On the Use of Three-Winding Current Transformers for Supplying Operating-Current Circuits

TROFIMOV R. F.

Associate Professor, Candidate of Technical Sciences

The use of storage batteries requires large initial expenditures and careful maintenance. Moreover, storage batteries poorly tolerate the temperature conditions of unheated rooms. If one takes into account that in the near future mine substations will be converted to automatic control, then at surface substations the required capacity of storage batteries will increase considerably, and the need will arise for storage batteries at underground substations. All this will greatly increase the cost of substations. The need to convert mine substations to alternating operating current is also caused by the fact that the existing protections, especially current protections built into KAM boxes, are imperfect. Mines cannot be regarded as non-critical consumers. At present there is a pressing need, even in high-voltage cubicles, to use more accurate relays, such as relays of the IT-81 type, etc.

The use of storage batteries can be abandoned. For this purpose, the closing and tripping devices of circuit breakers must be supplied from current transformers and from low-power power or lighting transformers. Lighting or power transformers, as a source of operating current for feeding relays with make contacts, have one substantial drawback: in the event of a short circuit between phases near the point where the circuit breaker is installed, the latter may fail to operate at all, since the voltage supplied to the closing coil will be insufficient. With relays having break contacts, the circuit breaker will operate for any voltage drop caused by any reason. In general it should be noted that in these cases selectivity is practically unattainable. Lighting and power transformers should be used for closing and tripping purposes, but at normal voltage.

For the purpose of tripping oil circuit breakers during short circuits, more advanced schemes are those that use current transformers as the source of operating current. One method was described by K. L. Plyusnin in the journal Electric Stations, No. 6, 1939. This method requires the use of main current transformers of high power (120–140 VA) with two cores and additional special fast-saturating current transformers of the TKB-1 type, manufactured by the Elektroapparat plant. A second method, simplifying the scheme and providing a large saving in iron and nonferrous metals, proposed by the author of the present article and published in the journal Electric Stations, No. 2, 1940, requires the use of current transformers with two secondary windings placed on one core: one for supplying the protection and the second for the tripping devices of the circuit breaker (Fig. 1).

This will make it possible to achieve selectivity and to have the auxiliary secondary winding normally open. In the study of e.m.f., no more than 40–50 V was obtained between the open ends of the additional winding of a TP-type current transformer. Such a current transformer, only when the protection operates (the operating-current circuit is closed),

Fig. 1

switches over to the mode of a three-winding current transformer. Both secondary windings, being seated on one core, influence one another through their leakage fields, as a result of which the internal resistances and the voltage drops in them increase.

Fig. 2

The equivalent circuit of a three-winding current transformer can be expressed by three complete impedances \(z_1\), \(z_2\), and \(z_3\), connected in a star at point 0 (Fig. 2). Such a circuit should be understood not as a vector diagram with the place of electrical connection at point 0, but as a single-conductor connection diagram in which points 1, 2, and 3 belong to one phase. The impedances \(z_1\), \(z_2\), and \(z_3\) can be determined experimentally by pairwise testing of the windings. In this case

\[ \dot z_{12}=\dot z_1+\dot z_2, \]

\[ \dot z_{23}=\dot z_2+\dot z_3, \]

\[ \dot z_{31}=\dot z_3+\dot z_1; \]

whence

\[ \dot z_1=\frac{1}{2}\,(\dot z_{12}+\dot z_{31}-\dot z_{23}),\qquad \dot z_2=\frac{1}{2}\,(\dot z_{12}+\dot z_{23}-\dot z_{31}), \]

\[ \dot z_3=\frac{1}{2}\,(\dot z_{23}+\dot z_{31}-\dot z_{12}). \]

The complete equivalent circuit is shown in Fig. 3.

When current transformers with two secondary windings are used, the current in the relay winding, after the latter has operated, must not be less than the return current, since otherwise the oil circuit breaker may fail to trip. The latter requires the use of low-power tripping coils for MV switches.

The existing maximum relays of automatic boxes KAM (VM-14) consume a tripping power of 50 VA. As a result of investigations and changes in the parameters of the tripping device, it proved possible to reduce the tripping power of the MV to 5.7 VA \((u = 9\ \text{V},\ I = 0.63\ \text{A})\). This made it possible (as verified experimentally) to supply simultaneously the maximum relay and the tripping coil from a single current transformer.

Fig. 3

For the tripping coil, the relation between current and voltage, as is known, is expressed by the following formula:

\[ I = \frac{u}{ \sqrt{ R^{2}+ \left[ \frac{0.4\pi w\omega 10^{-8}} {\,l\left(\dfrac{2}{k}+\dfrac{s}{l}\cdot\dfrac{1}{g_{e}+\dfrac{g\cdot l}{3}}\right)} \right]^{2} } }, \]

where: \(R\) is the active resistance of the coil, \(w\) is the number of turns in the coil, \(s\) is the cross-sectional area of the core in \(\text{cm}^{2}\), \(\omega\) is the angular frequency, \(l\) is the length of the coil, \(k=\dfrac{B}{H}\), \(g_{e}\) is the magnetic conductance of the working air-gap flux, and \(g\) is the conductance of leakage fluxes per 1 cm length of the magnetic circuit.

Knowing \(I\), one can find the value of the magnetic induction

\[ B = \frac{0.4\pi w I} {l\left[ \dfrac{2}{k}+ \dfrac{s}{l\left(g_{e}+\dfrac{g l}{3}\right)} \right]} \]

and the magnetic-field strength

\[ H = \frac{B} {\left(g_{e}+\dfrac{g l}{3}\right)\dfrac{l}{s}}. \]

Tangential forces

\[ P=\frac{(Hl)^2}{8\pi}\cdot \frac{dg_x}{db}\cdot \frac{1}{981}\ \text{grams}. \]

The values of the derivative of the permeance along the armature path and the parameters of the coil can be determined by the method of Doc. B. S. Sotskov, published in the journal Avtomatika i telemekhanika, No. 2, 1940.

Let us determine the current flowing through the tripping coil. From the equivalent circuit shown in Fig. 3 it follows that

\[ \dot E_0=\dot I_2(\dot z_2+\dot z_{\text{relay}}) \]

and

\[ \dot E_0=\dot I_3(\dot z_3+\dot z_{\text{coil}}); \]

equating these two equations,

\[ \dot I_2(\dot z_2+\dot z_{\text{relay}})=\dot I_3(\dot z_3+\dot z_{\text{coil}}), \tag{a} \]

where \(\dot I_3\) is the geometric difference between the currents of the primary winding and of the winding to which the relay is connected,

\[ \dot I_3=\dot I_1-\dot I_2; \]

we rewrite this relation in the following form:

\[ \dot I_2=\dot I_1-\dot I_3 \]

and substitute it into equation (a):

\[ (\dot I_1-\dot I_3)\cdot(\dot z_2+\dot z_{\text{relay}}) = \dot I_3(\dot z_3+\dot z_{\text{coil}}). \tag{b} \]

As indicated above,

\[ z_{\text{coil}} = R+j\left[ l\left( \frac{0.4\pi w s \omega\cdot 10^{-8}} {\frac{2}{k}+\frac{s}{l}-\frac{1}{g_e+\frac{g\cdot l}{3}}} \right) \right]. \]

\(\dot z_{\text{relay}}\), \(\dot z_2\), and \(\dot z_3\) can be determined by known methods.

Introduce the notation

\[ \dot z_2+\dot z_{\text{relay}}=\dot z'_2 \]

\[ \dot z_3+\dot z_{\text{coil}}=\dot z'_3 \]

and substitute into equation (b):

\[ (\dot I_1-\dot I_3)\cdot \dot z'_2=\dot I_3\cdot \dot z'_3 \]

or

\[ \dot I_1\cdot \dot z'_2=\dot I_3(\dot z'_2+\dot z'_3), \]

whence the current flowing through the tripping coil is determined as

\[ \dot I_3=\dot I_1\frac{\dot z'_2}{\dot z'_2+\dot z'_3}. \]

When switching to the mode of a three-winding transformer, current transformers acquire additional errors. To clarify this question, let us construct an ampere-turn diagram (Fig. 4). The ampere-turns \(AW'_2\) of the winding of the current transformer that feeds the relay protection, after its operation, decrease and may acquire an additional phase shift relative to the emf \(\dot E_2\). This will affect the errors of the current transformer.

Let us determine the current error for the winding feeding the protection. The difference between the primary and secondary ampere-turns of winding 2 can be expressed by the segment \(AD\), which is the sum of three segments: \(DE\), \(Eg\), and \(Ag\), i.e.

\[ AD = DE + Eg + Ag, \]

where: the segment \(DE = OC - OC \cdot \cos \delta = AW_1 - AW_1 \cdot \cos \delta = AW_1(1-\cos \delta),\)

the segment \(Eg = AW_0 \cdot \sin(\alpha+\varphi),\)

and the segment \(Ag = AW_3 \cdot \cos \gamma;\)

therefore, the segment

\[ AD = AW_1(1-\cos \delta) + AW_0 \cdot \sin(\alpha+\varphi) + AW_3 \cdot \cos \gamma. \]

Fig. 4

The current error can be expressed as the ratio of the difference between the primary and secondary ampere-turns of winding 2 to the primary ampere-turns, i.e.

\[ \Delta I\% = \left[ (1-\cos \delta) + \frac{AW_0}{AW_1}\cdot \sin(\alpha+\varphi) + \frac{AW_3}{AW_1}\cdot \cos \gamma \right]\cdot 100; \]

in view of the smallness of the angle \(\delta\), we take

\[ 1-\cos \delta \simeq 0, \]

then

\[ \Delta I\% = \left[ \frac{AW_0}{AW_1}\cdot \sin(\alpha+\varphi) + \frac{AW_3}{AW_1}\cdot \cos \gamma \right]\cdot 100. \]

The angular error is found from the following relations:

\[ \sin \delta = \frac{CE}{OC}, \]

where

\[ CE = CF + FE = AW_0 \cos(\alpha+\varphi) + AW_3 \cdot \sin \gamma, \]

and

\[ OC = AW_1. \]

Consequently,

\[ \sin \delta = \frac{ AW_0 \cdot \cos(\alpha+\varphi) + AW_3 \cdot \sin \gamma }{ AW_1 }. \]

Assuming, in view of the smallness of the angle, that \(\sin \delta \simeq \delta\),

\[ \delta = \frac{ AW_0 \cdot \cos(\alpha+\varphi) + AW_3 \cdot \sin \gamma }{ AW_1 }, \]

or, expressing \(\delta\) in minutes,

\[ \delta' = 3440\, \frac{ AW_0 \cdot \cos(\alpha+\varphi) + AW_3 \cdot \sin \gamma }{ AW_1 } \text{ minutes}. \]

To the known formulas for determining errors for two-winding current transformers it may be added that the errors of three-winding current transformers are directly proportional to the secondary ampere-turns of winding 3 and to $\cos$ (for current errors) or $\sin$ (for angle errors) of the angle between the vectors of the ampere-turns of secondary windings 2 and 3.

These errors will have to be taken into account, especially in schemes of differential and earth-fault protections.

To determine the electrodynamic stability of three-winding current transformers, one may use the known methods for determining the forces of interaction between the windings of three-winding transformers. These calculation methods are not given here. But from them it is known that in three-winding transformers the mechanical forces between windings are 2 or 4 times smaller than in two-winding transformers. The same is also true for three-winding current transformers.

Thermal stability is of no particular interest, since upon transition to the mode of a three-winding current transformer the current in the secondary winding feeding the relay decreases and in general quickly disappears, owing to the tripping of the oil circuit breaker. For this reason thermal stability is not considered.

Conclusion

1. Three-winding current transformers do not require an increase in dynamic and thermal stability.

2. For the purposes of supplying protection and tripping devices of oil circuit breakers, the rating of three-winding current transformers will be less than that of two-winding transformers with additional current transformers of the TKB-1 type.

3. With three-winding current transformers, the scheme of electrical connections is simplified, and the dimensions and weight of measuring transformers are reduced, which will make it possible to use them also in high-voltage cubicles.

4. Rewinding current transformers and changing the parameters of tripping devices can be carried out by local personnel at the sites, by calculation on the basis of the relations derived above.

5. In the future it is necessary to investigate the possibility of operation of three-winding current transformers in other protection schemes as well.