Accounting for the Influence of Core Parameters on Transformer Operation under Pulsed Load Conditions

D. I. Anderman

(Presented by the scientific seminar of the Department of Electrical Power Stations and Electric Networks and Systems)

In some power-supply circuits for electronic accelerator devices, transformers operate in a pulsed-load mode, when a sinusoidal voltage of industrial frequency is applied to the primary winding, while current flows in the load during only an insignificant part of the period [1]. The voltage of the transient process that arises when the load is switched on, with an appropriate selection of transformer parameters and sufficient capacitance of the secondary winding relative to ground, remains constant in magnitude, with a certain degree of accuracy, for some time. The less stringent the requirements imposed on voltage stability, the greater the possible pulse duration. In [2], results are given of a study of the influence of transformer parameters on the maximum possible pulse duration for a specified voltage stability.

In [1] and [2], the magnetic permeability of the core, $\mu$, was taken to be infinite. In reality, $\mu$ has a finite value and varies in time as a function of the magnetic-field intensity. The present article gives a comparison of the results of studying the influence of transformer parameters on the maximum possible pulse duration with and without taking the magnetization branch into account.

The equivalent circuit with lumped parameters of a transformer with a pulsed load, when the magnetization branch is taken into account, may be represented by Fig. 1.

Fig. 1

Here

$r_1$ and $L_1$ are the active resistance and leakage inductance of the primary winding;

$r_2'$ and $L_2'$ are, respectively, the active resistance and leakage inductance of the secondary winding referred to the number of turns of the primary winding;

\(C\) is the referred value of the output capacitance, including the capacitance of the secondary winding with respect to ground and the capacitance additionally connected at the transformer output;

\(R\) is the referred value of the load resistance;

\(L_\mu\) and \(R_{\mathrm{B}}\) are the inductance of the magnetizing branch and the active resistance accounting for losses due to eddy currents and remagnetization of the core.

\(R_{\mathrm{B}}\), like \(L_\mu\), is a function of the magnetic-field intensity and therefore varies with time.

The equations of electrical equilibrium of the circuit in Fig. 1, in differential form, are:

\[ \begin{aligned} \frac{di_1}{dt}={}& - i_1 \frac{R_{\mathrm{B}}L'_2+r_1L_\mu+r_1L'_2} {L_\mu L'_2+L_1L_\mu+L_1L'_2} + i'_2 \frac{L'_2R_{\mathrm{B}}-r'_2L_\mu} {L_\mu L'_2+L_1L_\mu+L_1L'_2} \\ &- u_C \frac{L_\mu} {L_\mu L'_2+L_1L_\mu+L_1L'_2} + u_1 \frac{L_1+L_\mu} {L_\mu L'_2+L_1L_\mu+L_1L'_2}; \\[6pt] \frac{di_2}{dt}={}& i_1 \frac{R_{\mathrm{B}}L_1-r_1L_\mu} {L_\mu L'_2+L_1L_\mu+L_1L'_2} - i'_2 \frac{r'_2L_\mu+R_{\mathrm{B}}L_1+r'_2L_1} {L_\mu L'_2+L_1L_\mu+L_1L'_2} \\ &- u_C \frac{L_1+L_\mu} {L_\mu L'_2+L_1L_\mu+L_1L'_2} + u_1 \frac{L_\mu} {L_\mu L'_2+L_1L_\mu+L_1L'_2}; \\[6pt] \frac{du_C}{dt}={}& i'_2\frac{1}{C}-u_C\frac{1}{RC}. \end{aligned} \tag{1} \]

The initial values of the functions \(i_{10}\), \(j_{20}\), \(u_{C0}\) depend on the parameters of the equivalent circuit and on the magnitude and phase of the primary voltage:

\[ \begin{aligned} i'_{10} &= \varphi_1(L_1,\ r_1,\ L'_2,\ r'_2,\ L_\mu,\ R_{\mathrm{B}},\ C,\ U_{1m},\ \psi_1),\\ i'_{20} &= \varphi_2(L_1,\ r_1,\ L'_2,\ r'_2,\ L_\mu,\ R_{\mathrm{B}},\ C,\ U_{1m},\ \psi_1),\\ u_{C0} &= \varphi_3(L_1,\ r_1,\ L'_2,\ r'_2,\ L_\mu,\ R_{\mathrm{B}},\ C,\ U_{1m},\ \psi_1). \end{aligned} \tag{2} \]

The following assumptions are adopted:

\[ r_1=r'_2=\frac{r_k}{2}, \tag{3} \]

\[ L_1=L'_2=\frac{L_s}{2}. \tag{4} \]

The calculation was carried out in per-unit quantities, where the load resistance during the current pulse \(R\) and the angular frequency of the primary voltage \(\omega\) were taken as base quantities. The amplitude of the applied primary voltage was taken equal to unity. The transformer parameters in per-unit quantities are determined as

\[ k_1=\frac{\omega L_s}{R},\qquad k_2=R\omega C,\qquad k_3=\frac{r_k}{R}, \]

\[ k_4=\frac{R_{\mathrm{B}}}{R},\qquad k_5=\frac{\omega L_\mu}{R}. \tag{5} \]

Solving the system of differential equations (1) with allowance for nonlinearity is rather complicated. In the regime under consideration, the load-switching time is much less than the current period, which makes it possible to take the inductance of the magnetizing branch as constant. The load is switched on at an instant close to the instant at which the voltage passes through its maximum. In this case the induction in the core is close to zero, and the magnetic permeability of the core \(\mu=dB/dH\) is close to its maximum value.

The inductance of the magnetizing branch can be determined by

known dimensions of the magnetic core and magnetic permeability. Thus, for example, for one of the experimental samples we manufactured, at a relative value of the maximum magnetic permeability \(\mu_a = 6000\), the ratio of the inductance of the magnetizing branch to the leakage inductance is

\[ \frac{L_b}{L_s} = 300 . \]

The active resistance of the magnetizing branch may be taken equal to its static value:

\[ R_b = \frac{P_c}{I_0^2}, \tag{6} \]

where

\[ P_c \]

is the loss in the steel, W;

\[ I_a \]

is the no-load current, A.

For ordinary power transformers, the ratio of the active resistance of the magnetizing branch to the input impedance of the transformer

\[ Z_{\mathrm{in}} = \frac{U_1}{I_1} \]

\[ \frac{R_b}{Z_{\mathrm{in}}} = \frac{P_c I_1}{I_0^2 U_1} = \frac{P_c}{i_0^2 S} \tag{7} \]

lies in the range \(0.85 \div 2.0\) and decreases as the transformer power decreases. Here \(i_0 = \frac{I_0}{I_1}\) is the relative value of the no-load current, \(S = U_1 I_1\) is the transformer power, VA. We specified the value \(k_4\) in the range \(k_4 = 0.15 \div 2.0\).

The calculation was carried out according to an algorithm analogous to that used in the study of the simplified circuit. As a result, for each combination of parameters of the selected equivalent circuit, the optimal phase was obtained, at which the maximum pulse duration \(\tau\), corresponding to the specified voltage stability, is attained. The calculation also provides for obtaining a numerical value of the load voltage in relative units and of the time \(\tau\). For a sample with equivalent-circuit parameters \(k_1 = 0.06\), \(k_2 = 2.0\), \(k_3 = 0.015\), at frequency \(f = 50\) Hz and \(\varepsilon = 0.1\%\), when calculating without taking the magnetizing branch into account, the maximum possible pulse duration was \(295\ \mu\text{s}\) at \(\psi_1 = -116^\circ\). Taking into account the parameters of the magnetizing branch, \(k_5 = 300 k_1\), \(k_4 = 0.15;\ 0.5;\ 1.0;\ 1.5;\ 2.0\), \(\tau = 315\ \mu\text{s}\) at the same value of the optimal switching phase \(\psi_1 = -116^\circ\).

Since the optimal switching phase differs substantially from the instant at which the voltage passes through its maximum value, the instantaneous value of the inductance of the magnetizing branch may differ significantly from the maximum. Values of \(\tau\) were found for a circuit with a magnetizing branch at \(k_5 = (7 \div 300) k_1\) and \(k_4 = 0.15 \div 2.0\), and at various combinations of \(k_1\), \(k_2\), \(k_3\).

A comparison of the obtained values of the maximum possible pulse duration with and without taking the magnetizing branch into account is given in Fig. 2. The dashed line shows the maximum possible pulse duration without taking the magnetizing branch into account. For \(k_5 \geq 30 k_1\), the transient-voltage curve with and without taking the magnetizing branch into account is approximately the same; consequently, the maximum possible pulse duration differs only insignificantly. Variation of \(k_4\) within the indicated limits has almost no effect on the processes under investigation.

Fig. 2

Fig. 3, a.

Fig. 3, b. Oscillograms (from top to bottom) of the primary voltage, the blanking signal, the current of the primary winding, the secondary voltage, and the current in the load for the cases when the induction in the core is: a) \(B=1.3\ \mathrm{T}\), b) \(B=1.73\ \mathrm{T}\).

Similar results were obtained in cases of increased capacitance \((k_1 = 0.06;\ k_2 = 3.0;\ k_3 = 0.015)\), increased active resistance of the windings \((k_1 = 0.05;\ k_2 = 2.0;\ k_3 = 0.05)\), and increased leakage inductance of the windings \((k_1 = 0.1;\ k_2 = 2.0;\ k_3 = 0.01)\).

It may be assumed that, at optimal switching phases, the instantaneous value of the inductance of the magnetizing branch differs from the leakage inductance by no less than a factor of 30. This makes it possible not to include the magnetizing branch in the equivalent circuit.

To verify the correctness of the theoretical results obtained, an experiment was carried out. In a laboratory model of a transformer, whose pulsed mode was produced by applying an unlocking signal to the grid of a triode serving as the load, oscillograms were taken of the transient voltage at an induction in the core of \(1.3\ \mathrm{T}\) (Fig. 3, a) and with a reduced core cross-section, induction \(B = 1.73\ \mathrm{T}\) (Fig. 3, b).

In the second case the magnetizing current increased considerably, the secondary voltage became nonsinusoidal, while the transient voltage upon switching on the load remained unchanged in comparison with the first case.

Conclusion

The study of transient processes in transformers that arise when a load is switched on for a small fraction of a period can be carried out using an equivalent circuit that takes into account the leakage inductance of the windings, their active resistance, and the capacitance of the secondary winding with respect to ground. Introducing into the circuit a magnetizing branch that accounts for the parameters of the core complicates the calculation and has little effect on the final results.

References

1. D. I. Anderman, I. D. Kutyavin. On the possibility of using high-voltage transformers as a power-supply source for electronic accelerating tubes. Izv. TPI, vol. 211, Tomsk, Publishing House of TSU, 1970.

2. D. I. Anderman, I. D. Kutyavin. On the ratio of parameters of transformers connected to a pulsed load. Ibid.