ANGULAR ERROR OF THE ZERO-SEQUENCE VOLTAGE OF AN NTMI-TYPE VOLTAGE TRANSFORMER

A. Kh. Musin

(Presented by the scientific seminar of the Department of Electric Power Stations)

The ground-fault signaling device for compensated networks, being developed at the Department of Electric Power Stations of the Tomsk Polytechnic Institute, as well as known directional devices [1], uses as one of its input quantities the zero-sequence voltage \(U_0\), obtained from the secondary winding of an NTMI-type voltage transformer connected in an open delta. The angular error of the voltage \(U_0\) may impair the operability of the device. However, the adverse effect of a constant angular error can be compensated quite simply, for example by introducing a correcting phase-shifting circuit at the input of the device. Of particular concern are fluctuations of the angular error, which are a function of external factors, in particular of the ambient temperature.

In the present article an attempt is made to estimate the influence of fluctuations in ambient temperature on the angular error of the zero-sequence voltage \(U_0\) and to outline measures for reducing such influence.

Figure 1 shows the circuit of the electrical connections of an NTMI voltage transformer when phase \(A\) of the network is grounded. Here the voltages \(\dot U_{AB}\) and \(\dot U_{AC}\) are shifted relative to one another by an angle of \(60^\circ\). The main secondary winding has a load connected in a star with an isolated neutral (or in delta), \(z_4'\), through which the load current \(\dot I_4'\) flows, and a load connected in a star with grounded neutral, \(R_5'\) (usually these are insulation-monitoring voltmeters), through which the load current \(\dot I_5'\) flows. The sum of the load currents flows through the windings:

\[ \begin{aligned} \dot I_{2a}' &= \dot I_{4a}' + \dot I_{5a}',\\ \dot I_{2b}' &= \dot I_{4b}' + \dot I_{5b}',\\ \dot I_{2c}' &= \dot I_{4c}' + \dot I_{5c}'. \end{aligned} \tag{1} \]

In the neutral conductor there flows the current \(3\dot I_0'\), equal to the sum of the winding currents of all phases:

\[ 3\dot I_0' = \dot I_{2a}' + \dot I_{2b}' + \dot I_{2c}'. \tag{2} \]

A current \( \dot I_3' \) flows through the additional secondary winding.

The currents \( \dot I_{2a}' \) and \( \dot I_3' \), flowing through the secondary windings of phase \(a\), are transformed into the primary winding of phase \(A\) and close through the circuit \(R_1, X_1\), and \(R_{\mathrm{p}}\). The currents \( \dot I_{1C} \) and \( \dot I_{1B} \) cannot branch off into the primary winding of phase \(A\), since a current source is connected into it.

Let us write the voltage-balance equation for the additional secondary winding:

Fig. 1. Diagram of the electrical connections of an NTMI voltage transformer for a phase-A-to-ground fault in the network: \(R_1\) and \(X_1\)—active resistance and leakage reactance of the primary winding; \(R_2'\) and \(X_2'\)—the same for the main secondary winding, referred to the primary side; \(R_3'\) and \(X_3'\)—the same for the additional secondary winding, referred to the primary side; \(Z_3'\), \(Z_4'\), and \(R_5'\)—load impedances referred to the primary side; \(R_{\mathrm{p}}\)—transition resistance at the fault point.

\[ j\omega w_1 \dot\Phi_C + j\omega w_1 \dot\Phi_B - \dot I_3'(3R_3' + 3jX_3') - \]

\[ - (\dot I_3' + \dot I_{2a}')(R_1 + jX_1) - (\dot I_{1C} + \dot I_{1B} + \dot I_{2a}' + \dot I_3')R_{\mathrm{p}} - 3\dot U_0' = 0, \tag{3} \]

where

\[ \omega \text{— angular frequency,} \]

\[ w_1 \text{— number of turns of the primary winding,} \]

\[ \dot\Phi_C \text{ and } \dot\Phi_B \text{— magnetizing magnetic fluxes in the cores of phases } C \text{ and } B. \]

We eliminate \( \dot\Phi_C \) and \( \dot\Phi_B \) from (3), expressing them in terms of the known parameters of the primary circuit. For this purpose, let us write the voltage-balance equations for the primary circuit:

\[ \dot U_{AC} - \dot I_{1C}(R_1 + jX_1) - j\omega w_1 \dot\Phi_C - (\dot I_{1C} + \dot I_{1B} + \dot I_{2a}' + \dot I_3')R_{\mathrm{p}} = 0, \]

\[ \dot U_{AB} - \dot I_{1B}(R_1 + jX_1) - j\omega w_1 \dot\Phi_B - (\dot I_{1C} + \dot I_{1B} + \dot I_{2a}' + \dot I_3')R_{\mathrm{p}} = 0. \tag{4} \]

From (4) we determine the sum

\[ j\omega w_1 \dot\Phi_C + j\omega w_1 \dot\Phi_B = \dot U_{AC} + \dot U_{AB} - (\dot I_{1C} + \dot I_{1B})(R_1 + jX_1) - \]

\[ - 2(\dot I_{1C} + \dot I_{1B} + \dot I_{2a}' + \dot I_3')R_{\mathrm{p}} = 0 \tag{5} \]

and substitute into (3):

\[ \dot U_{AC}+\dot U_{AB}-(\dot I_{1C}+\dot I_{1B})(R_1+jX_1) -2(\dot I_{1C}+\dot I_{1B}+\dot I'_{2a}+\dot I'_3)R_{\Pi} \]
\[ -\dot I'_3(3R_3+3jX_3)-(\dot I'_3+\dot I'_{2a})(R_1+jX_1) -(\dot I_{1C}+\dot I_{1B}+\dot I'_{2a}+\dot I'_3)R_{\Pi} -3\dot U'_0=0. \tag{3^1} \]

In \((3^1)\) the sum of the voltages \(\dot U_{AC}\) and \(\dot U_{AB}\) is three times the zero-sequence voltage:

\[ \dot U_{AC}+\dot U_{AB}=3\dot U_{01}. \tag{6} \]

The currents \(\dot I_{1C}\) and \(\dot I_{1B}\) are the sum of the magnetizing currents and the secondary load:

\[ \dot I_{1C}=\dot I_{\mu C}+\dot I'_{2C}+\dot I'_3, \]
\[ \dot I_{1B}=\dot I_{\mu B}+\dot I'_{2b}+\dot I'_3, \tag{7} \]

where

\[ \dot I_{\mu C}\ \text{and}\ \dot I_{\mu B} \]

are the magnetizing currents of phases \(C\) and \(B\).

The sum of the currents flowing through the main secondary windings is three times the secondary zero-sequence current \(3\dot I'_0\):

\[ \dot I'_{2a}+\dot I'_{2b}+\dot I'_{2c}=3\dot I'_0. \tag{8} \]

Then, after simple transformations, equation \((3^1)\) takes the form:

\[ 3\dot U_{01}-(\dot I_{\mu CB}+3\dot I'_0+3\dot I'_3)(R_1+3R_{\Pi}+jX_1) \]
\[ -3\dot I'_3(R_3+jX_3)=3\dot U'_0, \tag{9} \]

where

\[ \dot I_{\mu CB}=\dot I_{\mu C}+\dot I_{\mu B}. \]

Equation (9) corresponds to the equivalent circuit (Fig. 2) as for a single-phase transformer. The currents \(3\dot I'_3\) and \(3\dot I'_0\) are determined by the corresponding loads. Determination of the magnetizing currents \(\dot I_{\mu C}\) and \(\dot I_{\mu B}\) is associated with calculation of the branched nonlinear magnetic circuit of the voltage transformer (Fig. 3). The equivalent circuits of the magnetic circuits are different when there is a ground fault of the phase whose winding is located on the middle main limb, and of the phase whose winding is located on an outer main limb (Fig. 3, b and c). Therefore, the magnetizing currents will also be different for different cases. In the calculation, the specified quantities are the magnetic fluxes of the unfaulted phases; in the case considered by us these are the fluxes \(\dot\Phi_C\) and \(\dot\Phi_B\). The magnetic flux of the faulted phase is equal to zero. With sufficient accuracy it may be assumed that the fluxes \(\dot\Phi_C\) and \(\dot\Phi_B\) are specified by the voltages \(\dot U_{AC}\) and \(\dot U_{AB}\):

\[ \dot U_{AC}=j\omega w_1\dot\Phi_C, \]

\[ \dot U_{AB}=j\omega w_1\dot\Phi_B. \]

The magnetic resistance \(R_i\) is determined by the dynamic magnetic permeability \(\mu_i\), which is a nonlinear function of the magnetic flux in the \(i\)-th section. Therefore an exact calculation presents a certain difficulty. The required accuracy can be achieved by carrying out the calculation by the method of successive approximations. As a zeroth approximation, the magnetic permeability may be taken as equal to some average value, the same in all sections.

In accordance with the equivalent circuit of Fig. 2, the angular error of the zero-sequence voltage is determined [2] by the formula

\[ \delta=\delta_{\mu}+\delta_{0}+\delta_{3}, \tag{10} \]

where \(\delta_{\mu}\) is the error due to the magnetizing current \(I_{\mu CB}\), equal to

\[ \delta_{\mu}= \frac{I_{\mu CB}(R_1+3R_{\mathrm{п}})}{3U_{01}}\cos\alpha - \frac{I_{\mu CB}X_1}{3U_{01}}\sin\alpha = \]

\[ =\delta_{\mu a}-\delta_{\mu p}, \tag{11} \]

where \(\alpha\) is the loss angle of the magnetic-core material. Expression (11) consists of two terms. The first term is the angular error \(\delta_{\mu a}\), caused by the active resistance when the full magnetizing current \(I_{\mu CB}\) passes through it—hereinafter the error \(\delta_{\mu a}\) is called the active error due to the magnetizing current. The second term is the angular error \(\delta_{\mu p}\), caused by the reactive resistance when the full magnetizing current \(I_{\mu CB}\) passes through it—hereinafter the error \(\delta_{\mu p}\) is called the reactive error due to the magnetizing current.

Fig. 2. Equivalent circuit of an NTMI voltage transformer for determining the angular error of the zero-sequence voltage

\(\delta_0\) is the error due to the zero-sequence current \(3I'_0\) flowing through the main secondary winding, equal to

\[ \delta_0= \frac{3I'_0(R_1+3R_{\mathrm{п}})}{3U_{01}}\sin\varphi_0 - \frac{3I'_0X_1}{3U_{01}}\cos\varphi_0 = \]

\[ =\delta_{0a}-\delta_{0p}, \tag{12} \]

where \(\cos\varphi_0\) is the cosine of the angle of the load connected in star with a grounded neutral; \(\delta_{0a}\) and \(\delta_{0p}\) are, analogously to (11), the active and reactive angular errors due to the current \(3I'_0\); \(\delta_3\) is the error due to the current \(I'_3\) flowing through the additional winding, equal to

\[ \delta_3= \frac{3I'_3(R_1+3R_{\mathrm{п}}+R'_3)}{3U_{01}}\sin\varphi_3 - \frac{3I'_3(X_1+X'_3)}{3U_{01}}\cos\varphi_3 = \]

\[ =\delta_{0a}-\delta_{0p}, \tag{13} \]

where \(\cos \varphi_3\) is the power factor of the load connected to the additional secondary winding;

\(\delta_{3a}\) and \(\delta_{3p}\) are, analogously to (11), the active and reactive angular errors due to the current \(I'_3\).

Fig. 3. For determining magnetizing currents:
a) configuration of the magnetic core of an NTMI;
b) equivalent circuit of the magnetic circuit of an NTMI when the phase whose winding is on the middle main limb is grounded;
c) the same, but for the phase whose winding is on an outer main limb.
\(\Phi_A,\ \Phi_B,\ \Phi_C\) and \(F_A,\ F_B,\ F_C\) are the magnetic fluxes and magnetizing forces in phases \(A,\ B\), and \(C\) of the transformer;
\[ R_i=\frac{l_i}{\mu_i S_i} \]
is the magnetic resistance of the \(i\)-th section; \(l_i,\ S_i\), and \(\mu_i\) are the length of the mean line of force, cross-sectional area, and dynamic magnetic permeability of the \(i\)-th section.

Thus, the total angular error \(\delta\) according to (10) can be represented as the difference between the active and reactive angular errors:

\[ \delta=\delta_a-\delta_p, \tag{10} \]

where

\[ \delta_a=\delta_{\mu a}+\delta_{0a}+\delta_{3a}, \qquad \delta_p=\delta_{\mu p}+\delta_{0p}+\delta_{3p}. \tag{14} \]

Having achieved equality of the active and reactive angular errors, the total angular error can be reduced to zero. This can most simply be implemented by a corresponding choice of the magnitude and character of the load connected to the additional secondary winding.

A change in ambient temperature causes additions to appear in the values of the active resistances of the equivalent circuit of Fig. 2:

\[ \Delta R_i=R_i\gamma_i t, \tag{15} \]

where \(\gamma_i\) is the temperature coefficient of resistance, equal for copper to \(0.004\ 1/\text{degree}\); the value of \(\gamma\) for the contact resistance \(R_p\) depends on many factors.

\(t\) is the magnitude of the temperature change, degrees.

A change in the active resistances according to (15) will not cause a change in the magnitudes of the currents in the equivalent circuit, since these currents are determined by the loads (the active resistances of the equivalent circuit are incommensurably smaller than the load resistances). However, a change in the active resistances according to (15) will cause the appearance of the corresponding addition to the angular errors:

\[ \Delta\delta_\mu= \frac{I_{\mu}(\,)}{3U_{01}} \left(R_1\gamma_1+3R_p\gamma_p\right)t\cos\alpha, \tag{16} \]

\[ \Delta\delta_0= \frac{3I'_0}{3U_{01}} \left(R_1\gamma_1+3R_p\gamma_p\right)t\sin\varphi_0, \tag{17} \]

\[ \Delta\delta_3= \frac{3I'_3}{3U_{01}} \left[(R_1+R'_3)\gamma_1+3R_p\gamma_p\right]t\sin\varphi_3. \tag{18} \]

where \(\gamma_1\) is the temperature coefficient of the resistance of copper, and \(\gamma_{\mathrm{p}}\) is the same coefficient for the transition resistance. The total increment of the angular error is equal to the sum:

\[ \Delta \delta = \Delta \delta_{\mu} + \Delta \delta_0 + \Delta \delta_3 . \tag{19} \]

The reactive angular error \(\delta\) gives no increment to the angular error.

From (17) and (18) it is evident that the increments of the angular errors \(\Delta \delta_0\) and \(\Delta \delta_3\) can be reduced to zero by ensuring the active character of the currents \(3I_0'\), \(3I_3'\). In this case \(\sin \varphi_0 = 0\) and \(\sin \varphi_3 = 0\), and \(\Delta \delta_0 = 0\), \(\Delta \delta_3 = 0\). It should be noted that in reality the current \(3I_0'\) is active, since the zero-sequence load consists of insulation-monitoring voltmeters, for which \(\cos \varphi = 1\).

When \(\Delta \delta_0\) and \(\Delta \delta_3\) are reduced to zero, the total increment of the angular error reaches a minimum:

\[ \Delta \delta_{\min} = \Delta \delta_{\mu} \tag{20} \]

and is determined only by the active component of the magnetizing current.

It should be noted that ensuring the active character of the currents \(3I_0'\) and \(3I_3'\) also reduces to zero the active angular errors \(\delta_{0a}\) and \(\delta_{3a}\), and ensures the maximum value of the reactive angular errors \(\delta_{0p}\) and \(\delta_{3p}\) (see (12) and (13)). The total angular error then reaches its maximum.

The results presented in the article were obtained for the case of a ground fault in phase \(A\). In the event of a ground fault in any other phase, the equivalent circuit of Fig. 2 and its parameters do not change. The magnetizing current in the circuit \(\dot I_{\mu \mathrm{c}}\) is the geometric sum of the magnetizing currents of the unfaulted phases, which depends, as already indicated, on the limb of the magnetic core on which the winding of the faulted phase is located.

Conclusions

Fluctuations in ambient temperature cause fluctuations in the angular error of the zero-sequence voltage of an NTMI voltage transformer. Fluctuations in the angular error can be reduced to a minimum by ensuring the active character of the load connected to the secondary auxiliary winding, and of the load connected in star with grounded neutral, connected to the main secondary winding.

References

1. Signaling of ground faults in compensated networks. Collection of articles edited by V. I. Iozelson. GEI, 1962.
2. Electrical Engineering Handbook. Vol. I, “Energiya,” 1971.