Influence of Magnetic-Circuit Saturation on the Current of a Shock Generator

G. A. Sipailov, A. V. Loos

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

Variable saturation of sections of the magnetic circuit causes a change in the inductive reactances of the generator. The change in inductive reactances due to saturation occurs for two reasons [1, 2]:

1. Saturation of the magnetic circuit by the main magnetic flux.
2. Saturation of sections of the magnetic circuit by leakage fluxes.

In a shock generator, the first dependence is determined by the voltage at the terminals and leads mainly to a decrease in mutual inductance and to a reduction in the permeances of the leakage paths immediately before the short circuit. Under the condition that the main magnetic flux remains practically constant at the first instant of the short circuit, this dependence can be taken into account, for a specified voltage, by selecting saturated values of the parameters.

More significant for a shock generator is the need to take into account the saturation of sections of the magnetic circuit by leakage fluxes. At the first instant of the short circuit, a considerable part of the generator flux rushes along the leakage paths of the generator and causes additional saturation of sections of these paths, which brings about a corresponding decrease in the transient and subtransient parameters of the machine and leads to an additional increase in the short-circuit current.

The principal influence on the leakage inductances of the generator is exerted by the magnitude of the short-circuit current. The dependence of the generator leakage inductance on the magnitude of the short-circuit current can be determined from the distribution of the leakage magnetic fields individually for each design. However, this task is laborious, since the magnetization characteristics of the individual parts of the machine are so different that it is hardly meaningful to strive for an exact accounting of the specific saturation characteristics of each machine [2]. It is more expedient to use experimentally obtained characteristics for calculations. The saturation coefficient for the subtransient and transient inductances and for the negative-sequence inductance of the phases is determined by the ratio of the saturated value at a given stator current to the value determined without accounting for saturation:

\[ \kappa_{\mathrm{н}}''=\frac{x_{d\mathrm{н}}''}{x_d''}, \qquad \kappa_{\mathrm{н}}'=\frac{x_{d\mathrm{н}}'}{x_d'}, \qquad \kappa_{\mathrm{н}}^{2}=\frac{x_{2\mathrm{н}}}{x_2}. \]

Figure 1 shows the dependences of the saturation coefficients \(K_{\mathrm{H}}''\), \(K_{\mathrm{H}}'\), \(K_{\mathrm{H}}^2\) and of the inductive reactances \(x_d''\), \(x_d'\), \(x_2\) as functions of the stator current for constructed shock generators.

Fig. 1. Dependence of the parameters of a shock generator on saturation.

Studies of the influence of variable saturation on the current of a shock generator were carried out on the analog computer MNB-1 according to a specially developed method [3]. From the dependences given in Fig. 1 one can find the dependence

\[ \frac{1}{(x_{s1}+x_2)\,0.5}=F(\Psi_{s1}^{2}), \]

which must be introduced into the solution circuit. The obtained dependence is set up in the nonlinearity unit by the method of piecewise-linear approximation with the aid of diode elements.

Using this method, calculations were made of the current curve for various values of \((x_{s1}+x_2)\,0.5+x_{\mathrm{H}}\) without taking saturation into account. The results of these calculations are presented in Fig. 2 (curve 1). The maximum value of the current at \(x_{\mathrm{H}}=0\) was taken as the unit of current.

Then calculations were made of the current curve with allowance for variable saturation for various values of \((x_{s1}+x_2)\,0.5+x_{\mathrm{H}}\). The results of these calculations are presented as curve 2 in Fig. 2. Comparison of these two curves makes it possible to estimate the influence of saturation on the maximum value of the current for various values of the load inductance.

In comparative calculations of various variants of power-supply circuits, in calculations of the current of a shock generator operating in a system with other energy-storage devices, in approximate determination of the maximum value of the current when operating into an inductive load, etc., the complete system of differential equations for the circuits of the shock generator is usually not used, since this is rather difficult and impractical. By analogy with three-phase machines, the sudden short circuit of a shock generator is commonly considered as the connection of a certain equivalent inductance \(L_{\mathrm{r}}\) and resistance \(r_{\mathrm{r}}\) to a sinusoidal emf \(E_m\sin(\omega t+\psi)\) [4]:

\[ E_m\sin(\omega t+\psi)=L_{\mathrm{r}}\frac{di_{\mathrm{r}}}{dt}+i_{\mathrm{r}}r_{\mathrm{r}} \tag{1} \]

where \(L_{\mathrm{r}}=\dfrac{L_d''+L_2}{2}\) is the equivalent inductance of the generator in

Fig. 2. Results of calculations on an analog computer.

in the supertransient regime, taking into account the shielding effect of the excitation winding and the damping system of the rotor;

\(r_{\mathrm{g}}\)—active resistance of the stator winding.

When the generator is connected to an inductive load, the load parameters \(L_{\mathrm{n}}, r_{\mathrm{n}}\) are added to \(L_{\mathrm{g}}\) and \(r_{\mathrm{g}}\):

\[ E_m \sin(\omega t+\psi)=(L_{\mathrm{g}}+L_{\mathrm{n}})\frac{di_{\mathrm{n}}}{dt}+i_{\mathrm{n}}(r_{\mathrm{g}}+r_{\mathrm{n}}). \tag{2} \]

Calculation of the current from equations (1) and (2) gives, for practical calculations, acceptable agreement with the results of complete calculations for the first half-wave of the current.

The investigations carried out on the influence of the change in saturation make it possible to introduce into equations (1) and (2) a correction factor that takes into account the change in the generator inductance under saturation of the magnetic circuit:

\[ E_m \sin(\omega t+\psi)=(L_{\mathrm{g}}K_{\mathrm{n}}+L_{\mathrm{n}})\frac{di_{\mathrm{n}}}{dt}+i_{\mathrm{n}}(r_{\mathrm{g}}+r_{\mathrm{n}}). \tag{3} \]

The maximum value of the current calculated from equation (3) coincides with the results of complete calculations taking into account variable saturation. The values of the saturation coefficient \(K_{\mathrm{n}}\) can be found from Fig. 2 by comparing the values of inductances corresponding to the same current with and without taking saturation into account. The graph of the dependence of the saturation coefficient \(K_{\mathrm{n}}\) as a function of the ratio of the load inductance to the generator inductance is presented in Fig. 3. Using this graph, one can determine the values of the generator inductance

…of the generator, which must be selected for the given load in order to obtain the maximum current value corresponding to the allowance for variable saturation.

Fig. 3. Dependence of the generator inductance saturation coefficient on \( \dfrac{L_{\mathrm{n}}}{L_{\mathrm{g}}} \).

As a result of the investigations carried out, it may be concluded that, with accuracy sufficient for most practical problems, the effect of variable saturation, determined by the leakage fluxes around the slots, on the current amplitude of a shock generator can be taken into account by selecting such a value of the generator inductance as corresponds to the maximum current value. The paper gives the dependence of the saturation coefficient as a function of the ratio of the inductances of the load and the generator.

REFERENCES

1. R. A. Luther. Parameters of synchronous machines with allowance for saturation. Sb. Elektrosila, 1951, No. 10.

2. A. I. Vorontsov, G. M. Khutoretsky. A test generator for apparatus laboratories. Proceedings of the All-Union Scientific Research Institute of Electromechanics, 1963, vol. 14.

3. A. V. Loos, G. A. Sipailov. Mathematical modeling of a shock generator with allowance for saturation and variation of rotational speed. Izvestiya TPI, vol. 160, 1966.

4. G. A. Sipailov, K. A. Khorkov. Power, energy, and machine constant of a shock generator. Izvestiya TPI, vol. 132, 1965.