Analysis of Errors in Measuring the Voltage Distribution by the Two-Measurement Method

V. S. Dmitrevsky

(Presented by the Scientific Seminar on Dielectrics)

The accuracy of measuring the voltage distribution by the previously described method [1] depends on the following factors:

a) the accuracy of the measuring instrument used,
b) the length of the connecting wires,
c) changes in the conductivity of individual sections of the specimen (for example, due to corona) under different circuit arrangements for connecting the instrument.

Thus, the error in measuring the voltage distribution over the surface of the insulation may be written as:

\[ a = a_1 + a_2 + a_3, \tag{1} \]

where
\(a\) is the error of the measurement method;
\(a_1\) is the error introduced by the measuring instrument;
\(a_2\) is the error introduced by the connecting wires;
\(a_3\) is the error due to changes in the conductivity of individual sections of the specimen when the instrument is connected.

Let us determine the magnitudes of the individual errors and ways of reducing them.

Influence of the Accuracy of the Instrument Used on the Measurement Error

To calculate the influence of the accuracy of the measuring instrument on the error in measuring the voltage distribution, we use the procedure described in [2]. We shall calculate the error in measuring the voltage distribution on the basis of the following formula, obtained by us earlier [1]:

\[ U_2 = U \frac{U'_2}{U'_1 + U'_2}. \tag{2} \]

Taking the logarithm of formula (2), we obtain:

\[ \ln U_2 = \ln U + \ln U'_2 - \ln \left(U'_1 + U'_2\right). \]

Differentiate the expression obtained:

\[ \frac{dU_2}{U_2}=\frac{dU}{U}+\frac{dU_2'}{U_2'}-\frac{d(U_1'+U_2')}{U_1'+U_2'}. \]

In the equation obtained, replace the differentials by finite increments:

\[ \frac{\Delta U_2}{U_2}=\frac{\Delta U}{U}+\frac{\Delta U_2'}{U_2'}-\frac{\Delta U_1'+\Delta U_2'}{U_1'+U_2'}. \]

In this expression, \(\Delta U\) with the corresponding indices is the absolute error in measuring the quantity. Replace the absolute measurement errors by relative ones according to the formula:

\[ a=\pm\frac{\Delta U}{U}. \]

Then the relative measurement error will be written in the form:

\[ \pm a_2=\pm a \pm a_2' - \frac{\pm a_1' U_1' \pm a_2' U_2'}{U_1'+U_2'}. \tag{3} \]

When measurement is carried out with one instrument, the relative error remains approximately constant in magnitude. Carrying out a detailed analysis of equation (3), we come to the conclusion that the maximum measurement error will occur under the following conditions: \(a_1'<0\); \(a_2'>0\); \(a>0\); \(U_1'\gg U_2'\). In this case the relative measurement error will be equal to:

\[ a_{\text{max}}=3a. \tag{4} \]

In all other cases the error will be smaller and will vary within the limits from \(a\) to \(3a\).

Influence of the Length of the Connecting Wires on the Measurement Error

Figures 1 and 2 show complete equivalent circuits for measuring the voltage distribution, taking into account the conductance of wire \(AB\) (Fig. 3). For these equivalent circuits one may write the following equations:

\[ I_a=U_2'(y_2+y_2'+y_{\text{meas}}'')=(U-U_2')(y_1+y_1'), \tag{5} \]

\[ I_b=U_1'(y_1+y_1'+y_{\text{meas}}')=(U-U_1')(y_2+y_2'). \tag{6} \]

In the absence of the measuring system,

\[ I=U_2y_2=(U-U_2)y_1. \tag{7} \]

In these equations

\[ y_2' \]
— the total conductance of wire \(AB\) with respect to ground,

\[ y_1' \]
— the total conductance of wire \(AB\) with respect to the high-voltage electrode,

\[ y_{\text{meas}}' \]
— the total conductance of the measuring instrument when it is connected between the probe and the high-voltage electrode,

\[ y_{\text{meas}}'' \]
— the total conductance of the measuring instrument when it is connected between the probe and the grounded electrode.

Solving the system of equations (5), (6), and (7) for the sought quantity \(U_2\), we obtain:

\[ U_2 = \frac{ U U'_2 y''_{\text{meas}} + U'_1 U'_2 \left(y'_{\text{meas}} - y''_{\text{meas}}\right) - U\left(U - U'_1 - U'_2\right)y'_1 }{ U'_1 y'_{\text{meas}} + U'_2 y''_{\text{meas}} - \left(U - U'_1 - U'_2\right)\left(y'_1 + y'_2\right) }. \tag{8} \]

Fig. 1. Equivalent circuit for measuring the voltage distribution when the instrument is connected between the probe and the grounded electrode.

Fig. 2. Equivalent circuit for measuring the voltage distribution when the instrument is connected between the probe and the high-voltage electrode.

Fig. 3. Connection diagram of the instrument when measuring the voltage distribution.

Let us calculate the measurement error obtained when computing \(U_2\) by formula (2), which does not take into account the influence of the total conductance of the wire \(AB\):

\[ \alpha'_2 = \frac{\Delta U_2}{U_2} = \frac{U_2^{(2)} - U_2^{(8)}}{U_2^{(8)}} = \]

\[ = \left\{ \left[ U U'_2 y''_{\text{meas}} + U'_1 U'_2 \left(y'_{\text{meas}} - \right. \right. \]

\[ \left. - y''_{\text{meas}} \right] \left[ U'_1 y'_{\text{meas}} + U'_2 y''_{\text{meas}} - \right. \]

\[ \left. \left. - \left(U - U'_1 - U'_2\right)\left(y'_1 + y'_2\right) \right] \right\} : \]

\[ : \left\{ \left(U'_1 y'_{\text{meas}} + U'_2 y''_{\text{meas}}\right) \left[ U U'_2 y''_{\text{meas}} + \right. \right. \]

\[ \left. \left. + U'_1 U'_2\left(y'_{\text{meas}} - y''_{\text{meas}}\right) - U\left(U - U'_1 - U'_2\right)y'_1 \right] \right\}. \tag{9} \]

Since for most measuring instruments the total conductivity changes little within the range of measurement on a single scale, i.e., one may assume \(y'_{\text{meas}}=y''_{\text{meas}}=y_{\text{meas}}\), formula (9) takes the following form:

\[ a'_2= \frac{(U-U'_1-U'_2)(U'_1y'_1-U'_2y'_2)} {U'_2(U'_1+U'_2)y_{\text{meas}}-(U'_1+U'_2)(U-U'_1-U'_2)y'_1}. \]

As an example, let us calculate the error in measuring the voltage distribution along an insulator string for different lengths of wire AB. The measurements were made with an electrostatic voltmeter S-96, having a capacitance \(C_{\text{meas}}=12\) pf. In the calculation we shall take into account only the capacitive component of conductivity, since with alternating voltage the active component will be small.

1st case. Wire AB is \(45\) cm long, diameter \(1\) mm. When the probe was placed on the first element from the wire of a six-element insulator string, the following results were obtained: \(U'_1=3.7\) kV, \(U'_2=10.7\) kV, \(U=17.8\) kV. The capacitance of wire AB with respect to ground may be determined as the capacitance of a cylindrical capacitor, whose inner plate is the surface of wire AB and whose outer plate is the walls, floor, and ceiling.

\[ C'_2=\frac{l}{2\ln\frac{R}{r}} =\frac{45}{2\ln\frac{200}{0.05}} =2.7\ \text{cm}=3\ \text{pf}, \]

where \(R=200\) cm is the distance from the wire to the floor and walls, \(r=0.05\) cm is the radius of the wire, and \(l=45\) cm is the length of wire AB. The capacitance of wire AB with respect to the high-voltage electrode may be determined as the capacitance of two parallel wires by the formula

\[ C'_1=\frac{l}{2\ln\frac{S^2_{1,2}}{r_1\cdot r_2}} =\frac{45}{2\ln\frac{(25)^2}{0.05\cdot 1}} =2.39\ \text{cm}=2.66\ \text{pf}, \]

where \(S_{1,2}\) is the distance between wire AB and the power conductor,
\(r_1\) is the radius of wire AB,
\(r_2\) is the radius of the power conductor.

Let us calculate the measurement error caused by wire AB \(45\) cm long:

\[ a'_2= \frac{(U-U'_1-U'_2)(U'_1y'_1-U'_2y'_2)} {U'_2(U'_1+U'_2)y_{\text{meas}}-(U'_1+U'_2)(U-U'_1-U'_2)y'_1} = \]

\[ = \frac{(17.8-3.7-10.7)(3.7\cdot 2.66-10.7\cdot 3)} {10.7(3.7+10.7)12-(3.7+10.7)(17.8-3.7-10.7)\cdot 2.66} =-0.0013. \]

2nd case. The length of wire AB is \(120\) cm, diameter \(1\) mm. When the probe was placed on the first element of a six-element insulator string, the following results were obtained:

\[ U=17.8\ \text{kV}, \qquad U'_1=4.35\ \text{kV}, \qquad U'_2=10\ \text{kV}. \]

Capacitance of wire AB with respect to ground

\[ C_2'=\frac{l}{2\ln \frac{R}{r}}=\frac{120}{2\ln \frac{200}{0.05}}=7.25\ \text{cm}=8.05\ \text{pF}. \]

Capacitance of wire AB with respect to the high-voltage electrode

\[ C_1'=\frac{l}{2\ln \frac{S_{1,2}^{2}}{r_1\cdot r_2}} =\frac{120}{2\ln \frac{(25)^2}{0.05\cdot 1}} =6.4\ \text{cm}=7.1\ \text{pF}. \]

The error introduced into the measurements by wire AB of length \(120\ \text{cm}\) will be:

\[ a_2'=\frac{(U-U_1'-U_2')(U_1'y_1'-U_2'y_2')} {U_2'(U_1'+U_2')y_{\text{meas}}-(U_1'+U_2')(U-U_1'-U_2')y_1'}= \]

\[ =\frac{(17.8-4.35-10)(4.35\cdot 7.1-10\cdot 8.05)} {10(4.35+10)\cdot 12-(4.35+10)(17.8-4.35-10)\cdot 7.1} =-0.126. \]

We conducted experiments in order to determine the influence of wire AB on the accuracy of measuring the voltage distribution along a string of insulators. The voltage distribution along the insulator string was measured by the method of constant separation of spheres, with spheres of diameter \(0.5\ \text{cm}\) and a distance between them of \(0.08\ \text{cm}\). The experiments were carried out on a six-unit insulator string of type G-4.5, to the cap of the first unit of which a wire of various lengths was connected. Figure 4 presents a photograph of such measurements.

Fig. 4. Photograph of measuring the voltage distribution along an insulator string with determination of the influence of wire AB.

Figure 5 shows the dependence of the error in measuring the voltage distribution, introduced by wire AB, when measuring the voltage drop on individual elements of a string of insulators (curve 1—the theoretical curve, curve 2—the experimental curve). In these experiments and calculations the length of wire AB was 46 cm and its diameter 7.5 mm. It is seen from Fig. 5 that, as the number of the element increases, counting from the wire, the measurement error increases. The theoretical dependence of the measurement error turns out to be greater than the experimental one. This discrepancy is apparently explained by an inaccurate calculation of the conductances \(y'_1\) and \(y'_2\) and by neglecting the influence of the conductance between wire AB and the individual elements of the insulator string.

Fig. 5. Measurement error introduced by wire AB.
1—theoretical curve, 2—experimental curve.

Table 1 gives the error in measuring the voltage drop on the first element from the wire of a six-element insulator string for various lengths of wire AB.

Table 1

It is seen from Table 1 that, as the length of wire AB is increased, the measurement error grows very rapidly; therefore, in making measurements, the length of wire AB should not be taken greater than 0.5 m, in order to avoid large errors. In cases where wire AB cannot be made very short (for example, when measuring the voltage distribution over insulating structures immersed in oil), it should be shielded. The shield is connected according to the circuits shown in Fig. 6.

The installation of the shield must not affect the voltage distribution along the surface of the insulation on which the measurement is being made. On this basis, the shield should be installed as far as possible from the object. Experiments showed that a shield installed at a distance of 15 cm from the axis of a six-element insulator string gives an error of \(+27.6\%\). When the shield is installed at a distance of 50 cm, the error decreases to \(+3\%\).

In these experiments the shield consisted of a hollow steel tube 50 mm in diameter and 1 m long, through the center of which wire AB, 1 mm in diameter, was passed strictly along the center. Using shield-

of wire AB, it is possible to increase considerably the length of the lead-in wires without reducing the accuracy of the measurement. This circumstance is extremely important when measuring the voltage distribution along the surface of a solid dielectric immersed in a vessel with liquid.

Fig. 6. Method of connecting the shield installed on the wire between the instrument and the probe.

The influence of wire CD and of the instrument housing on the error in measuring the voltage distribution along the dielectric surface was investigated by us experimentally, because of the difficulty of carrying out theoretical calculations. The experiment was performed as follows: a wire CD of a definite length was connected to the high-voltage electrode. The opposite end of wire CD was connected to the instrument housing. The voltage distribution along a string of insulators was measured by the constant ball-spacing method. Then wire CD was connected to the grounded end of the string, and the measurements were carried out again. After this, wire CD and the instrument were removed, and the voltage distribution along the string of insulators was measured. The error in measuring the voltage distribution was calculated from the formula

\[ a_i''=\frac{U_{2\mathrm{ism}}-U_{2\mathrm{ist}}}{U_{2\mathrm{ist}}}, \tag{10} \]

where

\(U_{2\mathrm{ist}}\) is the voltage drop measured in the absence of wire CD;

\(U_{2\mathrm{ism}}\) is the voltage drop recalculated by formula (2).

Table 2 gives data on the calculation of the measurement error for different lengths of wire CD. The diameter of the wire was equal to \(1\ \mathrm{mm}\). Figure 7 shows a photograph of the measurements.

Table 2

It is seen from Table 2 that, with an increase in the length of wire CD (i.e., with an increase in the distance to the instrument housing), the measurement error decreases.

The error in measuring the voltage distribution introduced by wire CD and by the instrument is positive.

The total measurement error introduced by the connecting wires will be found as

\[ a_2 = a_2' + a_2'' . \tag{11} \]

Fig. 7. Photograph of the measurement of the voltage distribution along an insulator string in determining the influence of the connecting wire.

Table 3 gives the values of the errors obtained when measuring the voltage distribution along a six-element insulator string with the distance between the instrument and the insulator string equal to 45 cm. The diameter of the connecting wires is 1 mm.

Table 3

Influence of changes in the conductivity of individual sections of the specimen on the accuracy of measuring the voltage distribution along the insulation surface

The most frequent cause producing a change in the conductivity of individual sections of an insulating structure is corona. Therefore, in what follows, changes in conductivity in individual sections will be regarded as occurring only as a result of corona inception at the high-voltage electrode.

Figs. 8 and 9 show equivalent circuits for measuring the voltage distribution in the presence of corona at the high-voltage electrode.

Fig. 8. Equivalent circuit for measuring the voltage distribution in the presence of corona at the high-voltage electrode (the instrument is connected between the probe and the high-voltage electrode).

Fig. 9. Equivalent circuit for measuring the voltage distribution in the presence of corona at the high-voltage electrode (the instrument is connected between the probe and the grounded electrode).

For the equivalent circuits given, the following equations may be written:

\[ I_1 = y_2 (U - U_1') = (y_{k1} + y_1 + y_{pr}) U_1', \tag{12} \]

\[ I_2 = (y_2 + y_{pr}) U_2' = (y_{k2} + y_1) (U - U_2'), \tag{13} \]

\[ I = y_2 U_2 = (y_k + y_1)(U - U_2). \tag{14} \]

Solving the system of equations jointly with respect to \(U_2\), we find

\[ U_2 = \frac{ U(U - U_1' - U_2')(y_{k2} - y_k) + U_1' U_2'(y_{k2} - y_{k1}) - U U_2' y_{pr} }{ (U - U_1' - U_2')(y_{k2} - y_k) + U_1'(y_{k2} - y_{k1}) - (U_1' + U_2') y_{pr} }. \tag{15} \]

Let us calculate the error in measuring the voltage distribution when using formula (2):

\[ \delta_3 = \frac{ U_1'(U - U_1' - U_2') \left[ U_2'(y_{k2} - y_{k1}) - U(y_{k2} - y_k) \right] }{ (U_1' + U_2') \left[ U U_2' y_{pr} - U(U - U_1' - U_2')(y_{k2} - y_k) - U_1' U_2'(y_{k2} - y_{k1}) \right] }. \]

Under the condition that the probe does not fall within the corona sheath, the change in conductivities may be considered directly proportional to the magnitude of the voltage change (taking into account that these changes are small) in the high-voltage section of the specimen. Since, with different connections,

of the instrument \(U_{1\text{inst}} - (U - U'_2)\) differs little from the quantity \(U'_1 - U_{1\text{inst}}\), then it may be assumed approximately that \(y_{\kappa 2} - y_\kappa = y_\kappa - y_{\kappa 1}\).

The introduction of such an approximation will be entirely legitimate if one takes into account that, together with the corona-producing section of the sample, which has a low resistance, a section having a high resistance will be included. Then the measurement error due to corona will vary from a positive value, when \(U'_2 > \dfrac{1}{2}U\), to negative values, when \(U'_2 < \dfrac{1}{2}U\); that is, the change in measurement error will

Fig. 10. Distribution of the error along the length of the sample in the presence of corona at the high-voltage electrode.

have the form shown in Fig. 10. Owing to the absence of data on the exact values of \(y_{\kappa 2}\), \(y_\kappa\), and \(y_{\kappa 1}\), it is not possible to determine the numerical value of the error.

The total error in measuring the voltage distribution along the dielectric surface in the absence of corona may be calculated from formula (13).

The magnitude of the instrument error when using measuring spheres will be \(a_1 = \pm 3\%\), and the error \(a_2\) we take from Table 3 as equal to \(+9.1\%\). Then \(a\) will be equal to

\[ a = \pm 3.3 + 9.1 = (-0.1) \div (+18.1). \]

The analysis carried out has shown that the measurement error \(a = (-0.1) \div (+18.1)\) is no higher than the error of known measurement methods. In some cases the measurement error of the proposed method [1] will be lower than the measurement error of known methods (for example, when measuring the voltage distribution along the surface of an insulating structure immersed in a vessel).

REFERENCES

1. Dmitrevskii V. S. On a method for measuring voltage distribution over insulation. Izv. Tomsk. politekhn. in-ta, vol. 76, 142, 1954.

2. Bogdanova T. N., Subbotina E. P. Manual for Practical Exercises in Physics, Part I, “Sovetskaya Nauka” Publishing House, 1949.

ERRATA