Mathematical Modeling of Electromechanical Transients in an Electromechanical Pulsed Power Source

G. A. Sipailov, A. V. Loos, A. A. Shchegolev

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

An increase in the energy transferred to the load during a single current pulse is possible by using the effect of a voltage rise that occurs during a two-phase short circuit of a synchronous machine with an asymmetric rotor on the remaining open third phase of the stator [1].

An electromechanical pulsed power source [2] that uses this effect contains an impact synchronous generator with an asymmetric rotor and a two-phase stator, shown in Fig. 1. The stator windings A and B are displaced relative to one another by 90 electrical degrees. The working winding B may be made more powerful than the auxiliary winding A.

Fig. 1. Schematic diagram of an electromechanical pulsed power source

A detailed study of the electromechanical processes of the electromechanical power source under consideration requires solving a nonlinear system of differential equations with a large number of periodic coefficients. The presence of periodic coefficients is due to the simultaneous asymmetry of the stator and rotor, while the nonlinearity of the system is caused by the need to take into account saturation of the magnetic circuit and changes in the rotor speed.

Additional difficulties are introduced into the analysis of transients by the change in the structure of the synchronous machine, since initially, with the aid of...

of the switching apparatus K-1 the winding A is closed. This occurs at the instant when the emf of the auxiliary winding \(e_A\) passes through zero. Then, at the instant corresponding to the beginning of the positive half-wave of the emf in winding B, the load is switched on by switch K-2.

The presence of periodic coefficients and nonlinearity in the system, taking into account its structural change, makes its analytical investigation practically impossible and brings computing machines to the fore for this purpose [3].

Attempts to model the system of equations describing the electromechanical transients of an electromechanical pulsed energy source in the \(\alpha, \beta, o\) and \(d, q, o\) axes do not make it possible to eliminate variable coefficients and lead to the necessity of performing the differentiation operation.

The possibility of obtaining, on an analog computer (AVM), a stable model composed according to the complete system of differential equations with periodic coefficients for a single-phase synchronous generator was shown in [4]. This approach is applicable in compiling a mathematical model for an electromechanical energy source.

The voltage-balance equations, written in the physical axes in a form convenient for modeling on an AVM, have the form:

\[ \frac{d\Psi_A}{dt}=-r_A i_A, \tag{1} \]

\[ \frac{d\Psi_B}{dt}=-(r_{\mathrm{H}}+r_B)i_B, \tag{2} \]

\[ \frac{d\Psi_f}{dt}=U_f-r_f i_f, \tag{3} \]

\[ \frac{d\Psi_{Dd}}{dt}=-r_{Dd}i_{Dd}, \tag{4} \]

\[ \frac{d\Psi_{Dq}}{dt}=-r_{Dq}i_{Dq}. \tag{5} \]

Flux linkages of the stator and rotor circuits:

\[ \Psi_A=x_A i_A+x_a i_f\cos\gamma+x_a i_{Dd}\cos\gamma+x_a i_{Dq}\sin\gamma, \tag{6} \]

\[ \Psi_B=(x_{\mathrm{H}}+x_B)i_B+x_a i_f\sin\gamma+x_a i_{Dd}\sin\gamma-x_a i_{Dq}\cos\gamma, \tag{7} \]

\[ \Psi_f=x_a i_A\cos\gamma+x_a i_B\sin\gamma+x_f i_f+x_a i_{Dd}, \tag{8} \]

\[ \Psi_{Dd}=x_a i_A\cos\gamma+x_a i_B\sin\gamma+x_a i_f+x_{Dd}i_{Dd}, \tag{9} \]

\[ \Psi_{Dq}=x_a i_A\sin\gamma-x_a i_B\cos\gamma+x_{Dq}i_{Dq}, \tag{10} \]

where \(i_A, i_B, i_f, i_{Dd}, i_{Dq}\) are the currents, respectively, of stator windings A and B, the excitation winding, and the damper circuits along the direct and quadrature axes;

\(r_A, r_B, r_{\mathrm{H}}, r_f, r_{Dd}, r_{Dq}\) are the active resistances of stator windings A and B, of the load, of the excitation winding, and of the damper circuits along the direct and quadrature axes, respectively, and \(x_A, x_B, x_{\mathrm{H}}, x_f, x_{Dd}, x_{Dq}\) are inductive resistances;

\(U_f\) is the voltage of the excitation winding;

\[ \gamma=\int_0^t \omega t \]

is the angle between the axis of the stator winding and the direct axis of the rotor.

From the simultaneous solution of equations (8) and (9), we determine the currents \(i_f\) and \(i_{Dd}\):

\[ i_f = \frac{x_{Dd}}{x_f x_{Dd}-x_a^2}\Psi_f - \frac{x_a}{x_f x_{Dd}-x_a^2}\Psi_{Dd} - \frac{x_a(x_{Dd}-x_a)}{x_f x_{Dd}-x_a^2} i_A \cos\gamma - \frac{x_a(x_{Dd}-x_a)}{x_f x_{Dd}-x_a^2} i_B \sin\gamma, \tag{11} \]

\[ i_{Dd} = \frac{x_f}{x_f x_{Dd}-x_a^2}\Psi_{Dd} - \frac{x_a}{x_f x_{Dd}-x_a^2}\Psi_f - \frac{x_a(x_f-x_a)}{x_f x_{Dd}-x_a^2} i_A \cos\gamma - \frac{x_a(x_f-x_a)}{x_f x_{Dd}-x_a^2} i_B \sin\gamma . \tag{12} \]

From equation (10) we determine the current \(i_{Dq}\)

\[ i_{Dq} = \frac{1}{x_{Dq}}\Psi_{Dq} - \frac{x_a}{x_{Dq}} i_A \sin\gamma + \frac{x_a}{x_{Dq}} i_B \cos\gamma . \tag{13} \]

Substituting the values of the currents \(i_f\), \(i_{Dd}\), \(i_{Dq}\) into the stator flux-linkage equations (6), (7), after simple transformations we obtain expressions for the currents \(i_A\) and \(i_B\):

\[ \begin{aligned} i_A={}& \frac{2}{x''_{qA}+x''_{dA}}\Psi_A - \frac{2x_a(x_{Dd}-x_a)} {(x''_{qA}+x''_{dA})(x_f x_{Dd}-x_a^2)} \Psi_f \cos\gamma \\ &- \frac{2x_a(x_f-x_a)} {(x''_{qA}+x''_{dA})(x_f x_{Dd}-x_a^2)} \Psi_{Dd}\cos\gamma - \frac{2x_a}{(x''_{qA}+x''_{dA})x_{Dq}} \Psi_{Dq}\sin\gamma \\ &+ \frac{x''_{qA}-x''_{dA}}{x''_{qA}+x''_{dA}} i_A \cos 2\gamma + \frac{x''_{qA}-x''_{dA}}{x''_{qA}+x''_{dA}} i_B \sin 2\gamma , \end{aligned} \tag{14} \]

\[ \begin{aligned} i_B={}& \frac{2}{2x_H+x''_{qB}+x''_{dB}}\Psi_B - \frac{2x_a(x_{Dd}-x_a)} {(2x_H+x''_{qB}+x''_{dB})(x_f x_{Dd}-x_a^2)} \Psi_f \sin\gamma \\ &- \frac{2x_a(x_f-x_a)} {(2x_H+x''_{qB}+x''_{dB})(x_f x_{Dd}-x_a^2)} \Psi_{Dd}\sin\gamma + \frac{2x_a}{(2x_H+x''_{qB}+x''_{dB})x_{Dq}} \Psi_{Dq}\cos\gamma \\ &- \frac{x''_{qB}-x''_{dB}}{2x_H+x''_{qB}+x''_{dB}} i_B \cos 2\gamma + \frac{x''_{qB}-x''_{dB}}{2x_H+x''_{qB}+x''_{dB}} i_A \sin 2\gamma , \end{aligned} \tag{15} \]

where \(x''_{dA}\), \(x''_{qA}\), \(x''_{dB}\), \(x''_{qB}\) are the subtransient inductive reactances along the \(d\)- and \(q\)-axes of stator windings \(A\) and \(B\).

On the basis of the general laws of electrodynamics, the equation of motion of the rotor has the form:

\[ M_m = H_j \frac{d\omega}{dt} + M_\psi, \tag{16} \]

where \(M_m\) is the moment of mechanical forces,
\(H_j\) is the inertia constant of the rotor,
\(M_\psi\) is the moment of electromagnetic forces.

The electromagnetic torque is found as the partial derivative of the electromagnetic energy with respect to the rotor rotation angle \(\gamma\).

\[ M_\psi = \frac{\partial W}{\partial \gamma} = x_a\left[ (i_{Dq}\cos\gamma - i_f\sin\gamma - i_{Dd}\sin\gamma)i_A + (i_{Dq}\sin\gamma + i_f\cos\gamma + i_{Dd}\cos\gamma)i_B \right]. \tag{17} \]

Taking the latter expression into account, we solve equation (16) with respect to the derivative

\[ \frac{d\omega}{dt} = \frac{1}{H_j} M_m + \frac{x_a}{H_j} \left[ (i_f \sin \gamma + i_{Dd}\sin\gamma - i_{Dq}\cos\gamma)i_A - (i_f \cos\gamma + i_{Dd}\cos\gamma + i_{Dq}\sin\gamma)i_B \right]. \tag{18} \]

The block diagram of the analog model, compiled from equations 1–5, 11–15, and 18, is shown in Fig. 2.

Using nodes I–V of the model, the currents and flux linkages in the stator and rotor circuits of the synchronous machine are determined; node VI realizes equation (18). The model makes it possible to take into account saturation of the synchronous generator along the main path of the magnetic flux, which is achieved by changing the coefficients in the equations as a function of the total flux in the air gap.

In order to simplify the electronic model, the normalized magnetization curve (Fig. 3, curve 1) is approximated by several linear sections (Fig. 3, curve 2). For these sections, the mutual-induction reactance \(x_a\) is found graphically as a function of the square of the total flux linkage in the air gap (Fig. 3, curve 3).

The program for transition from one section to another is specified by node VII, which determines the value \(\Psi_\delta^2\) from the formula

\[ \Psi_\delta^2 = (\Psi_{Dd} - x_{\sigma Dd} i_{Dd})^2 + (\Psi_{Dq} - x_{\sigma Dq} i_{Dq})^2, \tag{19} \]

where \(x_{\sigma Dd}\), \(x_{\sigma Dq}\) are the leakage inductive reactances of the damper circuits along the \(d\) and \(q\) axes.

When \(\Psi_\delta^2\) reaches the value found for the break point of the approximating curve \(x_a=f(\Psi_\delta^2)\), the operational relays OPR-1, OPR-2, OPR-4 operate; through their contacts they change the structure of the model. Since the differentiation operation on an analog computer is critical with respect to noise, in order to eliminate it the insertion of winding B is performed by means of relay OPR-3 at the instant when the flux linkage \(\Psi_B\) reaches its maximum value. On the analog machine (node VIII), it is more convenient to obtain the flux linkage \(\Psi_B\) from equation (15), taking into account that the current \(i_B\) in the open working winding is equal to zero.

\[ \Psi_B = \frac{x_a(x_{Dd}-x_a)}{x_f x_{Dd}-x_a^2}\,\Psi_f \sin\gamma + \frac{x_a(x_f-x_a)}{x_f x_{Dd}-x_a^2}\,\Psi_{Dd}\sin\gamma - \]

\[ - \frac{x_a}{x_{Dq}}\,\Psi_{Dq}\cos\gamma - \frac{x''_{qB}-x''_{dB}}{2}\,i_A\sin 2\gamma . \tag{20} \]

The harmonic functions of the sine and cosine of the double angle are formed by node IX of the model. Figure 4 shows one of the oscillograms of the solution obtained for the currents \(i_A\), \(i_B\) of an electromechanical energy source with the parameters:

\[ x''_{dA}=x''_{dB}=0.0885,\qquad x''_{qA}=x''_{qB}=0.6,\qquad x_A=x_B=x_{Dd}=1.05, \]

\[ x_f=1.2,\qquad r_A=r_B=r_{Dd}=r_f=0.005,\qquad r_{Dq}=0.5. \]

It follows from the phase-current curves (Fig. 4) that, in the presence of auxiliary winding A and rotor asymmetry, the current in working winding \(i_{B1}\) increased by a factor of 2.5 in comparison with the current \(i_{B2}\), which was recorded for the case when winding A is absent and the rotor is symmetrical. Comparison of the solution results obtained from the mathematical model on the analog computer with experimental studies [5] confirmed the correctness of the chosen modeling procedure.

The block diagram of the developed mathematical model contains no more than 50 computing blocks, which makes it possible to carry out a large

Fig. 2. Block diagram of the mathematical model of a pulsed power source.

Fig. 3. Normalized and approximating magnetization curves and the dependence \(X_a=f(\Psi_\delta^2)\)

Fig. 4. Current curves in the phase windings

the program for investigating the transient processes of a pulsed source even on a medium-power computer.

The model makes it possible to take into account the change in the rotor rotation speed and the influence of saturation on transients in a synchronous machine.

The analysis is carried out in physical axes, which significantly simplifies the processing of the solution results.

Thus, the correctness of the principles underlying the electromechanical energy source considered here, which makes it possible to substantially increase pulse power without additional energy sources, has been confirmed theoretically and experimentally.

REFERENCES

1. K. P. Kovach, I. Rats. Transients in AC Machines. TEI, 1963.

2. G. A. Sipailov, A. V. Loos, Yu. A. Romanov, V. F. Sergeev. Electromechanical source of pulses. Author’s certificate No. 304681.

3. I. I. Treshchev. Methods for Investigating AC Machines. “Energiya,” 1969.

4. G. A. Sipailov, A. V. Loos, E. I. Sobko. Mathematical modeling of a single-phase synchronous generator. Elektrotekhnika, 1972, No. 10.

5. G. A. Sipailov, A. V. Loos, A. A. Shchegolev. Electromechanical pulsed energy source. Izv. TPI, vol. 242, 1972.