Circuit Models of Q(t) Data and Analyses of Saturation Time Dependency on Delay Parameters

In recent years, there has been a considerable increase in the use of DC power equipment in the industry, making it crucial for insulation materials to withstand high DC electric fields. The current integration method (Q(t) measurement) is one of the most useful and effective methods to measure the time dependency of charges. However, the mathematical/physical model of Q(t) measurement is not advanced because obtained data are not completely analyzed, particularly the methods in which delay parameters affect the charge saturation time of insulators. In this study, we analyze the delay property of Q(t) data by solving differential equations with some approximations. In addition, sample sheets of polyimide and polyethylene terephthalate are tested for experimental purposes. The results indicate that circuit properties can be estimated from the analysis of experimental Q(t) data based on the assumption of a circuit model. The proposed analysis method can be used for further discussions regarding the analytical estimation of timing of charge saturation, and the relationship between circuit models and the space charges phenomenon.

SECTION I. Introduction

In recent years, an increasing number of electric power devices has been associated with high DC voltages, high electric fields, and complex environments. Therefore, the reliability and durability of electric power equipment need to be improved. These circumstances have increased the requirement for research on the diagnoses of electrical insulators.

Q(t) measurement is a diagnostic technique used for electrical insulators in power devices, cables, and other equipment [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. Q(t) measurement was first demonstrated in the 1970s by Takada et al. [1], and the mechanism and structure of the circuit are almost the same as those of a leakage current detector, which can detect an extremely small electric current that indicates the incompleteness and/or the defects of electrical insulation inside insulators. In contrast to partial discharge (PD) diagnosis [17], [18], whose main target is oscillating voltage, Q(t) measurement applies to DC voltage.

Several studies have been reported on Q(t) measurement. Hanazawa et al. demonstrated the relationship between the temperature and leakage current around a power device [4], [5]. Wang et al. applied Q(t) measurement to detect water treeing [6], Wang et al. [7], Fujii et al. [8], [9], and Iwata et al. [10] demonstrated that the leakage current increases with electrical treeing. Uehara et al. [14] demonstrated transition of the ratio of charge amount during Q(t) measurement of several polymeric materials at both room temperature and high temperature. Kadowaki et al. [15] demonstrated simultaneous measurements of both Q(t) and space charges.Sekiguchi et al. [16] demonstrated the correlation of time-temperature and time-electrical field.

On the contrary to partial discharge [17], [18], few studies have considered the circuit models. Previous research assumes the circuit model shown in Fig. 1(a). This is because the main purpose of previous studies has been verifying the difference between samples or their treatment conditions. However, to analyze the results of Q(t) measurement and compare other experimental/simulation results, for example, the pulse electro-elastic acoustic emission method (PEA) [19], or quantum chemical calculation and molecular dynamics simulation (QM/MD), a mathematical model of Q(t) measurements and its analysis are required. Moreover, mathematical/physical models that describe the precise separation of the transition from absorbed to leakage current based on experimental results do not exist, though the separation is considered a merit of using Q(t) measurement [14], instead of other leakage current detection methods. This situation indicates substantial risks in actual diagnoses owing to the lack of proper understanding/treatment of experimentally obtained values from Q(t) measurement.

FIGURE 1.

(a) Circuit model from a previous study, (b) a typical experimental result, and (c) the difference between an experimental result and the circuit model of (a).

Therefore, as the first step in our research, we focus on the delay property of Q(t) measurement. Fig. 1(b) presents a typical result pattern of Q(t) measurements. To analyze such data, a previous study [10] used the Q(t) circuit model represented in Fig. 1(a); However, it does not contain the delay property. Fig. 1(c) shows a schematic representation of a comparison of typical experimental data of Q(t) measurements and simulation data created from the formula of the model depicted in Fig. 1(a) (note that the position of the start time (t = 0) of the two plots shown in Fig. 1(c) are slightly shifted intentionally to visually distinguish the steep rising process of each plot). For the steep rising of the initial charging around t = 0 s and the gradient of linear part after t = 200 s, the simple circuit representation of Fig. 1(a) matches the experimental results well. However, this circuit representation loses information of the time constant that indicates the transient process between the steep rising and the linear part. Thus, detailed analyses of the difference between the absorbed and leakage currents have not been performed from the results of the Q(t) measurement.

In this study, we propose a new circuit model and compare it with the previous one. First, we set up and solve a differential equation from a circuit model of Q(t) measurements to obtain the main parameters as a parallel component of one capacitor and one resistor (the same model as that shown in Fig. 1(a)), and then to similarly obtain the delay parameters as a parallel component of one resistor and a series capacitor–resistor connection (the new model). Second, using the solution and its approximation, an estimation method of the parameters is introduced and numerical testing is conducted. The analysis reveals a relation between the charges contained inside the Q(t) meter and the ones estimated inside the insulation material. Third, regressions of experimental data obtained from Q(t) measurements of polyimide (PI) and polyethylene terephthalate (PET) sheet samples are conducted, and the performance of this estimation method is evaluated. Finally, from the results, the physical meaning and practical use of Q(t) measurement are discussed.

SECTION II. Methods

The outline of the methods is as follows: First, the circuit model is established. Second, the differential equation of the model is solved. Third, parameters are estimated. Fourth, numerical tests are conducted. Finally, Q(t) measurement and regressions to experimental data are performed.

A. Circuit Model and the Ordinary Differential Equation of the Single-Condenser and Single-Resistor Model

Hereinafter, the single-condenser-and-single-resistor-circuit model of Q(t) measurement, which is shown in Fig. 1(a), is referred to as Model $1. C_{0}$ and $C_{3}$ denote the capacitances of the integral condenser and a material, respectively, and $V_{0}$ denotes the DC voltage. $R_{0}$ indicates the safety resistor for the circuit, and $R_{1}$ indicates the main resistance of the material. Using Kirchhoff’s laws, the differential equation to be solved is derived as follows:\begin{align*}& V_{0} -\frac {Q_{3}\left ({t }\right)}{C_{3}} - \frac {Q_{0}\left ({t }\right)}{C_{0}} - I_{0}R_{0} = 0 \tag{1}\\ {}\smash {\left \{{\vphantom {\begin{matrix}.\\.\\.\\.\\.\\.\\ \end{matrix}}}\right.}& \frac {Q_{3}\left ({t }\right)}{C_{3}} = I_{1}R_{1} \tag{2}\\ & I_{1}+I_{3} = I_{0} \tag{3}\end{align*} Here, $Q_{0}(t)$ and $Q_{3}(t)$ represent the charges inside the capacitance with respect to each of the above $C_{0}^{}$ and $C_{3}$ . $I_{0}$ , $I_{1}$ , and $I_{3}$ indicate currents through $R_{0} (C_{0})_{,}R_{1}, $ and $C_{3}$ , respectively. The model is assumed to represent an insulator with the simplest charge accumulation mechanism (e.g., voids). Though there are limitations in dividing the parameters (for example, insulators, the gap of electrodes and the effect of surface roughness cannot be separated), the model is chosen primarily owing to its outlook and simplicity.

Equations (1), (2), and (3) can be written as the following time-dependent differential equation:\begin{align*} \frac {d^{2}Q_{3}\left ({t }\right)}{dt^{2}}{C_{0}R}_{1}+\frac {dQ_{3}\left ({t }\right)}{dt}\left ({\frac {{{C_{0}R_{0}+C}_{0}R}_{1}+{C_{3}R}_{1}}{{C_{3}R}_{0}} }\right) \\ +\frac {Q_{3}\left ({t }\right)}{C_{3}R_{0}} &= 0 \\{}\tag{4}\\ \therefore \frac {d^{2}Q_{3}(t)}{dt^{2}} +A_{1}\frac {dQ_{3}\left ({t }\right)}{dt} +B_{1}Q_{3}\left ({t }\right) &= 0 \\{}\tag{5}\end{align*} Here, $A_{1}=\left ({R_{0} C_{0}+R_{1} C_{0}+R_{1} C_{3}}\right) / R_{0} R_{1} C_{0} C_{3}$ , $B_{l}=1 / R_{0} R_{l} C_{0} C_{3}$

In a normal case, $Q_{3}(t)$ contains two exponential functions, such as:\begin{equation*} Q_{3}\left ({t }\right) = k_{0}\left \{{ \mathrm {exp(}\alpha _{1}t\mathrm {) - exp(}\beta _{1}t) }\right \} \tag{6}\end{equation*} $Q_{0}(t)$ is defined as:\begin{align*} Q_{0}(t) & = C_{0}V_{0}{-{ C}_{0}R_{0}k}_{0}\left ({D +\alpha _{1} }\right)\mathrm {exp}\left ({\alpha _{1}t }\right) \\ &\quad \mathrm {-(}D+\beta _{1}\mathrm {)exp(}\beta _{1}t\mathrm {)\} } \tag{7}\end{align*} Here, $\alpha _{1} \mathrm {= }\frac {-A_{1}\!+\!\sqrt {A_{1}^{2} \mathrm {- 4}B_{1}} }{2}, \beta _{1}\mathrm { = }\frac {-A_{1}\!-\!\sqrt {A_{1}^{2} \mathrm {- 4}B_{1}} }{2}$ , $k_{0} = \frac {V_{0}}{R_{0}\sqrt {A_{1}^{2}\mathrm { - 4}B_{1} } }$ , and $D$ = ($R_{0}+R_{1}) / R_{0}R_{1}C_{3}$ .

$\vert \alpha \vert \mathrm {\ll }\vert \beta \vert $ , and $\alpha $ and $\beta $ are normally real values.

B. Circuit Model and Ordinary Differential Equation of Single-Condenser-Single-Resistor and Single-Resistor Model

The new circuit model of the Q(t) measurement (Model 2) is illustrated in Fig. 2. $C_{0}^{}$ and $C_{2}$ denote the capacitances of the integral condenser and an insulation material, respectively, and $V_{0}$ denotes the DC voltage. $R_{1}$ denotes the main resistance of the material, and $R_{2}$ denotes the delay resistance ($R_{1} \gg R_{2}$ ). Note that $R_{0}$ is omitted in the modification from Fig. 1 to Fig. 2. $R_{0}$ is omitted for two reasons. First, $R_{0}$ is necessary in Fig. 1 and not necessary in case of Fig. 2 as the circuit safeties. In Fig. 1, if $R_{0}$ does not exist, it causes a short circuit at an early stage just after applying the voltage because $C_{3}$ is initially equal to a conductor component in circuit calculation. However, in case of Fig. 2, the circuit does not become short circuit owing to $R_{1}$ and $R_{2}$ . Moreover, the values of $R_{1}$ and $R_{2}$ are much larger than $R_{0}$ , so that $R_{0}$ can be omitted as shown in Fig. 3.

FIGURE 2.

Circuit model containing main resistance ${R} _{1}$ of the material, a series circuit consisting of material capacitance ${C} _{2}$ , delay resistance ${R} _{2}$ , and the integral capacitor of the Q(t) meter.

FIGURE 3.

Circuit model approximation about omitting a safety resistor in model 2.

Using Kirchhoff’s laws, the differential equation can be expressed as follows:\begin{align*}& V_{0} -{ I}_{2}R_{2} - \frac {Q_{2}\left ({t }\right)}{C_{2}} - \frac {Q_{0}\left ({t }\right)}{C_{0}}\mathrm { = 0} \tag{8}\\ {}\smash {\left \{{\vphantom {\begin{matrix}.\\.\\.\\.\\.\\.\\ \end{matrix}}}\right.}& I_{2}R_{2} + \frac {Q_{2}\left ({t }\right)}{C_{2}} = I_{1}R_{1} \tag{9}\\ & I_{1} +{ I}_{2} = I_{0} \tag{10}\end{align*} Here, $Q_{0}(t)$ and $Q_{2}(t)$ represent the charges inside the capacitance with respect to each of the above $C_{0}$ and $C_{2}$ . Note that $I_{0}$ , $I_{1}$ , and $I_{2}$ indicate the currents through $C_{0}, R_{1}, $ and $R_{2}$ .

Model 2 is assumed to behave like an insulator with voids, and it also contains a charge-delaying parameter. Though limitations exist in dividing the parameters (for example, insulators, the gap of electrodes, and the effect of surface roughness cannot be separated), the model is chosen primarily owing to its outlook and simplicity in mathematical analysis. The detailed discussion of the assumed model is presented in Section III-C.

In a normal case, $Q_{2}(t)$ has two exponential functions:\begin{align*} &\frac {d^{2}Q_{2}(t)}{dt^{2}}R_{2}C_{0} + \frac {dQ_{2}\left ({t }\right)}{dt}\left ({\frac {C_{0}}{C_{2}}+\frac {R_{1}+R_{2}}{R_{1}} }\right) +\frac {Q_{2}\left ({t }\right)}{R_{1}C_{2}} \\[1pt]& = 0 \tag{11}\\ \therefore &\frac {d^{2}Q_{2}(t)}{dt^{2}} + A \frac {dQ_{2}(t)}{dt} + BQ_{2}(t) \\ & = 0 \tag{12}\\[1pt] &Q_{2}\left ({t }\right) \\ & = F_{1}\exp \left ({\alpha t }\right)-{ F}_{1}\exp {\left ({\beta t }\right).} \tag{13}\end{align*} $Q_{0}(t)$ also contains two exponential functions as followings:\begin{equation*} Q_{0}\left ({t }\right) = G_{1}\exp \left ({\alpha t }\right)+G_{2}\exp \left ({\beta t }\right)+C_{0}V_{0}. \tag{14}\end{equation*} Here, $F_{1}=\frac {1}{(\alpha -\beta)} \frac {V_{0}}{R_{I}} $ , \begin{align*} G_{I}&=-C_{0} F_{l}\left ({\frac {1}{C_{2}}+\alpha R_{I}}\right), G_{2}=C_{0} F_{l}\left ({\frac {1}{C_{2}}+\beta R_{I}}\right), \\ \alpha &=\frac {-A+\sqrt {A^{2}-4 B}}{2^{2}}, \beta =\frac {-A-\sqrt {A^{2}-4 B}}{2}, \\ A&=\frac {R_{l} C_{0} C_{2}+R_{2} C_{2}}{R_{I} R_{2} C_{0} C_{2}} { \text {and }} B=1 / R_{I} R_{2} C_{0} C_{2}.\end{align*} $\vert \alpha \vert \ll \vert \beta \vert $ , and $\alpha $ and $\beta $ are normally real values.

C. Approximation of the Solution and the Estimation of Parameters of Model 1

To understand the equations of Model 1, we approximated them. The approximation requires some hypotheses. Note that for ease of explanation, a good outlook is prioritized before mathematical rigidity in the following sections.

The resistance of insulators is extremely high, and their capacitance is low. In the case of $R_{0} \ll R_{1}$ , $C_{3} \ll C_{0}$ and $1 \gg B_{1} / A_{1}$ , the value of $Q_{0}(t)$ is inversely proportional to $R_{1}$ in the latter part of charging (= linear part). Using $C_{mix}, $ the composite capacitance of $C_{0}$ and $C_{3}$ , the approximation of $k_{0}$ is expressed as:\begin{equation*} k_{0 } \cong \frac {V_{0}}{R_{0}A\mathrm {(1 - 2}B\mathrm { /}A^{2})} \cong \frac {C_{0}C_{3}}{C_{0}+C_{3}}V_{0}=C_{mix}V_{0} \tag{15}\end{equation*} Then, by applying Taylor expansion, we obtain the following:\begin{align*} Q_{0}\left ({t }\right) &\cong C_{0}V_{0}-{ C}_{mix}V_{0} \left\{\frac {C_{0}}{C_{3}}\mathrm {exp}(\alpha _{1}t)+\mathrm {exp}(\beta _{1}t)\right\} \tag{16}\\ &\cong \frac {V_{0}}{R_{1}}t+{ C}_{mix}V_{0} \tag{17}\\ { Q}_{3}\left ({t }\right) &\cong C_{mix}V_{0} \big({\mathrm {exp}}(\alpha _{1}t) - {\mathrm {exp}}(\beta _{1}t) \big) \tag{18}\\ &\cong C_{mix}V_{0} \left(1 -\frac {1}{C_{0}R_{1}}t \right) \tag{19}\end{align*}

The outline of the schematic relation between the graph plot and approximated solution is shown in Fig. 4. Fig. 5 shows the obtained $Q_{0}$ and $Q_{3}$ from Circuit Model 1 and the parameters of $C_{0}$ = $\mathrm {1.0\times }{10}^{-7}$ F, $C_{3}$ = $\mathrm {6.0\times }{10}^{-12}$ F, $V_{0}$ = 1 kV, $R_{0}$ = $\mathrm {1.45\times }{10}^{5}\mathrm { \Omega }$ , and $R_{1}$ = $\mathrm {1.5\times }{10}^{13}\,\,\mathrm {\Omega }$ . The results shown in Fig. 4 and Fig. 5 have two meanings:

1. We can estimate $R_{1}$ from the latter part of the experimental result (e.g., $t >$ 100 s), and the linearity of the latter part is almost identically based on the value of $V_{0} / R_{1}$ .

2. $C_{3}$ is obtained by the first steep change of $Q_{3}$ (at approximately $t$ = 0 s, and its approximated form is represented as $Q_{up} $ in Fig. 4).

FIGURE 4.

Image of Q(t) of Circuit Model 1, which consists of the material’s main resistance ${R} _{1}$ , material capacitance ${C} _{3}$ , safety resistor ${R} _{0}$ , and DC power supply ${V} _{0}$ .

FIGURE 5.

Numerical test results of Q(t) from Circuit Model 1.

D. Approximation of the Solution and Estimation of Parameters of Model 2

In the case of Model 2, directly estimating parameters from the aforementioned solution of (13) and (14) (e.g., with the biexponential function regression) is difficult because the number of parameters is increased and one parameter can be excessively high, whereas the other can be exceedingly low: For example, the order of $R_{1}$ is $10^{13}\mathrm {\Omega }$ and that of $C_{2}$ is $10^{\mathrm {-10}}\text{F}$ . Therefore, it is better to use approximation results from $\mathrm {\vert }\alpha \vert \ll \vert \beta \mathrm {\vert }$ and a simple regression form.

From (10), when ${t}$ is sufficiently small to regard “exp $(\alpha t)$ ” as being $\mathrm {\cong } 1$ (compared to exp $\left ({\beta t }\right))$ , $Q_{0}(t)$ is approximated as:\begin{equation*} Q_{0}\left ({t }\right) \cong G_{1} + G_{2}\exp {\left ({\beta t }\right) }+ C_{0}V_{0}. \tag{20}\end{equation*} This approximation similarly means $Q_{0}\left ({t }\right)\mathrm {} Q_{2}\left ({t }\right)$ in the early stages of Q(t) measurements.

Then, let $\Delta Q_{0}(t\mathrm {) }$ be $Q_{0}\left ({t\mathrm { + \Delta }t }\right) -Q_{0}(t)$ .\begin{equation*} \Delta Q_{0}(t) = G_{2} \mathrm {exp(}\beta t\mathrm {) (- 1+}\exp {(\beta \Delta t)}) \tag{21}\end{equation*}

From experimental results obtained from Q(t) measurements, $\mathrm {\Delta }t$ indicates the time step of the measurement; therefore, considering the ratio of $\mathrm {\Delta }Q_{0}\left ({t\mathrm { + \delta }t }\right)\mathrm { and \Delta }Q_{0}\left ({t }\right)\mathrm {, }$ \begin{equation*} \frac {\Delta Q_{0}\left ({t\mathrm { + }\delta t }\right)}{\Delta Q_{0}\left ({t }\right)}\mathrm { = exp}\left ({\beta \delta t }\right) \tag{22}\end{equation*} and introducing logarithms on both sides, \begin{equation*} \beta \delta t\mathrm { = ln }\frac {\Delta Q_{0}(t\mathrm { + }\delta t)}{\Delta Q_{0}(t)}. \tag{23}\end{equation*} $\mathrm {\delta t}$ is also the time step of the measurement, but it is not necessarily the same value as $\mathrm {\Delta }t$ . From the exponential regression from (22) or linear regression from (23), the estimated value of $\beta ^{}$ is obtained. Note that, in general, the results of the regression of (22) and (23) are not equal. To simplify the explanation, we chose the regression of (23).

The estimated value of $\beta ^{}$ is denoted as $\widehat {\beta }$ in the following sentences. Thus, substituting $\widehat {\beta }$ in (21), the estimated value of $G_{2}$ is also obtained from (24) in the same manner as the estimation of $\widehat {\beta }$ (written as $\widehat {G_{2}}$ ,) \begin{equation*} \ln \frac {\Delta Q_{0}\left ({t }\right)}{\left ({\mathrm {- 1+}\exp \left ({\widehat {\beta }\Delta t }\right) }\right)}=\ln {\vert G_{2}\vert }+\widehat {\beta }t \tag{24}\end{equation*} Finally, the obtained values of $\widehat {\beta }$ and $\widehat {G_{2}}$ are substituted to (14) and obtain (25) and (26):\begin{align*} \widehat {R_{2}}&\cong \frac {-C_{0}S}{T\left ({\frac {C_{0}K}{T}+C_{0}\widehat {\beta }+\frac {\widehat {G_{2}}\widehat {\beta }}{V_{0}} }\right)} \tag{25}\\ \widehat {C_{2}}&\cong \frac {T}{KR_{2}+S}\mathrm {, } \tag{26}\end{align*} where $K=\widehat {\beta }\mathrm { (1 + }\widehat {\beta } C_{0}R_{1}\mathrm {),}\,\,S\mathrm { = }\widehat {\beta }R_{1}$ , and

$T \mathrm {= 1 - }\widehat {\beta } C_{0}R_{1}\mathrm {}(\vert \alpha \mathrm {\vert \ll \vert }\beta \mathrm {\vert }$ is used for the approximation of $F_{1}$ and $R_{2} $ such that $\frac {1}{\beta - \alpha }\mathrm { \cong }\frac {1}{\beta }$ ).

As $C_{0}$ and $V_{0}$ are given, and $R_{1}$ is easily estimated from the gradient of the latter part of Q(t) data, the estimated values of $R_{2}$ and $C_{2}$ are obtained ($\widehat {R_{2}}$ and $\widehat {C_{2}}$ .)

E. Numerical Testing

In this section, numerical tests are conducted to evaluate the appropriateness of the estimation method shown in Section II-D. The testing is a “restoration test,” which means using imaginary data yielded by the pre-given values of the circuit constant. This test checks the difference between the original values of $R_{2}$ and $C_{2}$ and their estimated values. In this case, referring to [10], $C_{0}$ and $V_{0}$ are $4.7\mathrm {\times }{10}^{-7}$ F and 1 kV, and $R_{1}$ and $C_{2}$ are set to $\mathrm { 2\times }{10}^{13}\mathrm { \Omega }$ and $1\mathrm {\times }{10}^{-10}\text{F}$ , respectively (note that the values of $C_{0}$ , $V_{0}$ , and $R_{1}$ are different from those used to plot the data of Model 1). Generally, these parameters were considered/referred from our previous experimental/simulation data; $C_{0}$ was decided referring to capacitance values of the integral capacitor used in experiments and $R_{0}$ was decided referring to the safety resistor of the circuit that was implemented on our Q(t) measurement setup, $R_{1}$ was decided by referencing the pre-test data.

This research used python [20], numpy [21] and pandas [22] in coding of data processing phase, and Scipy [23] in the optimization phase. The fitting process is performed in the following steps; First, from equation (24), \begin{equation*} \frac {1}{\widehat { \beta }}\ln \frac {\Delta Q_{0}\left ({t }\right)}{\left ({\mathrm {-1+}\exp \left ({\widehat {\beta }\Delta t }\right) }\right)}=(ln{\vert G_{2}\vert)}\mathrm { / }\widehat {\beta }\mathrm { + }t\mathrm {, } \tag{27}\end{equation*}

The data are then fitted to (27) as linear function t + b, where b is an intercept and is the target of estimation here.

Solver is “scipy.optimize.curve_fit” function, which uses minimizing least squares algorithm.

Finally, the fitting condition is written as follows:\begin{equation*} \mathrm {minimize}\sum \limits _{t=t_{start}}^{t_{end} }{\left ({\frac {1}{\widehat { \beta }}\ln \frac {\mathrm {\Delta }Q_{0}\left ({t }\right)}{\left ({\mathrm {-1+}\exp \left ({\widehat {\beta }\Delta t }\right) }\right)} - b }\right)^{2} } \tag{28}\end{equation*} under the restriction of $\mathrm {b \le 0}$ (because it is assumed to $\widehat {\beta } < 0$ to prevent the explosion of exponential function.) and initial b = 0.

F. Regression of “Experimental Results” With Models

To apply the proposed method in an actual case, the regression of the experimental data obtained by the Q(t) measurement of PI and PET are conducted. This regression estimates the values of the delay parameters of an insulation material and help the specification of detailed charging-saturation time under the model. The flowchart of the regression is represented as Fig. 6.

FIGURE 6.

Flowchart of estimation process.

Experimental conditions and circuit parameters are listed in Table 1 and the outlook of the experimental setup is shown in Fig. 7. A DC voltage supplier, a Q(t) measurement device (Q(t) meter) and an electrical Insulation material (sample) were connected serially. A heater was set under the sample and the temperature was set to 50°C. Note that the temperature was decided referencing for [11]. Samples were PI and PET seats, and each seat was cut to the size of “$100\times 100$ mm”. The humidity was not controlled actively, and the values moved from 33%RH to 57%RH in case of PI, and from 51%RH to 74%RH in case of PET. The upper electrode was a $\phi 50$ cylinder, and the lower electrode was a $\phi 100$ cylinder. Around the upper electrode, guard electrode (inner-$\phi 60$ cylinder) was set so that the distance between the upper electrode and guard electrode was set to 10 mm (These electrodes set up was based on IEC 62631-3-1 and 62631-3-2, which are standard of measurement of volume/surface resistivity.)

TABLE 1 Experimental Conditions and Circuit Parameters

FIGURE 7.

Experimental setup of Q(t) measurements of PI and PET. (a) Outline of the actual setup and (b) circuit outline.

When t = 0, DC voltage switch turned on and the set value was 1 kV. When turning on DC voltage, charges started to be accumulated on the sample($Q_{2}$ ) and the Q(t) meter ($Q_{0}$ ).

Note that humidity and temperature of the sample should be ideally included in the equations as parameters, but from [11] and [14], the relation appears to be quite complicated, thus their effects are not considered in this study.

SECTION III. Results and Discussions

A. Results of Numerical Testing on Model 2

Table 2 presents the results of the restoration test when $\mathrm {\Delta }t$ = 7 s and $\delta t\mathrm { = 12 s}$ . Fig. 8(a) shows the Q(t) data created as an imaginary data. Fig. 8(b) shows the plot of $\mathrm {\Delta }Q_{0}\left ({t }\right)$ , and Fig. 8(c) shows the ratio of $\mathrm {\delta }Q_{0}$ , which is the plot of the lefthand side of (22). The time resolution is 1 s. From $C_{2}$ and $R_{2}$ listed in Table 2, the estimated values are approximately equal to the true values (input data).

TABLE 2 Parameters Obtained From the Restoration Test

FIGURE 8.

Results of numerical testing (restoration test.) (a) Original Q(t) data and its approximated form, (b) ${\Delta }{Q}_{0}$ , and (c) Ratio of ${\delta }{Q}_{0}$ (from lefthand side of Equation (22).)

B. Results of the Regression of “Experimental Results”

Fig. 9 shows the regression results of experimental data of PI samples when $\mathrm {\Delta }t$ = 10 s and $\mathrm {\delta }t\mathrm { = }22$ s. The number of samples is nine. Fig. 9(a) shows the Q(t) measurement data, $\mathrm {\Delta }Q_{0}$ , and the ratio of $\mathrm {\delta }Q_{0}$ of all the experimental data and their sample-averaging.

FIGURE 9.

Result of experimental tests of PI when ${\Delta }{t}{ = 10 s}$ and ${\delta }{t}{ = 22 s}$ . (a) ${Q}_{0}$ , ${\Delta }{Q}_{0}$ and ratio of ${\delta }{Q}_{0}$ , (b) ${\beta }{, } {G}_{2}{,} {R}_{2}$ , and ${C}_{2}$ values, and (c) reversely calculated ${Q} _{0}$ and ${Q} _{2}$ from the substitution of the obtained ${R} _{2}$ and ${C} _{2}$ .

The first graph of Fig. 9(a) indicates that initially the condenser $C_{0}$ (and the sample) intake charges steeply (during less than 1 s), and then $Q_{0}(t)$ behaves as if it had a very small linear coefficient against time. Note that the coefficient actually has time dependency [14] so that its value changes slightly as time passes by. Main differences between numerical test and experimental PI data are the ratio of $Q_{0}$ (${t}$ = 1) and $Q_{0}$ (${t}$ = 300), and the steepness of the transient step between $t$ = 1 and ${t}$ = 300 s. In other words, experimental PI data show high-speed responsibility in the early stage of charging, and high resistance in the latter stage. The second graph of Fig. 9(a) indicates the moving summation of current (A) through $C_{0}$ during $\Delta t$ ($\Delta t \mathrm {=10 s}$ ) when the current value was defined as {$Q_{0}(t$ + 1) - $Q_{0}(t)$ }. The moving summation is equal to {$Q_{0}(t$ + 1) - $Q_{0}(t)$ } + {$Q_{0}(t$ + 2) - $Q_{0}(t$ + 1)} +…+ {$Q_{0}(t$ + 10) - $Q_{0}(t$ + 9)} here, and then finally equal to $Q_{0}(t$ + $\Delta t$ ) - $Q_{0}(t)$ = $\Delta Q_{0}(t)$ , which correspond to (21). Generally, $\Delta Q_{0}(t)$ is extremely at the start of charging, so the data indicates a type of inrush/transient current occurred in the circuit (but because of the high resistance of the insulator, the absolute value of the current is extremely low.) The third graph of Fig. 9(a) indicates the ratio of the moving summation of current through $C_{0}$ during $\delta t$ ($\delta t\mathrm { = 22 s}$ ), which correspond to (22). Note that the value of $\Delta t$ and $\delta t$ was chosen to make the fitting calculation stable.

Fig. 9(b) shows the estimated ${\beta, G}_{2}\mathrm {, }R_{2}\mathrm {, and}{ C}_{2}$ . Fig. 9(c) shows the results of the estimated values of $Q_{0}$ and $Q_{2}$ , which were obtained by substituting estimated values of $R_{2}$ and $C_{2}$ using the method proposed in this study. All timesteps of the experimental setting were set to 1 s, and the actual measuring timestep contains a small error within a range of −0.07 to 0.07 s. However, to calculate the averages from the data, the time values are rearranged and rounded to integer values. Note that $R_{1}$ is assumed to be obtained by the same manner as Model 1, and the precise value is not required because of the dependency on $R_{2}$ and $C_{2}$ . In addition, the estimation process needs fitting-start time and fitting-end time as fitting parameters. In case of estimation of Fig. 9, fitting-start time and fitting-end time are set to 0 s and 40 s, respectively.

Fig. 10 (a)-(c) shows data of PET that are same types as Fig. 9 (a)-(c), respectively. Note that $\mathrm {\Delta }t$ = 8 s and $\delta t\mathrm { = 11 s}$ respectively, and fitting-start time and fitting-end time are set to 0 s and 20 s respectively. It is because the shape of the initial curve (and the dependent parameters) are different from the case of polyimide samples. In comparing the results of Fig. 9 and 10, the response speed during the first 10 s of reconstruction data of Fig. 10 (c) is higher than Fig. 9 (c).

FIGURE 10.

Result of experimental tests of PET when ${\Delta }{t}{ = 8 s}$ and ${\delta }{t}\mathrm { = 11 s}$ . (a) ${Q}_{0}$ , ${\Delta }{Q}_{0}$ and ratio of ${\delta }{Q}_{0}$ , (b) ${\beta }{, } {G}_{2}{,} {R}_{2}$ , and ${C}_{2}$ values, and (c) reversely calculated ${Q} _{0}$ and ${Q} _{2}$ from the substitution of the obtained ${R} _{2}$ and ${C} _{\mathbf {2}}$ .

Though there is a difference between the estimated charges and experimental results to some extent, the tendency and order are similar, such that the estimated $R_{2}$ and $C_{2}$ explain the property of obtained experimental data within the range (and the limitation) of the circuit model.

Note that the difference of the steepness of estimated charges and experimental results at the early stage is caused by the limitation of mathematical expression of two exponential functions: If it requires more precise expression (e.g. for strict fitting of initial steep charging during $t < 1$ s), the introduction of time-dependent circuit-component(s) is required. Furthermore, the deeper analysis also needs the comparison of PEA data because the time-dependency appears to be related to space charge accumulation inside polymeric materials [19].

C. Practical and Physical Meaning of Obtained Results and Calculated Parameters

From the previous sections, $\beta $ , $G_{2}$ , $R_{2}$ , and $C_{2 -}$ the delay parameters of Q(t) measurement—are obtained, and $\alpha $ also can be obtained from $R_{2}$ and $C_{2}$ . The meanings of $\alpha $ and $\beta $ are self-explanatory from the definition, such that the time-constant of Q(t) measurement can be revealed, as shown in Fig. 11 ($\mathrm {\vert }\alpha \vert \ll \vert \beta \mathrm {\vert }$ ), which is crucial for determining whether “$Q(t)/Q$ (0) > 2”. (originally, as in the situation shown in lower center of Fig. 11, the value of “2” of the righthand side is considered the states in which charges are no more accumulated inside an insulator [14], and the value is easily obtained by solving the Poisson equation:\begin{equation*} - \Delta V\left ({x }\right) = \nabla \cdot E\left ({x }\right) = \frac {\rho }{\varepsilon } \left ({\mathrm {b.c. }E\left ({d }\right) = 0 }\right) \tag{29}\end{equation*} in this situation [1], where x-axis is along with the space between electrodes and d is the distance. However, owing to the space limitation, the detailed introduction is omitted here.)

FIGURE 11.

Schematic representation of the relation between physical phenomena and the analysis.

Q(t) measurement is treated as a method to decide whether $Q(t)/Q$ (0) > 2 [2]. However, the decision is difficult as there is no unique way for determining the value of Q(t) for use in practical measurements, which contains fuzzy boundaries in the transition process. Thus, using the circuit model and the method proposed in this study, the charge saturation point can be analytically estimated/identified from the experimental results. Moreover, $Q_{0}(t)$ is known, but $Q_{2}(t)$ cannot be obtained directly; when Model 2 is applied, it means using the estimated $Q_{2}(t)$ as the numerator in $Q(t)/Q$ (0), instead of $Q_{0}(t)$ . In other words, unless we do not apply circuit models on the discussion of the threshold between absorbed and leakage current, the result of $Q(t)$ measurement roughly indicates only the summation of all currents and does not classify the charge property as a circuit component. This is not only considered as the result of measurement errors, but also as modeling design errors, such that the misunderstanding may even cause severe problems of theoretical over-estimations and under-estimations of the value of Q(t), as shown in Fig. 12.

FIGURE 12.

Images of risks of over-estimation and under-estimation of Q(t) / Q(0) without adequate understanding of circuit models. (a) Over-estimation and (b) under-estimation (overlooking saturation before the minimum time resolution).

On the contrary to $\alpha $ and $\beta $ , the physical mean of $R_{2}$ and $C_{2} $ is unclear. Fig. 13 shows the determination of the physical mean from the shape of the circuit model. From an experimental perspective, the delay is observed with increasing Q(t); however, the appropriateness of the modeling method has not been discussed well in the Q(t) measurement.

FIGURE 13.

Schematic representation of the relation between physical phenomena and the analysis.

Importing the model of partial discharge appears to be good. One example of a circuit model that contains both a resistor and a capacitor is shown in [18]. Fig. 14 shows the circuit with resistors and capacitors. Adding the resistance component to the circuit model improves the simulation results and then improves the correlation between the actual experimental values and those obtained in PD simulations. It is thought to be same as in DC cases, such that the circuit shown in Fig. 13 can be regarded as the similar (or simpler) form of the with-resistance model presented in literature.

FIGURE 14.

Image of the circuit model of PD measurements consisting of resistors and capacitors [18].

From another perspective, contrary to PD cases, the delay of Q(t) shown in this study cannot be measured without the series components: resistor and capacitor, though applying an AC voltage indicates the back-and-forth movement of charges and that the total delay is dominated by a capacitance component. Therefore, in the DC circuit, the series component of $R_{2} $ and $C_{2} $ should be considered.

D. Limitation of This Research and Issues

The model in [18] or more complicated ones are analyzed by solving the more complex differential equations, but we considered a simpler method because the analytical solution can be found relatively easily. Then, complicated circuit models should be applied when a deeper discussion regarding the physical mean is required (for example, the regression of the Q(t) data in high temperature, which contains curves that are not expressed by the combination of just two exponential functions). Note that if the circuit model becomes complicated (for example, one-more-parallel-condenser and safety resistor are added to the circuit Model 2), it becomes a third-order differential equation. To solve the problem, Cardano’s method, which is a cubic formula, is required so the solution has to be treated as “complex” form.

Moreover, the relationship between the results of Q(t) data and pulse electro-elastic acoustic emission (PEA) measurement should be considered [13], [15], [19].

The proposed method in this study also contains parameter vulnerability, which depends on the regression process. $\mathrm {\Delta }t, \mathrm {\delta }t,^{}$ and the regression parameters/method could be decided arbitrarily to some extent (which means it is crucial to verify the results of the estimated values by substituting them into the equations/solutions of the model, particularly for practical use).

In terms of the data variance, this research did not define the error factor. However, error terms should be considered (for example, stochastic trend) as the countermeasure of actual usage cases. To analyze such effects, a statistical and numerical method for analyzing stochastic models (for example, the auto regression and/or moving average models) should be introduced.

Therefore, extensive research on the analysis methods should be performed to improve Q(t) measurements.

SECTION IV. Conclusion

To analyze the results of Q(t) data, hypothesizing a circuit model is crucial for a deeper understanding of obtained data. From the shape of experimentally obtained data, we first consider the circuit model as a circuit containing an insulator comprising a capacitor in parallel with one resistor, and then as a parallel circuit comprising one resistor in parallel with a series capacitor–resistor connection. After that the differential equations are solved as the circuit models of Q(t) measurements. Subsequently, we presented the estimating method from the solution of the circuit model and conducted a numerical test of estimation of the parameters. Finally, we performed the simulation of actual Q(t) data of PI and PET samples and obtained the delay parameters, $R_{2}$ and $C_{2}$ .

The results revealed that:

1. Solving the differential equation determines the parameters of the circuit model, which causes the main circuit properties and the delay of initial charging of Q(t) data.

2. From the results of numerical testing and regression of experimental data, the actual experimental data could be analyzed to some extent and the estimated values of delay parameters can be obtained.

3. The method can be used to determine the timing of charge saturation analytically, and for further discussing of estimating “$Q(t)/Q$ (0)”.

   The unique contribution of this study is proposing a practical analysis method of Q(t) data using circuit properties, focusing on the delay property of Q(t) data. Solving the ordinary differential equation established from a circuit model, the delay property can be approximately expressed as a simple combination of the circuit properties, so that estimation method of the circuit properties from Q(t) data is proposed. Such an approximated expression as circuit model is useful for designers/diagnostic-engineers of electrical insulation materials to apply what Q(t) measurement indicates owing to the simpleness. Moreover, the approach of this study is expected to reveal the charge characteristics such as space charges/electromagnetics models to the circuit properties through the estimated parameters in the circuit model with further analysis.

   Conversely, based on the limitations of modeling and mathematical manipulations, this method contains some limitations that must be addressed in the future as follows:

4. When applying this method to more complicated circuit models/Q(t) data, a higher order of differential equations must be solved.

5. The relationship among circuit constants must be considered to obtain Q(t) data and/or PEA data.

6. More statistically/numerically stable methods for estimating the parameters must be found.

In addition, this research explained and performed the significance of considering a circuit model as a basic treatment/literacy of understanding the results of Q(t) measurements. The origin of expressed circuit/material models must be determined with the results of theoretical electromagnetics and/or QM/MD to create high-endurance materials.