ON THE METHOD OF EQUIVALENT LINEARIZATION IN CALCULATING THE CURRENT OF A SHOCK GENERATOR TAKING SATURATION INTO ACCOUNT

A. V. Loos, G. A. Sipaylov

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

The processes occurring during operation of a shock generator are greatly influenced by saturation of the magnetic circuit. In this case, as the investigations carried out have shown, the principal role is played by saturation of the magnetic circuit along the stator leakage paths, which leads to a change in the shock inductive reactance \(X_{\text{sh}}\) depending on the magnitude of the current [1].

One of the most obvious ways of solving the nonlinear problem of determining the current is to replace the nonlinear elements by linear ones. The parameters of the linear elements are chosen in such a way that the solution of the resulting linear system is close to the solution of the original nonlinear system. This approach to the solution was adopted in [1], where, by investigating the complete system of differential equations of the shock generator, written in the \(d\) and \(q\) axes, and taking into account the stator-winding leakage inductance varying as a function of current, a correction factor was obtained whose introduction into the differential equation of the circuit by which the shock generator is conventionally represented gives a solution in good agreement with experiment.

However, the application of the results obtained is somewhat limited by the fact that in these investigations the dependence of the leakage inductance as a function of the stator current was introduced into the differential equations of the generator for executed shock generators. In the case of determining the current for generators in which the dependence of the leakage inductance as a function of the stator current differs from the one introduced, the data obtained are not applicable. In this connection there arises the problem of obtaining a general analytical method suitable for generators of different dimensions.

To solve the problem posed, it is expedient to apply the method of equivalent linearization of N. M. Krylov and N. N. Bogolyubov [2]. The meaning of this linearization is characterized by the following provisions. If an unknown current passes through a nonlinear element, then it is necessary to replace this element by a linear one. The magnitude of the equivalent linear element is chosen in such a way as to reproduce the nonlinear element as accurately as possible. It is most expedient to choose the given nonlinear element as varying according to a certain law as a function of the current. This constitutes the main advantage and distinction of the considered method of equivalent linearization from other methods of linearization, which makes it possible to use it to describe explicitly nonlinear phenomena.

Let us consider the application of this method in connection with the problem of calculating

of the current when a shock generator operates into an inductive load. Initially we shall neglect current damping. Such an assumption is permissible in determining the equivalent inductance, since the harmonic components of the voltage drop across the inductance are determined mainly by the periodic component of the current. In this case, the expression for the current can be represented, in simplified form, as

\[ i=I_m(1-\cos \omega t). \tag{1} \]

If this current passes through a nonlinear element, then the voltage at the terminals of the nonlinear element—in our case, the inductance—will be expressed by a Fourier series with fundamental frequency \(\dfrac{\omega}{2\pi}\):

\[ U_L=\sum_{n=0}^{\infty}(U_{cn}\cos n\omega t+U_{sn}\sin n\omega t). \tag{2} \]

The voltage across the inductance is, in the general case, a function not only of the current but also of the derivatives of the current. Let us denote the voltage across the inductance as follows:

\[ U_L=f(\omega t). \tag{3} \]

In this case the coefficients of series (2) are determined by the expressions

\[ U_{cn}=\frac{1}{\pi}\int_{0}^{2\pi} f(\omega t)\cos n\omega t\,d(\omega t); \tag{4} \]

\[ U_{sn}=\frac{1}{\pi}\int_{0}^{2\pi} f(\omega t)\sin n\omega t\,d(\omega t), \tag{5} \]

where

\[ 2\pi=T\omega . \]

The equivalent linear inductance is determined according to the principle of harmonic balance, which consists in requiring that the fundamental harmonics of the voltage across the equivalent linear and nonlinear inductances be equal.

The voltage drop across the nonlinear inductance is equal to the time derivative of the flux linkage:

\[ U_L=\frac{d}{dt}[\psi(i)]=\frac{d}{dt}\left\{\psi\left[I_m(1-\cos \omega t)\right]\right\}. \tag{6} \]

The flux linkage, in turn, is some function of the current. The experimental dependence of the flux linkage \(\psi\) on the current \(i\), which is an odd function, is most simply approximated by an expression of the form

\[ \psi=a_1 i-a_2 i^3. \tag{7} \]

To determine \(a_1\) and \(a_2\), it is most advisable to apply the method of least squares.

Then the fundamental sinusoidal harmonic of the voltage will be written in the form

\[ U_{s1}=\frac{1}{\pi}\int_{0}^{2\pi}\left\{\frac{d}{dt}\left[\psi\left(I_m(1-\cos \omega t)\right)\right]\sin \omega t\right\}\,d(\omega t). \tag{8} \]

The fundamental cosine harmonic of the voltage

\[ U_{c1}=-\frac{1}{\pi}\int_{0}^{2\pi}\left\{\frac{d}{dt}\left[\psi\left(I_m(1-\cos\omega t)\right)\right]\cos\omega t\right\}d(\omega t). \tag{9} \]

Substituting into (8) and (9) the expression for the flux linkage

\[ \psi=a_1 I_m(1-\cos\omega t)-a_2 I_m^3(1-\cos\omega t)^3, \tag{10} \]

we obtain

\[ U_{s1}=\omega a_1 I_m-3.75\omega a_2 I_m^3, \tag{11} \]

\[ U_{c1}=0. \tag{12} \]

In turn, for the equivalent linear inductance the voltage is determined by the expression

\[ U_L=L_{\mathrm{eq}}\frac{di}{dt}=\omega L_{\mathrm{eq}}\cdot I_m\sin\omega t. \tag{13} \]

Then, according to the principle of harmonic balance, using expressions (2) and (13), we obtain

\[ \omega L_{\mathrm{eq}} I_m\sin\omega t = \omega I_m(a_1-3.75a_2 I_m^2)\sin\omega t, \tag{14} \]

whence

\[ L_{\mathrm{eq}}=a_1-3.75a_2 I_m^2. \tag{15} \]

The solution of the differential equation for the generator current, without taking saturation into account when switched onto an inductive load, has the form [3]

\[ i_g=I_m\left[\sin(\omega t-\varphi_{ng})+ e^{-\frac{t(r_g+r_n)}{L_g+L_n}}\cdot\sin\varphi_{ng}\right], \tag{16} \]

where

\[ I_m=\frac{E_m}{\sqrt{(r_g+r_n)^2+(\omega L_g+\omega L_n)^2}}. \]

Substituting into expression (13), instead of \(L_g\), the expression obtained for \(L_{\mathrm{eq}}\), we obtain the following relation for the current amplitude:

\[ I_m= \frac{E_m} {\sqrt{(r_g+r_n)^2+\left[\omega(a_1-3.75a_2 I_m^2)+\omega L_n\right]^2}}. \tag{17} \]

By solving expression (17) with respect to \(I_m\), we find the current amplitude taking into account saturation of the magnetic circuit. Substitution of the obtained value \(I_m\) into expression (16) gives the equation for the current taking into account saturation of the magnetic circuit. Calculations carried out by this method for the completed shock generators showed good agreement with the results obtained in [1]. This indicates that the given method of accounting for saturation can be extended to generators of any dimensions. In Fig. 1 the current curves are presented, calculated with allowance for saturation of the magnetic circuit using the equivalent-linearization method and using a method based on introducing a correction factor into the differential equation. As can be seen from the curves, both methods give practically the same result.

Fig. 1. Short-circuit current curves of the stator: \(a\) — according to the method under consideration, \(b\) — taking into account the correction factor that accounts for saturation.

REFERENCES

1. G. A. Sipailov, A. V. Loos. Taking into account saturation of the magnetic circuit and changes in the speed of rotation in determining the current of a shock generator, Elektrichestvo, 1969, No. 1.

2. N. M. Krylov, N. N. Bogolyubov. Introduction to Nonlinear Mechanics, Kiev, 1937.

3. G. A. Sipailov, A. V. Loos. Generator of large impulse powers with a stepped or trapezoidal current waveform. Elektrichestvo, 1967, No. 5.