Current Transformers with an Air Gap

I. D. Kutyavin

Current transformers with an air gap, within core inductions up to 8000–10000 gauss, have a constant error, since the main part of the magnetizing ampere-turns is accounted for by the air gap, which has a constant magnetic reluctance. The presence of an air gap in the cores of current transformers can also eliminate the influence on their operation of residual magnetism and saturation of the iron.

Current transformers with the indicated properties may find application for certain relay protections (for differential, distance, etc.), for which it is desirable to have identical accuracy in measuring an electrical quantity under all operating conditions.

The author has investigated the possibility of using such transformers for differential busbar protection; as a result it has been established that this measure can increase the sensitivity of this protection severalfold and simplify its adjustment.

As current transformers with an air gap one may use ordinary mass-produced current transformers of the types TPOF, TPSHF, TV and TFN, in the core of which an air gap must be made of such magnitude that the induction in the iron does not exceed 8000–10000 gauss. Below are given the elements of the calculation of the magnitude of the air gap of the transformers described for differential busbar protection.

1. Determination of the air gap of transformers for differential busbar protection with current balance

The following assumptions are adopted as the basis of the calculation:

1) the design condition is a through short circuit with the greatest aperiodic component of the current;

2) the calculation is to be made for the current transformer carrying the greatest short-circuit current;

3) the error of the transformer in transforming the periodic component of the current under the design condition must not exceed a specified value, for example, \(K = 0.05\text{–}0.10\);

4) when an air gap is made in ordinary current transformers, the core cross section \(q\), the number of turns of the windings, and the resistance of the secondary winding are preserved. These quantities, when determining the air gap, may be regarded as given and may be taken from factory data.

To determine the magnitude of the air gap it is necessary to find the maximum amplitude of the magnetizing current of the transformer under the calcu-

in steady-state operation; for this one may use the formula of A. B. Chernin,^1 which has the following form:

\[ I_{nm}=\frac{\sqrt{2}\,I''}{n_T} \left[ \frac{1}{\omega T_2} + \left(\frac{T_1}{T_2}\right)^{\frac{T_2}{T_2-T_1}} \right], \tag{1} \]

where \(I''\) is the supertransient current of the primary circuit, in amperes,
\(n_T\) is the transformation ratio of the current transformer,
\(T_1\) is the time constant of the primary circuit, and
\(T_2\) is the time constant of the transformer circuit consisting of the resistances \(z_n\) and \(z_2\) (see Fig. 1).

The first term in the square brackets of expression (1) represents the relative magnitude of the periodic component of the magnetizing current, while the second term represents the relative magnitude of the total aperiodic current.

Fig. 1

However, in order to use formula (1), it is necessary to know the constant \(T_2\), which depends on the magnitude of the air gap of the transformer and therefore can be determined in advance only indirectly.

As the initial datum for determining \(T_2\), we shall use the permissible error in the transformer current \(K\) when transforming only the periodic component. Then the primary and secondary currents of the transformer will be related by

\[ i_2=i_1(1-K)=\frac{I_1}{n_T}(1-K). \tag{2} \]

In addition, on the basis of the vector diagram and the equivalent circuit of the transformer (Fig. 1), all three currents \(i_1\), \(i_2\), and \(i_n\) are related by the following relations:

\[ i_1=\sqrt{(i_2\cos\varphi)^2+(i_2\sin\varphi_2+i_n)^2}, \tag{3} \]

\[ i_n=i_2\,\frac{z_2}{z_n}. \tag{4} \]

^1 A. B. Chernin. “Magnetizing Currents of Current Transformers.” Electrical Stations, No. 3, 1937.

Solving (2), (3), and (4) simultaneously, we obtain the expression for \(z_{\mu}\):

\[ z_{\mu} = \frac{z_2^2} {\sqrt{\left(\dfrac{z_2}{1-K}\right)^2-r_2^2}-x_2}, \tag{5} \]

where \(r_2\) and \(x_2\) are the components of the resistance \(z_2\).

The time constant of the transformer secondary circuit is:

\[ T_2=\frac{x_2+x_{\mu}}{\omega r_2} =\frac{x_2+z_{\mu}}{\omega r_2}, \tag{6} \]

since the active resistance of the transformer magnetizing branch is equal to zero.

The magnitude of the air gap can be determined from the maximum magnetizing ampere-turns from the following expression:

\[ \delta=\frac{i_{\mu m} w_2}{0.8 B_{m\partial}}, \tag{7} \]

where \(w_2\) is the secondary number of turns of the transformer and
\(B_{m\partial}\) is the maximum permissible induction (8000–10000 gauss).

Expression (1) gives the value of the magnetizing current, in good agreement with experiment. Thus, for example, Fig. 2 shows an oscillogram in which curve (1) represents the primary overcurrent, and curve (2) the magnetizing current of a transformer with an air gap of \(2.5\) mm. The time constant is \(T_1=0.02\) sec. Resistances: \(z_{\mu}=2.0\) ohms, \(x_2=0\); \(r_2=0.3\) ohms. The steady-state primary current is \(20\) A rms. The transformation ratio of the transformer is \(n_T=1\).

The maximum amplitude of the magnetizing current calculated on the basis of (1) is \(13.4\) A, and the steady-state effective current is \(3\) A; according to the oscillogram, the maximum amplitude is \(11.5\) A and the steady-state current is also \(3\) A.

Fig. 2

Example of transformer calculation.
For differential busbar protection it is proposed to use bushing transformers of type TV, built into MKP-160, with transformation ratio \(n_T=150=w_2\), core cross-section \(q=60\ \text{cm}^2\). The resistance of the secondary winding is \(x_2=0.14\) ohms and \(r_2=0.61\) ohms. The resistance of the connecting wires is \(r_n=0.1\) ohms. The transient calculated overcurrent is \(I''=7.5\) kA. The time constant of the primary circuit is \(T_1=0.05\) sec. The permissible error of the transformer under the calculated condition is \(K=0.1\).

The total resistance of the secondary circuit is:

\[ z_2=\sqrt{(r_2+r_n)^2+x_2^2} =\sqrt{(0.61+0.1)^2+0.14^2} =0.724\ \text{ohms}. \]

The resistance of the magnetizing branch on the basis of (5) is

\[ z_{\mu}= \frac{0.724^2} {\sqrt{\left(\dfrac{0.724}{1-0.1}\right)^2-0.61^2}-0.14} =1.36\ \text{ohms}. \]

The time constant \(T_2\), on the basis of (6):

\[ T_2=\frac{0.14+1.36}{314\cdot 0.71}=0.0067\ \text{sec}. \]

The maximum magnetizing current of the transformer under the calculation conditions, on the basis of (1):

\[ I_{\mu m}=\frac{7500\sqrt{2}}{150}\left[\frac{1}{314\cdot 0.0067} +\left(\frac{0.0067}{0.05}\right)^{\frac{0.0067}{0.0067-0.05}}\right]=85\ \text{A} \]

The magnitude of the air gap, on the basis of (7),

\[ \sigma=\frac{85\cdot 150}{0.8\cdot 10000}=1.6\ \text{cm}. \]

II. Determination of the Air Gap of Transformers for Bus Differential Protection with Balance of Ampere-Turns

In the circuit under consideration, with a through primary overcurrent, only an unbalance current appears in the secondary circuit of the transformers, whose demagnetizing action may be neglected. Then the primary calculated overcurrent will be entirely the magnetizing current; therefore, the required magnitude of the air gap is determined very simply from the following expression:

\[ \sigma=\frac{2.55\,I''_{\text{av}}w_1}{0.8\,B_{md}}, \tag{8} \]

where the numerator contains the greatest instantaneous primary ampere-turns at the calculated overcurrent, and the denominator contains the maximum permissible induction.

For the data of the preceding example, but for the circuit under consideration, the magnitude of the air gap, on the basis of (8), is

\[ \sigma=\frac{2.55\cdot 7500}{0.8\cdot 10000}=2.4\ \text{cm}. \]