On the Long-Term Operation of Current Transformers with an Open Secondary Winding

L. V. Baginskii

(Presented by Professor, Doctor of Technical Sciences
I. D. Kutyavin)

In connection with the question of applying a simplified scheme of longitudinal differential protection with balancing of the secondary e.m.f.’s (without a current-circulation circuit) on generators, there arises the need for long-term operation of current transformers under no-load conditions. There are many indications that such an operating mode is inadmissible for current transformers. However, all these indications refer to transformers with a large volume of core steel. Since the current transformers of the above-mentioned protection do not require a large volume of steel, it is possible to find such conditions under which the mode of long-term operation of current transformers with an open secondary winding is permissible.

Fig. 1.

As is known, the mode under consideration is associated with the occurrence in the secondary winding of considerable peak voltages, dangerous for the insulation of the secondary circuits and also for the operating personnel, and with strong heating of the core, which may disable the insulation of the secondary windings. In addition, the suggestion has been made that an increase in the core temperature may lead to a loss of the magnetic properties of the steel [1]. Investigations by L. Sh. Kazarnovskii [2] refuted this assumption. Let us also note that the losses in modern transformer steel are 2–3 times smaller than those cited in [1], and that reducing the cross section of a ring-shaped core at unchanged internal diameter leads to an increase in the ratio of surface area to volume, improving the cooling conditions of the transformer.

To estimate the degree of heating of cores at various currents, several experiments were carried out with both spiral and laminated cores. Here a description is given of an experiment conducted under the most unfavorable conditions. A core with a comparatively large volume of steel, \(q = 17.5 \text{ cm}^2\), taken from a current transformer of type TPOF without any alterations, was tested. The cross section of the core was deliberately taken as oversized relative to the needs of the protection. Core 1 (Fig. 1) was enclosed in a compacted casing of sheet iron 2 and mounted on a cardboard sleeve 3. The sleeve was fastened on a wooden stand 4, which was intended to reduce the influence of the thermal field of the primary winding 5, containing 325 turns.

The ambient temperature was \(+25^\circ\mathrm{C}\). Measurements were made with the aid of two differential thermocouples, the hot junctions of which were placed between the strips of the core and on the inner surface of the torus. The steady-state temperature regime was established \(6\)—\(7\) h after the magnetizing current was applied. The results of the experiment are presented in Fig. 2 in the form of the dependence of the core temperature \(t_c\) and of the inner surface of the torus \(t_n\) on the specific magnetizing force.

In this experiment it was not possible to create operating conditions for the current transformer under test that were sufficiently close to normal service conditions. In view of the high inductance and nonlinearity of the transformer under test, the magnetizing force was nonsinusoidal. Thus, at the time of measurement, for \(\alpha w_n = 51\) ampere-turns/cm, the content of the first harmonic in the magnetizing current was \(77.5\%\), and of the third, \(20.8\%\). In addition, it was not possible to eliminate the influence of the thermal field of the primary winding. If under service conditions the temperature of the primary winding does not exceed \(70^\circ\mathrm{C}\), in the experiment it reached \(280^\circ\mathrm{C}\). However, both of these factors act in the direction of overestimating the temperature relative to the actual value and create a certain margin of reliability.

As can be seen from Fig. 2, at \(\alpha w = 60\) ampere-turns/cm the temperature of the core surface did not exceed \(125^\circ\mathrm{C}\). Consequently, when the secondary winding is made of PSD-grade wire with a briefly permissible temperature of \(180^\circ\mathrm{C}\), current transformers with an open secondary winding can operate for a long time on any generators currently being manufactured. In fact, the rated current of modern generators does not exceed \(I_n < 6000\) A, and the length of the mean magnetic line along the core for modern generator-voltage current transformers reaches \(l_{\mathrm{av}} \approx 100\) cm.

Experience has shown that a brief significant increase in magnetizing current (of a duration on the order of several seconds) does not affect the temperature of the core. When current flows through the secondary winding (the case of a short circuit in the protection zone), it heats up starting from the temperature of the preceding regime. The excess of the permissible temperature over the actual core temperature by \(55^\circ\mathrm{C}\) ensures the thermal stability of the secondary winding in this regime. For the same reasons, at \(I_n \leq 3000\) A it is permissible to use the cheaper PDA-grade wire. Since the heating of laminated cores is considerably less than that of spiral cores, when the former are used it is permissible to make the secondary winding with PDA-grade wire for any rated currents.

Fig. 2.

Thus, from the standpoint of the temperature regime of a current transformer intended for long-term operation with an open secondary winding, no changes in the design of existing transformers are required. The use of more expensive wire for making the secondary winding¹ is more than offset by the shorter length of the turn, as well as by the smaller number of turns of the secondary winding (see below).

The amplitude of the secondary emf of ordinary current transformers with an open secondary winding at the moment when a short-circuit current flows may reach tens of kilovolts [1]. The theoretical possibility of reducing the amplitude of the secondary emf, \(E_{2m}\), to any desired value consists in reducing the total flux penetrating the secondary winding, with a corresponding change in the quantities \(q\) and \(w_2\), i.e.,

\[ E_{2m}=q w_2 e_{2m}, \tag{1} \]

where \(e_{2m}\) is the specific amplitude of the secondary emf per one turn of the secondary winding and per \(1\ \text{cm}^2\) of core cross section.

However, \(q\) and \(w_2\) cannot be decreased without limit, since, on the one hand, a sufficiently high output of the transformers into the secondary circuit during internal short circuits must be ensured, and, on the other hand, it is difficult to make cores of very small cross section in view of the considerable dimensions of the primary winding and its insulation, which necessitates large values of \(l_{\mathrm{cp}}\). Thus, for a specified permissible value of \(E_{2m}\), the possibility of operating current transformers with an open secondary winding is determined by the value of \(e_{2m}\).

As a briefly permissible value of \(E_{2m}\), one may take \(1000\ \text{V}\) amplitude. In fact, there is extensive operating experience with longitudinal differential protection of generators using class D transformers with added resistances of \(10\text{–}20\ \Omega\) in the relay circuit, which inevitably must have caused a sharp increase in voltage (more than \(1000\ \text{V}\) amplitude) in the protection circuits during a short circuit in the protected zone, when the effective value of the current in the relay often exceeds \(50\ \text{A}\). At the same time, in the technical literature there are no data indicating that this has led to damage to the insulation of the protection circuits or to an accident. Moreover, for secondary protection circuits (according to the Electrical Installation Rules, PTE), a sinusoidal voltage of \(1000\ \text{V}\) effective is a test voltage.

The value of \(e_{2m}\) depends on the magnetizing current. However, in the technical literature at present there is very little information on the nature of the dependence \(e_{2m}=f(aw_{\mu})\), where \(aw_{\mu}\) is the amplitude of the specific magnetizing force of the core under sinusoidal magnetizing current.

A. A. Voskresenskii [3] believes that this dependence is easily expressed analytically:

\[ E_{2m}=5.32\,\frac{1}{a}cf\left(\ln \frac{n!}{m_1}-1.63\right), \tag{2} \]

where \(c=2\pi q w_2 10^{-8}\), and \(f\) is the frequency of the magnetizing current.

Expression (2) was obtained by summing the amplitudes of the higher-harmonic emfs determined on the basis of harmonic analysis of a function of the form:

\[ B=\frac{1}{a}\ln \frac{H}{m}, \tag{3} \]

which replaces the main magnetization curve.

It follows from this that the main magnetization curve and the dependence \(e_{2m}=f(aw_{\mu})\), on a certain scale, must repeat one another, since both of these dependences are logarithmic. Such a character of the dependence \(e_{2m}=f(aw_{\mu})\), when \(e_{2m}\) increases considerably more slowly than \(aw_{\mu}\), is very advantageous from the standpoint of the possibility of attaining permissible values of \(E_{2m}\). In Fig. 3, for one and the same specimen, the results of determining the dependence \(e_{2m}\) are presented

experimentally (curve 1) and by calculation from expression (2) (curve 2). The reason for the enormous underestimation of the calculation results lies in the inaccurate approximation of the principal magnetization curve in the region of small values of \(H\). Indeed, at \(H=0\), according to equation (3), \(B=\frac{1}{a}\), whereas it should be \(B=0\). At the same time, it is precisely in the region of small values of \(H\) that the maximum value of \(\frac{d\psi}{dt}\) occurs, and hence also the maximum value of \(E_{2m}\). As a result, the use of such a method of replacing the magnetization curve in determining the amplitude of the secondary e.m.f. is fundamentally erroneous.

L. A. Bessonov [4] finds the amplitude of the secondary e.m.f. of a peak transformer operating with sinusoidal current by replacing the magnetization curve with the function

\[ B=\frac{1}{\beta}\operatorname{Arsh}\frac{H}{\alpha}. \tag{4} \]

Then the secondary e.m.f. is equal to:

Fig. 3.

\[ e_2=-\frac{d\psi}{dt} = -\frac{w_2 q \omega 10^{-8}}{\beta}\cdot \frac{H_m}{\alpha}\, \frac{\cos \omega t}{\sqrt{1+\frac{H_m^2}{\alpha^2}\sin^2 \omega t}} . \]

The author assumes that \(e_2=E_{2m}\) at \(t=0\), i.e.

\[ E_{2m}=\frac{w_2 q \omega H_m 10^{-8}}{\beta \alpha}. \tag{5} \]

It follows from this that the dependence \(e_{2m}=f(a w_m)\) must be linear.

In Fig. 3 this dependence is shown by straight line 3. In modern current transformers the value of the amplitude of the specific magnetizing force can reach \(a w_m \simeq 700\) ampere-turns/cm during short circuits in the primary circuit. From Fig. 3 it follows that at this value \(e_{2m}\) may reach \((80\text{--}100)\) V/turns·cm\(^2\). Consequently, the product of the variable parameters should be:

\[ q w_2 \leq (10\text{--}12)\ \text{turns}\cdot\text{cm}^2, \]

which is completely inadmissible. Therefore, current transformers intended for operation with an open secondary winding would in many cases be impossible to manufacture without the use of special materials or a change in design.

However, the table of values of the function \(E_{2m}=f(I_{1m})\) [1], found experimentally under operating conditions, as well as the experimental results represented by curve 1 in Fig. 3, testify to the nonlinearity of the function under consideration. It follows from this that either the experiment was performed incorrectly, or some substantial error was made in deriving expression (5).

Particular mistrust is caused by the possibility that a strictly sinusoidal magnetizing force existed in this experiment. A deviation of the form of the magnetizing force from sinusoidal can cause a substantial change in the magnitude of \(e_{2m}\), since \(e_2\) has a sharply peaked character (the 11th harmonic is contained only in a slightly smaller quanti-

...than the first). The non-sinusoidal nature of the magnetizing force may be caused by the non-sinusoidal nature of the primary current owing to the nonlinearity of the current transformer resistance, and also by the presence of a secondary non-sinusoidal current consumed by the voltmeter.

In the experiment, the possible influence of these factors was taken into account. Therefore an electronic voltmeter was adopted as the measuring instrument. The current was regulated by means of a powerful choke with an adjustable air gap, and the cross section of the current-transformer core was \(q = 0.875\ \text{cm}^2\) with \(\omega_1 = 182\) turns. At the same time, however, the transformer resistance due to the maximum value of the dynamic magnetic permeability, \(\omega M_{\partial m}\), reached 10% of the total inductive resistance of the circuit at considerable \(a\omega_m\). Nevertheless, at small \(a\omega_m\) a deviation of the dependence under consideration from linearity is also observed, although to a lesser degree. In addition, the above-mentioned data from [1] refer to a transformer with \(\omega_1 = 1\), as a result of which the influence of the transformer on the primary circuit was, undoubtedly, negligible. Therefore the deviation of the dependence under consideration from linearity cannot be explained solely by distortion of the primary-current waveform.

To clarify the true character of the dependence \(e_{2m} = f(a\omega_m)\), it is necessary to show that equation (5) does not take into account some factor which determines its deviation from linearity. In deriving equation (5), the phenomenon of hysteresis was not taken into account. To estimate the influence of this factor on the result, let us find \(E_{2m}\) without using an approximation of the magnetization curve:

\[ e_2 = -\frac{d\psi}{dt} = -\frac{d\psi}{di_1}\cdot \frac{di_1}{dt} = -\omega M_\partial I_{1m}\cos \omega t, \tag{6} \]

since \(i_1 = I_{1m}\sin \omega t\). The amplitude value of the secondary e.m.f. will be observed approximately at \(t = 0\), when the dynamic magnetic permeability assumes its maximum value, i.e. \(M_\partial = M_{\partial m}\). Then

\[ E_{2m} = \omega M_{\partial m} I_{1m}. \tag{7} \]

To prove the latter, let us examine expression (6) for a maximum, which reveals the following condition for the existence of a maximum value of the secondary e.m.f.:

\[ t_m = \frac{1}{\omega}\arccos \frac{\omega M_\partial} {\sqrt{\left(\dfrac{dM_\partial}{dt}\right)^2 + (\omega M_\partial)^2}}. \tag{8} \]

Expression (8) shows that, strictly speaking, \(t_m \ne 0\), since this would be possible only if the instants \(i_1 = 0\) (since \(t = 0\)) and \(M_\partial = M_{\partial m}\) coincided (because \(\dfrac{dM_\partial}{dt} = 0\)). The latter condition is not fulfilled because of the presence of hysteresis.

Figure 4 gives a graphical construction of the factors of expression (7). From this construction it follows that \(t_m \le t_g\), where \(t_g\) is the time determined by the hysteresis angle \(\varphi_g\). In view of the smallness of \(\varphi_g\) at high inductions, one may take \(\cos \omega t_m \cong 1\).

Thus, since the quantity \(\mu_{\partial m}\) is a constant for the given grade of steel, the identity of expressions (7) and (5) is entirely obvious. Consequently, allowance for the phenomenon of hysteresis cannot cause the deviation of the dependence \(e_{2m} = f(a\omega_m)\) from linearity. It therefore remains only to assume that, with a sinusoidal primary current, the magnetizing force is not sinusoidal.

In Fig. 5, the numerals 1 and 2 denote magnetization curves taken with a sinusoidal magnetizing current for two identical in size...

measures of the laminated cores of current transformers. The first of them was built up from steel with a sheet thickness of 0.5 mm, and the second—0.35 mm. The numerals 1′ and 2′ denote the dependences \(e_{2m}=f(aw_m)\), determined for these transformers. Despite the fact that the quality of the core steel of the first specimen is considerably higher than that of the second, curve 2′ deviates from a straight line considerably less than curve 1′. As a result, the values of \(e_{2m}\) of the first specimen exceed the corresponding values of the second specimen only at small \(aw_m\). Since the first specimen differs from the second by increased eddy currents in the core, it may be assumed that it is precisely the eddy currents that have the decisive influence on the character of the dependence \(e_{2m}=f(aw_m)\).

Fig. 4.

This assumption is confirmed by the following experiment. In Fig. 6, a and v are photographs of the hysteresis loops of the same core that was used in the two preceding experiments (sheet thickness 0.35 mm) at specific magnetizing forces \(aw_m=11.8\) ampere-turns/cm and \(aw_m=27\) ampere-turns/cm, respectively. It might have been expected that the maximum values of \(\dfrac{dB}{dH}\) in both cases would be the same, since the maximum value of the dynamic magnetic permeability is a constant of the given steel. However, these values proved to be proportional to 7.5 and 5.5, respectively¹. Such a result could not have been obtained through the influence of the nonsinusoidal magnetizing force of the secondary winding or through nonsinusoidality of the primary current. In fact, the amplitude value of the magnetizing force of the secondary winding did not exceed 0.04 ampere-turns/cm, and the nonsinusoidality of the primary current cannot affect the form of the hysteresis loop at all, since the hysteresis loops were photographed from the screen of an electronic oscilloscope—

Fig. 5.

oscilloscope, connected in the usual circuit with \(R\) and \(C\). Any change in the shape of the signal of the magnetizing force on the deflecting system will, through the hysteresis loop, cause a corresponding change in the shape of the current signal on the deflecting system, so that the shape of the hysteresis loop will remain unchanged. But the primary-current shunt is not traversed by eddy currents. Therefore, without affecting the shape and magnitude of the primary magnetizing force, the eddy currents somehow affect the resultant magnetizing force, equal to the difference between the magnetizing force of the primary winding and the magnetizing force of the eddy currents, causing distortion of the current signal. These considerations are confirmed by the results of the following experiment. On the core used in the preceding experiment an additional winding was placed, closed on an active resistance. The presence of current in this winding is equivalent (from the point of view of its effect on the shape of the hysteresis loop) to an increase of the eddy currents. Oscillograms of the hysteresis loop, taken under these conditions and at \(aw_m = 11.8\) ampere-turns/cm and \(aw_m = 27\) ampere-turns/cm, are shown in Fig. 6, б and г, respectively. From the oscillograms it is seen that the maximum value of \(\dfrac{dB}{dH}\) in the presence of such a current in the additional winding, for which the specific magnetizing force of the additional winding is \(aw_z = 0.15\) ampere-turns/cm effective, decreased to 5.5 and 4, respectively, while the values of \(e_{2m}\) proved to be equal to \(0.152\) V/turns·cm\(^2\) and \(0.26\) V/turns·cm\(^2\), whereas according to curve 2 of Fig. 5, at \(aw_z = 0\), these values were \(0.191\) V/turns·cm\(^2\) and \(0.37\) V/turns·cm\(^2\).

One more detail attracts attention, indicating the close connection of the eddy currents with the character of the dependence \(e_{2m} = f(aw_m)\) through the shape of the hysteresis loop. The change in the value of \(e_{2m}\) approximately corresponds to the change in the maximum value of \(\dfrac{dB}{dH}\). Indeed, when the primary current is increased by a factor of 2.3 (the case \(aw_z = 0\)), the maximum value of \(\dfrac{dB}{dH}\) decreases by a factor of 1.36. Therefore one would expect \(e_{2m}\) to increase by \(2.3 : 1.36 \simeq 1.7\) times, whereas in reality \(e_{2m}\) increases by a factor of 1.93. Similarly, when the additional winding is connected and \(aw_m = 11.8\) ampere-turns/cm, \(e_{2m}\) should decrease by \(7.5 : 5.5 = 1.36\) times, but it decreased by \(0.191 : 0.152 = 1.25\) times. At \(aw_m = 27\) ampere-turns/cm it should be \(5.5 : 4 \simeq 1.37\) times, while the observed value is \(0.37 : 0.26 = 1.42\) times.

а)             б)

в)             г)

Fig. 6.

Despite all that has been said, it remains unclear how the very small magnetizing force of the eddy currents can reduce the value of \(e_{2m}\) to such a large extent. The point is that the eddy currents will have the form and phase of an emf \(e_2\), since the resistances of the eddy-current circuits are active, and the currents themselves are caused by an emf induced by the same flux as \(e_2\). This means that, at very small effective values, the amplitude of the magnetizing force of the eddy currents will be sufficiently large to distort

the form of the resultant magnetizing force of the current transformer. At the same time, the instants at which the maxima of the magnetizing force of the eddy currents and of the emf \(e_2\) occur coincide, i.e., the greatest distortion of the resultant magnetizing force will occur precisely at the instant when the maximum emf \(e_2\) occurs. This explanation is fully confirmed by experiment, the results of which are the oscillograms shown in Fig. 7. The experiment was carried out in the same manner as the preceding one.

Curves 1 show the variation with time of the magnetizing force of the current transformer, taking into account the influence of eddy currents simulated by the circuit of an additional winding with number of turns \(w_3=91\). For this purpose the shunt of the loop recording these curves is connected to the difference of currents \(i_1-i_3\), with \(w_1=w_3\). Curves 2 were recorded by a highly sensitive loop connected to a separate winding with number of turns \(w_2=20\), and show the form of the emf \(e_2\). Oscillogram \(b\) in Fig. 7 was recorded at \(I_1=5\) A and a magnetizing force simulating eddy currents of \(Iw_3=20\) ampere-turns, while oscillogram \(a\) was taken under the same conditions, but with \(Iw_3=0\). On curve 1 of oscillogram \(b\) it is clearly seen that, when \(e_2=E_{2m}\), the form of the magnetizing force is sharply distorted, whereas at other instants this distortion is absent. When \(I_3=0\), the magnetizing force is purely sinusoidal (see oscillogram \(a\) in Fig. 7). Consequently, the distortion of the form of the magnetizing-force curve is caused by the flow of the eddy current \(I_3\). As a result, the value \(E_{2m}\) when \(I_3=0\) is 1.45 times greater than when \(Iw_3=20\) ampere-turns.

The experiments described above show that the nonlinear relation between the amplitude of the secondary emf and the primary sinusoidal current, given in [1], and also obtained in the author’s experiments, is quite regular.

Owing to the presence of eddy currents in the core of current transformers, the dependence \(e_{2m}=f(aw_m)\) cannot be linear.

Since the calculation methods for \(E_{2m}\) considered above do not give satisfactory results, the parameters of current transformers should be calculated using the dependence \(e_{2m}=f(aw_m)\), determined experimentally for a sinusoidal primary current. In doing so, it should be borne in mind that such dependences will differ from one another not only for cores made of different grades of steel, but also for cores made of steel of different thicknesses, or for cores of different types (laminated, spiral).

Fig. 7.

It should also be noted that, even for laminated cores made of 0.35 mm steel, the values of \(e_{2m}\) do not exceed \((2—2.5)\) V/turn·cm\(^2\) at \(aw_m\) up to 400 ampere-turns/cm. Therefore, at the greatest short-circuit currents possible for generators, the maximum permissible value of the product \(qw_2\) will be not less than \((250—300)\) turns·cm\(^2\). Such a value of the product of the desired parameters will not constitute an obstacle in manufacturing current transformers for differential protection of generators.

When designing current transformers for operation with an open secondary winding, it is desirable to attain permissible values of \(E_{2m}\), first of all, by reducing \(q\), since this reduces not only \(E_{2m}\), but also the interturn voltages and lowers the temperature of the core, which leads to increased reliability of operation of the current transformer.

Thus, current transformers intended for long-term operation with an open secondary winding can be manufactured for installation on any generators. In this case, the cross section of the spiral cores should not exceed \(17.5\ \text{cm}^2\), and the secondary winding, for \(I_n \geq 4000\ \text{A}\), should be made of PSD-grade wire. At lower values of \(I_n\), as well as with laminated cores, the use of PDA-grade wire is permissible. The somewhat higher cost of the winding wire is substantially offset by the reduction in the values of \(q\) and \(w_2\), owing to which such current transformers will be considerably cheaper than class D current transformers. The consumption of steel and copper is reduced by approximately a factor of 5–10.

Literature

1. S. S. Gorodetskii, Measurements at High Voltage, Energoizdat, 1934.

2. L. Sh. Kazarnovskii, “The effect of heating on the electromagnetic properties of laminated electrical steels,” Vestnik elektropromyshlennosti, 1955, No. 3.

3. A. A. Voskresenskii, “Equations of harmonic complex current curves in a circuit with iron,” Elektrichestvo, 1948, No. 1.

4. L. A. Bessonov, Electrical Circuits with Steel, Gosenergoizdat, 1948.

1. This refers to determination of the value of \(\dfrac{dB}{dH}\) by measuring the tangent of the angle of inclination of the tangent to the hysteresis loop. ↩↩