Experimental Investigation of a Scalable Dimensionless Model of an AC Circuit with a Nonlinear Rectifier Load

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper presents a well-motivated and scalable dimensionless model of an AC circuit with a nonlinear rectifier load, enabling clear assessment of harmonics, reactive power exchange, and distortion across different grid strengths and load conditions. The experimental validation supports the claims, so I recommend the manuscript for acceptance for publication.

Author Response

We thank the Reviewer for the positive evaluation and for recommending the manuscript for publication. We appreciate the constructive feedback provided throughout the review process.

Reviewer 2 Report

Comments and Suggestions for Authors

The paper investigates modeling uncontrolled rectifiers, a common nonlinear load. It effectively merges simulation and experimental results. However, the manuscript needs revisions, and comments are listed below.

1. The authors should revise the introduction and abstract to be more concise, review the cited articles, and clearly mention their contributions and gaps, such as in References [14] and [15].
2. The authors claim to address gaps, such as ignoring series resistance/inductance and reactive power conversion, but this is overstated. Notably, some cited references have already discussed similar issues in rectifier modeling, i.e., [21], [22], and others.
3. The author needs to improve the abstract and overall writing: The abstract mentions "energy-saving strategies," but the paper hardly addresses them—mostly descriptive, not prescriptive.
4. I noticed some errors in the paper:
5. Selected experimental characteristics of the circuit with an uncontrolled rectifier

• To verify the analysed circuit model, experimental tests were performed. Based onthe initial measurements of current and voltage waveforms presented in Chapter 2, …..? Chapter 2?
• Also in the same section, the author mentioned/cited Figure 1.

5. The authors should improve their results and discussion section, especially since the key observations from Figures 7, 8, 9, and 11 are merely described. 
6. The single-phase focus restricts relevance to three-phase systems commonly used in industry. The author should include a discussion on varying supply frequency, distorted sources, or multiple loads.

Author Response

We would like to thank the Reviewer for the careful reading of our manuscript and for the constructive and valuable comments. The remarks helped us significantly improve the clarity, structure, and scientific completeness of the paper. Below we provide detailed responses to each point, together with a description of the revisions introduced in the manuscript.

Comment 1
The authors should revise the introduction and abstract to be more concise, review the cited articles, and clearly mention their contributions and gaps, such as in References [14] and [15].
Response 1
We thank the Reviewer for this comment. Both the abstract and the introduction have been revised to make them more concise and to clarify the contributions of the cited works. In the revised introduction, we now explicitly identify the scope and limitations of Refs. [14] and [15], as well as other key studies.
In particular:
•    Ref. [14] is now described as a study that includes supply inductance but relies on a sinusoidal approximation of the supply current, thereby neglecting higher-order harmonics relevant in practical operation.
•    Ref. [15] is identified as analysing the role of supply inductance only in continuous conduction mode and under the assumption of constant DC output voltage, which restricts applicability to real low-power rectifiers with variable load and conduction patterns.
The introduction has been streamlined by removing redundant background content and reorganising the literature review to emphasise the progression from classical models to recent developments and finally to the open gap addressed by our work.
The abstract has also been rewritten to:
•    better reflect the actual scope of the study,
•    clearly state the contributions of our model,
•    remove overly general statements unrelated to the core results.
These changes improve the clarity, focus, and scientific context of the manuscript.

Comment 2
The authors claim to address gaps, such as ignoring series resistance/inductance and reactive power conversion, but this is overstated. Notably, some cited references have already discussed similar issues in rectifier modeling, i.e., [21], [22], and others.
Response 2
We appreciate this observation. In the revised manuscript, the statements concerning gaps in the literature have been moderated and clarified. Instead of suggesting that these aspects are entirely absent from prior work, the revised text now acknowledges that:
•    Ref. [21] (now Ref. [16]) addresses interharmonic interactions in rectifiers,
•    Ref. [22] (now Ref. [17]) analyses harmonic amplification in the presence of active filters,
•    several other studies consider selected elements of AC-side impedance.
However, these works typically focus on specific scenarios, rely on restrictive assumptions, or do not provide a generalised and scalable formulation applicable over a broad range of grid strengths and rectifier load conditions.
To avoid overstating novelty, we now explicitly state that the contribution of the present study lies in:
•    introducing a dimensionless, compact model capturing the joint influence of AC-side inductance, resistance, and rectifier operating mode;
•    providing experimental validation across a wide parameter range;
•    quantifying phenomena (such as reactive power redistribution across harmonics) that are mentioned in prior literature but not analysed systematically in the context of uncontrolled single-phase rectifiers.
The revised introduction reflects these clarifications and more precisely positions our work relative to the existing literature.
We also note that, due to reorganisation of the introduction and literature review, the numbering of some references has changed in the revised version of the manuscript.

Comment 3
The author needs to improve the abstract and overall writing: The abstract mentions "energy-saving strategies," but the paper hardly addresses them—mostly descriptive, not prescriptive.
Response 3
We thank the Reviewer for this observation. We agree that the original wording in the abstract could suggest that the manuscript proposes specific energy-saving strategies, which is beyond the intended scope of the study. In accordance with the Reviewer’s suggestion, this phrase has been removed and the abstract has been rewritten to more accurately reflect the analytical and experimental contributions of the work.
To clarify the practical relevance of the model, a concise explanation has been added in the Conclusions section. The new sentence states:
“The model enables prediction of current and voltage harmonic components as a function of grid and load parameters, which is directly relevant for tasks such as assessing the need for filtering, selecting compensators, or estimating additional losses caused by nonlinear loads.”
This addition does not extend the scope of the paper but explains how the quantitative characteristics obtained from the model (harmonic content, reactive power exchange, dependence on source impedance) are commonly used in engineering analyses related to power quality and loss assessment. We believe that the revised abstract and conclusions now accurately reflect the contribution of the work and resolve the Reviewer’s concern.

Comment 4
“Selected experimental characteristics…
• ‘Based on the initial measurements… presented in Chapter 2…’ — Chapter 2?
• Also in the same section, the author mentioned/cited Figure 1.”
Response 4
We thank the Reviewer for pointing out these inconsistencies. The wording “Chapter 2” has been corrected to “Section 2”, as the manuscript is organised by sections rather than chapters.
In addition, the description of Figure 1 has been revised for clarity. A short explanation has been added to indicate that the same experimental setup shown in Figure 1 was used both for the initial qualitative observations of current and voltage waveforms and for the subsequent, more extensive measurement campaign performed to validate the mathematical model. This clarification explains the purpose of citing Figure 1 in this section and ensures consistency with the methodological description provided earlier in the manuscript.
These corrections improve the accuracy and clarity of the experimental description.

Comment 5
The authors should improve their results and discussion section, especially since the key observations from Figures 7, 8, 9, and 11 are merely described. 
Response 5
We thank the Reviewer for this valuable suggestion. In the revised manuscript, the Section 5 has been expanded by adding concise interpretative remarks to Figures 7, 8, 9 and 11. These additions explain the physical mechanisms behind the observed trends (changes in conduction interval, harmonic generation, equivalent inductance behaviour and reactive power redistribution) and highlight their relevance for power quality assessment. The discussion remains concise while providing clearer insight into the significance of the results.

Comment 6
The single-phase focus restricts relevance to three-phase systems commonly used in industry. The author should include a discussion on varying supply frequency, distorted sources, or multiple loads.
Response 6 
We thank the Reviewer for this valuable comment. The present study focuses on the single-phase rectifier because it provides a fundamental building block for analysing nonlinear load behaviour and allows the proposed dimensionless model to be introduced and experimentally validated in a controlled and transparent manner. We fully agree that extending the framework toward three-phase systems, distorted supply conditions, and networks containing multiple nonlinear loads is of high practical importance.
To address this remark, a short discussion has been added to the Conclusions section. We now explicitly state that the modelling approach developed here is not restricted to the single-phase case and can be generalised to more complex supply configurations. Preliminary simulations performed by the authors indicate that the dimensionless formulation remains applicable when supply frequency varies or when multiple rectifiers operate within the same network, but these topics require dedicated analysis and will be the subject of future work.
We believe that this clarification strengthens the manuscript while keeping the scope of the present study well-defined.

Reviewer 3 Report

Comments and Suggestions for Authors

1- The reason for using higher inductance as mentioned in text related to figure 1 is not clear. Introducing higher inductance will maintain continuous current with ripple minimization, which in practical cases is not available.  Revision is needed.

2- The choice of the capacitance value (680 uF) is not obvious. Usually, it is implemented for eliminating voltage ripples which, consequently, affects the reading according to its time constant.  The transferred impedance will be affected by this value, so its clear design is essential.

3- Authors need to add a power balance analysis through the rectifier part to properly represent the resistance seen from the left side in circuit of figure 1.

4- Performance shown in figure 7 is based on experimental results according to experimentally verified circuit, not the simulation referred to figure 4. Revision is needed. Furthermore, where is chapter 2 listed in text included in section 5?

5- Although the implemented AC supply is programmable (both voltage and frequency are adjustable), no reading was conducted for different frequencies to assess its effect on results.

6- The significance of results in figure 2 (Either it was derived from simulation or calculated) is missing.  Deeper discussion and elucidation may be presented about hysteresis of the rectifier circuit.  In addition, how odd were harmonic values extracted from these characteristics?

7- The attitude of current signals shown in figures 3 (a) and 3 (b) is somewhat contradicting in the interval from 0.03 to 0.04 sec. Illustration is required. 

Author Response

We would like to thank the Reviewer for the careful assessment of our manuscript and for the constructive comments provided. All remarks have been addressed in detail, and the corresponding clarifications and improvements have been incorporated into the revised version of the paper.

Comment 1

The reason for using higher inductance as mentioned in text related to figure 1 is not clear. Introducing higher inductance will maintain continuous current with ripple minimization, which in practical cases is not available. Revision is needed.

Response 1

We thank the Reviewer for this observation. The relatively high value of the series inductance in the laboratory setup was not intended to represent practical distribution-network impedances. Instead, it was required due to the current and power limitations of the programmable AC source used during the experiments. At low values of R_L, the rectifier generates highly pulsed currents with a large peak-to-RMS ratio, which would exceed the instantaneous current capability of the source. Increasing the inductance significantly reduces the peak current and ensures safe and stable operation of the test setup across the entire measurement range.

This adjustment concerns only the experimental implementation and does not influence the generality of the proposed model. In the mathematical formulation, the supply inductance enters the model exclusively through the dimensionless parameter r_L = R_L/ωL, which allows the results to be scaled to any practical supply impedance, regardless of the specific inductance value used in the laboratory environment.

To clarify this point, an explanatory note has been added to Section 2 of the manuscript.

 

 

Comment 2

The choice of the capacitance value (680 µF) is not obvious. Usually, it is implemented for eliminating voltage ripples which, consequently, affects the reading according to its time constant.  The transferred impedance will be affected by this value, so its clear design is essential.

Response 2

We thank the Reviewer for the insightful comment. We agree that the value of the DC-side capacitance is an important parameter in rectifier behaviour, as it affects the output-voltage ripple and therefore influences the current waveform. In the experimental setup, the capacitance C = 680 µF was selected as a representative value for low-power rectifier applications and to ensure stable operation over a wide range of load resistances R_L. Because the purpose of the study was to analyse how rectifier behaviour changes with varying load, it was neither practical nor meaningful to adjust the capacitor for each operating point, which reflects real-world conditions where the DC-side filter is fixed.

Importantly, the specific numerical value of the capacitor does not limit the generality of the proposed dimensionless model, because the model uses the normalised parameter c=ω²LC, which represents the combined effect of the supply inductance L and capacitance ?.

Any practical capacitor value can therefore be mapped to the appropriate value of c, and the qualitative conclusions of the model remain independent of the specific capacitance used in the laboratory setup.

A clarifying note has been added to Section 2 of the manuscript.

 

 

Comment 3

Authors need to add a power balance analysis through the rectifier part to properly represent the resistance seen from the left side in circuit of figure 1.

Response 3

We thank the Reviewer for this valuable comment. We would like to clarify that the power balance at the rectifier input can be directly obtained from the quantities already defined in the manuscript. In particular, the equivalent resistance R_eq introduced in (30) inherently represents the fundamental-frequency power balance, as it is derived from the first-harmonic voltage, current, and their phase shift.

To improve clarity for the reader, a short explanatory sentence has been added in Section 5 immediately after the definition of the equivalent parameters. No additional equations were required, because the power-balance interpretation follows directly from the existing formulation.

 

 

Comment 4

Performance shown in figure 7 is based on experimental results according to experimentally verified circuit, not the simulation referred to figure 4. Revision is needed. Furthermore, where is chapter 2 listed in text included in section 5?

Response 4

Thank you for pointing out this ambiguity. Figure 7 indeed presents a comparison between the results obtained from the mathematical model and the experimental measurements, serving as a validation of the proposed approach. To make this clear, the caption of Figure 7 has been revised to explicitly state that both simulated and measured harmonic components are shown. In addition, a clarifying sentence has been added in Section 5.

The incorrect reference to “Chapter 2” has also been corrected to “Section 2”.

 

 

Comment 5

Although the implemented AC supply is programmable (both voltage and frequency are adjustable), no reading was conducted for different frequencies to assess its effect on results.

Response 5

We appreciate the Reviewer’s remark. Although all measurements were carried out at 50 Hz, the dimensionless formulation used in the paper is directly scalable with respect to supply frequency. A clarifying sentence has been added to the Conclusions section, and the analysis of variable-frequency supply conditions will be considered in future work.

 

 

Comment 6

The significance of results in figure 2 (Either it was derived from simulation or calculated) is missing.  Deeper discussion and elucidation may be presented about hysteresis of the rectifier circuit.  In addition, how odd were harmonic values extracted from these characteristics?

Response 6

Thank you for this valuable remark. A clarification has been added to the manuscript. The small loop-like deviations observed in Figure 2 arise from minor dynamic effects during measurement and do not represent static hysteresis of the rectifier characteristic. The purpose of Figure 2 is solely to illustrate the qualitative change of the U_r(I) relationship with different values of R_L. The harmonic components discussed later in the manuscript are obtained directly from the measured time-domain waveforms, not from these static characteristics. The added explanation also notes that the underlying rectifier behaviour preserves approximate half-wave symmetry, which is why only odd-order harmonics appear in subsequent analyses.

 

 

Comment 7

The attitude of current signals shown in figures 3 (a) and 3 (b) is somewhat contradicting in the interval from 0.03 to 0.04 sec. Illustration is required. 

Response 7

We thank the Reviewer for this remark. The current waveforms in Figures 3(a) and 3(b) originate from experimental measurements, and their absolute time reference is determined by the trigger instant of the data-acquisition system. For this reason, the two plots are not aligned to a fixed phase of the supply voltage, which may give the impression of a discrepancy in the interval between 0.03 and 0.04 s. This effect is purely related to the measurement procedure and does not indicate any inconsistency in the rectifier behaviour. A brief explanatory note has been added to the manuscript to clarify this point.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have addressed my concerns, and the revised paper has been improved.

Reviewer 3 Report

Comments and Suggestions for Authors

No additional comments. Authors replied to all concerns