Analysis of the Dynamic Properties of the Rogowski Coil to Improve the Accuracy in Power and Electromechanical Systems

Abstract

This paper presents an analysis of the dynamic properties of the Rogowski coil, primarily by determining the dynamic errors for several selected test signals and the upper bound of the dynamic error for two quality criteria: the integral-square error and the absolute error. A procedure for filtering and reproducing these signals is also presented. The foundation of the presented research is an equivalent circuit model of the Rogowski coil, developed primarily for applications in electrical power and electromechanical systems. Two novel aspects of this work are the determination of dynamic errors for the Rogowski coil and a graphical and quantitative comparison of their values. The research results presented in this paper may serve as a foundation for enhancing the accuracy and dynamic reliability of both the Rogowski coil and other devices (e.g., transformers and current transformers) used in the power industry and mechanical engineering, particularly in the condition monitoring of a broad range of power equipment and in the experimental analysis of electromechanical systems operating under variable load conditions. The findings also highlight the importance of accurate current measurement in modern energy systems, where transient and high-frequency components increasingly affect performance and reliability. Consequently, the presented methodology provides a useful framework for guiding sensor selection and signal processing strategies in advanced monitoring and control applications.

1. Introduction

Rogowski coils are coreless current transformers that are used to measure alternating-phase and pulsed currents in situations where galvanic isolation is required between a high-current circuit and the corresponding measuring system. Current measurements can therefore be performed without the need for a physical connection between the coil and the circuit being measured. Rogowski coils are non-invasive, flexible measurement loops that can be opened and closed around a conductor [1,2]. These coils are widely used in the energy industry, in particular for monitoring current values in transformer stations and in electricity distribution systems, for accurate current measurements in protection systems, and in the testing of electrical devices and power supply systems [3]. They can also be widely used in mechanical engineering, particularly in the monitoring and control of electromechanical systems, where accurate current measurement using the Rogowski coil supports the assessment of their dynamic behaviour and operational reliability. An example is the phenomena that occur in power transformers due to discharges. In specific cases, this can lead to transformer failure [4]. A comparison of different Rogowski coil geometries to assess their ability to detect discharges in such cases is presented in [5]. In [4], two different materials for the Rogowski coil core are compared (ferrite and Teflon). Moreover, a simulation based on a mathematical model of the coils was carried out, and the electrical parameters were obtained. A short-time Fourier transform (STFT) [6] was used in the analyses [4] to detect repetitive discharges. The analyses show that in a thorough analysis, both the Rogowski coil circuit itself and the Rogowski coil-monitored system with current flow are electromechanical systems with interactions. These coils are characterised by a wide measurement band and significant linearity of the magnetisation characteristics over a wide range of measured currents, resulting from the limited effect of core saturation [7].

Due to the dynamic development of power grids and the concomitant increasing demand for electrical power consumption, sudden and dynamic changes in load currents occur. It is therefore necessary to analyse, among other things, the impact of the dynamic properties of the Rogowski coils on the conditions of electricity measurement, as only static (steady-state) errors meet the requirements of this type of measurement. Since it is increasingly impossible to ensure steady-state operating conditions for the Rogowski coils in current power systems, it is important to analyse the dynamic properties of this type of device [8]. A steady-state error occurs when the measured quantity is no longer subject to change and the measuring device has reached a steady state; in other words, a static error occurs when there is a consistent difference between a steady-state (measured) value and the actual value. A dynamic error, in turn, occurs when the measured quantity is subject to change and the device has not reached a steady state [9,10]. Figure 1 shows a graphical illustration of static and dynamic errors, where the function $y\left(t\right)$ represents the stabilisation of the measurement result [11].

Analysis of the dynamic properties of Rogowski coils is generally omitted from accuracy studies, and research in this area has mainly focused on modelling the BH magnetisation curve [12], for example, by using a polynomial approximation [13]. Accuracy analyses have also been carried out with the Jiles–Atherton hysteresis model to improve the hysteresis of ferromagnetic materials using the Langevin function [12,14]. The influence of angular and amplitude errors on the operation of current transformers was analysed in [15], using only the steady-state error as an example. However, the influence of the shape of the test signal was not considered, and no mathematical model of such a signal was developed to study the dynamic properties of Rogowski coils. No research has been conducted on the influence of the shape or parameters of the input signal on the dynamic properties of the coil, despite the fact that this is an extremely important measuring device in the energy and electromechanical industries, and particular attention should be paid to its accuracy. According to the definition of a dynamic error, its value is influenced not only by the dynamic properties of the measuring device under consideration but also by the shape and parameters of the measurement signals (measurands), meaning that a study of the properties of such signals seems fully justified. A comparative analysis of the dynamic error values resulting from the processing of such signals is also necessary.

The following theoretical aspects are considered in Section 2 of this paper for the purpose of analysing the dynamic properties of Rogowski coils:

• An equivalent circuit and mathematical model (transfer function and corresponding impulse response) of a Rogowski coil are presented [8].
• Methods for determining dynamic errors for certain types of test signals are discussed [16,17].
• Mathematical formulae are introduced that enable the upper bound of the dynamic error to be determined for two quality criteria (the integral-squared error and the absolute error) [16,17,18] and for two constraints (time and magnitude) [19] on the input signals obtained by a simulation of the Rogowski coil.
• Procedures for filtering and reproducing the input signal of the Rogowski coil are developed [20,21,22].

The transfer function is obtained from an analysis of the corresponding equivalent circuit and provides a basis for determining the impulse response, which in turn is a function of time and is necessary for determining the dynamic errors of the Rogowski coil. The second function necessary for determining this type of error is a mathematical description of the simulated input signal of the coil, which is in the form of an equation based on the values of the associated parameters [16,17]. The time function of the dynamic error is the convolution integral, which combines a mathematical description of the input signal with the impulse response of the Rogowski coil [23]. The upper bound of the dynamic error determined for the assumed quality criterion [16,17] can be treated as equivalent to the accuracy class that can be determined for devices intended to measure static quantities (signals with a precisely defined input shape, such as, constant signals or harmonic ones) [24]. The relevant numerical procedures primarily involve determining a mathematical description of the constrained simulation input signal and then, based on this signal, obtaining the upper bound of the dynamic error. The constrained simulation signal found in this way corresponds to the worst-case scenario of the actual input signal in the sense that any other signal can only generate a lower value of the dynamic error [16,17].

Rogowski coils can largely be compared to Hall-effect transducers, which are also designed for current measurement. Rogowski coils are characterized by a wide measurement bandwidth, high linearity, and resistance to magnetic core saturation [1,2,13], making them particularly suitable for high-frequency transient measurements. Hall-effect sensors, on the other hand, can measure both AC and DC currents but are characterized by limited bandwidth and thermal drift [14,15].

The remainder of this paper is organized as follows. Section 2 introduces the equivalent circuit model of the Rogowski coil and the adopted simulation framework. Section 3 presents the analysis of dynamic errors for the selected test signals and the determination of the upper error bound and discusses the filtering and reconstruction procedure and summarizes the main findings. Finally, Section 4 provides the conclusions and outlines directions for future work.

2. Materials and Methods

The principle of operation of the Rogowski coil is based on the phenomenon of electromagnetic induction. It measures the alternating magnetic flux generated by the flow of electric current through the measuring wire. The basis for the analysis of the dynamic properties of the Rogowski coil is the equivalent circuit shown in Figure 2 [8].

$e\left(t\right)$ is the voltage induced in the coil, $R_0$ is the resistance of coil, $L_0$ is the self-inductance of the coil, $C_0$ is the stray capacitance, $i_1\left(t\right)$ is the measured current, $i_2\left(t\right)$ is the secondary current, $v_0\left(t\right)$ is the voltage at the output of the coil, $M$ is the mutual inductance of the coil, and $R_L$ is the load resistor.

According to Faraday’s law, the voltage $e\left(t\right)$ induced in the coil is proportional to the change in current over time. Hence, for the circuit shown in Figure 2, we have the following:

$$e(t)=-M{\displaystyle \frac{d}{dt}}{i}_1(t)=L_0{\displaystyle \frac{d}{dt}}{i}_2(t)+R_0{i}_2(t)+v_0(t)$$

(1)

The current $i_2(t)$ is connected with the voltage $v_0(t)$ by the following differential equation:

$$i_2(t)=C_0{\displaystyle \frac{d}{dt}}{v}_0(t)+{\displaystyle \frac{1}{R_L}}{v}_0(t).$$

(2)

The transfer function associated with Figure 2 is given by the following:

$$K_R(s) ={\displaystyle \frac{U_0(s)}{I_1(s)}}={\displaystyle \frac{-\frac{M}{a}s}{s^2+\frac{b}{a}s+\frac{c}{a}}},$$

(3)

where $a=L_0{C}_0,$ $b=L_0/R_L+R_0{C}_0$, $c=R_0/R_L+1,$ and $s=\mathrm{j}\omega .$ The symbol $\mathrm{j}=\sqrt{-1}$ denotes the imaginary unit, while $\omega =2\mathsf{\pi }f \left[rad/s\right]$ denotes the angular frequency [25,26].

The impulse response can be obtained as a transient function using the following formula:

$${ k}_R\left(t\right)={\mathcal{L}}^{-1}\left[K_R\left(s\right)\right]={\sum }_{i=1}^n\left[{}_{s=p_i}{}^{res}{K}_R\left(s\right) e^{p_{i}t}\right],$$

(4)

where

$${}_{s=p_i}{}^{res}{K}_R\left(s\right) ={}_{s\to p_i}{}^{lim}\left(s-p_i\right)K_R\left(s\right) ,$$

(5)

and ${\mathcal{L}}^{-1}$ and $p_i$ denote the inverse Laplace transform and the poles associated with the denominator of the transfer function given in Equation (1), while $t=0, {\Delta }_t,\dots , T$ [27]. The variables ${\Delta }_t$ and $T$ are the time quantisation step and the time constraint on the input signal $i_1\left(t\right),$ respectively.

The mathematical formula that involves the input signal and the impulse response ${ k}_R\left(t\right)$ of the Rogowski coil is as follows:

$$y\left(t\right)={\int }_0^t{i}_1\left(\tau \right){ k}_R(t-\tau )\mathrm{d}\tau ,$$

(6)

which represents the convolution integral [23]. The signal $y\left(t\right)$ is a function of time, and in order to determine the dynamic error, it is necessary to apply the relevant dynamic error criterion. In this work, two types of criteria are used for the quantitative analysis of the dynamic error: the integral-square error and the absolute error. The integral-square error criterion [16,17] is defined by the following formula:

$$\mathrm{I}={\int }_0^T{⌈y\left(t\right)⌉}^2\mathrm{d}t,$$

(7)

where $\alpha$ denotes the magnitude constraint of the input signal $i_1\left(t\right).$ The absolute error criterion [16] is defined as follows:

$$\mathrm{D}=\mathrm{m}\mathrm{a}\mathrm{x}\left|y\left(t\right)\right|.$$

(8)

The parameter $T$ corresponds to the steady state of the impulse response $k\left(t\right)$.

Typical test signals, presented below in Equations (9)–(19), are proposed for determining the dynamic error of the Rogowski coil.

The sine-wave signal is as follows [24]:

$${i_1}^1\left(t\right)=\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{e} \mathrm{W}\mathrm{a}\mathrm{v}\mathrm{e}= \alpha ·\mathrm{s}\mathrm{i}\mathrm{n}\left(2\mathsf{\pi }ft\right),$$

(9)

where $f$ denotes the signal frequency [Hz]. The parameter $\alpha$ is determined based on Equation (3) and is defined as follows:

$$\alpha ={\displaystyle \frac{M}{a}}={\displaystyle \frac{M}{L_0{C}_0}},$$

(10)

and denotes the magnitude constraint.

It should be noted that the parameter $\alpha ,$ although introduced in this paper as the amplitude constraint of the input signal, is inherently a scaling factor determined by the physical parameters of the Rogowski coil, as expressed in Equation (10). In this study, this coil-derived parameter is consistently used as the amplitude constraint for all test signals. This dual interpretation emphasizes that $\alpha$ reflects the intrinsic gain of the coil model while also serving as the imposed input constraint in the dynamic error analysis.

The multisine signal is defined as follows [28]:

$${i_1}^2(t)=\mathrm{M}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}={\sum }_{n=n_1}^N{A}_{n}sin\left({\displaystyle \frac{2\pi nt}{T_1}}+{\varphi }_n\right),$$

(11)

where $n_1,$ $N$, and $T_1$ denote the first component of the series, the total number of components of the series, and the observation time for the harmonic component, respectively. $A_n$ and ${\varphi }_n$ are defined by the following formulae:

$$A_n=\alpha \left[1-2{\left({\displaystyle \frac{n^2}{2N}}\right)}_{mod2}\right]$$

(12)

and

$${\varphi }_n=2\mathsf{\pi }\left[{\displaystyle \frac{n^2+n-{n_1}^2+n_1}{2\left(N-n_1+1\right)}}\right],$$

(13)

where the symbol $\mathrm{m}\mathrm{o}\mathrm{d}2$ denotes the modulo 2 function and is expressed as follows:

$${\left({\displaystyle \frac{n^2}{2N}}\right)}_{mod2}=\left\{\begin{array}{c}0 if integer {\displaystyle \frac{n^2}{2N}} is even\\ 1 if integer {\displaystyle \frac{n^2}{2N}} is odd\end{array}\right. .$$

(14)

In the case of $n_1=1,$ Equation (13) simplifies to the following:

$${\varphi }_n=\mathsf{\pi }\left[{\displaystyle \frac{n^2+n}{N}}\right].$$

(15)

The OOK (On–Off Keying) signal is defined as follows [29]:

$${i_1}^3(t)=\mathrm{O}\mathrm{O}\mathrm{K}=\left\{\begin{array}{c}\begin{array}{c}\alpha ·sin\left(2\pi ft\right) if 0<t\le t_1 ,\\ 0 if { t}_1<t\le t_2, \end{array}\\ \alpha ·sin\left(2\pi ft\right) if { t}_2<t\le t_3,\\ \begin{array}{c}0 if { t}_3<t\le t_4,\\ \alpha ·sin\left(2\pi ft\right) if { t}_4<t\le t_5=T,\end{array}\end{array}\right.$$

(16)

where the times $t_1,$ $t_2,$ $t_3,$ $t_4$, and $t_5=T$ denote the periods of the signal ${i_1}^3\left(t\right)$.

The up-chirp signal is defined as follows:

$${i_1}^4\left(t\right)=\mathrm{U}\mathrm{p}-\mathrm{C}\mathrm{h}\mathrm{i}\mathrm{r}\mathrm{p}= \alpha ·\mathrm{s}\mathrm{i}\mathrm{n}\left[2\mathsf{\pi }\left(f_{0}t+{\displaystyle \frac{1}{2}}\beta t^2\right)\right],$$

(17)

where $f_0$ and $\beta$ denote the starting frequency of the chirp at $t=0$ and the chirp rate at which the frequency increases with time [30], respectively.

The down-chirp signal is defined as follows:

$${i_1}^5\left(t\right)=\mathrm{D}\mathrm{o}\mathrm{w}\mathrm{n}-\mathrm{C}\mathrm{h}\mathrm{i}\mathrm{r}\mathrm{p}= \alpha ·\mathrm{s}\mathrm{i}\mathrm{n}\left[2\mathsf{\pi }\left(f_{0}t-{\displaystyle \frac{1}{2}}\beta t^2\right)\right].$$

(18)

Equations (17) and (18) represent a sinusoidal signal where the frequency $f\left(t\right)=f_0+\beta t$ increases linearly with time.

The PRBS (Pseudo-Random Binary Sequence) signal [31] is defined as follows:

$${i_1}^6\left(t\right)=\mathrm{P}\mathrm{R}\mathrm{B}\mathrm{S}=\alpha ·{\sum }_{n=0}^{\infty }{a}_n·rect\left({\displaystyle \frac{t-n{\Delta }_T}{{\Delta }_T}}\right),$$

(19)

where $a_n$ and ${\Delta }_T$ denote binary values of the pseudo-random sequence—which can be generated, for example, by a linear feedback shift register (LFSR)—and the sampling period (duration of the PRBS bits), respectively. $\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left(·\right)$ is a rectangular function of width ${\Delta }_T$.

The upper bound of the dynamic error $I^{UB}$ for the integral-square error and for a doubly constrained input signal (simultaneously in time and magnitude) can be calculated using the following formula [16,17]:

$$I^{UB}={\int }_0^T{\left[{\int }_0^t{ k}_R\left(t-v\right){i_I}^{UB}\left(v\right)dv\right]}^{2}dt.$$

(20)

The signal ${i_I}^{UB}\left(t\right)$ with constraints is obtained using the fixed-point algorithm, which processes the impulse response in successive iteration steps [17].

The upper bound of the dynamic error $D^{UB}$ for the absolute error can be obtained as follows [16,17]:

$$D^{UB}=\alpha ·{\int }_0^T\left|{ k}_R\left(t\right)\right|dt,$$

(21)

while the corresponding signal ${i_D}^{UB}\left(t\right)$ is defined as follows:

$${i_D}^{UB}\left(t\right)=\alpha ·\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left[{ k}_R(T-t)\right].$$

(22)

The errors ${y_I}^{UB}\left(t\right)$ and ${y_D}^{UB}\left(t\right)$, which are functions of time, are determined based on Equation (6), where the signal $i_1\left(t\right)$ is replaced by the signals ${i_I}^{UB}\left(t\right)$ and ${i_D}^{UB}\left(t\right),$ respectively. These errors are follows:

$${y_I}^{UB}(t)={\int }_0^t{i_I}^{UB}(\tau ) { k}_R(t-\tau )\mathrm{d}\tau$$

(23)

and

$${y_D}^{UB}(t)={\int }_0^t{i_D}^{UB}(\tau ) { k}_R(t-\tau )\mathrm{d}\tau .$$

(24)

If the signal, $y\left(t\right),$ is disturbed by an additive disturbance, $d\left(t\right)$, i.e.,

$$y_d\left(t\right)=y\left(t\right)+d\left(t\right),$$

(25)

the reduction in the disturbance $d\left(t\right)$ in the signal $y\left(t\right)$ consists of subjecting the disturbed signal $y_d\left(t\right)$ to an integration over the window $w\left(t\right)$ in the interval $2\delta ,$ which corresponds to the window width [20,32,33]. The window $w\left(t\right)$ is a special window that must fulfill the standardisation condition, i.e.,

$${\int }_{-\delta }^{+\delta }w\left(t\right)\mathrm{dt}=1.$$

(26)

It should reach its maximum value in the middle of the analysis interval, while at the ends of this interval, its value and the values of its subsequent derivatives must be equal to zero. Taking these conditions into account, this special window is defined as follows:

$$w\left(t\right)={\displaystyle \frac{\frac{\mathsf{\pi }}{4}{\left\{{\int }_0^{\frac{\mathsf{\pi }}{2}}{\cos }^r\left(v\right)\mathrm{dv}\right\}}^{-1}}{\delta }}{\left\{cos\left[{\displaystyle \frac{\pi }{2\delta }}\left(\tau -t\right)\right]\right\}}^r, r=1, 2, 3,\dots ,$$

(27)

where $r$ denotes the window order.

The reduction in the disturbance $d\left(t\right)$ in the signal $y\left(t\right)$ can be achieved using the following formula:

$$y^{rd}(t)={\int }_{-\delta }^{\delta }{y}_d(t+\tau )w(\tau )\mathrm{d}\tau$$

(28)

or, equivalently, as follows:

$$y^{rd}(t)={\displaystyle \frac{1}{{\int }_{t-\delta }^{t+\delta }w(\tau -t)d\tau }}\left[{\int }_{t-\delta }^{t+\delta }{y}_d(\tau )w(\tau -t)d\tau \right].$$

(29)

The reconstruction of the signal $i(t),$ based on the disturbed signal $y_d\left(t\right)$ and the corresponding transfer function, can be achieved using the following formula:

$$i^r(t)={\int }_{t-\delta }^{t+\delta }{y}_{\mathrm{d}}(t+\tau )\left[{\sum }_{m=0}^M{\left(-1\right)}^m{\mathrm{a}}_m{w}^{(i)}(\tau )\right]\mathrm{d}\tau ,$$

(30)

where $i^r(t)$ and ${\mathrm{a}}_m$ denote the reconstructed signal and the constant coefficients of the corresponding transfer function of order $m$ [20,32,33]. Taking into account the transfer function given in Equation (3), we obtain the following formula:

$$i^r\left(t\right)={\int }_{t-\delta }^{t+\delta }{y}_{\mathrm{d}}(t+\tau )\left[{\displaystyle \frac{c}{M}}w(\tau )-{\displaystyle \frac{b}{M}}{\displaystyle \frac{d}{dt}}w(\tau )+{\displaystyle \frac{c}{M}}{\displaystyle \frac{d^2}{{dt}^2}}w(\tau )\right]\mathrm{d}\tau$$

(31)

or, equivalently, the following:

$$i^r(t)={\displaystyle \frac{1}{{\int }_{t-\delta }^{t+\delta }w(\tau -t)d\tau }}{\int }_{t-\delta }^{t+\delta }{y}_{\mathrm{d}}(\tau )\left[{\displaystyle \frac{c}{M}}w(\tau -t)-{\displaystyle \frac{b}{M}}{\displaystyle \frac{d}{dt}}w(\tau -t)+{\displaystyle \frac{c}{M}}{\displaystyle \frac{d^2}{{dt}^2}}w(\tau -t)\right]\mathrm{d}\tau .$$

(32)

Positive analysis results can be obtained for values of the coefficient $r$ in Equation (27) that are higher than four. The main difficulty when implementing the procedure given in Equations (28)–(32) is the need for optimal selection of the coefficient $\delta$; however, this problem can be solved using optimisation methods for selecting the width of the time window [32,33].

3. Results and Discussion

An analysis of the dynamic properties of the Rogowski coil was conducted by determining the dynamic errors for six typical test signals (sine wave, multisine, OOK, up-chirp, down-chirp, and PRBS), as presented below. The shapes of these signals are shown together with the corresponding dynamic errors vs. time and the associated error values for the integral-square and absolute criteria. We also summarise the results of research on the upper bound of the dynamic error for the above criteria under the assumption of two constraints (time and magnitude) on the simulated input signals and describe procedures for filtering and reconstructing the disturbed OOK signal. The summary of the dynamic error studies includes graphical comparisons for both quality criteria.

The following parameter values associated with the equivalent circuit of the Rogowski coil shown in Figure 2 were adopted as the basis for accuracy tests: ${\Delta }_t=100\times {10}^{-9}$ s, $T=5\times {10}^{-5} \mathrm{s},$ $L_0=1.489\times {10}^{-3}$ H, $C_0=23\times {10}^{-9}$ F, $R_0=72$ $\mathsf{\Omega }$, $R_L=200$ $\mathsf{\Omega }$, and $M=0.496\times {10}^{-6}$ $\mathrm{F}$ [25,26]. The adopted values are representative of the typical electromagnetic parameters of an LPCT (Low Power Current Transformer) coil. The Rogowski coil considered in this study, in addition to the equivalent circuit parameters listed above, is specified with a dielectric withstand voltage of 2 kV (rms, 1 min), which ensures safe operation when clamped around high-voltage conductors.

The numerical calculations and simulations were performed using MathCad 15. A consolidated overview of the adopted parameters is provided in

Table 1. Simulation parameters.

Parameter	Value
$T$	50 μs
${\Delta }_t$	0.1 μs
$a$	14.48 kA

.

Figure 3 shows the impulse response $k_R\left(t\right),$ given in Equation (4), obtained as the inverse Laplace transform of the transfer function given in Equation (3).

The impulse response $k_R\left(t\right)$ reaches the steady state of $t\approx$ $4\times {10}^{-5}$ s. Hence, a value a time value of $4\times {10}^{-5}$ s was taken as the time constraint $T$. The value of the magnitude constraint $\alpha$ was calculated, based on Equation (10), as $14.5$ kA.

Figure 4 shows the sine-wave signal ${i_1}^1\left(t\right)$ given in Equation (9) and the corresponding dynamic error $y^1\left(t\right)$ given in Equation (6). The frequency $f$ is equal to 0.4 MHz.

The error $y^1\left(t\right)$ shows instability up to about $0.5\times {10}^{-5}$ s and then becomes harmonic. The maximum deviation from zero appears for the first magnitude, and this is the value ${\mathrm{D}}_1$ given in Equation (8). The value of the integral-square error ${\mathrm{I}}_1,$ given in Equation (7), is 0.169 $\mathrm{V}{\mathrm{s}}^2$, while the value of the absolute error ${\mathrm{D}}_1$ is 117.8 $\mathrm{V}\mathrm{s}.$

Figure 5 shows the multisine signal ${i_1}^2\left(t\right)$ given in Equation (11) and the corresponding dynamic error $y^2\left(t\right).$ The signal ${i_1}^2\left(t\right)$ was generated for the following parameters: $n_1=0,$ $N=5$ and $T_1= {10}^{-5}$ s.

Both the signal ${i_1}^2\left(t\right)$ and the error $y^2\left(t\right)$ show repeatability of $0.5\times {10}^{-5}$ s. The value of the integral-square error ${\mathrm{I}}_2$ is 1.931 $\mathrm{V}{\mathrm{s}}^2$, while the value of the absolute error ${\mathrm{D}}_2$ is 370.9 $\mathrm{V}\mathrm{s}.$

Figure 6 shows the OOK signal ${i_1}^3\left(t\right)$ given in Equation (16) and the dynamic error $y^3\left(t\right)$. The frequency $f$ is equal to 0.4 MHz.

The signal ${i_1}^3\left(t\right)$ is an alternating harmonic function with zeros at intervals of $1\times {10}^{-5}$ s. The error $y^3\left(t\right)$ increases with a constant frequency in the intervals where the signal ${i_1}^3\left(t\right)$ is harmonic and decreases exponentially in the intervals where this signal is equal to zero. The value of the integral-square error ${\mathrm{I}}_3$ is 0.124 $\mathrm{V}{\mathrm{s}}^2$, while the value of the absolute error ${\mathrm{D}}_3$ is 85.6 $\mathrm{V}\mathrm{s}.$

Figure 7 shows the up-chirp (a) and down-chirp (c) signals ${i_1}^4\left(t\right)$ and ${i_1}^5\left(t\right)$ given in Equations (17) and (18) as well as the corresponding errors $y^4\left(t\right)$ and $y^5\left(t\right).$ The signals ${i_1}^4\left(t\right)$ and ${i_1}^5\left(t\right)$ were generated with the following parameters: $f_0=1$ Hz and $\beta ={10}^{10}$.

The properties of the signals ${i_1}^4\left(t\right)$ and ${i_1}^5\left(t\right)$ are an increase and a decrease in frequency, respectively, with increasing time. Over the considered time period $T$, the signal ${i_1}^4\left(t\right)$ is fast-changing, while the signal ${i_1}^5\left(t\right)$ is slow-changing. The values of the integral-square errors ${\mathrm{I}}_4$ and ${\mathrm{I}}_5$ are 0.627 $\mathrm{V}{\mathrm{s}}^2$ and 1.902 $\mathrm{V}{\mathrm{s}}^2,$ while the values of the absolute errors ${\mathrm{D}}_4$ and ${\mathrm{D}}_5$ are 262.0 $\mathrm{V}\mathrm{s}$ and 257.9 $\mathrm{V}\mathrm{s}.$ This means that higher values of both errors were obtained for the down-chirp signal than for the up-chirp signal.

Figure 8 shows the PRBS signal ${i_1}^6\left(t\right)$, generated using the function in Equation (19), and the corresponding dynamic error $y^6\left(t\right)$. The duration ${∆}_T$ of the PRBS bits is equal to $8\times {10}^{-8}$ s.

The error $y^6\left(t\right)$ is fast-changing at times when the signal ${i_1}^6\left(t\right)$ is fast-switching. The value of the integral-square error ${\mathrm{I}}_6$ is 3.175 $\mathrm{V}{\mathrm{s}}^2$, while the value of the absolute error ${\mathrm{D}}_6$ is 484.6 $\mathrm{V}\mathrm{s}$. This means that the highest dynamic error values were obtained for the PRBS signal for both quality criteria considered in this paper.

The upper bound of the dynamic error $I^{UB}$ obtained for the integral-square criterion given in Equation (20) is 3.913 $\mathrm{V}{\mathrm{s}}^2.$ Figure 9 shows the constrained signal ${i_I}^{UB}\left(t\right)$, obtained by applying the fixed-point algorithm, and the corresponding error ${y_I}^{UB}\left(t\right)$, calculated using Equation (23).

The signal ${i_I}^{UB}\left(t\right)$ with constraints $T$ and $\alpha$ has one time-switch at $t =$ $2.68\times {10}^{-5}$ s. The signal ${y_I}^{UB}\left(t\right)$ decreases initially (up to $0.4\times {10}^{-5}$ s) and then increases linearly until the signal ${i_I}^{UB}\left(t\right)$ switches. It then increases exponentially until $t \approx$ $3.1\times {10}^{-5}$ s and then decreases exponentially until time $T$.

The upper bound of the dynamic error $D^{UB}$ obtained for the absolute error criterion using Equation (21) is 525.0 $\mathrm{V}\mathrm{s}.$ Figure 10 shows the constrained signal ${i_D}^{UB}\left(t\right)$, which was determined by applying Equation (22) and the corresponding error ${y_D}^{UB}\left(t\right)$, calculated using Equation (24).

The constrained signal ${i_D}^{UB}\left(t\right)$ has only one time switch, similar to the signal ${i_I}^{UB}\left(t\right)$ but at a higher value of time $t =$ $4.54\times {10}^{-5}$ s. The signal ${y_I}^{UB}\left(t\right)$ decreases to the same instant in time as the signal ${y_I}^{UB}\left(t\right)$ and then increases slightly nonlinearly until the signal ${i_I}^{UB}\left(t\right)$ switches. Finally, the signal ${y_I}^{UB}\left(t\right)$ increases significantly until time $T$.

Table 2. Value of errors obtained for the integral-square error and absolute error.

$\mathbf{Integral}-\mathbf{Square} \mathbf{Error} \mathbf{I} \left[{\mathbf{V}\mathbf{s}}^2\right]$
${i_1}^1\left(t\right)$	0.169
${i_1}^2\left(t\right)$	1.931
${i_1}^3\left(t\right)$	0.124
${i_1}^4\left(t\right)$	0.627
${i_1}^5\left(t\right)$	1.902
${i_1}^6\left(t\right)$	3.175
${i_I}^{UB}\left(t\right)$	3.913
Absolute error $\mathbf{D} \left[\mathbf{V}\mathbf{s}\right]$
${i_1}^1\left(t\right)$	117.8
${i_1}^2\left(t\right)$	370.9
${i_1}^3\left(t\right)$	85.6
${i_1}^4\left(t\right)$	262.0
${i_1}^5\left(t\right)$	252.9
${i_1}^6\left(t\right)$	484.6
${i_D}^{UB}\left(t\right)$	525.0

summarizes the value of errors obtained for the signals ${i_1}^1\left(t\right), ..., {i_1}^6\left(t\right),$ ${i_I}^{UB}\left(t\right)$ and ${i_D}^{UB}\left(t\right),$ for both error criteria.

Figure 11 shows a graphical comparison of the dynamic error values obtained for the integral-square and absolute error criteria for the six types of test signals (sine wave, multisine, OOK, up-chirp, down-chirp, and PRBS) presented in Table 1.

From Figure 11, it can be seen that the order of errors $\mathrm{I}$ and $\mathrm{D}$ is the same except for those corresponding to the up-chirp and down-chirp test signals, where the former gives a lower value of the integral-square error and a higher value of the absolute error, and the relationship is reversed for the latter.

The percentage relationships of the errors $\mathrm{I}$ generated by the sine-wave, multisine, OOK, up-chirp, down-chirp, and PRBS signals to the error $I^{\mathrm{U}\mathrm{B}}$ are as follows: 4.32, 49.35, 3.17, 16.02, 48.61, and 81.14, respectively. The percentage ratios between the errors $\mathrm{D}$ and $D^{\mathrm{U}\mathrm{B}}$ are 22.44, 70.65, 16.30, 49.90, 49.12, and 92.31, respectively.

From Figure 11, it follows that the highest possible value of the dynamic error can be obtained only in response to special signals that generate the upper bound of the dynamic errors. The upper bound of the dynamic error can serve as an additional comparative criterion for evaluating the dynamic accuracy of coils with similar parameters but originating from different manufacturers [16,17]. This bound is determined through simulations using dedicated computational procedures. In experimental tests, values approaching the upper bound can be obtained when applying PRBS excitation signals, as demonstrated above. Consequently, PRBS signals may effectively substitute the constrained simulation signals traditionally used to generate the upper bound of the dynamic error.

To verify the procedure given in Equations (26)–(29), we add to the OOK signal in Equation (16) a disturbance $d(\mathrm{t})$ described by the following function:

$$d(\mathrm{t})= 6·\mathrm{s}\mathrm{i}\mathrm{n}(f_{x}t)+14·\mathrm{c}\mathrm{o}\mathrm{s}(f_{y}t),$$

(33)

where the frequencies $f_x$ and $f_y$ are equal to $2\times {10}^6$ Hz and $7\times {10}^6$ Hz, respectively.

Figure 12 shows the disturbed signal $y_d\left(\mathrm{t}\right)$ as given in Equation (25).

As a result of applying the filtering procedure using the special time window in Equation (27), an estimate ${\tilde{y}}^3\left(t\right)$ was obtained for the signal $y^3\left(t\right)$. The relative integral-square error when fitting signals ${\tilde{y}}^3\left(t\right)$ and $y^3\left(t\right)$ was 1.17%. The values of the coefficients $p$ and $\delta$ adopted for the analysis were 4 and 0.1, respectively. In the presented validation, the order of the window function was set to $p=4,$ in accordance with recommendations in [20,32,33], which indicate that values above four guarantee adequate boundary smoothness. The window width was selected as $\delta =0.1$ based on preliminary empirical trials, since this value provided a reasonable balance between reducing the disturbance and maintaining the fidelity of the reconstructed signal.

Applying the procedure in Equations (30)–(32) to the disturbed signal shown in Figure 12 and the transfer function in Equation (3) gives the reconstructed signal ${i_1}^{3r}\left(t\right)$ of the input signal ${i_1}^3\left(t\right)$ along with filtering of the disturbed signal $y_d\left(\mathrm{t}\right)$. The relative integral-square error when fitting the signals ${i_1}^{3r}\left(t\right)$ and ${i_1}^3\left(t\right)$ is 3.42%.

4. Conclusions

This paper has presented an analysis of the dynamic properties of the Rogowski coil by determining the dynamic errors for six typical test signals, the upper bound of the dynamic error for the integral-square and absolute criteria, and the filtering and reconstruction of one selected type of signal (OOK).

Test results show that the dynamic error values differed significantly depending on the type of input signal. These results were shown graphically and subjected to a quantitative comparison. The lowest dynamic error values for both quality criteria considered here were obtained for the OOK signal, with values of 0.124 $\mathrm{V}{\mathrm{s}}^2$ for the integral-square error and 85.6$\mathrm{V}\mathrm{s}$ for the absolute error. This means that this signal offers a useful method of testing Rogowski coils in industrial settings. The up-chirp and down-chirp signals generated error values that depended on the quality criterion under consideration, indicating the sensitivity of the Rogowski coil to the direction of frequency changes.

The values of the dynamic errors (the upper bound of the dynamic error) determined using the special test signals obtained with the proposed calculation procedures were 3.913 $\mathrm{V}{\mathrm{s}}^2$ (integral-square criterion) and 525.0 $\mathrm{V}\mathrm{s}$ (absolute criterion). These are the highest dynamic error values that can be obtained using the simulation test signals. Simulation-derived signals with constraints corresponding to the upper bound of the dynamic error have the property that any other signal within their constraints can only generate an error with a lower value. The errors that were closest to the upper bound of the dynamic error (81.14% and 92.31% for the integral-square and absolute error criteria, respectively) were obtained for the PRBS signal, since its random properties provide comprehensive frequency coverage for the dynamic analysis of the Rogowski coil. The application of the filtering and reconstruction procedures for the case of a simulated OOK signal resulted in a reproduction of the original signals at the level of 1.17% and 3.42%, respectively, which can be considered optimal results.

The mathematical formulae and research results presented in this paper can provide a basis for improving the accuracy and dynamic reliability of Rogowski coils and other measuring devices used in the power industry and mechanical engineering. They can also serve as a basis for practical applications in power and electromechanical system measurements and particularly for transformer substations and protection systems, where accurate current measurements during transient conditions are crucial.

The research methodology presented in this paper can be extended to other types of current transformers and measuring devices operating under dynamic conditions. Future research should focus on experimental verification of the proposed methodology using physical prototypes of Rogowski coils, an investigation of the environmental factors affecting dynamic properties, and the development of real-time dynamic error compensation algorithms. Furthermore, extending the analysis to a wider range of coil parameters could provide a more comprehensive insight into the dynamic behaviour of this type of measuring device.

Beyond traditional power system applications, accurate transient current measurement with Rogowski coils is increasingly relevant in electrified transportation and energy storage systems. Recent studies have shown that high-fidelity current data are essential for real-time energy management in connected plug-in hybrid electric vehicles [34], as well as for advanced thermal management strategies in cylindrical and prismatic lithium-ion batteries [35,36]. In these contexts, Rogowski coils can provide valuable support for both electrical accuracy and safe thermal operation.

Future work will focus on experimental verification of the proposed methodology using physical Rogowski coil prototypes and selected test signals. This step will complement the simulation-based results presented here and provide direct validation of the dynamic error analysis under practical measurement conditions. Future research will also include a comparison of the proposed methodology on physical Rogowski coil prototypes and its comparison with alternative approaches to dynamic error assessment.