A method for tracing impulse voltage peak value using weighted frequency components

This paper studies a method for synthesizing impulse voltage peak error based on frequency energy weight. The research designed three capacitive voltage dividers with a voltage rating of 1 kV, 10 kV, and 200 kV, respectively, and measured their response characteristics using the step wave response test. The scale factor and linearity of the 1 kV capacitor divider were measured at different frequencies and impulse voltages to verify the equivalence of voltage coefficients under power frequency and impulse. Using boundary conditions obtained from AC voltage standard sources and impulse voltage standard generators, the equivalence of 200 kV power frequency voltage coefficient and impulse voltage coefficient was derived.


Introduction
The impulse voltage is a high-voltage signal with a short duration and a very rich frequency content ranging from 10 Hz to several MHz.For measuring devices, the impact of distributed parameters on measurement errors and response characteristics is significant.Therefore, it is not possible to use the additive principle to determine voltage coefficients as with DC and power frequency voltages.[3] Therefore, the traditional traceability method is to use an impulse voltage standard generator below 1 kV as the source of the peak value of the impulse voltage. This article proposes a method for evaluating the measurement error of impulse voltage based on multi-frequency component weighted iteration.A voltage divider device based on an upright shielded gas standard capacitor is used as the transfer standard to realize the traceability of the voltage peak error under high impulse voltage.

Analytical Representation of Impulse Voltages
According to the source of the standard instrument, wideband testing methods can be divided into those based on a standard AC/DC source and those based on a standard current shunt.This paper adopts the latter for the research on wideband testing methods.
The waveforms of one-time and continuous signals can be represented in the time domain or by their spectrum in the frequency domain.Both forms of representation are equivalent. Specifications for the correct measurement of a signal can be derived from the waveform in the time domain as well as from the spectrum in the frequency domain.0] The double exponential function below is best suited for theoretical investigations with respect to the waveform and the spectrum of switching and lightning impulse voltages or of the transfer behavior of voltage dividers: 0 ( ) where k is a coefficient with which both the exponential functions are to be multiplied so that the impulse voltage attains its peak value at the peak time; α and β are the time constants, and U 0 is the peak value.
For lighting and switching impulses, β> α, which means that the waveform of the impulse voltage in the tail region is affected predominantly by the exponential term with α in Equation (1).
Performing a Fourier transform on the function yields the frequency spectrum of the lightning impulse, and the amplitude spectrum is given by the modulus of Equation ( 1).As shown in Figure 1, it can be seen that the lightning full-wave is formed by the superposition of waveforms of different frequencies, each with its own amplitude height.The amplitude of the lightning full wave is mostly concentrated in the low-frequency range, so it cannot be regarded as a purely high-frequency signal.
Figure 1 shows the amplitude and phase density of the impulse voltage referred to the corresponding DC component.The amplitude density of any voltage impulse is nearly constant up to a limiting frequency determined by the impulse form and then decreases more or less fast with increasing frequency.

Error Synthesis Method
According to Parseval's theorem, the energy of a signal is equal in both the time and frequency domains.Therefore, the weight of each frequency band can be calculated using the spectral information of the lightning full-wave impulse voltage waveform.
 A Fourier transform of the lightning full-wave impulse voltage is conducted to extract its spectral information. The spectrum information is divided into several frequency bands, which can be customized based on the required analysis. The power of each frequency band (i.e., the sum of the squares of amplitudes corresponding to all frequencies within the band) is calculated and the weight of each frequency band is determined based on the total power.It is important to note that due to the unique properties of the lightning full-wave impulse voltage waveform, its spectrum typically exhibits an exponential decay shape.As such, it may be necessary to apply special processing to the frequency band division and weight calculations to achieve more precise results.
Since the frequency spectrum of lightning full-wave impulse voltage typically exhibits an exponential decay shape, direct division of the spectrum into equally spaced frequency bands may overestimate some frequency bands and underestimate others.To solve this problem, logarithmic scales can be used to divide the frequency spectrum, i.e., the frequency axis is marked according to a logarithmic scale, and the spectrum is divided into several logarithmic frequency bands according to actual needs, with the weight of each frequency band calculated accordingly.This approach can better reflect the characteristics of energy concentration in the low-frequency band, resulting in more accurate results.
Assuming there are two lightning full-wave impulse voltage waveforms, with known amplitude and phase differences at different frequencies.The amplitude and phase differences between the two waveforms in each frequency band are calculated, then they are multiplied by the weight of that frequency band.Finally, the weighted errors of all frequency bands are added up to obtain the total error of the two waveforms.
Specifically, according to the properties of definite integrals, the energy of each frequency band is equivalent to the area enclosed by the corresponding frequency curve and the coordinate axis, namely the trapezoidal area formed by two frequency points on the amplitude-frequency curve and the x-axis.As the Figuire 2 shows, the frequency range is divided into the following frequency bands based on the sparseness of lightning spectra: below 1 kHz with intervals of 10 Hz, above 1 kHz with intervals of 1 kHz, and above 1 MHz with intervals of 1 MHz; frequency components above 100 MHz are ignored.The weights of different frequency bands are calculated as follows: where Wi is the total energy of each frequency band, F (i) and F (i-1) are the amplitudes of the two frequency endpoints, and Δf is the total frequency between the two frequency endpoints.Ai is the percentage of energy for each frequency band, also known as weight.
where f1 (t) and f2 (t) are the two impulse voltage waveforms.
According to the Parseval theorem, the ratio of peak value errors between two different impulse voltage waveforms can be expressed as Equation ( 5).

Standard Capacitor Divider
In order to realize the source tracing of the peak impulse voltage, it is necessary to establish a voltage divider based on an upright fully shielded standard capacitor for 1 kV, 10 kV and 200 kV. Figure 3 shows the high voltage capacitors of the 1 kV, 10 kV and 200 kV.The electrode system consists of a high-voltage electrode, a low-voltage electrode and a shielding electrode.The structures of the three high-voltage standard capacitors are the same, the materials and surface roughness requirements for each component are the same, and the distributed parameters and electric field distribution trends are consistent.
Figure 4 shows the distributed parameter equivalent circuit of a standard capacitor voltage divider.L 1 represents lead inductance, L 2 represents high-voltage rod inductance, L 3 represents transmission cable inductance, R 1 represents external damping resistance, R 2 represents internal damping resistance, R 3 represents matching resistance, C 1 represents lead capacitance, C 2 represents equalizing ring-toground capacitance, C 3 represents high-voltage electrode-to-shield capacitance, C 4 represents highvoltage electrode-to-enclosure capacitance, C 5 represents high-voltage rod-to-ground capacitance, C 6 represents low-voltage electrode-to-shield capacitance, C  According to Equation ( 5) and the calculation process in Figure 5, a calculation model for the measurement error of the peak impulse voltage can be established. represents the combined error of amplitude error and phase error, as shown in Equation ( 6). ( The scale factor and linearity of a 1 kV capacitor divider at different frequencies and impulse voltages are measured using AC voltage standard sources and impulse voltage standard generators, and the equivalence of voltage coefficients under power frequency and impulse is verified.Boundary conditions such as insulation, distributed capacitance, damping coefficient, and waveform of the measured voltage for the 1 kV and 10 kV capacitor dividers are obtained by using a 10 kV power frequency voltage standard generator, a 10 kV harmonic voltage standard generator, and a 10 kV impulse voltage standard generator.Based on the boundary conditions and measurement results of the scale factor and impulse scale factor at 10 kV power frequency, the equivalence of 200 kV power frequency voltage coefficient and impulse voltage coefficient is derived.
The 1 kV standard capacitor and the 10 kV standard capacitor have the same structure, use the same materials for their components, and have the same surface roughness requirements.They have consistently distributed parameters and electric field distribution trends.Therefore, their transfer characteristics are equivalent, meaning that the voltage coefficients of impulse voltage and power frequency voltage in the 1 kV standard capacitor can be extended to the 10 kV standard capacitor.Similarly, the equivalence of the 200 kV standard capacitor divider is also valid.This validates the conclusion that the impulse calibration and linearity of the device based on the upright full-shielded standard capacitor can be traced back to the power frequency voltage.

Conclusions
This paper proposes a traceability method for impulse voltage peak value.By using the 1 kV and 10 kV capacitive voltage dividers based on fully-shielded standard capacitors as carriers, the measurement error of impulse voltage peak value is traced back to the measurement error of multifrequency sinusoidal voltage.The equivalence between the impulse voltage coefficient and the AC voltage coefficient at different frequencies is verified using the 1 kV voltage divider.By solidifying the distributed parameters of the voltage dividers, an electrical circuit model of the dividers is established, and equivalent boundary conditions for 1 kV and 10 kV are obtained and verified by impulse testing.This method can achieve high-voltage traceability of impulse voltage peak value and greatly reduce the uncertainty components introduced by the linearity of the standard measurement system.

Figure 1 .
Figure 1.Spectrum characteristics of a lightning full wave.
7 represents high-voltage capacitance, C 8 EEICE-2023 Journal of Physics: Conference Series 2625 (2023) 012049 represents low-voltage capacitance, C 9 represents transmission cable capacitance, and C 10 represents input capacitance of the digital recorder.Validation of the measurement model.The response characteristics of 1 kV, 10 kV and 200 kV capacitor voltage dividers were obtained by means of a step waveform response test.Standard AC voltage waveform data of different frequencies can be constructed using mathematical analysis, and the error between the output voltage and the input voltage at different input AC frequencies can be calculated using the convolution method.