Accurate Circuit Parameter Determination of a Resonant Power Frequency Converter for High-Voltage and Partial Discharge Tests

Abstract

For high-voltage (HV) and partial discharge (PD) tests on high-voltage equipment, a resonant power frequency converter has recently been developed. A single-phase power frequency converter with a resonant tuning and filter circuit and an HV testing transformer comprise the developed system. A difficulty in the tuning and filter circuit design is the unknown testing system circuit parameters, including unknown parasitic inductance, capacitance, and internal resistance. In this paper, a system with a voltage rating of 75 kV_rms, apparent power of 40 kVA, and an operating frequency from 50 Hz to 200 Hz is considered for determination of the equivalent circuit parameters. From the determined circuit parameters, the appropriate resonant tuning and filter circuit was designed effectively. The transfer functions of the input and output testing voltages, along with the transfer impedance of the input voltage and signal voltage of the PD measuring port, were analyzed. The system design was verified by experiments with a voltage transformer. The gain of the transfer impedance was about 15 and 4 at the testing frequencies of 50 Hz and 200 Hz, respectively. With the proper design, it is possible to generate an output voltage waveform that is almost entirely sinusoidal and has a background noise level of under 1 pC. According to the experimental results, the system design of the resonant converter and the method for determining the equivalent circuit are very helpful for the HV and PD tests of voltage transformers in actual practice.

1. Introduction

It is well known that one of the main problems in the HV and PD tests of high-voltage equipment is the applied testing voltage, which is higher than the transformer rating voltage [1,2]. If such a voltage level with a rating frequency of 50 or 60 Hz is utilized in the tests for some iron-core devices, such as transformers and reactors, core saturation of the devices under test will occur, resulting in the distortion of the applied testing voltage. Furthermore, testing with such voltages also consumes a high current. To avoid the aforementioned conditions, the international standard defines the quality of the testing voltage by the different voltage (the root-mean-square value and peak value divided by the square root of two) and total harmonic distortion, which are not higher than 5%. Under the test conditions, the applied voltage in such tests is no more than two times the rating voltage of the transformer, so the applied voltage should have a frequency higher than two times the rating frequency. For safety purposes, the frequency used in the test should be in the range of 100 Hz to 400 Hz. Generally, it is found that the frequency of 200 Hz is sufficient in all tests of iron-core devices. Therefore, the frequency range from 50 Hz to 200 Hz was utilized in all considered simulations and experiments. In the past, the motor and generator test set was applied in the test; however, the test set has a high cost, and requires a large space for installation.

The high-voltage (HV) test is essential for confirming the HV equipment’s performance during design and construction procedures. Such tests comprise partial discharge measurement, power loss measurement, dielectric loss measurement, and insulation voltage withstand tests.

The PD measurement is the most crucial test for assessing the performance and lifetime of insulation. Partial discharges (PDs) are regional electrical discharges that only partially bridge the insulation between electrodes. They result from localized electrical stresses in the insulation or on its surface. Such discharges often manifest as pulses with a duration shorter than 1 µs. There are three main categories for PDs: Corona discharges, the first type of PD, can happen in places with sharp edges and intense electric field stress. The second type is caused by voids and bubbles in the interior insulation material. If there is sufficient electrical field stress and these faults have lower insulation levels than the main insulation material, a PD can arise. The final type of PD, known as surface discharge, takes place at the boundary of two materials.

The simplest technique [1] to produce HV for HV testing is using an HV testing transformer connected with a voltage regulator. The power supply of the regulator can be a generator or a low-voltage grid.

However, the nonlinear characteristics of iron-core and switching devices lead occasionally to issues with power quality, such as voltage distortion and interferences affecting the background noise in PD and HV tests. IEC 60060-1 [2] specifies that the testing voltage must have a nearly pure sinusoidal waveform, where the difference in voltage (DV; peak voltage divided by √2 and RMS voltage) and the total harmonic distortion voltage (THD_v), given by Equation (1), must not be greater than 5%.

$${\mathrm{THD}}_{\mathrm{v}}=\frac{1}{V_{p1}^{}}\sqrt{{\displaystyle \sum _{i=2}^{50}{V}_{pi}^2}}$$

(1)

where V_p1 and V_pi denote the peak voltages of the fundamental frequency and of the ith harmonic frequency, respectively.

For better understanding, a test case should be considered. A testing voltage of about 40 kV (much higher than the rating voltage of 24 kV) was applied to an iron-core device, and the PD characteristic is shown in Figure 1. It was found that the testing voltage deviated from the sinusoidal waveform due to the core saturation. The DV and THD_v were higher than 5%. In addition, the background noise level was higher than 2.5 pC (i.e., the acceptable background noise for the HV and PD tests of the power cable and the voltage transformers) [3,4,5]. To avoid the voltage distortion and the background noise level, and to satisfy the standard requirements, an additional measure—such as a voltage filter—must be applied to the system.

To confirm the insulation performance of the HV equipment in HV tests, the testing voltage must be increased to a level greater than the rating voltage. The power frequency (50 Hz or 60 Hz) of the testing voltage can be used in the HV test for capacitance loads such as insulators, bushings, cables, etc. For example, in the HV and PD testing of a power cable [4] (U₀ = 12 kV) for the pre-stress condition, a voltage of 2U₀ (24 kV) is applied to the cable, and then the testing voltage is reduced to the level of 1.73U₀ (20.8 kV) to record the PD activity. Figure 2a shows the procedure of applying voltage in the PD test for the cable.

However, in the case of iron-core test objects such as voltage transformers (VTs), reactors, etc., to prevent the core saturation and distortion of the applied voltage, the applied testing voltage must have a frequency that is higher than the power frequency. In the HV and partial discharge test [5] of a VT (with a rating voltage (U_r) of 24 kV), 80% of the withstand voltage (U_t = 50 kV) is applied to the VT for the pre-stress condition, and then the testing voltage is reduced to 1.2U_r (28.8 kV) to record the PD activity. Figure 2b shows the procedure of applying voltage in the PD test. Since the testing voltage in the PD test is higher than the rating voltage, the saturation of the iron core can occur. Therefore, to avoid such saturation, a voltage with a frequency is higher than the rated frequency of the VT is employed in the test. The saturation effect of the VT influences the applied voltage distortion, as shown in Figure 1, leading to DV and THD_v of more than 5% if a voltage with a power frequency of 50 Hz is applied to the HV side of the VT. Aside from the nonlinear voltage, the core saturation can cause thermal runaway and the explosion of the PT. In order to prevent the core saturation effect, a test voltage of twice the rating voltage must have at least twice the rating frequency to prevent the magnetic flux in the iron core from exceeding the rating magnetic flux. For most iron-core devices with ratings of 50 and 60 Hz frequencies, a frequency of 200 Hz is an appropriate candidate.

Power electronics technology is now widely used in practical industries, such as home appliances, traction and transportation vehicles, renewable energy sources, and HV transmission and distribution systems. The power electronics converter is also a powerful and reliable tool for AC/DC, DC/AC, and AC/AC power conversion for HV generation in testing. Nonetheless, the interference signal caused by the fast switching of power electronic devices is the main issue with the implementation of the power converter in the PD test. Such an interference signal causes the background noise in the PD detection system to be far greater than is acceptable. For instance, the allowable PD level in VTs insulated with oil and power cables is just 5 pC [4,5]. Moreover, the acceptable background noise in the testing system for the PD tests [3] should be less than 50% of the allowable PD level (2.5 pC). However, most commercial power converters with an extra filter can remove undesirable harmonic voltage and produce a voltage waveform that is close to a pure sinusoidal wave, but the background noise in the PD test is still higher than the acceptable level in the HV and PD tests.

Lately, there have been attempts to develop the HV and PD tests of HV equipment. In [6,7,8,9], a power frequency converter based on pulse-width modulation (PWM) techniques was developed as a low-voltage source for the HV testing transformer for the HV and PD tests. It was found that the commercial and self-developed frequency converters with a voltage filter can generate an almost-pure sinusoidal voltage waveform, but the PWM switching generates a high noise level, which is higher than the acceptable level (2.5 pC) of the PD test. Therefore, an additional filter or voltage filter must be redesigned with consideration of the noise generated by the converter switching. In [10,11], the analysis of the effect of the additional filter to reduce the background noise in the HV and PD tests was presented. In [12], the analysis of a PWM converter with an additional filter was presented, and the best switching frequency in terms of the quality of the generated voltage and background noise level was investigated. The achievement of the developed system in terms of the generated voltage quality and background noise was determined. The voltage generated had a nearly pure sinusoidal waveform, the DV and THDv were less than 5%, and the background noise level was less than 2.5 pC. Nevertheless, it was found that the developed system in [12] required an additional filter with fairly large capacitance and high power consumption. In [13], a system based on a resonant power converter was introduced for HV and PD tests on voltage transformers. Instead of the PWM voltage waveform, a square wave was utilized as the input voltage of the HV testing transformer. The developed system provided promising performance in terms of the output voltage waveform quality and low background noise level. The DV, THDv, and background noise levels were lower than 2%, 3%, and 1.5 pC, respectively. The tuning and filter circuit must be implemented with the system for the achievement of the required resonant frequency. To design the proper tuning and filter circuit, the circuit parameters of the system, along with the parasitic and stray inductance and capacitance, are required. However, some parameters cannot be measured accurately in real practice. In the design process of [14], the parameters in the simulation were adjusted to obtain results that were consistent with the experimental ones. It would be advantageous for the system design if the circuit parameters could be determined accurately.

In this paper, using the preliminary experimental results, an improved Prony method [15] was applied to determine the crucial equivalent circuit parameters of the testing system. The trial-and-error approach for the determination of the unknown circuit parameters in the previous approach was not necessary. Using the proposed method, the unknown circuit parameters were determined precisely, and the appropriate resonant tuning and filter circuit was designed effectively. The transfer functions of the input and output testing voltages, along with the transfer impedance of the input voltage and signal voltage of the PD measuring port, were analyzed. The system design was verified by experiments of the HV and PD tests on a voltage transformer and a power cable. The gain of the transfer impedance was over 3 at the frequency of 200 Hz for testing the VT, and over 10 at the frequency of 50 Hz for testing the cable. The output voltage waveform was almost a pure sinusoidal function, and a background noise level of below 1 pC was achieved as per the design and the standard requirements [1,2,3,4,5]. The validity of the proposed method was confirmed by the experimental results, and the system design of the resonant converter and the approach for the equivalent circuit determination are very useful for the HV and PD tests of voltage transformers in real practice. This paper is organized into four parts: The first section is an introduction, and the second is a review of the developed system for HV and PD testing systems. In the third section, the determination of the circuit parameters of the testing system is presented. Then, the determined parameters are used for the selection of the circuit parameters to obtain the desired output testing voltage and background noise level in the HV and PD tests on the HV apparatuses. Finally, conclusions are presented in the last section.

2. HV and PD Testing System Based on the Resonant Power Frequency Converter

In Figure 3, the developed system represented by the equivalent circuit is composed of a power frequency converter, an additional inductor, an additional capacitor, an HV testing transformer, and a partial discharge detection system. The terms Z_f, C_k, CD, CC, and MI refer to the internal impedances of a testing transformer, a coupling capacitor, a coupling device (the measuring impedance), a coaxial measuring cable, and a measuring instrument, respectively.

2.1. Partial Discharge Measuring System

In the developed PD measuring system, the coupling capacitor (C_k) is connected in series with the measuring impedance, of which the equivalent circuit is depicted in Figure 4. The measuring impedance was designed with a band-pass characteristic, and the capacitance C_k was chosen to be 1 nF. Figure 5 expresses the transfer impedance characteristic with low and high cutoff frequencies of 30 kHz and 20 MHz, respectively. For avoiding the undesired noise signal in HV testing environments, the standard [3] recommends using the band-pass filter for a quasi-integration of the charge determination, and the acceptable range of cutoff frequencies of the filter is also provided [3]. In this paper, a digital band-pass filter with cutoff frequencies of 100 kHz and 400 kHz is utilized as the standard requirement. The transfer function of the filter and the transfer impedance with the considered filter are also presented in Figure 5.

2.2. Resonant Power Frequency Converter

In this paper, the power frequency converter previously developed in [6,7,8,9,10,11,12,13,14] with the H-bridge configuration, as depicted in Figure 6, was utilized in experiments for validation of the proposed method. The simple H-bridge converter with an RC snubber circuit was designed as per the recommendations in [16]. The resistance and capacitance of the snubber circuit were 39 Ω and 75 nF, respectively. The insulated-gate bipolar transistor (IGBT) type of IXXN110N65B4H1—of which the specifications [17] are V_CBS = 650 V, I_C110 = 110 A, V_CE(sat) ≤ 2.1 V, and t_fi(typ) = 85 ns—was selected for the H-bridge converter. From the voltage specification, the maximum input voltage of the converter is 1300 V, which is sufficient for the applied DC voltage from the rectifier (V_p = 537 V). The IGBTs in the converter can be programmed to produce either square-wave or pulse-width modulation (PWM) waveforms.

Based on the unipolar PWM approach [16,18,19], the operation uses four control switches (S₁₊, S₁₋, S₂₊, and S₂₋) and four diodes (D₁₊, D₁₋, D₂₊, and D₃₋), as indicated in

Table 1. Switch conditions in the full-bridge single-phase inverter.

State	Switch Conduction Status		Conduction Status of IGBT and Diode		Vo
	ON	OFF	Io > 0	Io < 0
1	S1+, S2−	S1−, S2+	S1+, S2−	D1+, D2−	+Vdc
2	S1+, S2+	S1−, S2−	S1+, D2+	D1+, S2+	0
3	S1−, S2+	S1+, S2−	D1−, D2+	S1−, S2+	−Vdc
4	S1−, S2−	S1+, S2+	D1−, S2−	S1−, D2−	0

. One of the following three voltage levels (+V_dc, −V_dc, or 0) is represented in the AC output voltage waveform.

The reference voltage signal, the controlled signal, and the output voltage waveform for the unipolar PWM approach are shown in Figure 7.

For generation of the PWM waveform, the triangular waveform frequency is set to between 1.2 kHz and 20 kHz. However, it was found in [12] that the best performance of the PWM converter occurs at a switching frequency of 3.2 kHz. In the proposed resonant circuit, the frequency of the triangular waveform is set to be the same as that of the control waveform.

In the case of the resonant converter, the resonant condition is required, and the square wave is the best choice of the generation from the converter, because the component of the square waveform at the fundamental frequency is higher than that of the PWM one. The tuning and filter circuit in Figure 3 must be implemented with the testing system, of which the equivalent circuit is expressed in Figure 8. All impedances can be transferred to the low-voltage (LV) or HV side for simplicity in the circuit analysis. For this paper, all impedances were transferred to the HV side. The adjustable inductance and the additional capacitance were selected to determine the resonant condition at the desired testing frequency. To design the proper tuning and filter circuit, the circuit parameters of the system, along with the parasitic and stray inductance and capacitance, are required. Nonetheless, some of these parameters cannot be measured accurately using a conventional impedance meter, because the equivalent circuit of the testing system cannot be represented by the simple configurations from which the meter can extract the circuit components. In the design process [13,14], the parameters in the simulation were adjusted to obtain results that were consistent with the experimental ones. It would be advantageous in the system design if the circuit parameters could be determined accurately.

From the equivalent circuit in Figure 8, the circuit can be approximated as a series resonant circuit, as expressed in Figure 9, where the equivalent resistance (R_eq) is the summation of the internal resistance (R_int) of the system, the total series resistance (R_tr) of the HV testing transformer, and the internal resistance (R_add) of the adjustable inductance. The equivalent inductance (L_eq) is the summation of the internal inductance (L_int) of the system, the total series inductance (L_tr) of the HV testing transformer, and the internal inductance (L_add) of the adjustable inductance. The equivalent capacitance (C_eq) is the total capacitance, including the effect of the stray capacitance of the system. The series resonant condition is described by Equation (2). If L_int and L_tr are known, the required additional inductance (L_add) for the required testing frequency can be calculated by Equations (2) and (3). The voltage gain at the resonant condition is equivalent to the quality factor (Q) as expressed in Equation (4).

$$L_{eq}=\frac{1}{{\omega }^2{C}_{eq}}$$

(2)

$$L_{eq}=\left(L_{int}+L_{tr}\right)+L_{add}$$

(3)

$$Q=\frac{\sqrt{L_{eq}/C_{eq}}}{R_{eq}}$$

(4)

In this paper, the required Q was set to be no less than 3, which is adequate to obtain the almost-pure sinusoidal output voltage. It should be noted that a quality factor of over 3 is confirmed to be sufficient by the simulation results in the next section. In the designed inductor, the internal resistance must be controlled to obtain the desired Q. The maximum internal resistance (R_int) can be calculated by Equation (5) if R_int and R_tr are known.

$$R_{eq}=\left(R_{\mathrm{int}}+R_{tr}\right)+R_{add}$$

(5)

From the equations above, with the conditions of the approximate C_eq of below 2 nF and the resonant frequency from 50 Hz to 200 Hz, the required adjustable inductance should be varied from 12 mH to 300 mH. This requirement was declared to the manufacturer for the design and construction of the adjustable inductors. Two variable inductors with inductance ranging from 3 mH to 30 mH and from 30 mH to 300 mH, respectively, were constructed for use in the experiments to confirm the validity of the proposed method.

3. Circuit Parameter Determination of the HV and PD Testing System

As mentioned in Section 2.2, the equivalent circuit parameters of the testing circuit must be known for the proper selection of L_add, R_add, and C_add in the tuning and filter circuit. To design the proper tuning and filter circuit, the circuit parameters of the system, along with the parasitic and stray inductance and capacitance, are required. However, some parameters cannot be measured precisely in practical ways. In the design process [13,14], the parameters in the simulation were adjusted to obtain results that were consistent with the experimental ones. It would be an advantage in the system design if the circuit parameters could be determined exactly. Therefore, in this paper, an improved Prony method [15] was applied to determine the system circuit parameters from the preliminary experimental results. After that, the determined parameters were used in the design and selection of L_add, R_add, and C_add.

3.1. Improved Prony Method

The Prony method decomposes the waveform in terms of multiple real or complex exponential functions. However, the method is very sensitive to noise. To overcome this problem, the improved Prony method was developed. In the improved method, the integration of the considered waveform is employed for the accurate determination of the exponent terms of the considered waveform. In [15], the improved Prony method was proposed to determine the base curve of the full lightning impulse voltage and current waveforms, in the form of two real or complex exponential functions. From the results of the test cases provided by the standard [20], the performance of the method in terms of accuracy and computational time in the waveform parameter determination has been confirmed to be comparable with the standard recommended method. The derivation of the method is repeated here for better understanding.

For clarifying and considering the waveform (y(t)), which is fitted well with two exponential functions (f(t)) as given in Equation (6), the definite integration of the waveform from 0 to t can be fitted well with two exponential functions and a constant term, as expressed in Equation (7), where A₁, A₂, and A₃ are constant coefficients. This can be expressed in the discrete-time domain as shown in Equation (8), where the ith exponential term ($e^{{\alpha }_i\Delta t}$) is equal to r_i. It should be noted that α₁ and α₂ can be real or complex numbers.

$$y(t)\approx f(t)=a{e}^{{\alpha }_{1}t}+b{e}^{{\alpha }_{2}t}$$

(6)

$${\displaystyle \underset{\tau =0}{\overset{\tau =t}{\int }}f(t)d\tau }=g(t)=A_1{e}^{{\alpha }_{1}t}+A_2{e}^{{\alpha }_{2}t}+A_3$$

(7)

$$g(n\Delta t)=A_1{e}^{{\alpha }_{1}n\Delta t}+A_2{e}^{{\alpha }_{2}n\Delta t}=A_1{\left(e^{{\alpha }_1\Delta t}\right)}^n+A_2{\left(e^{{\alpha }_2\Delta t}\right)}^n+A_3$$

(8)

One of the solutions to the homogeneous difference equation in Equation (9) is the sequence in Equation (8), where C is a constant.

$$f[n+2]-(r_1+r_2)f[n+1]+(r_1{r}_2)f[n]=C$$

(9)

Because there are many more data points than there are unknown coefficients for r₁ + r₂ and r₁r₂ in Equation (9), a linear least squares regression can be used to compute these coefficients. The parameters of r₁ and r₂ can be calculated from the solution (r) of the auxiliary equation provided in Equation (10).

$$z^2-(r_1+r_2)z+(r_1{r}_2)=0$$

(10)

It is simple to calculate the exponential terms in Equation (7) once the r₁ and r₂ parameters are known. Additionally, linear least squares regression can be employed to calculate the coefficients in Equation (7) from the considered waveform.

3.2. Application of the Improved Prony Method for the Circuit Parameter Determination

For the application of the improved Prony method for the circuit parameter determination, the square waveforms with low frequency and amplitude were utilized in the testing system with variable additional inductance. The output waveforms from the HV side of the HV testing transformer were recorded. In the first case, there was no additional inductance, and in the second case, the inductor with inductance of 5 mH and internal resistance of 1.3 Ω was connected at the input port of the system. It should be noted that the internal resistance of the adjustable inductor is almost constant, because the inductance is adjusted by adjusting the air gap of the iron core of the inductor. Considering the equivalent circuit in Figure 8 and Figure 9, the circuit is a second-order RLC circuit. The response of the signal at the HV side of the HV testing transformer can be calculated in the form of a second-order differential, as presented in Equation (11), and it can be rewritten in another form with the exponent terms of the solution in Equation (12). The solutions of Equations (11) and (12) can be represented well as a damped sinusoidal function or a two-complex exponential function, as shown in Equation (13).

$$\frac{d^2{V}_{out}(t)}{d{t}^2}+\left(\frac{R_{eq}}{L_{eq}}\right)\frac{d{V}_{out}(t)}{dt}+\left(\frac{1}{L_{eq}{C}_{eq}}\right)V_{out}(t)=C$$

(11)

$$\frac{d^2{V}_{out}(t)}{d{t}^2}+\left({\alpha }_1+{\alpha }_2\right)\frac{d{V}_{out}(t)}{dt}+\left({\alpha }_1{\alpha }_2\right)V_{out}(t)=C$$

(12)

$$V_{out}(t)=A{e}^{-\alpha t}\sin (\omega t+\varphi )+B=A_1{e}^{{\alpha }_{1}t}+A_2{e}^{{\alpha }_{2}t}+A_3$$

(13)

Therefore, the improved Prony method can be employed to determine the waveform parameters. Figure 10 shows the response voltages when the square-wave voltages were taken as the input voltages of the system. As shown in Figure 11, the integral waveforms of the measured waveform in a period associated with a frequency of 50 Hz when removing the DC component were selected for the determination of the circuit parameters using the improved Prony method.

In the first and second cases, the circuit parameters and the exponent terms in Equations (14)–(17) have relations as expressed in Equations (14) and (15), where α_i^(j) is the ith exponent term of the jth case, R_s is a summation of R_int and R_tr, and L_s is a summation of L_int and L_tr.

$$\frac{R_s}{L_s}={\alpha }_1{}^{(1)}+{\alpha }_2{}^{(1)}$$

(14)

$$\frac{1}{L_s{C}_{eq}}={\alpha }_1{}^{(1)}{\alpha }_2{}^{(1)}$$

(15)

$$\frac{R_s+R_{add}}{L_s+L_{add}}={\alpha }_1{}^{(2)}+{\alpha }_2{}^{(2)}$$

(16)

$$\frac{1}{\left(L_s+L_{add}\right)C_{eq}}={\alpha }_1{}^{(2)}{\alpha }_2{}^{(2)}$$

(17)

The exponent terms can be calculated by the improved Prony method, and R_add and L_add can be readily measured. From Equations (14) to (17), L_s, R_s, and C_eq can be calculated by Equations (18)–(20).

$$L_s=\frac{L_{add}}{\left(\frac{{\alpha }_1{}^{(2)}{\alpha }_2{}^{(2)}}{{\alpha }_1{}^{(1)}{\alpha }_2{}^{(1)}}-1\right)}$$

(18)

$$R_s=\left({\alpha }_1{}^{(1)}+{\alpha }_2{}^{(1)}\right)L_s$$

(19)

$$C_{eq}=\frac{1}{L_s\left({\alpha }_1{}^{(1)}{\alpha }_2{}^{(1)}\right)}$$

(20)

With the application of the improved Prony method to the waveforms in Figure 9, the exponents can be calculated, and α₁⁽¹⁾, α₂⁽¹⁾, α₁⁽²⁾, and α₂⁽²⁾ are 1.8952 × 10³ + j4.2783 × 10³, 1.8952 × 10³ − j4.2783 × 10³, 5.1858 × 10² + j2.1045 × 10³, and 5.1858×10² − j2.1045 × 10³, respectively. With the known circuit parameters (i.e., R_add of 34.56 kΩ and L_add of either 0 or 132.92 H on the HV side), and by transferring all impedance to the HV side, it can be determined by Equations (18)–(20), and L_s, R_s, and C_eq are 36.310 H, 137.63 kΩ, and 1.257 nF, respectively. The fitting curve using the determined parameters was in good agreement with the measured waveforms, as also shown in Figure 12.

3.3. Proper Selection of the Circuit Parameters of the Tuning and Filter Circuit

The objectives of the tuning and filter circuit are for tuning the additional inductance to obtain the required resonant frequency for the HV test, as well as for filtering the noise signal generated by power electronic switching. With the accurate circuit parameters of the HV and PD testing system, the proper selection of the circuit parameters of the tuning and filter circuit can be selected effectively.

For the first objective, the required testing frequency is from 50 Hz to 200 Hz. Using Equations (2) and (3), L_add can be calculated by Equation (21). With the required testing frequency, L_add should be adjustable from 360 H to 8000 H on the HV side (13.52 mH to 300 mH on the LV side). The inductance range (3 mH to 300 mH on the LV side) is consistent with the preliminary selection in Section 2.2. With the known R_s and R_add of the gain of the transfer function of the input and output voltage, the gain can be calculated by Equation (4) and plotted with the required frequency, as shown in Figure 13.

$$L_{add}=\frac{1}{{\omega }^2{C}_{eq}}-\left(L_{int}+L_{tr}\right)$$

(21)

The lower the required frequency, the higher the gain. At frequencies of 50 Hz, 150 Hz, and 200 Hz, the gain is about 15, 5, and 3.5, respectively. It should be noted that the testing frequency of 50 Hz is utilized in the testing of power cables, and the proper testing frequency is in the range of 150 Hz to 200 Hz, as utilized in most VT testing cases. In Figure 14, the normalized transfer functions (V_out/V_in) in the frequency domain with different additional inductances, as shown in

Table 2. The additional inductances and resistances.

Case	Transfer to the LV Side		Transfer to the HV Side
	Radd	Ladd	Radd	Ladd
1	0 Ω	0 mH	0 kΩ	0 H
2	1.3 Ω	5.0 mH	34.56 kΩ	132.92 H
3	1.3 Ω	17.2 mH	34.56 kΩ	457.23 H
4	1.8 Ω	31.5 mH	47.85 kΩ	837.37 H
5	1.8 Ω	300 mH	47.85 kΩ	7974.95 H

, are presented. The gains in Figure 13 and Figure 14 are in good agreement.

For the second objective, the additional capacitor is connected to the system for filtering the noise signal, which interferes with the PD detection. Considering the equivalent circuit in Figure 8, the parameters associated with the resonant frequency of 200 Hz and all circuit parameters in Figure 15 are transferred to the HV side of the testing transformer, the transfer function in the form of the attenuation factor (V_PD/V_in) can be determined as per the results in Figure 15a, and the attenuation factor with the band-pass filtering recommended by the standard are expressed in Figure 15b. The filter according to the standard requirement [3] in this paper is a band-pass type with lower and higher cutoff frequencies of 100 kHz and 400 kHz, respectively. It was found that the additional capacitance (C_add) of 1 μF associated with the maximum attenuation factor of below 10⁻⁸ is sufficient for attenuation of the interference signal in the PD measurement. This is also confirmed by the results in the time domain. Figure 16 shows the input square wave of 19.6 kV transferred to the HV side (the red line), and the output voltage of 100 kV (the black line). Figure 17 shows the comparison of the PD calibrated signal of 1.0 pC (the black lines) and the interference signal of the output peak voltage of 100 kV, or about 70 kV_rms (the blue lines). It can be seen that the interference signal level is lower than that of the calibrated signal.

4. Experiments

As shown in the experimental setup in Figure 18, some tests were carried out to see how well the designed system performed in the HV and PD tests in terms of the output voltage quality and background noise. The testing system is composed of a test object (i.e., a VT or a power cable), an HV testing transformer, a PD detection system, an additional capacitor, an adjustable inductor, and the developed converter. The performance of the developed system was investigated in terms of the different voltage, the total harmonic distortion of the output voltage, and the background noise level.

A commercial EO/OE converter (Omicron) [21] with fiber-optic cables was used with the developed system to transmit the measured PD signal and prevent interference signals from electromagnetic coupling in the HV testing area of the HV laboratory, which was shielded from external electromagnetic fields by metallic fences and ground plane. In the calibration process of the PD test, a standard PD current with a charge of 5 pC was employed. Without the operation of the converter, the background noise was approximately 0.8 pC.

For presenting the problem of interference generated by the typical power frequency converter with the pulse-width modulation topology, the PWM converter was applied to the testing system. As shown in Figure 19, and at a testing voltage of about 14.4 kV, the noise level of 162.4 pC was very high—much higher than the acceptable level of 2.5 pC.

To represent the performance of the developed system using the selected parameters, a resonant converter with additional inductance of 17.2 mH (LV side) was set to obtain the resonant frequency of 200 Hz. The experiments without a test object were performed to investigate the voltage quality and background noise level. The standard PD pulse with a charge of 5 pC was used for calibration in the PD test. When the additional and adjustable inductor was connected to the LV side of the HV transformer but there was no additional capacitor connected to the system, at only the testing voltage of about 5 kV, the background noise reached the acceptable level (2.5 pC), as shown in Figure 20a. When the additional capacitor was connected to the system, the background noise level was reduced significantly. At the pre-stress testing voltage of 40 kV_rms, the noise level was only 0.77 pC, as shown in Figure 20b. At the output voltage of 40 kV_rms, the DC input voltage supplied to the converter was about 83 V, corresponding to a voltage gain of 4.5. This is consistent with the simulation results. The different voltage and THD_v were 0.61% and 0.35%, respectively.

In the second experiment, a voltage transformer was tested. The frequency of the input voltage was 200 Hz for examining the voltage gain (V_out/V_in), the quality of the testing waveform, and the background noise level. The voltage was raised to 40 kV_rms for the pre-stress condition, and decreased to 28.8 kV_rms for recording PD activity. In the calibration process for the PD test, the standard PD pulse of a 5 pC charge was employed. The experimental testing voltage of about 30 kV_rms is shown in Figure 21. The voltage gains at 30 kV_rms and 40 kV_rms were almost the same value of 3.8, which is consistent with the simulation results in Figure 13 and Figure 14. The difference voltage (peak voltage/√2 = 28.98 kV and RMS voltage 28.79 kV) was 0.65%, the THD_v was 0.35%, and the background noise was 0.871 pC.

In the final experiment, a power cable was tested. The frequency of the input voltage was 50 Hz for examining the voltage gain (V_out/V_in), the quality of the testing waveform, and the background noise level. The voltage was raised to 24 kV_rms for the pre-stress condition, and decreased to 20.8 kV_rms for recording PD activity. A 5 pC PD pulse was also employed in the calibration process. The experimental testing voltage of about 21 kV_rms is shown in Figure 22. The voltage gains at 21 kV_rms and 24 kV_rms were almost the same value of 12.8, which is fairly consistent with the simulation results in Figure 13 and Figure 14. The difference voltage (peak voltage/√2 = 20.92 kV and RMS voltage 20.85 kV) was 0.56%, the THD_v was 0.32%, and the background noise was 0.688 pC.

5. Conclusions

An effective approach for the circuit parameter determination of the HV and PD testing system for testing voltage transformers was developed. The testing system is composed of a single-phase power frequency converter with a resonant tuning and filter circuit and an HV testing transformer. The unknown testing system circuit parameters—including unknown parasitic inductance, capacitance, and internal resistance—were determined accurately. From the determined circuit parameters, the appropriate resonant tuning and filter circuit was designed effectively. The transfer functions of the input and output testing voltages, along with the transfer impedance of the input voltage and signal voltage of the PD measuring port, were analyzed. The system design was verified by experiments with tests on a voltage transformer and a power cable. From the simulation and experimental results, an output voltage waveform of almost-pure sinusoidal function was achieved. The difference voltage, THD_v, and background noise were less than 1%, 0.5%, and 1.0 pC, respectively. The gain of the transfer impedance was about 4 for the voltage transformer test at a frequency of 200 Hz, and about 15 for the cable test at a frequency of 50 Hz, and a background noise level of below 1 pC was achieved with the design. From the experimental results, the system design of the resonant converter and the approach for the equivalent circuit determination are very useful for the HV and PD tests of voltage transformers in real practice.