CALCULATION OF A COMPENSATING TRANSFORMER

I. P. Chuchalin, M. F. Filippov

(Presented by the scientific seminar of the Faculty of Physics and Technology)

In power-supply circuits for the electromagnet of a betatron with direct-current biasing, provision must be made for compensation of the emf induced in the direct-current winding. For compensation of the emf, a so-called compensating transformer is used, which differs from an ordinary transformer in that a direct current, sometimes reaching a considerable magnitude, flows through the turns of one of its windings.

The present work gives a method for calculating a compensating transformer. In this method, unlike known calculation methods, it is possible to make a more justified choice of all dimensions and parameters of the compensating transformer, including the magnitude of the air gap unavoidable in the transformer core.

The compensating transformer can be used not only in betatron power-supply circuits, but also in other circuits where direct-current biasing is employed. The calculation of the compensating transformer has been carried out as applied to one of the pulsed betatrons with direct-current biasing of the central accelerating magnetic flux.

The biasing is accomplished by means of an additional winding wound around the central core.

Fig. 1

Figure 1 shows a direct-current circuit with a compensating transformer (C.T.).

In the central core of the betatron electromagnet, as well as in the core of the compensating transformer, during biasing there will be two components of the magnetic flux: an alternating component and a direct component.

In the central core, the direct and alternating components of induction have opposite directions (Fig. 2).

To compensate the e.m.f. induced in the direct-current winding, the turns of the primary winding of the compensating transformer must be calculated so that in the turns of its secondary winding an e.m.f. arises equal to, and opposite in direction to, the e.m.f. of the direct-current winding. In this case, in the core of the compensating transformer the alternating and direct components of the induction prove to be directed in one and the same direction (Fig. 3).

Fig. 2

If the capacitor bank is recharged through the electromagnet winding, the core of the compensating transformer must be calculated for the maximum value of the induction (Fig. 4)

\[ B_m = B_{\sim} + B_{=}. \]

Fig. 3

If there is a substantial air gap in the path of the magnetic flux, then the magnetic resistance of the steel will be small in comparison with the resistance of the gap. To simplify the calculation, the magnetic resistance of the steel may be neglected.

Fig. 4

To create the direct component of the flux in the core of the compensating transformer, the required ampere-turns are

\[ AW_{=} = W_2 J \simeq 0.8 B_{=} \delta, \tag{1} \]

where \(W_2\) is the number of turns of the secondary winding of the compensating transformer,

\(J\) is the constant magnetizing current in A,

\(\delta\) is the air gap in cm.

To create the alternating component of the flux in the transformer core, the following ampere-turns are required:

\[ AW_{\sim}=W_1J_{1m}=0.8B_{\sim}\delta, \tag{2} \]

where \(W_1\) is the number of turns of the primary winding of the compensating transformer,

\(J_{1m}\) is the maximum value of the current in the primary winding of the transformer.

The number of turns of the secondary winding of the transformer can be determined from the expression:

\[ W_2=\frac{U_{2m}\cdot 10^8}{2\pi fB_{\sim}S}, \tag{3} \]

where \(U_{2m}\) is the maximum value of the emf in the secondary winding, in volts,

\(f\) is the equivalent frequency of the current pulse in Hz,

\(S\) is the cross-sectional area of the transformer core in \(\text{cm}^2\).

Similarly, the number of turns of the primary winding is determined as:

\[ W_1=\frac{U_{1m}\cdot 10^8}{2\pi fB_{\sim}S}, \tag{4} \]

where \(U_{1m}\) is the maximum value of the emf in the primary winding. From (1 and 3) and (2 and 4), respectively, we have:

\[ \frac{U_{2m}\cdot J_{=}\cdot 10^8}{2\pi fB_{\sim}S}=0.8B_{=}\delta, \tag{5} \]

\[ \frac{U_{1m}\cdot J_{1m}\cdot 10^8}{2\pi fB_{\sim}S}=0.8B_{\sim}\delta. \tag{6} \]

Dividing (6) by (5), we obtain:

\[ J_{1m}=J_{=}\frac{U_{2m}}{U_{1m}}\cdot\alpha, \tag{7} \]

where

\[ \alpha=\frac{B_{\sim}}{B_{=}}. \tag{8} \]

It is evident from formula (7) that, for selected values of \(J_{=}\) and \(\frac{U_{2m}}{U_{1m}}\), the current in the primary winding of the transformer is determined by the ratio of the amplitude of the alternating component to the direct component of the induction in the transformer core.

The maximum value of the induction in the transformer core (Fig. 4) is

\[ B_m=B_{\sim}+B_{=}. \tag{9} \]

From equations (5, 8, and 9), respectively, we have:

\[ S=\frac{U_{2m}\cdot J_{=}\cdot 10^8}{1.6\pi fB_{\sim}B_{=}\delta}, \tag{10} \]

\[ \left. \begin{aligned} B_{=}&=\frac{1}{1+\alpha}B_m,\\ B_{\sim}&=\frac{\alpha}{1+\alpha}B_m. \end{aligned} \right\} \tag{11} \]

The cross section of the core of the compensating transformer is selected according to the maximum value of the induction \(B_m\). Solving jointly (10) and (11), we find the cross section of the transformer core:

\[ S=\frac{U_{2m}J_{-}(1+\alpha)^2\cdot 10^8}{1.6\pi f B_m^2 \delta}. \tag{12} \]

Figure 5 shows the curve of the dependence of the cross section \(S\) of the transformer core on \(\alpha\), for constant \(U_{2m}\), \(J_{-}\), \(f\), \(B_m\), and \(\delta\).

The minimum cross section of the core of the compensating transformer is obtained at \(\alpha=1\), i.e., when the constant and variable components of the induction in the transformer core are equal.

Fig. 5

From formula (12) it follows that, as the size of the air gap increases, the cross section of the transformer core decreases hyperbolically. In the absence of an air gap in the core, its cross section must be very large. This makes it clear why an air gap is necessary in transformers of this type.

Example of calculation of a compensating transformer

Initial data:

1. Voltage on the electromagnet windings \(U_{1m}=3000\) V.

2. Ampere-turns of the constant component of the current required to create the calculated flux in the central core

\[ AW_{-}=2260. \]

1. Variable component of the flux in the central core

\[ \Phi_{\mathrm{ac}}=102\cdot 10^6 \text{ maxwells}. \]

1. Equivalent frequency of the current pulses \(f=50\) Hz.

2. Period of repetition of the pulses \(t_c=1\) sec.

The emf induced in one turn of the direct-current winding located on the central core is

\[ e_1=2\pi f\Phi_{\mathrm{ac}}\cdot 10^{-8} =2\pi\cdot 50\cdot 102\cdot 10^{-2} =320\text{ V}. \]

We choose the number of turns of the direct-current winding as

\[ W_{\mathrm{dc}}=8\text{ turns}. \]

The current and emf in the magnetizing winding will respectively be equal to

\[ J_{-}=\frac{AW_{-}}{W_{\mathrm{dc}}} =\frac{2260}{8} =283\text{ A}, \]

\[ U_{2m}=W_{\mathrm{dc}}\cdot e_1 =8\cdot 320 =2560\text{ V}. \]

Taking the value \(\alpha = 1\), then the maximum value of the current in the primary winding is

\[ I_{1m}=I\cdot \frac{U_{2m}}{U_{1m}}\alpha = 283\frac{2560}{3000}=242\ \text{A}. \]

The cross-sectional area of the transformer core, the ratio of the weights of copper and steel, and, consequently, the cost of the transformer depend on the selected value of the length of the air gap in the core.

Table

The table gives the results of calculating variants of the compensating transformer for different values of \(\delta\). For each variant the total cost \(C\) of the active materials (copper and steel) is indicated.

We settle on the air-gap value

\[ \delta = 10\ \text{cm}, \]

as corresponding to the lowest cost of the active materials.