHz to 1 Hz and for a maximum impedance of 125 kΩ, with a measured absolute error in both magnitude and phase remaining under 3% within the whole investigated frequency range.

3.2. Solutions Based on System Integration

Despite the need for high accuracy in general-purpose impedance-meters, a lot of attention is given to solutions aiming at low costs, high portability or wearability, possibility or need for in-situ measurements, and modularity, in order to design a single device for one specific application and/or adapt it to different environments.

This leads to the design of modular devices or all-in-one integrated solutions including all the components, i.e., for signal generation, impedance conversion, further signal processing, data storing, etc.

In ref. [78], the authors propose an extremely versatile solution, arranging commercial and external components in a modular device, which can be easily adapted to different needs. The measurement is performed by a standard impedance converter implemented with a transimpedance amplifier (TIA), while the signal generation and detection can be independent of the employed device. The functionality of the device is demonstrated by adapting a commercial audio board to generate the needed probe signals. With reference to Figure 6a, the two output channels are used to generate one carrier and one modulating frequency, which are mixed and filtered to obtain the voltage signal at a suitable frequency to be applied to the DUT. The stimulus voltage is taken at the DUT node, while the TIA provides a signal corresponding to the measured current. As the audio board is used to acquire the measurements as well, a further modulation is needed to adapt the frequency of the measured signals to the board bandwidth, retrieving the original frequency via ex-post digital processing.

Hardware architecture is mostly reduced by moving to software solutions to generate, acquire and process the signals, controlling the process by using standard libraries and a specifically designed software (by the same authors) to manage low-frequency signals [79].

Particular attention has to be given to the leakage signals coming from modulation/demodulation processes: indeed, it is shown that the measurement time, which is obviously affected by the number of acquisitions to be averaged, is inversely proportional to the equivalent noise bandwidth. The latter is calculated as a ratio of the frequency to be analyzed, and the reported factor 5 has demonstrated giving good results in the preliminary characterization of RC elements (Figure 6b). An advanced test is performed on a supercapacitor structure, where electrode–electrolyte processes need to be evaluated: the device demonstrates working at frequencies down to 10 mHz, extending the range reached with commercial instruments.

Despite the need for a certain software complexity, as said, the advantage of this approach is the high modularity of the system: in fact, the functional blocks (signals modulation/demodulation, front-end, etc.) are completely decoupled, thus allowing easy substitution of the front-end electronics when other components with customized performances are needed.

The most used method when hardware simplicity and integrability have to be prioritized is to use partially or completely embedded digital components, with microcontrollers representing an obvious choice when dealing with portability and modularity.

One option is to employ the microcontroller to only extract the impedance parameters after the measurements and to control the measurement procedure, as, for example, described in ref. [80], where the impedance of a Li-ion battery is measured with a dedicated AD5941 IC [81], with the system sketched in Figure 7a. The AD5941 is used to generate the voltage signal, which is converted into a current signal to probe the DUT, with the voltage-to-current conversion provided by an AD830 [82] difference differential amplifier in a servo-loop configuration and an AD8066 [83] to minimize the common mode and with a properly designed instrumentation amplifier to adapt the small measured signal to the AD5941 dynamic.

An analog switch module allows for the acquisition of the values from both the DUT and a reference resistor. The AD5941 acquires the signals after a pre-processing stage to adapt the voltage levels and sends the processed values to a STM32 microcontroller. The latter acts as a control unit to supervise the measurement process: it controls the AD5941 by setting the parameters for signal generation and acquisition, synchronizes the operations of the analog switch and the impedance meter during one measurement at a given frequency and along the whole frequency range, and it finally processes the raw data to give the module and phase of the measured impedance.

The measurements at a single frequency with the proposed components show an accuracy of 2.8% when compared to a commercial impedance meter, while the frequency span for the device is tested between 0.1 Hz and 2 kHz (Figure 7b), demonstrating excellent performances with respect to other homogeneous devices and allowing the derivation of the equivalent RC parameters of several batteries for different SoCs.

Microcontroller-based solutions can also achieve higher portability, employing a single device for the measurement (except for the front-end (analog) circuit) and designing and implementing the complete measuring system by software rather than using different circuits for each task. In this case, direct digital synthesis techniques [84,85] can be implemented using internal DACs for signal generation, while the data acquisition can rely on the internal ADCs, eventually taking advantage of the microcontroller DMA to optimize the data transfer and the signal processing.

The front-end circuitry can be tailored depending on the specific need. Faktorovà et al. [86] employ a simple TIA for impedance measurement and a programmable gain amplifier to adapt the signal to the ADC range, while Piasecki et al. [87] perform a 4-wire measurement with two ADCs, simultaneously acquiring the voltage and current signals from an instrumentation amplifier and a TIA, respectively, including stages for level shifting and range tuning as well, with the set-up represented in Figure 8a. In ref. [87], the system allows for impedance measurements from 10 Ω to 1 GΩ in the frequency range of 100 mHz to 1 MHz. The comparison with a commercial impedance analyzer showed an accuracy ranging from 0.1% to 10% for resistors from 1 Ω to 100 kΩ, the worst cases for the frequency range 100 mHz–1 Hz, demonstrating good performances for RC measurements as well (0.1–0.3% of accuracy for 100 Hz ÷ 100 kHz, shown in Figure 8b).

The advantage of this approach is very light hardware, relying on the firmware for easy and controllable signal generation and acquisition, retrieval of the impedance values (modulus and phase) from the acquired signals, and the setting of the measurement parameters, data storage and communication. Though representing an excellent solution due to higher portability and hardware simplification, these kinds of devices have the main disadvantage of needing calibration with reference resistors, which can be periodically repeated due to analog component aging and drift.

When high performances have higher priority with respect to low-cost alternatives, the design of a completely custom circuit can be the best option to meet the system requirements, with CMOS monolithic integration being the preferred option for impedance measurements in systems for tissue characterization [88], disease diagnosis [89] or biological processes control [90].

Restricting to low frequencies, the designed circuit can be only a part of the system: one of the critical elements, as already seen, is the quality of the current source used as a stimulus for the DUT, where CMOS technology can offer better performances in terms of output impedance or signal dynamics with respect to other general-purpose solutions [91].

On the other hand, the employment of CMOS technology can instead be extended over the whole system, allowing size, power consumption, noise level and accuracy optimization. One example of an all-in-one circuit is from Viswam et al. [92], through the design, realization and characterization of a complete impedance measurement system for tissue characterization in 0.18 μm standard CMOS technology.

As can be seen from the block diagram in Figure 9, the circuit is based on an electrode array controlled by a switch matrix [93], addressing over 59k pixels, allowing for EIS and imaging through parallel application of a voltage stimulus and sensing of the resulting current. The waveform generator is digitally implemented by a clock generator controlling an internal look-up table to handle the parameters of the desired sinusoidal values and a 10-bit DAC to obtain the final signal. The current flowing from the electrodes array is measured by a suitably designed LIA: the relative TIA is designed with selectable gain resistive-capacitive feedback to adapt the device to different ranges of impedance magnitude, while a ΔΣ-ADC is designed to optimize accuracy, linearity and noise reduction. A 32-channel parallel architecture allows for impedance imaging as well, although needing an initial calibration procedure to compensate for a relatively low (∼3%) mismatch in the 32 TIA feedback resistors.

The system provides good performance with respect to other parallel architectures in terms of measured impedance, current and frequency range, density and noise levels. The impedance range (1 kΩ–1 GΩ) is quite limited in the downward direction with respect to the other cited systems, but it must be underlined that it is a highly customized circuit, and the reported range is optimized for all the required specifics.

The system allows for the combination of several measurement methodologies in bio-impedance: the authors report an initial characterization with an RC network, and a further electrochemical characterization, allowing the study of electrode–electrolyte interfaces, further demonstrating excellent ability to identify the specific characteristics of tissues such as cardiac or brain cells.

4. Discussion

As a summary of the previous dissertation, in

Table 1. Comparison of the reported EIS systems. Portability is a qualitative parameter: “High” is used for systems having a high degree of integration, for instance, when realized as integrated circuits; “Medium” is for systems that are ready to be integrated; “Low” is for systems that are in a prototypal form and need to be further developed to be easily portable.

Reference	Frequency Range [Hz]	Measured Range [Ω]	Accuracy [%]	Portability
[65]	10−3 ÷ 107	100 ÷ 106	1	High
[67]	10−3 ÷ 106	102 ÷ 108	n.r.	Medium
[75]	10−2 ÷ 100	103 ÷ 104	1.6	High
[78]	10−2 ÷ 102	102 ÷ 105	n.r.	Low
[80]	10−1 ÷ 103	103 ÷ 104	2.8	High
[87]	10−1 ÷ 105	10−1 ÷ 109	0.1–3	High
[92]	10−3 ÷ 108	10−1 ÷ 109	0.1–10	High

, we selected a subset of characteristics from the systems reviewed in this paper. Specifically, we report the frequency range of operation and the range of impedance magnitude, the accuracy (where available) and the degree of portability. Some general considerations can be pointed out to give a correct framework of low-frequency impedance measurement.

The applications of EIS span a wide range of fields, even when restricting the analysis to the low- (or extremely low-) frequency domain. These include, for instance, the characterization of energy storage systems, fuel cell analysis, and tissue investigation for diagnostic or material studies. Each of these applications can present specific requirements in terms of accuracy, measurement setup, and wearability. Moreover, even within a single application, different frequency ranges may be employed to probe different properties, highlighting the importance of accessing multiple frequency ranges.

As a consequence, it is challenging to establish a classification that encompasses all, or even most, of the available solutions for a straightforward comparison. Nonetheless, in recent years, systems have increasingly aimed toward greater integration, with a preference for portability and the capability of in situ measurements as alternatives to commercial instruments or laboratory equipment. Microcontroller-based digital solutions, implemented at varying levels of complexity, contribute to portability and lightweight hardware architectures, although sometimes at the cost of reduced performance.

Indeed, costs can be considered a critical factor as well, particularly given the potential widespread adoption of EIS characterization systems. Conversely, when accuracy is a stringent requirement, customization of part or all of the system design may represent the only viable—albeit expensive—approach.

At the same time, the front-end analog electronics are closely linked to the selected measurement methodology, which can vary significantly across applications. In most cases, these systems rely on current/voltage measurements, requiring precise, stable, and controllable signals. In this context, galvanostatic systems are the most commonly employed, owing to their ability to generate high-quality currents and regulate current levels in compliance with safety standards when needed. Among these, the Howland’s circuit and its enhanced versions can be considered the most suitable solutions.

Providing a uniform comparison across the wide range of proposed systems and their characteristics is nearly impossible, as they cover an extremely broad design space. Nonetheless, all the reported circuits or architectures for EIS systems propose noticeable design solutions, each optimizing the parameter(s) that best suits the specific application.

Finally, it is noteworthy to consider that EIS is particularly susceptible to several sources of error, which may significantly distort the impedance data and complicate its interpretation, potentially leading to inaccurate conclusions about the system under study. Hence, the design of any measurement systems or circuit should take into account the most critical sources of error in low-frequency EIS, namely drift, random and deterministic noise, system non-linearity, and calibration errors.

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Drift. In applications requiring a mHz frequency range, the system’s state may evolve due to factors such as temperature fluctuations, changes in electrode surface properties, or shifting concentration gradients. Such drift can introduce artifacts into the impedance spectrum, potentially causing data unreliability. Averaging the measurement over successive acquisitions results in further time consumption to assess systematic temporal deviations. Time-saving alternatives include applying algorithms to compensate for drift in real time. For instance, first-order effects can be corrected via linear interpolation between consecutive DC values, while in cases of more pronounced variations, more advanced trend-removal techniques are recommended [94]. Alternative methods employ the Fourier Transform of input and output signals to correct for drift, providing a robust approach for systems with very long relaxation times [95]. Remarkably, Operando EIS [96] has been proposed as a promising technique for EIS under time-varying conditions, not only mitigating the limitations associated to drift but also enabling the study of dynamic processes occurring under realistic operating conditions.

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Noise. Extremely low-frequency measurements are particularly susceptible to both random and deterministic noise. The signal-to-noise ratio is inherently lower at frequencies 1 mHz–1 Hz due to the small amplitude of the applied perturbation and the extended acquisition time, which increases the likelihood of interference from environmental sources. To reduce the impact of noise, the experimental setup should be properly shielded and grounded, typically by employing a Faraday cage [65]. A good practice for the identification and elimination of noise sources is to use a test cell with known impedance values to evaluate the system’s susceptibility to noise and to calibrate the measurement setup.

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Calibration Errors. The accuracy of EIS measurements strongly depends on the proper calibration of the impedance analyzer. Calibration inaccuracies—particularly in the current and voltage ranges—can introduce systematic errors in the measured impedance, more pronounced at very low frequencies. Phase measurements are especially sensitive to such errors, as even small phase shifts can significantly impact data in the low-frequency regime. To reduce these errors, proper calibration using traceable standards (open, short, and known loads) is crucial, with special focus on current and voltage ranges. Compensation for cables and fixtures through open/short/load corrections near the device under test is also recommended. Also, in this case, drift correction methods—such as baseline adjustments over multiple sinusoidal periods—help mitigate phase errors caused by thermal or environmental instabilities [97]. Finally, repeated measurements with dummy cells enable verification of system stability and accuracy across frequencies [98].

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Non-linearity. EIS relies on the assumption that the system’s response is linear. However, at low frequencies, the applied voltage or current may induce larger-than-expected changes in the system’s state, driving it into a non-linear regime. For instance, a low-frequency signal can significantly alter the concentration of reactants at the electrode surface. Once the system exhibits non-linearity, the EIS data can no longer be accurately interpreted using standard equivalent circuit models.

To improve the quality of low-frequency measurements, it is essential to optimize experimental parameters. Reducing the amplitude of the AC perturbation helps maintain linearity in the system’s response. Additionally, increasing the integration time or averaging multiple cycles per frequency point can enhance the signal-to-noise ratio. However, this improvement comes at the cost of longer measurement durations, increasing the risk of drift. A careful balance between these parameters is therefore critical to ensure data reliability. In some cases—particularly in systems like Li-ion batteries—the system is inherently nonlinear and nonstationary during operation. Also in this case, operando EIS offers a valuable tool for tracking time-dependent impedance variations, providing insight into dynamic processes that are inaccessible to conventional EIS [99].

5. Conclusions

Performing low-frequency impedance measurements in EIS systems is an established method for materials and systems characterization with a broad range of applications. The investigated spectrum range (1 mHz–1 kHz) requires attention to some specifics, which can be fulfilled through suitable measurement methods, system architecture and customized solutions. Here, we commented on some recently proposed electronic circuits aiming to optimize performance and/or cost. We analyzed some examples of controllable current sources and their optimization in the framework of EIS, fundamental in bio-impedance measurements, and systems with different degrees of integration focused on adapting the measurement to different environments. As a general trend—and with the exception of highly specialized systems—we may identify versatility as the most valuable property in the design of EIS measurement devices, as it enables the use of a broader spectrum of applications, facilitates the adaptation of system properties to diverse environments, and ultimately contributes to reducing fabrication costs.

Moreover, the importance of EIS for probing the dynamic behavior of electrochemical systems requires challenging error sources such as drift and non-linearity, which can compromise data reliability. Architectural optimization, high-precision hardware components and system versatility have to be integrated with strategies such as careful calibration, drift correction algorithms, and real-time optimization of measurement parameters to ensure data accuracy under experimental conditions. Looking forward, operando EIS offers a promising path toward overcoming many of the limitations of conventional EIS. The integration of this approach with parameter-varying models, multisine excitation signals, and physics-based interpretation frameworks opens new possibilities for advanced diagnostics, system optimization, and the development of predictive tools such as digital twins.

High-speed electronics, fast processing units and low-power systems can certainly enable the use of principles and methods of operando EIS to apply to a wider range of electrochemical technologies. Continued research in this direction is expected to significantly expand the capabilities of EIS-based system analysis in both fundamental studies and practical applications.