Calculation of Ground-Fault Protection Implemented Using Cable Current Transformers with Biasing

I. D. Kutyavin

In recent years, protection against ground faults in generators, motors, and feeders using cable current transformers has begun to become widespread. However, there is still no sufficiently simple method for calculating and implementing such protection. The calculation of this protection proposed by E. G. Markvard¹ and A. B. Barzam² is based on an analytical expression for the magnetization curve, which is valid for inductions from 2 to 8 kG, whereas cable transformers operate at inductions up to 2 kG. In addition, the indicated analytical expression contains coefficients that depend on the grade of steel and vary within very wide limits. Meanwhile, relay engineers often have at their disposal a sufficient quantity of steel of unknown properties from old current transformers of the T. P. type, the use of which encounters difficulties for the reasons indicated above.

Below is given a very simple method for calculating protection implemented by means of individual cable current transformers.

Design of a Cable Transformer with Biasing

To obtain protection with a primary pickup current from 1 to 5 A and with acceptable transformer dimensions, the use of alternating-current biasing is unavoidable.³ For this purpose, the transformer core is made of two completely identical packets of steel (for example, steel from T. P. current transformers). A biasing winding with the same number of turns is wound on each packet, and these windings are connected in opposition in series. The packets with the biasing windings are covered with insulation and placed one on top of the other with a pressboard spacer between them. Similar spacers should be placed on both sides of the core. The secondary winding is wound on the finished core and is likewise covered on top with insulation. Each winding must be applied to the core strictly uniformly. An insulating terminal block with two pairs of terminals is installed on the core mounting bolts.

Calculation Data

To calculate the transformer it is necessary to have information on the quality of the steel. This information can be obtained if a trial

a prototype transformer with an arbitrary number of turns in the windings and with any core cross-section. For example, the core cross-section may be taken as \(20\ \mathrm{sq.\ cm}\) with \(10\ \mathrm{sq.\ cm}\) per pack, and the secondary and biasing windings may each be wound with 100 turns.

Since different old transformers, manufactured at different times, may have steel with unequal properties, the steel for the core of the test transformer must be selected after thorough mixing of the entire quantity of steel prepared for manufacturing the transformers.

Fig. 1

Fig. 2

To obtain the data, the test transformer is connected according to the circuit of Fig. 1. Instead of the primary winding, a straight conductor I must be passed through the window of the transformer so that it bends away no closer than \(25\text{--}30\ \mathrm{cm}\) from the core. An ammeter with a \(5\ \mathrm{A}\) scale and an adjustable resistance are connected in the circuit of this conductor. An electronic voltmeter, allowing measurement of voltages from \(0.1\) to \(5\ \mathrm{V}\), is connected in the circuit of the secondary winding II. It is recommended to supply the biasing winding III through an autotransformer AT, since the use of a sliding rheostat can distort the biasing-voltage curve.

At the beginning of the test, the residual e.m.f. \(E_{nb}\), created by the biasing winding, is recorded as a function of the biasing induction \(B_n\), and the curve \(E_{nb}=f(B_n)\) is plotted, as shown in Fig. 2. The magnitude of \(E_{nb}\) at \(B_n<3000\ \mathrm{G}\) should not exceed several hundredths of a volt.

Fig. 3

Then a series of magnetization curves is taken at biasing inductions equal to \(0;\ 1000;\ 2000;\ 3000\ \mathrm{G}\), and these curves are plotted in the form shown in Fig. 3, where the specific ampere-turns are plotted along the abscissa axis, and the induction in \(\mathrm{G}\) along the ordinate axis. As can be seen from the curves of Figs. 2 and 3, obtained for a test transformer with a core cross-section of \(20\ \mathrm{sq.\ cm}\) and windings of 100 turns each, it is not advisable to use a biasing induction above \(2000\text{--}3000\ \mathrm{G}\), since this may give an insigni-

a significant increase in the sensitivity of the protection, but may sharply increase \(E_{\mathrm{nb}}\) and lead to its false operation. The curves of Fig. 3 are used for calculating the protection. These curves are close to straight lines and therefore make it possible to obtain a simple analytical dependence of the induction on the specific magnetizing ampere-turns of the following form:

\[ B_p = kaw, \tag{1} \]

where \(k\) is an angular coefficient determined from the coordinates of some point lying on the magnetization curve with the selected value \(B_n\),

\(B_p\) is the operating induction of the transformer.

In order to obtain the greatest sensitivity of the protection at an unchanged primary operating current \(I_{PT}\), the transformer must deliver maximum power to the secondary circuit. This condition is satisfied when the no-load resistance of the transformer \(z_o\) is equal to the relay resistance\(^1\) \(z_p\):

\[ z_o = z_p. \tag{2} \]

In the case where the protection circuit contains \(m\) current transformers connected in parallel, expression (2) will have the form:

\[ z_o = mz_p. \tag{3} \]

The no-load current of the transformer \(i_o\) and the relay operating current \(i_{PT}\), on the basis of (3), will be in the following relation:

\[ i_{PT} = mi_o. \tag{4} \]

Using relations (1)--(4) and the magnetization curves of Fig. 3, the transformer and the protection can be calculated.

The appended Table 1 gives the technical data of highly sensitive current relays of type ETD-551, which can be used for calculating the protection.

Table 1

Calculation of the Protection

At the beginning of the calculation, using the data of Table 1, the relay type and the operating current \(i_{PT}\) are selected, and the value of the resistance of the relay \(z_p\) and of the connecting wires \(r_p\) is established. Then the total resistance of the secondary circuit of the transformer is

\[ z_{pn} = \sqrt{(r_n + r_p)^2 + x_p^2}, \tag{5} \]

where \(r_p\) and \(x_p\) are the components of the resistance \(z_p\). The value \(r_p\) can be determined with a bridge.

\(^1\) See G. V. Zeveke, “Differential Protections of Electrical Machines,” Electric Stations, No. 1, 1940.

The specific ampere-turns of magnetization can, taking (4) into account, be expressed through the pickup current as follows:

\[ a w=\frac{i_0 w_2}{l}=\frac{i_{pT} w_2}{m l}, \tag{6} \]

where \(w_2\) is the number of turns of the secondary winding and
\(l\) is the length of the mean line of force.

Then the operating induction \(B_p\), on the basis of (1) and (6), is

\[ B_p=\frac{k i_{pT} w_2}{m l}. \tag{7} \]

Let us now use the expression for the secondary e.m.f. at the relay pickup current

\[ E_2=222 q w_2 B_p \cdot 10^{-8}=i_{pT} z_{pn}. \tag{8} \]

Substituting the value of \(B_p\) from (7) into (8), we obtain the expression for the secondary number of turns:

\[ w_2=10^4 \sqrt{\frac{m l Z_{pn}}{222 k q}}. \tag{9} \]

Having specified the cross section of the core \(q\), the secondary number of turns \(w_2\) is calculated from expression (9). Then, from expression (6), the specific ampere-turns \(a w\) are determined and, using the magnetization curve of Fig. 3, the operating induction \(B_p\) is found. The calculation can be checked by determining the e.m.f. \(E_2\) from (8) and comparing it with the product \(i_{pT} z_{pn}\). If expression (8) is not satisfied, a correction is introduced into the calculation by selecting a new value of \(k\) corresponding to the induction \(B_p\) obtained in the calculation.

The number of turns of the biasing windings \(w_n\) is determined by the selected biasing induction \(B_n\), taking into account that half the voltage \((0.5\,U_n)\) is supplied from the current source to each core pack winding and the cross section of the pack is equal to \(0.5q\):

\[ w_n=\frac{0.5 U_n\,10^8}{222\cdot 0.5 q B_n} =\frac{U_n\,10^{-8}}{222 q B_n}. \tag{10} \]

If the protection contains \(m\) transformers connected in parallel, whose biasing windings are intended to be connected in series, then the number of turns of the biasing winding is

\[ w_n=\frac{U_n\cdot 10^8}{222 m q B_n}, \tag{11} \]

where \(U_n\) is the voltage of the source feeding the biasing windings.

On each core pack there must be wound \(w_n\) turns.

The maximum current in the relay, taking (3) into account, is

\[ i_{pм}=\frac{I_{з.м}}{w_2}\,\frac{z_0}{z_0+m z_p} =\frac{I_{з.м}}{2w_2}. \tag{12} \]

This current can be used as the basis for selecting the conductor cross section of the secondary winding. To determine the current consumed by the biasing winding, it is necessary to take the magnetization curve of the steel on alternating current up to \(B=B_n\). Having disconnected the biasing winding in the circuit of Fig. 1, the current in the primary winding is gradually increased and the voltage on the secondary is measured.

From the primary current, the magnetizing specific ampere-turns are found, and from the voltage—the induction; the magnetization curve is plotted. From this curve, for the selected induction \(B_n\), the ampere-turns \(aw_n\) are found. Then the biasing current is determined from the expression:

\[ i_n=\frac{aw_n l}{w_n}. \tag{13} \]

The power expended for biasing is

\[ S_n=u_n i_n . \tag{14} \]

Let us now determine the primary pickup current of the protection \(I_{PT}\). In doing so, we shall regard the no-load current \(i_0\) as purely reactive and take into account relation (4).

Then the primary pickup current is:

\[ \begin{aligned} I_{PT} &= w_2\sqrt{(i_{PT}\cos\varphi_p)^2+(i_{PT}\sin\varphi_p+mi_0)^2} \\ &= w_2 i_{PT}\sqrt{\cos^2\varphi_p+(\sin\varphi_p+1)^2} \\ &= w_2 i_{PT}\sqrt{2+2\frac{x_p}{z_{PT}}}. \end{aligned} \tag{15} \]

The dead zone of the protection:

\[ b_0=\frac{I_{PT}}{I_{z.m}}100, \tag{16} \]

where \(I_{z.m}\) is the maximum earth-fault current.

The current \(I_{z.m}\) in networks with isolated neutrals of a given voltage will be equal to the capacitive earth-fault current \(I_c\) of this network. If some neutrals of the network are grounded through an active resistance, then the current \(I_{z.m}\) will also contain an active component.

Then

\[ I_{z.m}=\sqrt{I_z^2 a+I_c^2}. \tag{17} \]

In networks with a compensated neutral, the current \(I_{z.m}\) will be equal to the residual current at the point of insulation breakdown.

Table 2 gives examples of calculating such protection, carried out according to circuits \(a\), \(d\), and \(c\) of Fig. 4. Each circuit is calculated in two variants: for relays ETD-551/40 and ETD-551/60.

Fig. 4

The ratio between the resistances \(r_p\) and \(x_p\) is taken as approximate. The transformer cores have \(l=45.5\ \mathrm{cm}\) and cross section \(q=20\ \mathrm{sq.\ cm}\). In the preliminary calculation, the angular coefficient \(k\) is taken equal to 15300 for all variants, which corresponds to the operating induction \(B_p=250\ \mathrm{Gs}\).

Table 2

It was found that the protection transformers in circuit (a) would have an inductance of 245 H for the first variant and 250 H for the second. The secondary emf for this circuit proved to be close to the product $i_p r_{\mathrm{T}p}$ (see paras. 9 and 17), and therefore the calculation of the transformers for protection according to circuit (a) proved successful with $k = 15300$. For protection according to circuits (b) and (c), the secondary emf proved to be less than $i_p r_{\mathrm{T}p}$; therefore, for these protections, the coefficient $k$ had to be taken in accordance with the inductances obtained in the preliminary calculation: for circuit b (at $B_p = 120$ G)—12600, and for circuit c (at $B_p = 80$)—11800. At these values, the emf $E_2$ proved to be close to $i_p r_{\mathrm{T}p}$ for the indicated circuits (compare paras. 9 and 22).

As can be seen from Table 2, the protection under consideration can be implemented for generators of all ratings. In terms of the number of secondary-winding turns, the second variant proved the most acceptable for all three circuits, since it gives the smallest value of $w_2$. In the second variant, the core cross-section could have been reduced to $10 \div 15\ \mathrm{cm}^2$, which would have led to some increase in $w_2$, but would have reduced the cost of the transformer.

The proposed method for calculating the protection under consideration is simple and sufficiently accurate. The use, for the calculation, of experimental data obtained for the available steel enables local relay engineers to use cores from old current transformers for making such transformers, which is of great importance under present conditions.

In conclusion, it should be noted that the sensitivity of the protection under consideration is quite high for generators with ratings up to 25 MW, as well as for motors and feeders of all ratings.

1. See Electrical Stations, Nos. 8–9, 1938. ↩

2. See Electrical Stations, No. 4, 1949. ↩

3. When manufacturing the transformer core from a special alloy, it is possible to obtain very high protection sensitivity even without biasing. See the article by Ya. S. Gefand et al. in No. 11 of Electrical Stations, 1951. ↩