On the Use of Current Transformers with a Small Steel Core Volume for Differential Protection of Generators

L. V. Baginskii

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

In the journal Elektricheskie stantsii, No. 1 for 1956, an article by B. A. Nechaev was published in which the author proposes using the property of convergence of the characteristics of current transformers with cores made of steel of the same grade in the region of high inductions [1]. For this purpose it is recommended that current-transformer cores be made of small cross section, so that under external short circuits they operate in the region of very high inductions, while under normal operation of the generator connection the induction would be no less than \(B_p=(5—6)\) kgs. At such operating induction the cores are incapable of retaining residual fluxes, which, as is known, are the main obstacle to achieving high selectivity of differential protection. These undoubted advantages are achieved simultaneously with a substantial reduction in the consumption of materials for the production of current transformers. Therefore, the expediency of using the mentioned properties of current transformers with a small steel volume is evident.

B. A. Nechaev recommends connecting such transformers according to a circulating-current scheme, including inductive resistances \((10—20)\) ohms [2] in the relay circuits. In this case selective operation of the protection will be ensured at primary operating currents \(I_{\mathrm{sr}}=(0.3—0.4)I_{\mathrm{ng}}\).

However, this method of using the above-described properties of current transformers with small core cross sections raises a number of doubts. In fact, inequality of the arm resistances will cause inequality of induction in the cores of the current transformers of the phase of the protection circuit even when their characteristics coincide. Deep saturation of the cores under external short circuits will then be the cause of a sharp increase in the unbalance current. Therefore, the reliability of operation of such a scheme will probably be lower than that of schemes with class D current transformers, since the former may operate even in the event of partial disturbance of contact connections in its arms.

The conclusion that the differential-protection scheme proposed by B. A. Nechaev possesses self-balancing is also doubtful. The author believes that the unbalance current can be eliminated by an appropriate selection of the resistance of the relay circuit, so that the equalizing current sent by the current transformer with the higher characteristic into the other transformer would be such that

\[ I_{2I}=I_{2II}+I_{ur}=I'_{2II}{}^{*}. \tag{1} \]

* In expression (1) the notation used by B. A. Nechaev in equation (3) of his article [1] has been adopted, and a typographical error has been corrected, because of which the latter did not correspond to Kirchhoff’s first law.

In this case the currents in the arms are completely identical, and therefore the current in the relay is equal to zero. However, the condition for the existence of an equalizing current is the presence of a voltage on the differential circuit of the scheme, which is conceivable without unbalance current only when \(z_p=\infty\). Undoubtedly, increasing \(z_p\) will lead to balancing of the scheme at the expense of an increase in the voltage drop on the differential circuit. But such balancing is inherent in any differential scheme and was described already by N. P. Potashev [3].

The presence of a large resistance in the relay circuit causes a sharp increase in the already high induction in the cores during a short circuit in the protection zone, as compared with the case of an external short circuit. This has a detrimental effect on the sensitivity factor \(k_4\) and on the speed of operation of the protection, making doubtful the advisability of using it on slow-speed hydrogenerators without damper circuits (the current of a two-pole short circuit \(I_1''{}^{(2)}=2.4I_{ng}\), the time constant of decay of the aperiodic component of the stator current \(I_1\) being up to \(0.5\) sec), although, judging by the recommended settings, such protection is intended precisely for them. To confirm these assumptions, a differential protection scheme was tested with current transformers having a toroidal core cross section¹ \(q=3.2\ \text{cm}^2\), an average magnetic path length \(l_{cp}=48.8\ \text{cm}\), with the number of turns of the primary and secondary windings \(w_1=112\) and \(w_2=100\), at \(\omega L_p=12\ \text{ohms}\). The scheme was adjusted to an induction \(B_p=6\ \text{kGs}\), with the nominal secondary current taken as \(I_1'=5\ \text{A}\). It turned out that, with fault current in the protection zone \(I_{1n}=0.4I_{ng}\), the current in the relay \(I_{2n}=0.45\ \text{A}\); with \(I_{1n}=0.6I_{ng}\), \(I_{2n}=0.51\ \text{A}\); with \(I_{1n}=I_{ng}\), \(I_{2n}=0.58\ \text{A}\); and with \(I_{1n}=2.4I_{ng}\), \(I_{2n}=0.64\ \text{A}\). If, in the first three cases, the protection pickup current is taken equal to \(I_{2n}\), then the values of \(k_4\) will be, respectively, 1.42; 1.25; 1.1. Such values of \(k_4\) do not ensure sufficiently reliable operation of the protection and completely exclude any possibility of increasing the relay settings. Moreover, the manufacturer guarantees the operating time of the ET-520 relay \(t_{cp}=0.15\ \text{sec}\) at \(I_p=1.2I_y\), which, in combination with the delay in operation when an asymmetrical fault current flows through the primary winding of the current transformer, can provide protection operating times exceeding the total tripping time of modern instantaneously acting circuit breakers. Such a situation is entirely impermissible for high-speed protection, since it is equivalent to introducing a time delay.

Figure 1 gives oscillograms characterizing delays in the operation of the protection due to the presence of an aperiodic component in the primary current. The adjustment of the scheme and the parameters of the current transformers were taken the same as in the preceding experiment. The characteristic of the relay used in the experiment has the following points: at \(I_p=1.2I_y\), \(t_{cp}=0.025\ \text{sec}\); at \(I_p=2I_y\), \(t_{cp}=0.01\ \text{sec}\). As can be seen from the oscillogram in Fig. 1,a, at \(I_{1n}=2.4I_{ng}\), \(t_{cp}=0.15\ \text{sec}\), \(\omega L_p=12\ \text{ohms}\), and \(I_{1cp}=0.6I_{ng}\), the delay in the operation of the protection is approximately \(0.12\ \text{sec}\). The oscillogram of Fig. 1,b shows that, when the setting is increased to \(I_{1cp}=1.2I_{ng}=5.7\ \text{A}\), even in the absence of additional resistance in the relay circuit, the delay in operation remains large (\(0.12\ \text{sec}\)), although the sensitivity factor thereby increases to \(k_4=1.61\).

Thus, a sharp increase in the load on the current transformers in the fault condition within the protection zone, aggravated by their small pow—

… leads to the fact that the positive properties of the generator differential-protection scheme proposed by B. A. Nechaev are achieved at the cost of a sharp deterioration in other necessary qualities of the protection—speed of operation, sensitivity factor, and reliability of the scheme’s operation.

Nevertheless, it would be erroneous to think that the use of the positive properties of current transformers with a small volume of steel is always associated with deterioration of other protection qualities. The differential-

Fig. 1.

protection scheme with balancing of the no-load e.m.f. of the current transformers (without a circulating-current circuit) does not have the above-noted shortcomings, while retaining all the positive properties of the scheme proposed by B. A. Nechaev. The latter is due to the reduction, inherent in the scheme with balancing of the no-load e.m.f., of the induction in the cores of the current transformers during internal short circuits as compared with the inductions during external short circuits.

Figure 2 shows the schematic diagram of differential protection of a generator with balancing of the no-load e.m.f. of the current transformers. It was proposed simultaneously with the scheme with circulating currents, but did not become widespread in relay-protection practice. The fundamental difference between it and the scheme with circulating currents is the series-opposed connection of the secondary windings of the current transformers. Owing to this, during normal operation the secondary windings carry only the unbalance current, which is not capable of exerting a noticeable influence on the magnetic state of the core. Therefore, the operating mode of the current transformers in the scheme under consideration is very close to the no-load mode, which is unacceptable for class D transformers, usually used in differential protection. In addition, because of slight saturations of the core, concerns arise regarding the impossibility of attaining high sensitivity factors and sufficient speed of operation of the protection. There are also opinions expressed with respect to the operation of the ASEA scheme with an open circulating-current circuit, asserting the inevitability, in such a mode, of the appearance of a significant unbalance current, which may increase still further as a result of temperature effects of the environment.

Fig. 2.

At the same time, there is a whole series of considerations allowing one to assume that the scheme under consideration has very valuable properties.

Operation of current transformers at high inductions eliminates the possibility of residual fluxes influencing the transient unbalance current and makes it possible to expect that the scheme will have the properties of a rapidly saturating transformer with respect to currents of external short circuits.

In addition, the operating reliability of such a scheme should be higher than that of a scheme with circulating currents, owing to its simplicity and insensitivity to breaks in the secondary circuits. In this connection, investigation of the properties of such a scheme is of definite interest.

It is known that, in the no-load mode, the cores of current transformers become strongly heated, and considerable peak-shaped e.m.f.’s appear in the secondary winding. Fears are expressed that these voltages may puncture the insulation in the secondary-connection scheme and are dangerous for the operating personnel, while heating of the core may be the cause of impairment of the thermal stability of the insulation of the secondary winding.

However, these fears are fully justified only with respect to current transformers having a large volume of steel. Experimental investigations carried out at the Department of Electric Power Stations of the Tomsk Polytechnic Institute, for the purpose of determining the admissibility of prolonged operation of current transformers with a small volume of steel with an open secondary winding, showed the following:

1. If the secondary winding is made with wire of the PSD grade, the cost of which, for diameters of the order of 2 mm and more, exceeds the cost of PBD-grade wire by approximately 15%, then the long-term permissible specific magnetizing force for current transformers with \(q = 17\ \text{cm}^2\) and a spiral core is \(a w_n = 60\) ampere-turns/cm. In this case the temperature of the core surface will not exceed \(125^\circ\text{C}\), with a briefly permissible temperature for PSD-grade wire of \(180^\circ\text{C}\). Heating of laminated cores is considerably less than that of spiral cores.

2. The process of establishing the core temperature is very long (4–5 h). Therefore the flow of overcurrents has practically no effect on the temperature state of the core.

3. The dependence of the amplitude of the secondary e.m.f. \(E_{2m}\) on the primary sinusoidal current differs considerably from a linear one because of the influence of eddy currents on the form of the resulting magnetizing force of the transformer core. This makes it possible to obtain any desired value of \(E_{2m}\) by decreasing \(q\) and \(w_2\). In this case, decreasing \(q\) is more advantageous, since it leads not only to a reduction of \(E_{2m}\), but also of the interturn voltages, and improves the thermal regime of the core.

4. Existing methods for calculating the value of \(E_{2m}\) with a sinusoidal magnetizing current, based on approximation of the magnetization curve, give unacceptable results either because of the inaccuracy of the approximation or because the influence of eddy currents is not taken into account.

5. For practical calculations it is convenient to use the dependences \(e_{2m}=f(a w_1)\), obtained experimentally with a sinusoidal magnetizing force (here \(e_{2m}\) is the specific amplitude of the secondary e.m.f. per 1 turn of the secondary winding and 1 \(\text{cm}^2\) of core cross section, and \(a w_1\) is the specific magnetizing force per 1 cm of length of the mean magnetic line in the core). These dependences will be different not only for different grades of steel, but also for different core designs (thickness of sheets or strips, type of core).

6. As a briefly permissible value of \(E_{2m}\) during the flow of overcurrents, 1000 V amplitude may be adopted.

Thus, current transformers intended for differential protection with balancing of the secondary no-load e.m.f.’s may have the same design as the class D transformers currently being manufactured.

Let us consider the basic properties of generator differential protection with equalization of the secondary e.m.f.’s of the no-load current transformers. For this purpose we shall use the expression for the instantaneous values of the transient currents in the relay under external and internal short circuits (\(i_2\) and \(i_{2n}\), respectively), found for the calculation circuit shown in Fig. 3, under the assumption of linear transformer characteristics, absence of leakage, iron losses, and damping of the periodic component of the current \(I_1\):

\[ i_2 = I_{1M}\left[ \frac{z_{MA}-z_{MB}}{z_{2M}}\sin(\omega t-\varphi_1-\varphi_2) -\frac{(M_A-M_B)e^{-\frac{t}{T_1}}}{r_2(T_1-T_2)} + \frac{(M_A-M_B)\omega^2T_1T_2 e^{-\frac{t}{T_2}}}{r_2(T_1-T_2)(1+\omega^2T_2^2)} \right]; \tag{2} \]

\[ i_{2n}=I_{1M}\left[ \frac{z_{MA}}{z_{2M}}\sin(\omega t-\varphi_1-\varphi_2) -\frac{M_Ae^{-\frac{t}{T_1}}}{r_2(T_1-T_2)} + \frac{M_A\omega^2T_1T_2e^{-\frac{t}{T_2}}}{r_2(T_1-T_2)(1+\omega^2T_2^2)} \right], \tag{3} \]

where

\[ T_2=\frac{M_A+M_B+L_2}{r_2} \]

is the time constant of the secondary circuit;

\[ z_{2M}=\sqrt{1+\omega^2T_2^2} \]

is the total impedance of the secondary circuit;

\[ \varphi_2=\operatorname{arctg}\omega T_2-\frac{\pi}{2} \]

is the phase angle of the circuit;

\[ z_{MA}=\omega M_A;\qquad z_{MB}=\omega M_B \]

are the mutual-induction reactances of current transformers \(A\) and \(B\).

In addition, it is assumed here that \(\omega^2T_1^2\gg 1\), \(\varphi_1\cong \dfrac{\pi}{2}\), and the switching angle of the generator e.m.f. \(\alpha=0\).

It is not difficult to see that the mathematical maximum of the last two components of equation (2) can exceed the amplitude value of the first component only when \(2T_2>T_1\). Moreover, if the damping of the last component is achieved within one half-period by reducing \(T_2\) to \(T_2=(0.003—0.005)\) sec, then the aperiodic unbalance current may generally be disregarded, since in this case

\[ \frac{z_{MA}-z_{MB}}{z_{2M}}\gg \frac{M_A-M_B}{r_2(T_1-T_2)}. \]

The latter is explained by the worsening of the transformation conditions for the aperiodic component of the primary current as the coefficient of mutual induction of the current transformers decreases. With such circuit parameters, an appreciable maximum of the aperiodic unbalance current can occur only at very small \(T_1\), i.e., with electrically remote short circuits.

Thus, with proper calculation of the parameters of the protection circuit, its selectivity will be determined by the magnitude of the periodic (steady-state) unbalance current. Moreover, in the presence of steel cores, their one-sided magnetization by the aperiodic component of the primary current will cause a reduction of the periodic unbalance current at the beginning of the transient process.

Fig. 3.

However, equation (3) shows that the character of the current in the relay for faults in the protection zone is the same as that of the unbalance current. Consequently, if in such a mode the values of \(T_2\) remained unchanged

in comparison with the corresponding values in the external short-circuit mode, a delay in the operation of the protection would be inevitable. But the value of \(T_2\) cannot remain unchanged. It approximately doubles owing to the demagnetizing action of the current \(I_{2n}\). This is what makes possible such an adjustment of the protection in which the unbalance current does not contain an aperiodic component and, at the same time, the current \(I_{2n}\) will not decrease at the beginning of the transient process on account of the presence in the primary current of an aperiodic component at large values of \(T_1\).

Figure 4 gives oscillograms of the currents \(I_2\) and \(I_{2n}\). Oscillogram \(a\) (Fig. 4) was taken in a circuit with current transformers having

Fig. 4.

\(q = 4 \text{ cm}^2\), \(w_2 = 76\) turns, \(\omega L_2 = 1.06\ \Omega\), \(r_2 = 1.12\ \Omega\), under conditions of an external short circuit with \(T_1 = 0.09\) sec and a magnetizing force of the primary winding \(AW_1 = 1000\) ampere-turns. The unbalance current has only one surge lasting less than one half-period, caused by remagnetization of the cores in the opposite direction, since the last half-period of the preceding process was opposite in sign to the one under consideration. Then a sharp decrease in the amplitude of the unbalance current is observed as a result of one-sided magnetization of the cores of the current transformers.

An analogous phenomenon is reflected in oscillogram \(b\) (Fig. 4), taken at \(q = 0.875 \text{ cm}^2\), \(w_2 = 91\) turns, \(l_{\text{av}} = 45.6 \text{ cm}\), \(r_2 = 1.12\ \Omega\), \(\omega L_p = 1.06\ \Omega\), \(AW_1 = 783\) ampere-turns, and \(T_1 = 0.12\) sec. Oscillograms \(v\) and \(g\) were taken with the same parameters of the current transformers and the circuit as oscillograms \(a\) and \(b\), respectively, but with \(AW_1 = 5000\) ampere-turns, \(T_1 = 0.03\) sec (oscillogram \(v\)), and \(AW_1 = 819\) ampere-turns, \(T_1 = 0.12\) sec (oscillogram \(g\)), under conditions of a fault in the protected zone and one-sided feeding of the fault point. Oscillogram \(v\) (Fig. 4) shows that the current \(I_{2n}\) at the beginning of the transient process not only does not decrease, but even increases somewhat. Unfortunately, under these conditions it was not possible to obtain a sufficiently large \(T_1\) to be able to assert that a decrease in \(I_{2n}\) is impossible at \(AW_1 = 5000\) ampere-turns. From oscillogram \(g\) it is evident that, with an increase in \(T_1\) and a decrease in \(q\), a decrease in \(I_{2n}\) at the beginning of the transient process is possible even at comparatively small \(AW_1\).

Thus, there exist such values of \(T_2\) for which the character of the transient processes in the circuits of the scheme described will be optimal—

... when there is no reduction of the current \(I_{2n}\) at the beginning of the transient process capable of causing a delay in operation in the event of a fault within the protected zone, and at the same time the aperiodic component of the unbalance current is such that it may be disregarded. The above-mentioned values, \(T_2=(0.003—0.005)\) sec, as experiments show, are the minimum permissible. Since at the beginning of the transient process the periodic component of the current \(I_2\) decreases, some exceeding of this value of \(T_2\) is permissible. In this case it is necessary to ensure that the values \(T_1\) and \(T_2\), at the maximum possible external short-circuit current, do not prove to be of the same order of magnitude.

In the following discussion it is assumed that the conditions for an optimal character of the transient processes in the protection circuits are satisfied.

It should also be noted, as positive properties of the circuit being described, that the periodic unbalance current is independent of \(q\) (owing to the independence of the magnetizing current from \(q\)), that it is impossible for the unbalance current to increase as a result of unequal loading of the current transformers of the circuit, and also that the protected generator cannot be tripped in the event of an open circuit in the secondary protection circuits.

Very great importance in the operation of the circuit being described is attached to the nature of the dependence of the unbalance current on the primary current. It is precisely this dependence that can prove or disprove B. A. Nechaev’s assertion about the convergence of the characteristics of current transformers, since the conclusion about convergence of the characteristics, made on the basis of comparing graphs of the dependences \(B=f(H)\), may be unreliable. The latter is explained by the fact that, in the region of very high inductions, even a small divergence of the characteristics may be the cause of a large unbalance current. Meanwhile, it is obvious that when the characteristics diverge, the unbalance current must increase faster than the increasing primary current. Consequently, the opposite result is possible only when the characteristics converge.

Fig. 5.

Numerous experiments to determine the dependences \(I_2=f(aw_1)\) for current transformers made of steel of the same grade showed that these dependences have the character illustrated by Figs. 5a and b, which present the curves \(I_2=f(aw_1)\), determined for the following parameters of the current transformers and circuit: \(q=4\ \text{cm}^2\); \(w_2=76\) turns; \(l_{\text{av}}=45.6\ \text{cm}\); \(r_2=1.05\ \Omega\); \(\omega L_2=0\), and \(q=10\ \text{cm}^2\); \(w_2=25\) turns; \(l_{\text{av}}=45.6\ \text{cm}\); \(r_2=1.46\ \Omega\); \(\omega L_2=0.9\ \Omega\). In the first case the difference of the characteristics from 1 to 9 kgs was 1–2%, and in the second, 16.5%. In view of the very large content of higher harmonics in the unbalance current, the measurements of \(I_2\) were carried out with a sensitive galvanometer connected into the secondary circuit through a type T102 thermoconverter,

which made it possible to measure the rms values of the unbalance current. Both curves testify to the convergence of the characteristics of the samples being tested. In addition, the graph in Fig. 5,b confirms B. A. Nechaev’s idea that the greatest discrepancies in the characteristics of current transformers occur at slight saturations of the cores (3–5 oersteds).

From consideration of the dependences presented in Fig. 5 there follows a conclusion of great practical importance. Since the largest component of the unbalance current is the periodic component, and it in turn depends only slightly on the primary current, it is possible to obtain a sufficiently reliable idea of the maximum possible unbalance current on the basis of measuring it at \(I_1 = I_{n2}\).

The dependence of induction on the temperature of the core may cause a significant increase in the unbalance current if the current transformers are at different ambient temperatures [4]. Such a case is often encountered in practice when current transformers on the line terminals of a generator are located in unheated switchgear.

Fig. 6.

For a quantitative assessment of the influence of the indicated effect on the unbalance current, the following experiment was performed. The current transformers of the circuit whose test results are presented in Fig. 5,a were placed under different temperature conditions, so that the difference in ambient temperatures was \(\Delta t = 10^\circ\text{C}\). To simulate the normal operating condition, an unchanged current was passed for a long time through the primary winding. Then the current was sharply increased and a measurement corresponding to an external short circuit was made. The results of the experiment are given in Fig. 6. Curve 1 shows the nature of the change in the unbalance current when the transformers are heated by different operating currents. Curves 2 represent the dependence of \(I_2\) on the primary currents of external short circuits. Comparison of curve 1 with the curve in Fig. 5,a shows that the unbalance current, with a temperature difference \(\Delta t = 25^\circ\text{C}\), may increase by a factor of 5–6 compared with the case \(\Delta t = 0\). However, such an effect is observed only when the characteristics of the current transformers coincide very closely. In attempting to obtain the same effect on the current transformers used to determine the characteristic presented in Fig. 5,b, it was found that at \(\Delta t = 15^\circ\text{C}\) the values of \(I_2\) deviate from the corresponding values at \(\Delta t = 0\) by \(\pm 5\%\), depending on which of the current transformers is at the lower temperature. The nature of curves 2 is probably explained by the fact that an increase in the temperature of the core reduces the induction in it differently: at small magnetizing forces to a greater extent than at large ones. As a result, the largest unbalance current is observed at \(aw_1 = (3–5)\) ampere-turns/cm. The experiment described shows that there is hardly any point in carefully selecting current transformers for a set, since temperature effects can nullify the results of such selection. At the same time, in the absence of careful selection of transformers, temperature effects can be disregarded.

Let us note that similar phenomena should also be inherent in the circuit proposed by B. A. Nechaev. In that case they should appear at least to no lesser degree than in the one described.

scheme, since in B. A. Nechaev’s scheme this effect will be aggravated by an increase in the load on the current transformer with reduced induction (owing to the temperature increase in the resistance of the connecting leads and secondary windings).

The principal characteristics of differential protection with balancing of the secondary no-load e.m.f.’s of the current transformers, as well as their dependence on the inductive resistance of the relay circuit, are shown in Fig. 7, a and b, which were plotted from the results of testing the same pairs of transformers as for Fig. 5, a and b, respectively. The dependences presented in Fig. 7, a were obtained under the following conditions: \(\Delta t = 15^\circ C\); \(w_1 = 100\) turns; \(w_2 = 76\) turns; \(r_2 = 1.6\ \Omega\). The current transformers were heated at \(I_1 = 15\ a\), while \(I_2\) was measured at the same current. The current \(I_{1cp}\) was determined when the current in the relay reached the value \(I_{2cp} = 5I_2\). The current \(I_{2n}\) was found at \(I_1 = 80\ a\), and \(k_4 = I_{2n}\cdot 5I_2\). The dependences shown in Fig. 7, b were determined similarly under the following conditions: \(\Delta t = 0\); \(w_1 = w_2 = 100\) turns; \(r_2 = 1\ \Omega\). The currents \(I_2\) and \(I_{2n}\) were determined at \(I_1 = 74\ a\); \(I_{2cp} = 3I_2\).

Fig. 7.

The results of these experiments show that, even with a considerable divergence of the current-transformer characteristics (Fig. 7, b) and with large reliability factors \((k_n = 3)\), very small primary operating currents are attainable in the absence of additional resistances in the relay circuit \((I_{1cp} = 5.9a)\), with a high sensitivity coefficient \((k_4 = 7.8)\). For example, the current \(I_1\), from which \(I_{2n}\) and \(k_4\) were determined in this experiment, corresponds to the rated current of a hydrogenerator without damper windings of magnitude \(I_{ng} = 7\,400 : 2.4 = 3\,080\ a\), and, consequently, the \(I_{1cp}\) obtained in the experiment corresponds approximately to \(0.15 I_{ng}\). Moreover, the nature of the dependence \(I_2 = f(I_1)\) gives grounds to hope (Fig. 5, b) that, with increasing \(I_1\), for unchanged parameters of the current transformers and of the circuit, the current \(I_2\) will change hardly at all. Therefore the current \(I_{1cp}\) will also remain almost unchanged, whereas the current \(I_{2n}\), and consequently \(k_4\), should increase. It follows from this that, with increasing \(I_{ng}\), the protection characteristics should improve.

From consideration of Fig. 7 it follows that, in cases where no requirement of increased sensitivity is imposed on the differential protection of a generator, there is no need to use additional resistances in the relay circuit. However, in individual cases the sensitivity of the protection may prove insufficient (for example, in the protection of low-power generators). In the event that it is necessary to increase—

to increase the sensitivity of the protection, additional inductive resistances in the relay circuit of the order of \(5\text{--}10\ \Omega\) should be used; these provide a sharp reduction in the values of \(I_{1cp}\) with an almost unchanged value of \(k_4\). In this connection it should be borne in mind that increasing the resistances of the relay circuit reduces the difference between the inductions in the cores in the modes of external and internal faults, which adversely affects the nature of the transient processes in the protection circuits.

The above-mentioned conditions that must be met by current transformers for differential protection of generators with balancing of the residual e.m.f. of no-load operation (the permissible values of \(E_{2m}\) and \(T_2\)) are sufficient for an approximate calculation of the quantities \(q\) and \(w_2\). If \(\omega L_2 = 0\) and \(M_A + M_B = 2M\) are assumed, then

\[ M = \frac{T_2 r_2}{2}. \tag{4} \]

Moreover,

\[ M = \frac{k q w_2 \mu_{\mathrm{eff}}}{l_{cp}}, \tag{5} \]

where \(\mu_{\mathrm{eff}} = \frac{B}{H}\), \(k = 0.4\pi 10^{-8}\).

The quantity \(\mu_{\mathrm{eff}}\) is determined from the magnetization curve at values of \(H\) corresponding to the supertransient current of a three-phase short circuit of the protected generator, \(I''_{1T}\).

The relation between the amplitude of the e.m.f. and the sought parameters will be:

\[ E_{2m} = q w_2 e_{2m}, \tag{6} \]

Solving equations (5) and (6) jointly with respect to \(q\) and \(w_2\), we obtain:

\[ q = \frac{E_{2m}^{2} k \mu_{\mathrm{eff}}}{e_{2m}^{2} l_{cp} M}. \tag{7} \]

\[ w_2 = \frac{M l_{cp} e_{2m}}{k \mu_{\mathrm{eff}} E_{2m}}. \tag{8} \]

Expressions (4), (7), and (8) are the calculation formulas. For the calculation it is necessary to know \(l_{tp}\), which must not exceed the values of \(l_{cp}\) of existing current transformers; the dependence \(e_{2m} = f(a w_1)\), obtained with sinusoidal current for the given grade of steel and core construction; and also the values \(E_{2m}\), \(T_2\), and \(r_2\).

The value \(e_{2m}\) should be determined at the maximum possible, for generators of various types on which the transformer being calculated may be installed, supertransient current of a three-phase short circuit \(I''_{1E}\). It is advisable to take the value \(E_{2m}\), when \(I''_{1E} < 20\ \mathrm{kA}\), as less than the maximum permissible, in order to avoid unjustified overestimation of core cross sections.

When choosing the calculated values of \(T_2\) and \(r_2\), it is necessary to take into account the possibility of different lengths of the conductors of the differential protection circuit on generators with approximately identical rated currents but of different types, as well as different values of \(I''_{1T}\). Therefore, it is advisable to take as calculated values \(T_2 = 0.005\ \mathrm{s}\) and \(r_2\) corresponding to the smallest cross section and length of the conductors at the given \(I_{n2}\), so that an increase in the length of the connecting conductors or in the current \(I''_{1T}\), relative to the calculated values, could be compensated for by increasing their cross section. For

for adjusting the protection it is desirable to have a specified dependence of the maximum permissible resistance of the connecting leads, \(r_{\text{pr.dop}} = f(I''_{1T})\), calculated from expressions (4) and (5) and the magnetization curve, as well as the dependence \(I_{2n} = f(0.87 I''_{1T})\), determined experimentally at values of \(r_{\text{pr.dop}}\) corresponding to the given \(I''_{1T}\). Adjustment of the protection is reduced to selecting the cross section of the connecting leads, determining the relay setting, and checking the values of the coefficients \(k_4\) and \(k_n\). For this purpose, after placing a short circuit in the protection zone, one should set such a current in the primary circuit as corresponds to the desired value of \(I_{1\text{sr}}\), and determine the relay setting. The ratio of the value of \(I_{2n}\), specified for the given \(I''_{1T}\), to the setting current must be \(k_n \ge 2\). After moving the short circuit outside the protected zone, we set the primary current \(I_1 = I_{n2}\) and measure the effective value of the unbalance current by means of an instrument requiring little power (of the thermomilliammeter type). It is necessary to have \(k_n = I_y : I_2 \ge 3\).

Using this method, current transformers were calculated for \(\omega_1 = 100\) turns, \(I''_{1T} = I''_{1E} = 50\ \text{a}\); \(r_2 = 1.2\ \text{ohm}\); \(E_{2m} = 300\ \text{v}\); \(\omega L_2 = 1.05\ \text{ohm}\); \(l_{\text{sr}} = 45.6\ \text{cm}\); \(T_2 = 0.003\ \text{sec}\). The calculated parameters proved to be \(q = 4\ \text{cm}^2\) and \(\omega_2 = 76\) turns. Oscillograms \(a\) and \(b\) in Fig. 4 show that the nature of the transient process in the secondary circuits of the circuit containing the calculated current transformers corresponds to the optimum.

Thus, differential protection of generators with balancing of the e.m.f. of the no-load current-transformers has improved characteristics in comparison with the scheme now in use, while at the same time retaining all the advantages of the scheme proposed by B. A. Nechaev. The described circuit can be applied to generators of any design and capacity.

Literature

1. Nechaev B. A., Current transformers with a small cross section of core steel in differential-protection circuits of generators, Elektricheskie stantsii, 1956, No. 1.

2. Nechaev B. A., On the question of the application of current transformers with a small volume of steel for differential protection of generators, dissertation, 1952.

3. Potashev N. P., Operation of current transformers during transient processes, Elektricheskie stantsii, 1935, No. 3.

4. Kazarnovskii L. Sh., Influence of heating on the electromagnetic properties of sheet electrical steels, Vestnik elektropromyshlennosti, 1955, No. 3.

1. Sheet steel from TTP-type current transformers served as the material for manufacturing the cores. ↩