Calculation of Fast-Saturating Current Transformers (FSCTs) for Differential Protection

I. D. Kutyavin

As is known, the idea of applying FSCTs is based on the use of a sharply pronounced dependence of their transformation ratio on the magnitude of the aperiodic component contained in the transformed current. This dependence of the transformation ratio for the periodic component is shown in Fig. 1, where along the ordinate is plotted the rms value of the periodic component of the primary operating current of the FSCT, \(I_{PT}\), in amperes, and along the abscissa—the aperiodic current \(I_a\), in relative units (referred to the amplitude \(I_{PT}\)). When the curves of Fig. 1 were taken, a current equal to the secondary operating current (the relay operating current) was maintained in the secondary circuit of the FSCT, while the action of the aperiodic current was simulated by passing direct current through an auxiliary winding. FSCTs with a primary calculated operating current of \(2 a\), with a calculated relay operating current of \(0.5 a\), and with a calculated induction of \(8\) and \(10\) kgs were tested.

The curves of Fig. 1, expressing the dependence of the transformation ratio of the periodic component on the aperiodic current, are at the same time the braking characteristics of the protection.

It is now of interest to trace the behavior of an FSCT with such a characteristic under various operating conditions of the protection. Under normal operating conditions, an unbalance current flows in the primary circuit of the FSCT; this current is much smaller than the operating current of the protection, and therefore this condition is of no interest. In the event of an interphase short circuit occurring in the zone of differential protection, a large current appears in the primary winding of the FSCT, containing an aperiodic component (if its decay is taken into account) \(I_a = 0\)—\(0.8\) of the amplitude of the periodic component. As can be seen from Fig. 1, at small values of the aperiodic component the protection in this case will operate with an operating current of \(2\) to \(4 a\). In the case when the aperiodic component has the greatest value, the operating current of the protection rises sharply. If, in this case, the fault current proves insufficient to actuate the relay, then, as the aperiodic component decays, the operating current of the protection will decrease, and it will operate with a certain time delay lying within the limits of the time constant of decay of the aperiodic current.

Fig. 1

In a through short circuit, the primary winding of the FSCT is traversed by the transient unbalance current. In this case the greatest danger for the [[unclear: word continues on next page]]

protections without an FST represent the maximum unbalance currents containing an aperiodic component \(I_{a*} \gg 1\).

For differential protection equipped with an FST, these unbalance currents are not dangerous, since with an aperiodic component \(I_{a*} \gg 1\) the pickup current of the protection increases so much that it will not operate.

To obtain an FST with such properties, it is necessary that, when a sinusoidal current equal to the relay pickup current \(I_{RT}\) flows in its primary circuit, the calculated induction in the core \(B_p\), the resistance of the no-load branch \(Z_x\) (see the equivalent circuit in Fig. 1), and the resistance of the secondary circuit \(Z_2 = Z_{TT} + Z_R\) be in a definite ratio.

Let us first establish the calculated induction \(B_p\). Fig. 2 shows the magnetization curve \(B\) in transformer steel (the core steel of TPO-type current transformers) and the resistance curve \(Z_x\), taken with alternating current as a function of the specific magnetizing effective ampere-turns. As can be seen from this figure, the resistance \(Z_x\) decreases very sharply beginning with an induction of \(8\ \text{kG}\), and this decrease continues at all higher inductions. If the magnetizing current also contains an aperiodic component \(I_{a*}=1\), then the resistance \(Z_x\) will additionally decrease by a factor of \(n\). The curve of the factor \(n\) of the decrease of \(Z_x\) is shown in the same figure as a function of the magnetizing alternating ampere-turns at \(I_{a*}=1\). As can be seen from the figure, the curve \(n\) first rises from 3.5 to 5.5, and at the end, beginning with an induction of \(18\ \text{kG}\), decreases.

Fig. 2

Study of the curves of Fig. 2 shows that the sharpest dependence of \(Z_x\) on the magnetizing current occurs at inductions from 8 to \(18\ \text{kG}\); therefore the calculated induction \(B_p\) in calculating the FST may be taken within these limits. However, at high calculated inductions the operation of the FST for a fault in the protected zone becomes unstable because of strong saturation of the core. In addition, it is always desirable to have the possibility of regulating the pickup current of the protection during operation; therefore it is desirable to use the calculated induction

\[ B_p = 8 \text{—} 12\ \text{kG}. \]

Between the calculated no-load resistance of the FST \(Z_x\) and the resistance of the secondary circuit \(Z_2\), the following relation must be satisfied: \(Z_x \leq (0.5)Z_2\). At larger values of the resistance \(Z_x\), nonselective operation of the protection is possible.

Core of the FST. The core is made of transformer steel. The cross section of the core is \(0.8\text{—}1\ \text{cm}^2\). The dimensions of the core laminations are not more than \(30 \times 40\ \text{mm}\), and the window dimensions are \(10 \times 20\ \text{mm}\). When the core is made of round laminations, the outside diameter may be taken as \(40\ \text{mm}\), and the inside diameter as \(20\ \text{mm}\). It is desirable to make the cores without joints, cutting out whole laminations from a steel sheet. During machining the magnetic properties of the steel deteriorate; therefore the fabricated laminations must be annealed at red-heat temperature, followed by slow cooling. After fabrication of the core, its magnetization curve on alternating current must be taken. For this purpose, two windings with an identical but arbitrary number of turns must be wound on the core (10–20 turns each).

kov). The measured alternating current is passed through one winding, and the e.m.f. is measured in the other, from which the induction is determined.

Selection of the relay. For joint operation with an FSCT, relays of the ET type are used, with a minimum pickup current of 0.3–0.5 A and, preferably, with a damping device. Before calculating the FSCT, the internal resistance of the relay with the operating contacts open is measured and the calculated resistance \(Z_2\) is established. If the FSCT is built into the relay, the resistance of the connecting wires may be neglected.

Determination of the numbers of turns of the windings. The number of turns of the secondary winding of the FSCT can be determined from the equation for the secondary e.m.f.:

\[ E_2 = I_{PT2} Z_2 = 4 k f w_2 q B_p \cdot 10^{-8}. \tag{1} \]

Then the number of turns of the secondary winding, if its resistance is neglected, will be equal to:

\[ W_2 = \frac{I_{PT2} Z_2 \cdot 10^8}{4 k f q B_p} \simeq \frac{S_p \cdot 10^8}{4 k f q B_p I_{PT2}}, \tag{2} \]

where \(I_{PT2}\) is the minimum pickup current of the relay,
\(k\) is the shape factor of the e.m.f. curve. According to M. I. Tsarev, this coefficient may be taken as equal to 1.15; \(B_p = 8\text{–}12\) kgf is the calculated induction at the minimum pickup current.

From the magnetization curve for the adopted \(B_p\), the specific magnetizing ampere-turns \(aw_x\) are determined, and from them the total no-load ampere-turns are found:

\[ AW_x = aw_x l, \tag{3} \]

where \(l\) is the length of the mean line of force in centimeters.

The ratio of the resistance \(Z_2\) to the resistance \(Z_x\) is:

\[ \frac{Z_2}{Z_x} = \frac{I_x}{I_{PT2}} = \frac{AW_x}{W_2 I_{PT2}} > 2. \tag{4} \]

The minimum primary pickup current \(I_{PT1}\) required for the protection (for the FSCT) is established. Then the number of turns of the primary winding can be determined from the following approximate expression:

\[ W_1 \simeq \frac{AW_x + W_2 I_{PT2}}{I_{PT1}}. \tag{5} \]

The primary number of turns of the FSCT, when it is tested in the laboratory together with the relay, can be changed in the desired direction.

The scale of the primary pickup currents of the FSCT is determined by the usual method. The pickup current according to the scale is set on the relay and the primary pickup current is determined. From the test data a calibration curve is constructed, which is used during operation. In these tests instruments must not be connected into the secondary circuit of the FSCT, since this will change the resistance \(Z_2\). When using the calibration scale, it should be borne in mind that the pickup current of the protection \(I_{PT1}\) can be increased only until the induction in the FSCT core reaches 14–15 kgf. For a rough determination of the maximum current \(I_{PT1m}\), it is necessary, from the magnetization curve, to find the maximum specific ampere-turns \(aw_{xm}\) for the maximum calculated induction \(B_{pm} = 14\text{–}15\) kgf and to use expression (5):

\[ I_{PT1m} \simeq \frac{aw_{xm} l + w_2 I_{PT2m}}{w_1}, \tag{6} \]

where \(I_{RT2M}\) is the maximum relay pickup current, determined from the following expression:

\[ I_{RT2M}=\frac{4kf\omega_2 q B_{PM}10^{-8}}{Z_2}. \tag{7} \]

Example of calculating an FSCT. It is required to calculate an FSCT for differential protection of a transformer with a minimum pickup current \(I_{RT1}=2\) A. As the relay, it is assumed that an ET-511/2 will be used, with a minimum pickup current \(I_{RT2}=0.5\) and winding resistance \(Z_2=0.4\ \Omega\). For the FSCT core, transformer steel from a TPO-type current transformer is used; its magnetization curve is shown in Fig. 2. The core dimensions are \(40 \times 35\) mm, the window \(20 \times 15\) mm; the core cross section is \(q=0.8\ \text{cm}^2\). The calculation has been made in two variants: for \(B_P=8\) kG and 10 kG.

Calculation of the FSCT at \(B_P=8\) kG. The number of turns of the secondary winding from (2):

\[ W_2=\frac{0.1\cdot 10^8}{4\cdot 1.15\cdot 50\cdot 0.8\cdot 8000\cdot 0.5}=14. \]

The winding is made with wire of \(d=0.7\) mm. The specific magnetizing ampere-turns from the curve in Fig. 2 for

\[ B_P=8\ \text{kG};\quad aw_x=1.6\ \frac{\text{A-turns}}{\text{cm}}. \]

The total magnetizing ampere-turns:

\[ AW_x=aw_x\cdot l=1.6\cdot 11=17.6\ \text{A-turns}. \]

The ratio of the resistances \(Z_2\) to \(Z_x\) (or the initial shunting coefficient):

\[ \frac{Z_2}{Z_x}=\frac{17.6}{14\cdot 0.5}=2.5. \]

The number of turns of the primary winding from (5):

\[ W_1=\frac{17.6+0.5\cdot 14}{2}=12. \]

The winding is made with wire of \(d=1.8\) mm.

The maximum secondary pickup current at \(B_{PM}=14\) kG,

\[ I_{RT2M}=\frac{4\cdot 1.15\cdot 50\cdot 14\cdot 0.8\cdot 14000\cdot 10^{-8}}{0.4}=0.9\ \text{A}. \]

The specific ampere-turns corresponding to \(B_{PM}=14\) kG,

\[ aw_{xM}=6.6\ \frac{\text{A-turns}}{\text{cm}}. \]

The maximum permissible primary pickup current for the FSCT:

\[ I_{RT1M}=\frac{6.6\cdot 11+0.9\cdot 14}{12}=7.0\ \text{A}. \]

The permissible range of pickup-current adjustment is \(2\text{--}7\) A.

The calculation of the FSCT for \(B_p = 10\ \text{kGs}\) was carried out analogously to the preceding one. The calculation results are as follows: \(W_2 = 11\); \(W_1 = 18\); \(Z_{\mathrm{e}} : Z_x = 5.6\); \(I_{\mathrm{rt}2\mathrm{m}} = 0.71\ \text{A}\); \(I_{\mathrm{rt}1\mathrm{m}} = 4.46\ \text{A}\), and the adjustment range of the pickup current is \(2\text{–}4.46\ \text{A}\).

Fig. 3

The braking characteristics of the calculated FSCTs are given in Fig. 1, and Fig. 3 shows a relay with a built-in FSCT. The left contact screws were used to fasten the FSCT, while the relay circuits were moved to the right side.