ON THE RELATIONSHIP BETWEEN THE PARAMETERS OF TRANSFORMERS CONNECTED TO A PULSE LOAD

D. I. Anderman, I. D. Kutyavin

In [1] the possibility of using high-voltage transformers for supplying power to electronic accelerator tubes is demonstrated. However, as experiment shows, obtaining a constant-voltage pulse is possible not for arbitrary parameters of the transformer and the load.

To determine the dependence of the magnitude and duration of the pulse on the parameters of the equivalent circuit and on the switching phase, a problem was formulated and calculated on the Minsk-1 digital computer in accordance with the equivalent circuit proposed in [1].

In this procedure, for each combination of parameters, the switching phase is found at which the possible time of voltage stability would be maximal. The problem was calculated for three values of stabilization accuracy:

\[ \varepsilon=\frac{u_c(0)-u_c(t)}{u_c(0)}, \tag{1} \]

where \(u_c(0)\) is the voltage across the load at the instant of switching on,
\(u_c(t)\) is the voltage across the load at the end of the pulse;
for \(\varepsilon = 0.01;\ 0.001;\ 0.0001\), and was compiled in relative units.

As base quantities, the load resistance \(R\), the frequency of the primary voltage \(f_1 = 50\ \text{Hz}\), and the amplitude value of the primary voltage, referred to the secondary winding, \(U_{1m}=1\), were taken. The leakage inductance was taken into account by the coefficient proportional to it,

\[ k_1=\frac{\omega L_k}{R}. \tag{2} \]

The capacitance was taken into account by the coefficient proportional to it,

\[ k_2=R\omega C. \tag{3} \]

The active resistance of the windings was taken into account by the coefficient

\[ k_3=\frac{r_k}{R}. \tag{4} \]

For the correct choice of the limits of variation of the coefficients \(k_1, k_2, k_3\), let us consider their physical meaning. In selecting the main dimensions of a transformer operating in a pulse regime, one must strive for the thermal loading of the transformer in the pulse regime to be the same as under continuous load. The useful active power \(P_{2u}\) transmitted to the load in the pulse regime can be determined from the expression:

\[ P_{2u}=\frac{1}{T}\int_0^\tau \frac{u^2}{R}\,dt =\frac{1}{T}\int_0^\tau \frac{(n_1U_{1m})^2}{R}\,dt =\frac{n_1^2U_{1m}^2}{R}\cdot\frac{\tau}{T}, \tag{5} \]

where \(U_{1m}\) is the amplitude value of the primary voltage, referred to the secondary circuit;

\(u\) is the voltage across the load;

\(n_1\) is a coefficient depending on the switching phase and the voltage drop in the windings. Usually \(n_1 = 0.85 \div 1\);

\(T\) is the period of the alternating voltage;

\(\tau\) is the duration of load connection.

The useful power of the transformer under long-term connection of a load whose resistance is equal to \(R_{\mathrm{dl}}\) is

\[ P_2=\frac{U_{2\mathrm{n}}^2}{R_{\mathrm{dl}}} =\frac{(n_2U_{1m})^2}{2R_{\mathrm{dl}}}, \tag{6} \]

where \(n_2\) is a coefficient taking into account the voltage drop in the windings;

\(U_{2\mathrm{n}}\) is the rms value of the voltage on the long-term connected load.

Equating expressions (5) and (6), taking \(n_1=n_2=1\),

\[ \frac{U_{1m}^2}{R}\cdot\frac{\tau}{T} = \frac{U_{1m}^2}{2R_{\mathrm{dl}}}, \quad \text{then} \quad R=R_{\mathrm{dl}}\cdot\frac{2\tau}{T}=mR_{\mathrm{dl}}, \tag{7} \]

where \(m\) is the load coefficient.

Then the coefficient \(\kappa_1\) can be written in another way:

\[ k_1=\frac{L_k\omega}{R} = \frac{L_k\omega\cdot I_{2\mathrm{n}}}{mR_{\mathrm{dl}}\cdot I_{2\mathrm{n}}} = \frac{U_p}{mU_{2\mathrm{n}}} = \frac{u_p}{m}, \]

where \(I_{2\mathrm{n}}\) is the rms value of the current in the long-term connected load.

The coefficient \(k_1\) is numerically equal to the relative value of the reactive component of the short-circuit voltage, divided by the load coefficient. Usually \(u_p=0.03 \div 0.15\). The load coefficient may vary, as calculation has shown, within very wide limits depending on the required accuracy of voltage stabilization and on the circuit parameters:

\[ \tau = 20\cdot 10^{-6} \div 2\cdot 10^{-3}\ \text{s}, \qquad m=0.002 \div 0.2,\quad \text{then} \quad \kappa_1=0.15 \div 75. \]

At large values of \(\kappa_1\), the voltage drops sharply when the load is switched on. At excessively small \(\kappa_1\), the transformer is thermally underloaded and the transient voltage varies with a frequency many times higher than the line frequency. In solving the problem, the range of variation of \(\kappa_1\) from 0.01 to 3 was investigated. The coefficient \(\kappa_3\) is chosen from the same considerations:

\[ k_3=\frac{r_k}{R} = \frac{r_k I_{2\mathrm{n}}}{mR_{\mathrm{dl}}I_{2\mathrm{n}}} = \frac{U_a}{mU_{2\mathrm{n}}} = \frac{u_a}{m}. \]

\(\kappa_3\) is numerically equal to the active component of the short-circuit voltage, divided by the load coefficient. Usually \(u_a=0.01\text{--}0.02\). The range of variation \(k_3=0.05 \div 0.5\) was investigated. The most acceptable results were obtained for \(\kappa_3=0.05 \div 0.2\). At larger \(k_3\), a significant part of the voltage drops across the winding resistances.

The value of the coefficient \(\kappa_2\) was investigated over a very wide range from \(0.001 \div 2.5\). A possible solution lies within the limits \(\kappa_2=0.5 \div 2.5\).

The calculations performed made it possible to analyze:

1) the influence of capacitance on the maximum pulse duration, which is shown in Fig. 1. For each circuit there is an optimum capacitance at which the possible pulse time is maximal. Usually \(\kappa_{2\mathrm{opt}} = 1.5 \div 2\);

2) the influence of leakage inductance on the maximum pulse duration at \(\kappa_2=\kappa_{2\mathrm{opt}}=1.5 \div 2\).

For small values of the inductance, \(\kappa_1=0.01 \div 0.30\), its change has little effect on the pulse time.

With further increase of \(\kappa_1\), the system approaches resonance and cannot operate in the required mode; a sharp decrease of the maximum pulse time occurs (unstable region). With further-

...as the inductance is increased, the maximum pulse time grows; however, the value of the initial voltage decreases, which is undesirable. Thus, for stable operation of a transformer in the pulse-load mode, \(k_1\) should lie within the range \(0.01 \div 0.3\). In this case it is possible—

Fig. 1. Effect of capacitance on the maximum possible pulse time at different voltage-stabilization accuracies.

—to obtain a voltage pulse constant in magnitude, with a stabilization accuracy of \(1\%\), for \(1.3 \div 2.7\) msec; with a stabilization accuracy of \(0.1\%\), for \(0.2 \div 0.7\) msec; and with a stabilization accuracy of \(0.01\%\), for up to \(0.040 \div 0.220\) msec;

3) the influence of the winding resistance on the maximum pulse time:

Fig. 2. Effect of the winding resistance on the maximum possible pulse time.

As can be seen from Fig. 2, as the winding resistance increases, the maximum pulse time decreases. However, \(r_k\) has a strong influence on the decay time of the switching transient process—with decreasing \(r_k\), it increases.

It is not advisable to design a transformer with an excessively small winding resistance;

4) the influence of voltage-stabilization accuracy on the maximum pulse time is clearly shown in Fig. 1.

Conclusions

The results obtained have shown that it is possible:

1) to design a transformer with parameters that make it possible to obtain the maximum duration of a constant-voltage pulse, with a stabilization accuracy of 1%, of up to 2.7 msec; with a stabilization accuracy of 0.1%, up to 0.7 msec; and with a stabilization accuracy of 0.01%, up to 0.22 msec;

2) to calculate the optimal capacitance that must be connected additionally to the transformer.

References

1. D. I. Anderman, I. D. Kutyavin. On the possibility of using high-voltage transformers as a power source for electron accelerator tubes (present collection).