OPERATION OF A SYNCHRONOUS SHOCK GENERATOR WITH A NONLINEAR ACTIVE LOAD

G. A. Sipaylov, A. V. Loos, E. I. Sobko

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

In physics and engineering there exists a whole class of pulsed energy consumers whose load has a nonlinear active character.

A pulsed power-supply mode is used in powerful generators of low-temperature plasma—the so-called plasmatrons [1]. The discharge current in a plasma arising between the electrodes of a discharger immersed in a liquid is also used in devices employing the electrohydraulic effect—for example, in installations for exciting seismic oscillations in geological exploration and in studying the structure of the earth’s crust. In installations for pulse-arc welding, powerful short-duration pulses of electrical energy are supplied to a steadily burning welding arc.

Pulsed power is used in optical quantum generators—lasers—capable of providing an exceptionally high energy density in a narrow, directed beam of electromagnetic radiation. For example, in ruby lasers a coherent beam is formed in a large crystal of synthetic ruby, which is excited by a powerful light pulse arising when current flows through a flash lamp. The efficiency of a laser, i.e., the ratio of the energy of the coherent beam to the energy of the light flash, is very small—from 0.1 to 0.5%. Therefore, even at the present time, to obtain powerful beams of electromagnetic radiation, pulsed power sources are required that are capable of transferring to the load, during the time of a single current pulse, energy exceeding 100 MJ [2].

A pulsed power system is necessary to ensure the operation of installations intended for testing models of aircraft in arc aerodynamic wind tunnels.

An analysis of existing trends in solving the problem of pulsed power supply at load energies of tens and hundreds of megajoules indicates that the most expedient solution is to be sought in the creation and application of energy sources of the electromechanical type.

One of the most promising pulsed sources is the synchronous shock generator. The use of shock generators for creating pulsed magnetic fields [3, 4, 5, 6] and for supplying a linear active load [7, 8] has been considered by many researchers. However, to date there have been no works devoted to theoretical and experimental studies of the operation of a shock generator with a nonlinear active load.

Let us consider some results of an investigation of the operation of a synchronous shock generator when supplying a nonlinear active load—an impulse arc. The processes occurring are most fully represented by the system of differential equations of the electromechanical transient process of a single-phase synchronous generator:

a) equations of voltage equilibrium of the windings

\[ 0=i_c r_c+\frac{d\Psi_c}{dt}+U_{\mathrm{a}}; \]

\[ U_{\mathrm{v}}=i_{\mathrm{v}}r_{\mathrm{v}}+\frac{d\Psi_{\mathrm{v}}}{dt}; \]

\[ 0=i_{\mathrm{Dd}}r_{\mathrm{Dd}}+\frac{d\Psi_{\mathrm{Dd}}}{dt}. \tag{1} \]

\[ 0=i_{\mathrm{Dq}}r_{\mathrm{Dq}}+\frac{d\Psi_{\mathrm{Dq}}}{dt}. \]

b) equations of rotor motion

\[ M_m=H_j\frac{d\omega}{dt}+M_{\mathrm{e}}; \tag{2} \]

c) equations of flux linkages of the windings

\[ \begin{aligned} \Psi_c &= i_c x_c+i_{\mathrm{v}}x_a\cos\omega t+i_{\mathrm{Dd}}x_a\cos\omega t+i_{\mathrm{Dq}}x_a\sin\omega t,\\ \Psi_{\mathrm{v}} &= i_c x_a\cos\omega t+i_{\mathrm{v}}x_{\mathrm{v}}+i_{\mathrm{Dd}}x_a,\\ \Psi_{\mathrm{Dd}} &= i_c x_a\cos\omega t+i_{\mathrm{v}}x_a+i_{\mathrm{Dd}}x_{\mathrm{Dd}},\\ \Psi_{\mathrm{Dq}} &= i_c x_a\sin\omega t+i_{\mathrm{Dq}}x_{\mathrm{Dq}}. \end{aligned} \tag{3} \]

where the indices \(c\), \(\mathrm{v}\), \(\mathrm{Dd}\), \(\mathrm{Dq}\) attached to variables and parameters refer respectively to the stator windings, excitation winding, and equivalent damper circuits along the direct and quadrature axes,

\(x_a\)—mutual-induction reactance when the winding axes coincide;

\(\omega\)—angular speed of rotation of the rotor.

\(M_m\)—torque of the mechanical forces.

\(H_j\)—inertia constant of the rotor.

\(M_{\mathrm{e}}=x_a i_c(i_{\mathrm{Dq}}\cos\omega t-i_{\mathrm{v}}\sin\omega t-i_{\mathrm{Dq}}\sin\omega t)\)—torque of the electromagnetic forces;

\(U_{\mathrm{a}}\)—voltage drop across the arc.

The impulse arc is dynamic. In it there is no single-valued dependence of voltage on current. When the current increases, this dependence differs from that when it decreases. The oscillogram in Fig. 1 shows characteristic changes in the current and voltage of an impulse arc when it is supplied from a model shock generator. For the characteristic current pulse of a shock generator, the arc voltage has a distinctive shape.

At the beginning of the pulse it rises rapidly and reaches a maximum value, called the arc ignition voltage. Then the voltage quickly falls to a minimum, called the minimum arc voltage, and remains constant until the end of the current pulse. At the end of the current pulse the voltage rises to the arc extinction voltage.

The processes of variation of the arc resistance as the current in it and the conditions of its burning change are most fully characterized by the differential equation of the arc [9]. The volt-ampere characteristics of the devices described above, which are based on the use of an impulse arc, are determined by their design and may have different forms. To clarify the qualitative picture, we adopt certain

assumption [10]. We shall assume that the voltage gradient in the arc column is constant. This means that we neglect the ignition and extinction voltage peaks of the arc and, consequently, that the arc voltage, for a constant

Fig. 1. Oscillogram of the voltage and current of a pulsed arc

length \(l = \mathrm{const}\), will be constant throughout the entire current pulse and equal to

\[ U_{\text{д}} = E_{\text{д}} l . \tag{4} \]

The assumptions adopted are entirely justified. Indeed, one of the main tasks of the investigations is to determine the optimum arc length at which the maximum energy will be released in it as a function of the parameters of the shock generator. In view of the large currents at which the shock generator operates, the ignition and extinction voltage peaks (Fig. 1) are small; moreover, even when they are significant, their duration is short and they have little effect on the magnitude of the energy.

The energy released in the arc during the current pulse of the shock generator is determined by the formula:

\[ W_{\text{д}} = \int_{0}^{t_{\text{и}}} U_{\text{д}} i_c\,dt = U_{\text{д}} \int_{0}^{t_{\text{и}}} i_c\,dt ; \tag{5} \]

where \(t_{\text{и}}\) is the duration of the current pulse.

The current \(i_c\) is determined from the solution of the system of differential and algebraic equations (1), (2), (3). A solution of these equations in general form is not available; therefore, in order to make it possible to carry out a large research program, a mathematical model based on the MN-14 analog computer was developed for the system of equations (1), (2), (3).

Figure 2 presents the energy characteristics of the shock generator when operating on an arc, as a function of the arc-burning voltage and the transient reactance of the shock generator \(x'_d\). It can be seen that, in this case, the maximum energy in the arc is released at an arc-burning voltage in relative units \(U_{\text{д}} = 0.32\), which

is the same for any values of \(x_d''\). As the base quantities, the amplitude of the rated phase voltage of the generator is taken,

\[ e_b = E_m \]

and the energy

\[ W_b = 0.5 e_b i_b t_b . \]

Using dependence (4), on the basis of the foregoing, one can determine, for each specific generator, the magnitude of the matched arc gap for a known voltage gradient \(E_a\).

For engineering calculations, analytical expressions are needed for determining the current and the energy in the arc. Let us determine them by making partial use of the theorem on the constancy of flux linkages [11]. The possibility of its application is confirmed by oscillograms of the flux linkages of the windings of a shock generator during operation on an arc (Fig. 3), from which it is evident that the flux linkages \(\Psi_s\), \(\Psi_{D d}\), \(\Psi_{D q}\) practically do not change during the current pulse. Neglecting the voltage drop across the active resistance of the stator winding in comparison with the arc voltage \(U_d\), the law of variation of the flux linkage \(\Psi_s\) can be represented in the form

\[ \Psi_s = \Psi_{s0} - U_d t, \tag{6} \]

Fig. 2. Energy characteristics of a shock generator during operation on an arc:
\(1\)—\(x_d'' = 0.015\); \(2\)—\(x_d'' = 0.02\); \(3\)—\(x_d'' = 0.03\); \(4\)—\(x_d'' = 0.035\); \(5\)—\(x_d'' = 0.05\); \(6\)—\(x_d'' = 0.06\); \(7\)—\(x_d'' = 0.07\).

Fig. 3. Nature of the variation of the flux linkages of the windings of a shock generator during operation on an arc load.

where \(\Psi_{s0}\) is the flux linkage of the stator winding at the initial instant.

Taking \(x_d'' = x_q''\), which is quite permissible for a shock generator, after transforming equations (3) with allowance for (6), the expression for the current is

\[ i_s = \frac{\omega}{x_d''} \left( \Psi_{s0} - U_d t - \frac{E_m}{\omega}\cos \omega t \right). \tag{7} \]

The energy released in the arc is determined in this case by the formula

\[ W_d = \frac{0.5 U_d \omega}{x_d''} \left( \Psi_{s0} t - \frac{t^2 U_d}{2} - \frac{E_m}{\omega^2}\sin \omega t \right). \tag{8} \]

The pulse duration is determined from the approximate solution of the transcendental equation (7):

\[ t_d = \frac{\pi}{\omega} - \frac{U_d}{E_m \omega} + \frac{1}{E_m \omega} \sqrt{ U_d^2 + 4E_m^2 + 2\pi U_d E_m }. \tag{9} \]

Figure 4 shows the dependences \(i_c=f(t)\) and \(W_d=f(t)\), calculated for a shock generator with parameters in p.u.
\(x_d''=0.0391,\ x_{\mathrm{B}}=1.2,\ x_{Dd}=x_{Dq}=1.02,\ r_c=0.00156,\ r_{\mathrm{B}}=r_{Dd}=r_{Dq}=0.001,\ U_{\mathrm{B}}=0.001,\ H_j=5.9\ \mathrm{s}\),
from the complete system of equations on the MN-14 analog machine (solid lines) and from the simplified relations (7), (8) (dashed lines). As can be seen from the figure, despite large discrepancies in the current calculations, the final value of the energy released in the arc gap agrees fairly well with the calculations by the complete equations and by the simplified formulas (7), (8).

Substituting the value of the arc burning voltage \(U_d=0.32\) into (8), taking (9) into account, we obtain a simple empirical formula for determining the energy released in the matched arc gap when it is supplied from a shock generator:

\[ W_{\mathrm{B}}=\frac{0.41E_m^2}{x_d''\omega}. \tag{10} \]

Fig. 4. Character of the change in current and energy in a pulsed arc: — calculation by the complete system of equations; — — — calculation by simplified formulas.

Thus, as a result of studying the operation of a shock generator on a nonlinear active load—an impulse arc—expressions have been obtained for the current (7) and the energy released in the arc (8), for the duration of arc burning (9), and recommendations have also been given for selecting the parameters of a matched arc gap. Comparison of the results of calculations by the complete system of equations (1), (2), (3) and by the analytical expressions obtained (7), (8), (9) shows their good agreement.

REFERENCES

1. A. G. Shashkov, O. I. Yas’ko, V. L. Sergeev, F. B. Yurevich. Electric-arc heaters for obtaining high-temperature jets. Elektrofizicheskii zhurnal, 1962, No. 1.

2. S. P. Kononov, B. A. Skrebnev. Application of pulsed electric-power systems. Impul’snaia elektroenergetika, Kazan, 1970.

3. V. V. Kaplan, V. M. Nashatyr’. On the question of the application of electromechanical energy accumulators for physical research. Elektrichestvo, 1966, No. 4.

4. G. A. Sipailov, K. A. Khor’kov. Specific energy of a shock generator. Izvestiya TPI, vol. 132, 1965.

5. F. K. Arkhangel’skii, A. I. Stolov. Application of generators for creating pulsed magnetic fields with large energy reserves. Elektrofizicheskaia apparatura, 1966, issue 4.

6. G. A. Sipailov, V. V. Ivashin, A. V. Loos. Generator of large pulsed powers with stepped or trapezoidal current waveform. Elektrichestvo, 1967, No. 5.

7. G. A. Sipailov, A. V. Loos, F. P. Zverev. Shock generator as a source of pulsed power with an active load. Devices for Power Supply and Electric Drive of Low Power, vol. 2, Energiya, 1970.

8. G. A. Sipailov, A. V. Loos, Yu. A. Romanov, F. P. Zverev. Autonomous source of pulsed power with a synchronous generator. Izvestiya TPI, vol. 212, 2, 1971.

9. O. Mayr. Beitrag zur Theorie der statischen und dynamischen lichtbogen, AfE, 1943, Bd. 37.

10. L. M. Zalesskii. Electrical Opening Arc. GEI, 1963.

11. I. I. Trushchev. Methods for Studying Electromagnetic Processes in AC Machines. Energiya, 1969.

12. Order 6523.