1 Introduction

Ferroresonance is a phenomenon characterized by its unpredictable and aggressive nature. It is a great threat especially when it occurs in electrical distribution systems. It appears after transient disturbances (transient overvoltage, lightning overvoltage or temporary fault) or switching operations (transformer energizing or fault clearing). Its effects are characterized by high sustained overvoltages and overcurrents with maintained levels of current and voltage waveform distortion. In high-voltage and current circuits (MV and HV networks), its consequences practically always include massive and expensive damage [1, 2].

Fig. 1
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Photograph of laboratory setup (a) and scheme (b)

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The reason why ferroresonance so widely affects power system circuits is due to dense accumulation of the transformers’ saturable inductances and the capacitive effects of power transmission and protective elements set in low resistance systems (low-loss transformers, unloaded transformers, low circuit losses) [3, 4].

Nevertheless, the ferroresonance phenomenon depends also on many other factors and circumstances such as initial conditions of the system, transformer iron core saturation curve, residual fluxes in the transformer core, type of transformer winding arrangement, capacitance of the circuit, point-of-wave switching operation or total losses. So its predictability may be considered as quite complex and difficult [1, 5,6,7].

The nature of a phenomenon described above and referred to in many research papers [4, 8,9,10,11,12,13,14,15,16] leaves no doubt as to the constant need of studying low-loss circuits containing capacitors and saturable inductors. Taking into account their tendency to become prone to ferroresonance condition, a new and interesting approach to the problem focuses on developing of new detection, diagnostic and monitoring procedures.

2 Characteristics of the test circuit

A specially designed laboratory setup has been used. It contains a voltage and current source, both controlled over a wide range. There is also a linear capacitor in the circuit, and its total value may be modified by applying different capacitors for series and parallel configurations. The main element of the circuit is iron core coil. The circuit configuration allows free access to terminals of each element.

For the test purpose, the series configuration has been chosen. Photograph of the setup and its scheme are depicted in Fig. 1.

Despite an apparent simplicity of the proposed test circuit, a ferroresonance effect may be induced in it. The series ferroresonant circuit implemented in the laboratory is equipped with carefully selected elements and supplied with a controlled sinusoidal voltage. The phenomenon of ferroresonance is induced in a controlled manner and on a safe scale, so there is no danger of damage to circuit elements. Moreover, a presented (R)LC circuit structure may be later adapted to many problems emerging when power system is modeled. The designed RLC circuit constitutes a fragment of simplified model of a power system [3, 4, 8, 13].

3 Characteristics of the measurement platform

The necessity of constructing a special system to diagnose the ferroresonant circuit has emerged during earlier ferroresonance analysis and computations. Previous investigations [17, 18] have shown that dealing with such problem like ferroresonance requires prior diagnosis of the ferroresonant circuit—performed separately for every circuit—in order to determine best scientific approach to the problem. Authors have assumed that the developed diagnostics unit should be a combination of all known tools and meters, using up-to-date data acquisition and processing technology. Moreover, some features should be accessible on real-time basis (online). Application of a universal way of communication (USB) is beyond discussion.

Among the various microcontrollers available on the market, the most popular systems come from manufacturers such as Microchip Technology, STMicroelectronics, Atmel and Freescale Semiconductor, Texas Instruments and Analog Devices. The developed measuring system uses Arduino solutions [19], namely Arduino Nano. It is equipped with the AVR microcontroller ATmega328 [20]. A photograph of the system is presented in Fig. 2.

Fig. 2
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Photograph of Arduino Nano system

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Short summary of its main possibilities and features is given in Table 1.

Table 1 Main parameters of Arduino Nano system
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Fig. 3
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Block scheme of signal rescale process

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Fig. 4
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Photograph of manufactured prototype of the additional separating unit during tests

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The chosen measuring unit operates in very specific and small range of voltages (Table 1). The tested circuit has to be separated from the Arduino system, and the signals need to be rescaled. Therefore, additional separating block has been developed and maximum of its operating range has been set at 400 V.

Figure 3 shows the block diagram of signal rescale process. The manufactured prototype of the additional separating unit is shown in Fig. 4.

All signal processing operations require verification, in particular those carried out in additional separating block. The certified industrial digital disturbance recorder RZ1 (Kared Gdańsk) [21] was used in calibration and authentication process. The photograph of complete diagnostics system FeD connected to the laboratory setup is depicted in Fig. 5.

Fig. 5
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Photograph of complete diagnostics system FeD connected to the laboratory setup

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Fig. 6
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View of the interface window in the SciLab environment

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4 Original control program

The use of computers in scientific research often requires specialized non-standard software. One of the basic applications is the computerized operation of measuring apparatus. Non-standard software is particularly useful where novel and original equipment is used. As a result, a computer with such software is used to receive electronic signals, collect data, record measurement results, create and run databases or perform specific procedures, calculations and other activities [22,23,24,25,26].

Lots of test results are obtained in particular by dynamic measurements (carried out on a continuous basis). Activities such as recording, calculating and interpreting result in a manual manner can be difficult or even impossible. Hence, the necessity of using IT solutions.

The interface of the authors’ original control program of the elaborated measurement system is shown in Fig. 6. The program has been written in the SciLab environment, which shows exceptional compatibility with the Arduino hardware platform.

Execution of the program is performed with the help of numerous buttons, and detailed descriptions are provided for each one. The offered options are previews of u–i waves for all circuit elements, True RMS measurements with their previews, preview of waveforms calculated from given patterns (i.e., flux, circuit and coil losses), performing advanced computing (STFT analysis) with visualization of results, data recording and exporting options, plotting characteristics and 2D/3D graphs from stored data, etc. The diagram of the functional structure of the program is shown in Fig. 7.

5 Measurements and computations

First of all, it was necessary to run through the full range of supply voltage. This made it possible to define the range of changes in current and voltages for all circuit elements and find the supply voltage value where the ferroresonance occurs. Maximum values of particular quantities obtained for all circuit elements are given in Table 2.

Fig. 7
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Block diagram (process flowchart) of functionality and capabilities of original control program

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Table 2 Maximum peak amplitude values of particular quantities obtained for all circuit elements
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The conducted measurements provided the information of voltage ‘location’ of the ferroresonance phenomenon. Ferroresonance occurs close to the supply voltage value corresponding to milliampere current circuit response.

A preview of the full range voltage–current circuit behavior made it possible to distinguish three circuit operating states: two steady states with different levels (low and high coil core saturation) and the transient state, when the ferroresonance occurs. Examples of three different circuit operating conditions are depicted in Fig 8.

Fig. 8
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Examples of three different circuit operating conditions

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5.1 Characteristics of the test circuit signals (STFT analysis)

Voltage–current signals measured and reviewed over a full supply voltage range have also been recorded. Recorded data were used to analyze the character of the signals, in particular in relation to frequency changes and harmonic distortions.

The traditional FFT analysis [27, 28] is used to obtain the characteristic frequencies present in the signals. This method allows us to examine the amount of distortion in the input signal by calculating the magnitude spectrum. The analysis of the signal has been divided into two different intervals. The voltage and current signals for each ferroresonance circuit steady state are shown alongside the corresponding magnitude spectrums in Fig. 9. One can observe that during the prolonged ferroresonance the voltage and current of two different ferroresonant circuit steady states have different magnitude spectrums.

Fig. 9
figure 9

Current for all three discussed circuit states (a–c) and voltage (d); steady-state, high-saturation conditions; waveforms and results of FFT analyses

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The presence of more than one characteristic frequency shows the multiplicity in periodicity, commonly present in some ferroresonant states. On account of nonlinearities, which are both high and variable (variability is progressive), a standard harmonic analysis is insufficient. In such situations, in addition to information about what components are present in a given signal, the course of their variability over time is also important. Therefore, spectral analysis has been used.

There are many methods of the so-called time–frequency analysis, which allow to visualize the course of spectrum variability of signals over time. One of the basic ones is the so-called short time Fourier transform (STFT) [29,30,31]. It is founded upon the fact that the analysis window is shifted along the time axis, and after each shift the FFT signal inside the window is calculated. Depending on how the analysis parameters are selected (window width, step with which we move them and FFT parameters), different results are obtained. The amplitude spectrum resulting from this analysis is called a spectrogram, and it can be presented as a 3D or 2D graph. The 3D example for investigated circuit is shown in Fig. 10.

It is worth to mention that it is important to correctly set the FFT parameters as well as the signal sampling parameters to obtain a signal which is free of the FFT artifacts like aliasing, spectrum leakage and poor frequency resolution [32, 33].

5.2 True RMS value

Fig. 10
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3D example of spectrogram (time–frequency representation) for investigated circuit

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Fig. 11
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Examples of coil current (a) and voltage (b) waveforms together with corresponding RMS values

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The root mean square mathematical operation is widely used in electrical engineering [34, 35]. For this reason, the computing possibilities of the developed system were also tested in measuring the RMS values of the circuit quantities. According to [36], the RMS value of the periodic waveform is defined by the formula:

$$\begin{aligned} F_{ \mathrm {RMS} } = \sqrt{ \frac{1}{T} \int _{0}^{T} (f(t))^2 \, \mathrm {d}t }, \end{aligned}$$
(1)

where \(F_\mathrm {RMS}\) is the RMS value, T is the waveform period and f(t) denotes instantaneous values of the waveform.

Computation results are shown in Fig. 11, and two waveforms: current and voltage, are demonstrated with corresponding RMS values.

The current and voltage waveforms recorded in the ferroresonance circuit, especially at the high saturation of the coil core, were strongly deformed (Fig. 9c, d).

Interesting result was obtained during analysis of the circuit in the steady state and with highly saturated coil core. Table 3 presents examples of measurement results; comparison of values measured with a universal multimeter with the ‘True RMS’ function and calculated by the developed measurement system (for the described high-saturation case). Due to the implementation of (1), the values calculated by Arduino should be recognized as identical with reference values (calculated according to the definition formula).

Table 3 Comparison of RMS values, example of results from two measurement devices: multimeter (a), Arduino-based system (b)
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The following formula was used to calculate relative percentage errors for the values measured with the multimeter:

$$\begin{aligned} \varepsilon = \frac{ | X_\mathrm {def} - X | }{ X_\mathrm {def} } \cdot 100\,\%, \end{aligned}$$
(2)

where \(\varepsilon \) is the relative error, \(X_\mathrm {def}\) is the RMS value calculated in accordance with definition (1), X is the RMS value measured with the multimeter.

5.3 Measurements and verification of capacitive reactance

The acquisition of waveforms over full range of available supply voltage and accessibility of RMS/amplitude data at any time instant have made it possible to analyze character and behavior of different circuit elements. The developed system uses the current and voltage of capacitor waveforms to calculate its capacitive reactance. Capacitance values are easily extracted, assuming that the element operates as an ideal capacitor. A single time period is selected from the waveform and subjected to a quasi-steady-state analysis (Fig. 12). Then, through a simple FFT analysis, fundamental voltage and current harmonic amplitudes can be obtained (Fig. 13).

Fig. 12
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The circuit current 3D spectrogram limited to the fundamental harmonic

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Fig. 13
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Extraction of the harmonic fundamental values from current and voltage waveforms; steady-state, high-saturation scenario

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The capacitance is obtained through the well-known formula [36]:

$$\begin{aligned} C = \frac{ I_{(\mathrm {1h})} }{ U_{(\mathrm {1h})} 2 \pi f } \end{aligned}$$
(3)

where I and U are, respectively, the current and voltage first time harmonic values and f is supply voltage frequency (i.e., 50 Hz).

Calculations have been conducted for two time intervals, with two different (low and high) coil saturations conditions; capacitance values have been obtained. These are: \(C = 29\,\upmu \mathrm {F}\) (for low saturation condition) and \(C = 29.1\,\upmu \mathrm {F}\) (for high coil saturation).

5.4 Computation of the nonlinear coil flux linkage

The flux linkage is defined as a time integral of the voltage across the coil [37]:

$$\begin{aligned} \varPsi (t) = \int _{t_0} ^{t} u_L(t)\,\mathrm {d}t + \varPsi (t_0) \end{aligned}$$
(4)

This quantity is not measurable using conventional engineering tools. However, it is required in order to perform the ferroresonance analysis.

In the FeD system, the flux linkage is obtained by means of numerical computations of the integral (4). The trapezoidal integration method is applied. Example of the obtained flux linkage is presented in Fig. 14 together with the investigated voltage waveform.

Fig. 14
figure 14

Examples of the voltage (a) and magnetic flux (b) waveforms of nonlinear coil for a high saturation of the core; supply voltage waveform added for comparison in a

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5.5 Iron core coil examination

To analyze the ferroresonance phenomenon, there is a need to model the nonlinear coil core characteristics. At first, previously recorded current waveform and the computed flux linkage waveforms have been used. This helped to plot hysteresis loops of the coil [37] (Fig. 15).

Fig. 15
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Exemplary plot of flux linkage versus current (the hysteresis curves)

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In many ferroresonance studies the nonlinear coil core is represented by a single magnetization curve [11, 38,39,40,41], not taking into account a hysteresis phenomenon. Because of its complexity, the hysteresis itself is then approximated through a determination of the power losses [42, 43]. The total core loss is represented by a constant resistance. In reality such a computed resistance value is not constant, but it decreases as saturation level goes up [44]. (This will be proved in the next subsection.)

In ferroresonance investigation, the shape of hysteresis curve significantly influences the ferroresonance phenomenon [45, 46]. This is true not only for the major loop, but also in relation to various minor loops [47]. These are often ignored, e.g., in many computational programs.

5.6 Iron core coil loss classification

The power loss in an iron core coil can be associated with several different properties and phenomena [48, 49]:

  • resistance of the windings—‘copper loss’,

  • magnetic friction in the core—‘hysteresis’,

  • electric currents induced in the core—‘eddy currents’,

  • physical vibration and noise of the core and windings,

  • electromagnetic radiation,

  • dielectric loss in materials used to insulate the core and windings.

As opposed to all other losses, copper losses are present even if the winding current does not change. All other losses drop down to zero for DC excitation. Hysteresis and eddy current losses are collectively known as iron loss or core loss (even if a ferrite core is used).

No matter how they are interpreted the total losses can be obtained directly from the coil waveforms. The instantaneous power is the product of the coil terminal voltage and current:

$$\begin{aligned} p(t)=u(t)\cdot i(t) \end{aligned}$$
(5)

Further analysis can be performed when computations of the active power are taken into account [36, 50, 51]:

$$\begin{aligned} P=\frac{1}{T}\int _{t}^{t+T}p(t)\text {d}t \end{aligned}$$
(6)

The above integral is computed numerically for every time instant \(t = t_{(\text {now})}\) in the FeD system [trapezoidal method is used for computing (4), as before]. The results are depicted in Fig. 16.

Fig. 16
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The time functions of instantaneous power and active power of the iron core coil

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Fig. 17
figure 17

Simple nonlinear coil models: series (a) and parallel (b)

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Fig. 18
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Examples of time variation of equivalent resistance values for series (a) and parallel (b) model and changes of this variation as the function of circuit current as well as supply and coil voltage

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Determination of equivalent resistance parameter in two simple models of the nonlinear coil is possible, when active power value is available (Fig. 17).

Accessibility of active power at every time instant is necessary, when equivalent resistance values in the described models are calculated.

The developed system provides a full access to the visualization of the waveforms. An example of how the equivalent resistance values vary in time (in the case of both models) is shown in Fig. 18.

Obtained core losses and characteristics \(\psi (i)\) and \(u_{R_\mathrm{L}}(i)\) will be used in future studies for making improvements to the nonlinear coil models.

6 Analysis of ferroresonance

There are several tools available to study the nonlinear dynamic systems [52,53,54,55]. Some of them are based on bifurcation theory [56,57,58,59]. This theory examines the evolution of the system solution by changing the value of a control parameter. However, in this paper the ferroresonant circuit is analyzed differently, using diagnostic methods. In the diagnostic approach the ferroresonance can be characterized by using either a spectral study method based on Fourier’s analysis or a stroboscopic analysis based on Poincaré maps [60,61,62]. These can be used to differentiate between ferroresonance states (e.g., a quasi-periodic state from a chaotic state).

Fig. 19
figure 19

Example of phase trajectory of the investigated system

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A dynamical, nonautonomous system can be described through the following system of differential state equations [58, 63]:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{c} \frac{\mathrm {d}x_1}{\mathrm {d}t} = f_1(x_1, x_2, \ldots x_N, t) \\ \frac{\mathrm {d}x_2}{\mathrm {d}t} = f_2(x_1, x_2, \ldots x_N, t) \\ \ldots \\ \frac{\mathrm {d}x_N}{\mathrm {d}t} = f_N(x_1, x_2, \ldots x_N, t) \\ \end{array} \end{array}\right. } \end{aligned}$$
(7)

In terms of discussed ferroresonance circuit, the following sets of equations depending on assumed mathematical model of nonlinear coil may be formulated:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{c} \frac{\mathrm {d}\varPsi }{\mathrm {d}t} = f_1(\varPsi , u_\mathrm{C}, t) \\ \frac{\mathrm {d}u_\mathrm{C}}{\mathrm {d}t} = f_2(\varPsi , u_\mathrm{C}, t) \\ \end{array} \end{array}\right. } \end{aligned}$$
(8)

Therefore, two separate cases can be found, where accordingly \(u_\mathrm{C}\) and \(\varPsi \) or \(i_\mathrm{L}\) are state variables.

State space is an illustration of time development of state variables [64, 65]. Due to the fact that the investigated system is nonautonomous, three-dimensional space is necessary. System of equations related to the model can be also modified to fit to the following form:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{l} \frac{\mathrm {d}x_1}{\mathrm {d}t} = f_1(x_1, x_2, x_3) \\ \frac{\mathrm {d}x_2}{\mathrm {d}t} = f_2(x_1, x_2, x_3) \\ \frac{\mathrm {d}x_3}{\mathrm {d}t} = 1 \\ \end{array} \end{array}\right. } \end{aligned}$$
(9)

Phase trajectory displaying two examples of steady states and transient state between them is presented in Fig. 19.

In order to reduce dimensionality of information about changes of state variables, it is possible to project the whole three-dimensional trajectory onto a horizontal plane. This operation results in obtaining a state plane of the system as depicted in Fig. 20.

Fig. 20
figure 20

Phase plane of the investigated system obtained from phase space presented in Fig. 19; bold line marks the steady-state circuit condition for low and high saturation of the coil core, respectively

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Due to the presence of a periodic input in the discussed system, a three-dimensional representation can be produced. This corresponds to a trajectory generated through a rotating \(\psi _\mathrm{r}\)–\(u_\mathrm{C}\) plane (Fig. 21), with:

$$\begin{aligned} \psi _\mathrm{r} = \psi + \psi _\text {ref}, \end{aligned}$$
(10)

where \(\psi _\text {ref}\) is an arbitrary value added, since \(\psi _\mathrm{r}\) should also be positive.

Fig. 21
figure 21

Three-dimensional representation of trajectory generated through the rotating \(\psi _\mathrm{r}\)–\(u_\mathrm{C}\) plane for two different saturation conditions of the ferromagnetic core coil: low saturation (a), high saturation (b)

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Figure 21 not only hints at the existence of steady states but also shows the effect of deformation of the waveforms of the state variables. Poincare map may be obtained through the intersection of a trajectory with a plane at a single angle \(\omega \mathrm {T}\) (Fig. 22).

Fig. 22
figure 22

Poincare maps of the system trajectory presented in Fig. 21—low saturation (a) and high saturation (b)

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A record of the intersection points from all time instants of trajectory results is expressed by the phase plane (Fig. 23).

The above analysis can be performed also for a dynamical condition of the system. The system trajectory on the rotating \(\psi _\mathrm{r}\)–\(u_\mathrm{C}\) plane for an example of a transient state is presented in Fig. 24.

A Poincaré map related to the presented dynamical trajectory is presented in Fig. 25.

In case of transient state the phase plane derived from Poincare maps can also be obtained (Fig. 26), in a manner similar to that used for steady states.

The above analysis is implemented in the FeD system. The instant values of the measured and calculated values are recorded, leading to the generation of the Poincaré map. The 2D visualization can be then easily interpreted by the user.

7 Simulation module

The FeD diagnostic system has been equipped with a simulation module used to model the nonlinear core coil. After a literature survey of many possible alternatives (as given in [66,67,68]) and basing on authors’ personal experiences (various studies [9, 10, 17, 18]), a fractional calculus model of the ferromagnetic coil has been chosen. From the field of fractional calculus, the Caputo fractional derivative is applied [69, 70]:

$$\begin{aligned} {}_{t_0}^{\mathrm {C}} \mathrm {D} _{t} ^\alpha f(t) = \frac{1}{\mathrm {\Gamma }(1-\alpha )} \int _{t_0}^t \frac{ f^{(1)}(\tau ) }{(t-\tau )^\alpha } \mathrm {d} \tau , \end{aligned}$$
(11)

where only the range of \(\alpha \in [0, 1]\) is considered.

Fractional derivatives have been applied with success in circuit analyses in modeling supercapacitors [71,72,73,74] and (important for this study) ferromagnetic core coils [9, 75,76,77]. Model containing a nonlinear fractional coil and a resistance connected in parallel (Fig. 27) has been used in this study.

The nonlinear fractional coil is defined by the fractional differential equation:

$$\begin{aligned} {}_{t_0}^{\mathrm {C}} \mathrm {D}_{t}^\alpha \varPsi = u_\mathrm{L} \end{aligned}$$
(12)

with \(\varPsi \) being an artificial variable with the unit \(\mathrm {Wb}\,\mathrm {s}^{(\alpha -1)}\) and a nonlinear function \(\varPsi (i)\). The nonlinear function is defined by (i, \(\varPsi \)) value pairs, where the i values are set as 15 values ranging from 0 to 42 mA and the \(\varPsi \) values are estimated. Both the fractional coil and resistance have a different effect on the shape of a resulting hysteresis curve. Estimation of model parameters is possible in FeD system. The obtained values are given in Table 4.

A comparison of the simulated hysteresis curve and the one obtained from measurements is presented in Fig. 28, while a comparison of the current waveforms is presented in Fig. 29.

Fig. 23
figure 23

Intersection points from Fig. 22 shown on a phase plane with the steady state for two analyzed saturation conditions of the coil core

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Fig. 24
figure 24

Three-dimensional representation of trajectory generated through the rotating \(\psi _\mathrm{r}\)–\(u_\mathrm{C}\) plane for an example of a transient state; bold line marks the steady-state trajectories for low- and high-saturation conditions of the coil core

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Fig. 25
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Example of Poincare map for the trajectory presented in Fig. 24

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Fig. 26
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The Poincaré map (dots) and phase plane (solid line) of the studied system (transient state)

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Fig. 27
figure 27

Applied ferromagnetic core coil model (the fractional order element is marked with a symbol introduced in [78, 79])

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Table 4 Estimated model parameter values
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Fig. 28
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Comparison of simulated and measured hysteresis curve (measurement results marked in gray)

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Fig. 29
figure 29

Comparison of simulated and measured current waveforms (measurement results marked in gray)

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In order to judge the accuracy of the results, the current waveforms have been compared. The following error formula was used:

$$\begin{aligned} \varepsilon =\sqrt{\sum _{i=1}^{n}(f_{i(\mathrm{m})}-f_{i(\mathrm{s})})^2}\frac{100}{nF_{\mathrm{max}}} \end{aligned}$$
(13)

where \(f_{i(\mathrm{m})}\), \(f_{i(\mathrm{s})}\) voltage or current value for the ith time instance, \(F_{\max }\) maximum value of the respective waveform. The index m denotes measured values, while s denotes those obtained by simulations. For the presented current waveforms, the computed value is \(\varepsilon = 0.04 \%\).

In future investigations, the model can be studied from the viewpoint of its dynamic behavior, as this is important for ferroresonance analyses [10, 47]. However, because of the complexity of such problems (fractional differential equation and nonlinear function), this requires numerical methods (e.g., as given in [81,82,83]).

8 Conclusions

System for measurements and diagnostics of ferroresonant circuits has been developed. It has been called FeD.

The initial part of the system is aimed at a series of signal transformations (Fig. 3 in Sect. 3). The processed signal is then sent to the Arduino module.

The Arduino module makes it possible to perform some single signal operations in online mode (e.g., RMS value calculation).

A unique feature of Fed system is original software developed by the authors. The program allows for a cooperation between the Arduino module and a PC, i.e., sending commands for measurements and data transfer.

The functionality of the program has been implemented specifically for the studies of a ferroresonant circuit. The following functions are present:

  • preview of the voltage and current waveforms,

  • True RMS measurements (and their visualizations),

  • computations of quantities that are not measured directly (e.g., flux, power losses),

  • advanced signal processing analyses, e.g., STFT (Sect. 5.1),

  • advanced ferroresonance analyses (Poincare maps, 3D trajectory visualizations—Sect. 6).

The FeD system has been equipped with a simulation module. The simulations are related to ferroresonant circuit, and a full description of circuit elements is required. For the ferromagnetic coil, a nonlinear fractional model is used. The capacitor is assumed to be a linear element. The module allows for an estimation of the parameters of each of these elements basing on their voltage and current waveform measurements.

To summarize, FeD system is equipped with a full range of functionality ranging from basics like previews of measured current and voltage waveforms to advanced analyses of a ferroresonant circuit. Each studied circuit provides unique u–i waveforms determined by individual properties of its elements (e.g., distinctive material properties of the iron core coil). For these waveforms, the FeD system performs a complete diagnostics, which yields a full spectrum of individual quantities and information on the behavior of circuit elements (e.g., magnetic hysteresis loop). The acquired features of the circuit and its components determine the approach for the analysis of the ferroresonance phenomenon and the manner in which the nonlinear coil is modeled.

Possibilities and needs for further research include:

  • the application of the FeD system and its algorithms for investigation of different types of ferroresonance modes, e.g., subharmonic and quasi-periodic modes; in this study, intentionally—only the fundamental mode was taken into account because of its predictability and deterministic behavior,

  • with an increase of the ferromagnetic core saturation level, the resistance decreases (what has been proven in Sect. 5.6); the obtained resistance data will be used for study of various ferromagnetic core coil models so that an optimal one can be selected (reflecting this behavior),

  • incorporation of simulations of dynamic behaviors of ferroresonant circuits (including cases where ferromagnetic core coils are modeled with the usage of fractional derivatives); this requires the implementation of specific solvers (e.g., there are ones basing on various numerical methods for fractional differential equations like the SubIval numerical method [80,81,82] and others [83, 84].