Behavior of Current Transformers with an Air Gap under Short Circuits in the Primary Circuit

I. D. Kutyavin

In the first part of the present work, published in this same volume of the Proceedings of the Institute under the title “Calculation of Current Transformers with an Air Gap,” we investigated the behavior of the transformers under consideration in the steady-state regime in the primary circuit and at inductions in the core up to 10 kGs. In that article the basic principles were also established for selecting the magnitude of the air gap and other transformer parameters.

The present article contains the results of an investigation of the behavior of the transformer under study during short circuits in the primary circuit. In doing so, the following tasks were set before the investigation:

1) study of the character of the change in the magnetizing and secondary currents during a short circuit in the primary circuit;

2) finding a sufficiently accurate method for the analytical determination of these currents;

3) determination of the maximum permissible induction in the core under the condition of preserving the rectilinearity of the transformer characteristic;

4) determination of the maximum permissible primary short-circuit current of the transformer from the point of view of core saturation.

Derivation of expressions for the magnetizing current and the secondary current of the transformer under consideration during a short circuit in the primary circuit

To derive the basic relations in the transient process, we shall use the circuit of the simplest primary circuit with the equivalent circuit of the current transformer included in it, as shown in Fig. 1. The total impedance of the primary circuit after the occurrence of the short circuit will be denoted by \(z_1\), and we shall assume that it includes the active resistance and the leakage reactance of the primary winding of the transformer. Steel losses are not taken into account, since the principal part of the magnetic resistance of the core is constituted by the air gap.

Fig. 1

All quantities indicated in the diagram of Fig. 1 are considered referred to the number of turns of the secondary winding of the transformer.

To derive expressions for the currents \(i_0\) and \(i_2\), let us consider the switching-on of a circuit consisting of resistances \(z_1(p)\), \(z_2(p)\), and \(z_0(p)\) to a sinusoidal emf

\(e = E_m \sin(\omega t + \alpha)\)

under the conditions that \(i_1(0) = 0\), \(i_2(0) = 0\), and \(i_0(0) = 0\).

Let us represent the indicated e.m.f. by the complex quantity \(\dot E(t)=\dot E_m e^{j\omega t}\), where \(\dot E_m=E_m e^{j\alpha}\), and take into account that

\[ \dot E(t)=\dot E(p)=\dot E_m \frac{p}{p-j\omega}. \]

Then the complex expression for the primary current is

\[ \dot I_1(t)=I_1(p)=\frac{E(p)}{z(p)}=\frac{p\dot E_m}{z_1(p)(p-j\omega)}. \tag{1} \]

In doing so, we assume that the total resistance of the transformer equivalent circuit may be neglected in comparison with the resistance of the primary circuit.

The complex expression for the magnetizing current of the transformer is

\[ \dot I_0(t)=I_0(p)=I_1(p)\frac{z_2(p)}{z_2(p)+z_0(p)} =\frac{p\dot E_m z_2(p)}{(p-j\omega)z_1(p)[z_2(p)+z_0(p)]} = \]

\[ =\frac{p\dot E_m(r_2+pL_2)}{(p-j\omega)(r_1+pL_1)[r_2+p(L_2+L_0)]} =p\frac{G_0(p)}{F(p)}, \tag{2} \]

where

\[ G_0(p)=\dot E_m(r_2+pL_2); \tag{3} \]

\[ F(p)=(p-j\omega)(r_1+pL_1)[r_2+p(L_2+L_0)]. \tag{4} \]

Analogously, the complex expression for the secondary current is

\[ \dot I_2(t)=I_2(p)= \frac{p\dot E_m pL_0}{(p-j\omega)(r_1+pL_1)[r_2+p(L_2+L_0)]} =p\frac{G_2(p)}{F(p)}, \tag{5} \]

where

\[ G_2(p)=\dot E_m pL_0. \tag{6} \]

The roots of the operational equation (4), \(F(p)=0\), are:

\[ p_1=-\frac{r_1}{L_1}=-\frac{1}{T_1};\quad p_2=-\frac{r_2}{L_2+L_0}=-\frac{1}{T_2};\quad p_3=j\omega. \tag{7} \]

The derivative of the operational function (4) is

\[ F'(p)=L_1(L_2+L_0)\left[(p-p_1)(p-p_2)+(p-p_1)(p-p_3)+\right. \]

\[ \left.+(p-p_2)(p-p_3)\right]. \tag{8} \]

The derivative of the operational function \(F'(p)\) at \(p=p_1\) is

\[ F'(p_1)=L_1(L_2+L_0)(p-p_2)(p-p_3). \tag{9} \]

The same at \(p=p_2\):

\[ F'(p_2)=L_1(L_2+L_0)(p-p_1)(p-p_3). \tag{10} \]

The same at \(p=p_3\):

\[ F'(p_3)=L_1(L_2+L_0)(p-p_1)(p-p_2). \tag{11} \]

Then, using the expansion theorem, we obtain an expression for the magnetizing-current complex:

\[ \dot{i}_0(t)=\sum \frac{G_0(p_k)}{F'(p_k)} e^{p_k t}\bigg|_{p_1,p_2,\ldots,p_k} = i_{M_1}\left[ \frac{z_2 e^{j(\omega t+\alpha-\varphi_1-\psi_0)}}{r_2\sqrt{1+\omega^2 T_2^2}} - \frac{(r_2T_1-L_2)e^{p_1t+j(\alpha-\varphi_1)}}{r_2(T_1-T_2)} + \frac{L_0\sqrt{1+\omega^2T_1^2}\,e^{p_2t+j(\alpha-\varphi_2)}}{r_2(T_1-T_2)\sqrt{1+\omega^2T_2^2}} \right]. \tag{12} \]

Passing from the magnetizing-current complex to its instantaneous value, we find the following expression:

\[ i_0=i_{M_1}\left[ a_0\sin(\omega t+\alpha-\varphi_1-\psi_0) -b_0e^{-\frac{t}{T_1}}\sin(\alpha-\varphi_1) + c_0e^{-\frac{t}{T_2}}\sin(\alpha-\varphi_2) \right], \tag{13} \]

where

\[ i_{M_1}=\frac{E_M}{z_1}=\frac{E_M}{r_1\sqrt{1+\omega^2T_1^2}} \]

is the amplitude of the primary steady-state current referred to the number of turns of the secondary winding;

\[ a_0=\frac{z_2}{r_2\sqrt{1+\omega^2T_2^2}}; \tag{14} \]

\[ b_0=\frac{r_2T_1-L_2}{r_2(T_1-T_2)}; \tag{15} \]

\[ c_0=\frac{L_0\sqrt{1+\omega^2T_1^2}}{r_2(T_1-T_2)\sqrt{1+\omega^2T_2^2}}; \tag{16} \]

\(\alpha\) is the angle characterizing the phase of the e.m.f. at the instant of short circuit;

\[ \varphi_1=\operatorname{arctg}\omega T_1; \tag{17} \]

\[ \varphi_2=\operatorname{arctg}\omega T_2; \tag{18} \]

\[ \psi_0=\operatorname{arctg}\frac{\omega L_0}{r_2+\omega^2L_2T_2}. \tag{19} \]

Analogously, one can find an expression for the instantaneous value of the secondary current, which has the following form:

\[ i_2=i_{M_1}\left[ a_2\sin(\omega t+\alpha-\varphi_1-\psi_2) -b_2e^{-\frac{t}{T_1}}\sin(\alpha-\varphi_1) -c_2e^{-\frac{t}{T_2}}\sin(\alpha-\varphi_2) \right], \tag{20} \]

where

\[ a_2=\frac{\omega L_0}{r_2\sqrt{1+\omega^2T_2^2}}; \tag{21} \]

\[ b_2=\frac{L_0}{r_2(T_1-T_2)}; \tag{22} \]

\[ c_2=\frac{L_0\sqrt{1+\omega^2T_1^2}}{r_2(T_1-T_2)\sqrt{1+\omega^2T_2^2}}=c_0; \tag{23} \]

\[ \psi_2=\operatorname{arctg}\left(-\frac{1}{\omega T_2}\right)=\frac{\pi}{2}+\varphi_2. \tag{24} \]

As is seen from (20) and (13), the secondary current \(i_2\) and the magnetizing current \(i_0\) under a short circuit in the primary circuit in the general case consist of three components.

The first term in the brackets of these expressions represents the relative value of the forced periodic current in the steady-state process.

The second term corresponds to the free aperiodic current of the primary circuit and therefore decays with the time constant \(T_1\).

The third term represents the aperiodic free current arising in the secondary circuit of the transformer during a short circuit in the primary circuit. The decay of this current is associated only with the constants of the transformer equivalent circuit and occurs with the time constant \(T_2\).

Determination of the greatest maximum of the magnetizing current \(i_0 = f(\alpha,t)\)

Expression (13) gives the magnitude of the magnetizing current in the most general form. For a practical assessment of the influence of this current on processes in relay-protection circuits, knowledge of the greatest maximum \(i_0\) is of great interest. This particular value of the current \(i_0\) is also necessary for determining the maximum permissible primary short-circuit current \(I_{\delta 1}\), at which the induction in the core reaches the limiting permissible value \(B_{m0}\). For this purpose, let us investigate expression (13).

In view of the fact that the first term of expression (13) represents the steady-state periodic current having a constant amplitude, the investigation for the maximum has been carried out only for the relative total aperiodic magnetizing current, which has the following expression:

\[ i_a=b_0 e^{-\frac{t}{T_1}}\sin(\alpha-\varphi_1)-c_0 e^{-\frac{t}{T_2}}\sin(\alpha-\varphi_2). \tag{25} \]

As is seen from (25), the total aperiodic current is a function of two independent variables, \(\alpha\) and \(t\); therefore the investigation of the indicated expression for a maximum should be carried out with respect to both variables.

The partial derivative of expression (25) with respect to time \(t\) is:

\[ \frac{\partial i_a}{\partial t} = -\frac{b_0}{T_1}e^{-\frac{t}{T_1}}\sin(\alpha-\varphi_1) + \frac{c_0}{T_2}e^{-\frac{t}{T_2}}\sin(\alpha-\varphi_2)=0. \tag{26} \]

The partial derivative of the expression under investigation with respect to the angle \(\alpha\) is:

\[ \frac{\partial i_a}{\partial \alpha} = b_0 e^{-\frac{t}{T_1}}\cos(\alpha-\varphi_1) - c_0 e^{-\frac{t}{T_2}}\cos(\alpha-\varphi_2)=0. \tag{27} \]

Solving the obtained equations (26) and (27) simultaneously, we find that

\[ \operatorname{tg}(\alpha-\varphi_1) = -\frac{T_1}{T_2}\operatorname{tg}(\alpha-\varphi_2); \tag{28} \]

\[ e^{\,t\left(\frac{1}{T_2}-\frac{1}{T_1}\right)} = \frac{c_0 T_1\sin(\alpha-\varphi_2)} {b_0 T_2\sin(\alpha-\varphi_1)}. \tag{29} \]

Replacing in (28) the tangents of the differences of angles by the tangents of these angles and taking into account (17) and (18), we obtain that the angle \(\alpha\) corresponding to the maximum of \(i_a\) will have two values:

\[ \alpha_1=0+n\pi, \tag{30} \]

\[ \alpha_2=\frac{\pi}{2}+\varphi_1+\varphi_2+n\pi. \tag{31} \]

Then the time \(t_1\) corresponding to the maximum, at \(\alpha=-\alpha_1\), from (29):

\[ t_1=\frac{T_1\cdot T_2}{T_1-T_2}\ln \frac{L_0(1+\omega^2T_1^2)}{(T_1r_2-L_2)(1+\omega^2T_2^2)}. \tag{32} \]

The time \(t_2\) at \(\alpha=-\alpha_2\):

\[ t_2=\frac{T_1\cdot T_2}{T_1-T_2}\ln \frac{T_1L_0}{T_2(T_1r_2-L_2)}. \tag{33} \]

In accordance with the obtained values of the angles \(\alpha_1\) and \(\alpha_2\) and the times \(t_1\) and \(t_2\), the maximum of the aperiodic current will have two expressions. The maximum of the relative value of the aperiodic current, upon substituting \(\alpha_1\) and \(t_1\) into expression (25):

\[ i_{aM_1}= \frac{\omega L_0\sqrt{1+\omega^2T_1^2}}{r_2(1+\omega^2T_2^2)} \cdot e^{-\frac{t_1}{T_2}}. \tag{34} \]

The same upon substituting the values \(\alpha_2\) and \(t_2\) into (25):

\[ i_{aM_2}= \frac{L_0e^{-\frac{t_2}{T_2}}}{(L_2+L_0)\sqrt{1+\omega^2T_2^2}}. \tag{35} \]

We are interested in the greatest maximum of the aperiodic current; therefore, from the two expressions (34) and (35), it is necessary to choose the one that gives the required maximum. To solve this problem, let us determine the ratio of \(i_{aM_1}\) to \(i_{aM_2}\):

\[ \frac{i_{aM_1}}{i_{aM_2}} =\omega T_2\sqrt{\frac{1+\omega^2T_1^2}{1+\omega^2T_2^2}} \cdot e^{\frac{t_2-t_1}{T_2}} \simeq \sqrt{1+\omega^2T_1^2}\, e^{\frac{t_2-t_1}{T_2}}. \tag{36} \]

For real circuits of electrical installations \(0<T_1<0.15\); therefore the factor on the right-hand side of (36) standing before the exponential function will be greater than unity:

\[ \sqrt{1+\omega^2T_1^2}>1. \tag{37} \]

The exponent of the second factor on the right-hand side of (36), on the basis of (32) and (33):

\[ \frac{t_2-t_1}{T_2} = \frac{T_1}{T_1-T_2} \ln \frac{T_1(1+\omega^2T_2^2)}{T_2(1+\omega^2T_1^2)} \simeq \frac{T_1}{T_1-T_2}\ln\frac{T_2}{T_1}. \tag{38} \]

It can easily be shown that, for \(T_2>T_1\), the exponent of the second factor will lie within the following limits:

\[ 0>\frac{t_2-t_1}{T_2}>(-1). \tag{39} \]

Then the second factor of relation (36) will have the following limits:

\[ 1>e^{\frac{t_2-t_1}{T_2}}>e^{-1}. \tag{40} \]

Therefore, the lower limit of relation (36) will be equal to \(\sqrt{1+\omega^2T_1^2}\,e^{-1}\), and consequently the greatest maximum will be \(i_{a}\), determined by expression (34).

Taking into account the foregoing for the greatest maximum of the magnetizing current on the basis of (13), (14), and (34), one may write the following expression:

\[ i_{м0}=\frac{i_{м1}}{r_2\sqrt{1+\omega^2T_2^2}} \left( z_2+\omega L_0 \sqrt{\frac{1+\omega^2T_1^2}{1+\omega^2T_2^2}}\, e^{-\frac{t_1}{T_2}} \right), \tag{41} \]

where \(z_2=\sqrt{r_2^2+\omega^2L_2^2}\) is the total impedance of the transformer secondary circuit.

If the unity under the radical sign is neglected and it is assumed that

\[ \frac{\omega L_0}{r_2}\simeq T_2, \]

then expression (41) may be simplified:

\[ i_{м0}=\frac{i_{м1}}{\omega T_2} \left( \frac{z_2}{r_2}+\omega T_1 e^{-\frac{t_1}{T_2}} \right). \tag{41a} \]

The instantaneous value of the magnetizing current at \(\alpha=\alpha_1\) and in the event of a short circuit in the primary circuit is:

\[ i_0=i_{м1}\left[ a_0\sin(\omega t-\varphi_1-\psi_0) +b_0e^{-\frac{t}{T_1}}\sin\varphi_1 -c_0e^{-\frac{t}{T_2}}\sin\varphi_2 \right]. \tag{42} \]

However, it should be borne in mind that expression (41) is valid only for the special case when the maximum of the periodic component of the magnetizing current coincides with the maximum of the aperiodic component. In all other cases the current \(i_{м0}\), determined from (41), will be greater than its actual value. In this connection it becomes necessary to determine the time corresponding to the maximum of the periodic component of the magnetizing current. This time can be calculated on the basis of Fig. 2, taking into account that the maximum \(i_0\) will repeat every \(0.02\) sec:

\[ t'=0.02\left[(n-1)+\frac{\varphi_1+\varphi_2+90}{360}\right], \tag{43} \]

where \(n\) is the ordinal number of the maximum, and \(\varphi_1\) and \(\varphi_2\) are angles in degrees, determined from (17) and (18).

Fig. 2

Then, for the maximum of the magnetizing current, the following expression can be obtained on the basis of (42):

\[ i_{м0}=i_{м1} \left( a_0+b_0e^{-\frac{t'}{T_1}}\sin\varphi_1 -c_0e^{-\frac{t'}{T_2}}\sin\varphi_2 \right). \tag{44} \]

The maximum value of the magnetizing current \(i_{м0}\) may be determined in the following manner. From expression (32) the time \(t_1\) is found, and from (43), by substituting \(n=1,2,3\ldots\), two values of \(t'\) nearest to \(t_1\) are determined (one nearest smaller value and the other larger). If, in this case, \(t_1\) coincides with one of the values \(t'\), then \(i_{м0}\) is found from (41) at \(t_1\). If \(t_1\) differs from both values \(t'\), then the current \(i_{м0}\) is determined from (44) for each value of \(t'\), and the larger value is taken for the calculation.

When evaluating formulas (41) and (44) for calculating \(i_{m0}\), it should be borne in mind that, with linear constant parameters of the transformer equivalent circuit, formula (44) must give the exact value of \(i_{m0}\), while formula (41) gives a somewhat exaggerated value of the current \(i_{m0}\), if \(t_1 \ne t'\).

However, the use of formulas (41) and (44) for determining the current \(i_{m0}\) is made difficult by the fact that the resistance of the magnetizing branch of the transformer equivalent circuit, \(\omega L_0\), depends not only on the magnitude of the maximum induction in the core, but also on the relative magnitude of the aperiodic component of the current \(i_{m0}\). This is seen from Fig. 3, which shows the dependence \(\omega L_0 = f(\beta, B)\), found experimentally. To obtain this dependence, two identical current transformers with an air gap \(\delta = 2.5\) mm were connected according to the circuit of Fig. 4. At a definite

Fig. 3

Fig. 4

constant biasing current \(I_p\), a quite definite alternating current was established, whose amplitude was \(\beta\) times smaller than the direct current; at the same time, the voltage drop across the alternating-current windings of the transformers was measured. Then the resistance of the magnetizing branch was calculated by the approximate formula:

\[ \omega L_0 \cong \frac{U_{\sim}}{2 I_{\sim}}, \tag{45} \]

and the relative value of the constant biasing current (aperiodic component) was found from the relation:

\[ \beta = \frac{W_1 I_p}{\sqrt{2}\, W_2 I_{\sim}}. \tag{46} \]

From the data obtained, the curves of Fig. 3 were constructed for \(\beta = 0;\ 3;\ 5\). In this case the maximum induction \(B_m\), plotted along the abscissa axis, was determined from the total maximum ampere-turns \(W_1 I_p + \sqrt{2}\, W_2 I_{\sim}\).

In order to calculate the current \(i_{m0}\) by one of expressions (41) or (44), it is necessary to determine from Fig. 3 the calculated resistance \(\omega L_0\) for the given maximum permissible induction \(B_{m0}\) and the greatest possible value of \(\beta\). This latter value (i.e., \(\beta\)) can be found from the ratio of \(i_{am}\) to \(a_0\) (see expressions 34 and 14):

\[ \beta = \frac{\omega L_0}{z_2} \sqrt{\frac{1+\omega^2 T_1^2}{1+\omega^2 T_2^2}}\, e^{-\frac{t_1}{T_2}} \cong \omega T_1 e^{-\frac{t_1}{T_2}}\cos\varphi, \tag{47} \]

where

\[ \cos \varphi=\frac{r_2}{z_2} \]

is determined from the data of the transformer secondary circuit.

The exponent \(t_1/T_2\) can be calculated by a simplified formula obtained from (32), since the value \(L_0\) is not yet known:

\[ \frac{t_1}{T_2}= \frac{T_1}{T_1-T_2}\ln \frac{L_0(1+\omega^2T_1^2)} {(T_1r_2-L_2)1+\omega^2T_2^2} = \frac{T_1}{T_1-T_2}\ln\frac{T_1}{T_2}. \tag{48} \]

This simplification amounts to taking the ratio

\[ \frac{L_0}{r_2}\simeq T_2 \]

and neglecting unity and the value \(L_2\) in comparison with \(T_1r_2\) in the denominator.

When calculating \(\beta\) from expression (47), one may use Fig. 5, where the dependence

\[ e^{-\frac{t_1}{T_2}}=f\left(\frac{T_2}{T_1}\right) \]

is given in the form of two curves, curve (1) being plotted for a larger scale of the abscissa axis.

Thus, to calculate the current \(i_{m0}\), it is necessary to find \(\omega L_0\), and to determine \(\omega L_0\) it is necessary to know the time constant \(T_2\), which itself depends on \(L_0\). Therefore the value \(\omega L_0\) can be determined only by the method of successive approximations, in which one should first specify the ratio of \(T_2\) to \(T_1\) and calculate \(\beta\), and then find \(\omega L_0\) from Fig. 3. The solution of this problem is accelerated by the fact that, in real installations, the time constant \(T_2 \gg T_1\), and under these conditions the exponential multiplier entering into (47) (as is seen from Fig. 5) changes very little.

Fig. 5

Experimental verification of the conclusions obtained

For the purpose of experimentally verifying the conclusions obtained above, a large number of oscillograms were taken in a short-circuit experiment, recording the currents \(i_1\), \(i_2\), and \(i_0\), one of which is shown in Fig. 6, while the remaining ones were used for plotting the curves in Fig. 7. The experiments were carried out for a current transformer with an air gap \(\delta=0.25\ \text{cm}\), a core cross section \(q=10\ \text{cm}^2\), and windings of 100 turns each. The total resistance of the secondary circuit was purely active and amounted to \(0.38\ \Omega\). The use, in the experiments described, of a reduced-power current-transformer model with an air gap was due to the fact that the use of a transformer of normal power and with a transformation ratio equal to unity distorted the results of the short-circuit experiments.

Figure 6 shows the variation of the currents \(i_1\), \(i_2\), and \(i_0\) during a short circuit, when the steady-state primary current \(I_1\) was \(15\ \text{A}\) and the constant

of the primary circuit \(T_1 = 0.025\). In this case the maximum amplitude of the magnetizing current \(i_{\mu 0}=12.6a\) occurs approximately after one and a half periods following the beginning of the transient process. The induction in the core corresponding to the current \(i_{\mu 0}\) is \(8.8\ \mathrm{kGs}\).

Further, Fig. 7 shows the dependence of the greatest maximum of the magnetizing current \(i_{\mu 0}\) on the rms value of the steady-state primary current \(I_1\) (curve 1), taken during the transient process and for \(a \simeq c_1\) and \(T_1 = 0.025\ \mathrm{sec}\).

Fig. 6

Here also is shown the dependence of the rms value of the steady-state magnetizing current \(I_{\mu 0}\) on the current \(I_1\)—curve (2).

As is seen from Fig. 7, the dependence \(i_{\mu 0}=f(I_1)\), for primary current \(I_1\) up to \(10a\), is close to a straight line, and then the current \(i_{\mu 0}\) begins to increase sharply, which indicates the onset of saturation of the core steel. This is clearly illustrated by Fig. 8, where the dependence of the current \(i_{\mu 0}\) on the induction in the core \(B\) during the transient process is given; it shows that the current \(i_{\mu 0}\) begins to increase sharply starting from an induction of \(8\text{–}9\ \mathrm{kGs}\). To determine the degree of deviation of the curve \(i_{\mu 0}\) from a straight line, an auxiliary dashed straight line has been drawn in Figs. 7 and 8.

Further, curve (3) of Fig. 7 gives the dependence \(i_{\mu 0}=f(I_1)\), calculated by formula (41a) for conditions coinciding with the experimental ones. Comparison of this curve with curve (1) shows that (as was to be expected) formula (41a) gives a value of \(i_{\mu 0}\) that is 20–30% too high compared with experiment. This is explained by the fact that, in calculating \(i_{\mu 0}\) in (41a), the resistance \(\omega L_0\) corresponding to the instant at which \(i_{\mu 0}\) occurs was substituted, whereas this resistance changes during the transient process and, at the period when \(i_{\mu 0}\) occurs, has its smallest value. This is due to the fact that the value of \(\beta\) during the transient process changes from values \(\beta < 1\) in the initial

moment to values \(\beta \gg 1\) at the instant when \(i_{\mathrm{mo}}\) occurs. The circumstance that formula (41a) gives an overestimated value of \(i_{\mathrm{mo}}\) does not reduce its value.

It should also be noted that formula (41a), being approximate, gives \(i_{\mathrm{mo}}\) overestimated by 3–5% in comparison with formula (41).

The numerous experiments carried out with the above-described model of a current transformer with an air gap show the possibility of creating a current transformer with linear constants.

From Fig. 8 it is seen that, in order to preserve the linearity of the constants of the transformers being studied, the induction in their cores under any operating conditions must not exceed 8–9 kGs.

The linearity of the constants of the transformers under consideration will make it possible to calculate the magnetizing currents and their derivatives (for example, the unbalance currents of differential protections) during short circuits analytically, which will simplify the study of the properties of protections and their adjustment under operating conditions.

The experiments carried out, as well as expression (41), show that the quantity \(i_{\mathrm{mo}}\) increases with the growth of the time constant \(T_1\); therefore, when determining the current \(i_{\mathrm{mo}}\) or its derivatives, it is necessary to take into account the greatest possible value of \(T_1\) for the given point of the system.

Determination of the Maximum Permissible Primary Short-Circuit Current for a Transformer

Let us find an expression for the maximum permissible primary short-circuit current for a transformer at which the induction in the core reaches the limiting permissible value. For this purpose we shall use Kirchhoff’s second law for the magnetic circuit of the transformer (neglecting leakage fluxes and losses in the steel):

\[ i_1 W_1 + i_2 W_2 = i_0 W_2 = \Phi R_m, \tag{49} \]

where \(\Phi\) is the instantaneous value of the magnetic flux in the core, and
\(R_m\) is the magnetic resistance of the core, the principal part of which is the resistance of the air gap. Taking into account the above assumptions, expression (49) may also be applied to the maximum values of the quantities entering into it, namely:

\[ i_{\mathrm{mo}} W_2 = \Phi_m R_m . \tag{50} \]

The maximum magnetic flux \(\Phi_m\) entering into (50) may be expressed through the limiting permissible induction:

\[ \Phi_m = q B_{\mathrm{mo}}, \tag{51} \]

where \(q\) is the cross section of the core in \(\mathrm{cm}^2\), and
\(B_{\mathrm{mo}} = 8\text{–}9\ \mathrm{kGs}\) is the limiting permissible induction.

To determine the magnetic resistance of the core one may use the formula for the inductance coefficient of a toroid, which has the following form:

\[ L = \frac{4\pi W^2 10^{-9}}{R_m}. \tag{52} \]

The magnetic resistance of the transformer core on the basis of (52) will have the following expression:

\[ R_m = \frac{4\pi W_2^2 10^{-9}}{L_0} = \frac{4\pi \omega W_2^2 10^{-9}}{\omega L_0}, \tag{53} \]

where \(L_0\) is the inductance coefficient of the magnetizing branch of the transformer equivalent circuit, and
\(\omega = 2\pi f\) (\(f\) is the number of current periods).

Then the maximum permissible rms value of the periodic component of the short-circuit current at the initial instant of time (supertransient current), on the basis of (11), (50), (51), and (53):

\[ I_{\partial 1}= \frac{q B_{\text{md}} R_m r_{\nu}\sqrt{1+\omega^2 T_2^2}} {\sqrt{2}\,W\left(Z_2+\omega L_0\sqrt{\frac{1+\omega^2 T_1^2}{1+\omega^2 T_2^2}}\,e^{-\frac{t_1}{T_1}}\right)} , \tag{54} \]

Expression (54) can be somewhat simplified if, under the radical sign, unity is neglected in comparison with \(\omega^2T^2\):

\[ I_{\partial 1}= \frac{q B_{\text{md}} R_m \omega(L_2+L_0)} {\sqrt{2}\,W\left(Z_2+\omega L_0\frac{T_1}{T_2}e^{-\frac{t_1}{T_1}}\right)} , \tag{55} \]

We shall now investigate the dependence of \(I_{\partial 1}\) on the quantities entering into the right-hand side of expression (55). For this purpose we shall use the data of the example given in the above-mentioned article by the author, “Calculation of Current Transformers with an Air Gap.” These data are as follows: the primary rated current of the transformer \(I_{n1}=400\) A, the secondary rated current \(I_{n2}=1\) A, the primary number of turns \(W_1=1\), the secondary number of turns \(W_2=400\), the cross-sectional area of the core \(q=25\ \text{cm}^2\), the air gap \(\delta=0.3\ \text{cm}\), the active resistance of the secondary winding \(r_{T2}=1\ \Omega\) and the reactance \(\omega L_{T2}=0\), the active resistance of the burden \(r_{H2}=1\ \Omega\) and the reactance \(\omega L_{H2}=3.2\ \Omega\), and the resistance of the magnetizing branch at \(B=9\ \text{kGs}\), \(\omega L_0=72\ \Omega\).

In addition, let us take \(B_{\text{md}}=9\ \text{kGs}\) and \(T_1=0.1\ \text{sec}\).

The magnetic resistance of the core from (53) is:

\[ R_m=\frac{4\pi\cdot314\cdot400^2\cdot10^{-9}}{72} =8.75\cdot10^{-3}\,\frac{a}{b\delta}. \]

The time constant of the secondary circuit of the transformer, on the basis of (7), is:

\[ T_2=\frac{72+3.2}{314(1+1)}=0.12\ \text{sec}. \]

The time corresponding to the maximum of the magnetizing current, from (27), is:

\[ t_1=\frac{0.1\cdot0.12}{0.1-0.12} \ln\frac{72\cdot0.1^2}{(314\cdot0.1\cdot2-3.2)0.12^2} =0.105\ \text{sec}. \]

The maximum permissible short-circuit current from (55) is:

\[ I_{\partial 1}= \frac{25\cdot9\cdot10^3\cdot8.75\cdot10^{-3}(72+3.2)} {2\left(3.8+72\frac{0.1}{0.12}e^{-0.88}\right)} =3640\ \text{A}. \]

The ratio of the current \(I_{\partial 1}\) to the rated current \(I_{n1}=400\) A is:

\[ K_{\text{md}}=\frac{I_{\partial 1}}{I_{n1}}=\frac{3640}{400}=9.1. \]

Similarly, the permissible short-circuit current ratio \(K_{\text{md}}\) was calculated for current transformers with primary rated currents of 1000, 2000, 3000, and 4000 A, and from the data obtained the curves of the dependence \(K_{\text{md}}=f(I_{n1})\) were plotted in Fig. 9 for \(T_1=0.05\) and \(0.1\ \text{sec}\). In determining the short-circuit current ratio for current transformers with rated primary current up to 4000 A it was assumed that the dimensions of the core, the magnitude of the air gap, and the resistance of the secondary burden—

... of the circuit do not change; the resistance \(r_{T2}\) changes proportionally to \(W_2\), and the resistance \(\omega L_0\)—proportionally to the square of \(W_2\).

From a comparison of the curves in Fig. 9 and an analysis of formula (55), it is seen that the permissible ratio \(K_{i\,\mathrm{dop}}\) increases as \(T_1\) decreases; therefore, when determining the maximum permissible value of \(K_{i\,\mathrm{dop}}\), it is necessary to use the largest value of \(T_1\) for the given conditions.

It is also seen from Fig. 9 that the permissible current ratio \(K_{i\,\mathrm{dop}}\) for transformers with a rated current up to \(1000\ \mathrm{A}\) may be insufficient; therefore there is interest in methods for increasing this parameter. For this purpose we investigate expression (55) at \(T_1 = 0.1\) for a transformer with primary rated current \(I_{n1}=400\ \mathrm{A}\), in order to determine the influence of the core area \(q\) and the magnetic resistance \(R_m\) on the value of \(K_{i\,\mathrm{dop}}\).

Fig. 9

Fig. 10

The data for this transformer were given above. The magnetic resistance of the transformer core with an air gap \(\delta = 0.3\ \mathrm{cm}\) and a cross section \(q = 25\ \mathrm{cm}^2\) was determined by us above and is \(8.75 \cdot 10^{-3}\). The value of this resistance is directly proportional to the calculated air gap \(\delta_p\):

\[ R_m = \frac{1}{q}\left(\frac{l}{\mu}+\delta_p\right), \tag{56} \]

where \(l\) is the length of the mean line of force in the steel of the core, and
\(\delta_p\) is the calculated air gap, taking into account the bulging of the lines of force and inaccuracies in assembling the gap strips. Consequently, the magnetic resistance of the core can be varied by changing the gap \(\delta_p\) or the cross section \(q\).

With a change in the magnetic resistance of the core, the inductive resistance of the magnetizing branch of the transformer equivalent circuit will change according to the following law:

\[ \omega L_{\omega2}=\omega L_{\omega1}\frac{R_{m1}}{R_{m2}}. \tag{57} \]

Figure 10 shows the dependence \(K_{i\,\mathrm{dop}} = f(q,R_m)\), constructed with allowance for relation (57) for \(q = 15, 25\), and \(35\ \mathrm{cm}^2\). As can be seen from this figure, the influence of \(q\) and \(R_m\) on the value of \(K_{i\,\mathrm{dop}}\) is very considerable and can be used in the design of transformers.

Secondary Current of the Transformer

Expression (20) is general for the secondary current as a function of the independent variables \(t\) and \(\alpha\). This expression, as well as the oscillogram (Fig. 6), shows that the amplitude of the periodic component \(i_2\), with the amplitude of the periodic component \(i_1\) constant, will remain practically constant throughout the entire transient process.

This conclusion is based on the linearity of the parameters of the equivalent circuit of the transformers under consideration, on the proportionality of the amplitudes of the periodic components of the currents, and is also confirmed by Fig. 7.

The distribution of the aperiodic component of the primary current between the secondary circuit and the magnetizing branch at the initial instant of time is characterized by the relation:

\[ -\frac{b_2}{b_0} = -\frac{L_0}{T_1 r_2 - L_2} \simeq -\frac{T_2}{T_1}. \tag{58} \]

The aperiodic current arising in the secondary circuit of the transformer has the same magnitude both in the current \(i_2\) and in the current \(i_0\).

The above investigation of the operation of current transformers with an air gap under a short circuit in the primary circuit shows that the manufacture of such transformers for the entire range of rated primary currents with the required value of \(K_{\mathrm{md}}\) is quite possible. The air gap, depending on the required value of \(K_{\mathrm{md}}\), will be on the order of \(2\)--\(5\) mm. It has also been established that the transformers under consideration, at inductions in the core up to \(8\)--\(9\) kilogauss, will retain linearity of parameters during a short circuit, and this will make it possible to calculate the currents \(i_2\) and \(i_0\) by an analytical method.