PROCESSES OF ELECTROMECHANICAL CONVERSION AND ENERGY TRANSFER FROM A SHOCK GENERATOR

G. A. Sipailov, A. V. Loos, V. F. Kulakov

(Presented by the seminar of the Department of Electrical Machines)

For a number of physical studies, pulsed input of electrical energy is required with a power considerably greater than can be taken from the network without disturbing its stable operation. In such cases an intermediate energy storage device is used, charged relatively slowly and then capable of transferring the stored energy to the load in a short interval of time. With energy-transfer times of several milliseconds and longer, the use of electrical-machine pulsed generators is promising; these represent a combination of a mechanical energy store and an electrical converter.

Among unipolar and bipolar energy converters, a number of advantages in specific power and in the conditions for commutation of the shock current distinguish bipolar machines, of which the synchronous machine is the most advantageous. High power amplification when operating in a supertransient mode, the absence of a sliding contact, energy transfer at high voltage, and other advantages create the prerequisites for the effective use of a synchronous-type shock generator.

In the present work, the processes of conversion of the kinetic energy stored in the rotor of a shock generator into the electromagnetic energy of a current pulse are analyzed. As an electromechanical converter, a single-phase synchronous-type shock generator may be described by the known equations for the electrical circuits of a single-phase synchronous machine:

\[ 0=(r_c+r_n)i_c+\frac{d\Psi_c}{dt}, \]

\[ u_b=r_b i_b+\frac{d\Psi_b}{dt}, \]

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

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

and by the equation of motion of the rotor

\[ H_j\frac{d^2\gamma}{dt^2}+M_\psi+M_0=0. \tag{2} \]

In these equations

\[ \begin{aligned} \Psi_c &= i_c(L_c+L_H)+i_b M\cos\gamma+i_{Dd}M\cos\gamma+i_{Dq}M\sin\gamma,\\ \Psi_b &= i_c M\cos\gamma+i_b L_b+i_{Dd}M,\\ \Psi_{Dd} &= i_c M\cos\gamma+i_b M+i_{Dd}L_{Dd},\\ \Psi_{Dq} &= i_c M\sin\gamma+i_{Dq}L_{Dq} \end{aligned} \tag{3} \]

are the flux linkages of the generator circuits,

\[ M_\psi=i_c M[i_{Dq}\cos\gamma-(i_b+i_{Dd})\sin\gamma] \]

is the electromagnetic torque,

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

is the angle of rotation of the rotor,

\(M_0\) is the no-load loss torque,

\(L_H, R_H\) are the inductance and active resistance of the load.

The system of equations (1, 2) has no general solution in the domain of ordinary algebraic and transcendental functions, and its solution was carried out on an analog computer [1].

The principal criterion of the character of the transient processes described by the system of equations (1, 2) may be the value of the critical inductance, determined from the condition of complete conversion of the kinetic energy of the rotor flywheel masses into the electromagnetic energy of the leakage fields of the generator and the load:

\[ L_{\mathrm{cr}}=\frac{4\Psi_{c0}^{2}}{I\omega_0^{2}}, \tag{4} \]

where

\(I\) is the moment of inertia of the rotor,

\(\omega_0\) is the initial rotational speed of the rotor,

\(\Psi_{c0}\) is the initial flux linkage of the stator winding.

Expression (4) was obtained under the condition of a very powerful symmetric damper system on the rotor, when \(\Psi_b, \Psi_{Dd}, \Psi_{Dq}\) remain constant during the transient process.

If

\[ L=L_H+L_d''=L_H+(L_c-M)+ \frac{1}{\dfrac{1}{M}+\dfrac{1}{L_{Dd}-M}+\dfrac{1}{L_b-M}} > L_{\mathrm{cr}}', \]

then, when suddenly switched on at zero emf, we have an aperiodic discharge of the generator, during which the rotor speed decreases in the course of operation, reaching a minimum not equal to zero at the moment of the greatest surge of current, and then, as the current decreases, rises again. Oscillogram-

Fig. 1. Oscillograms of currents and rotor rotational speed for different discharge characteristics into the load.

of such a discharge is presented in Fig. 1a. Here and below, in the oscillograms the shock current is given in fractions of the maximum possible value

\[ I_m=\frac{2\Psi^2 r_0}{L_d''}. \]

From the physical standpoint, the operation of the generator may be represented as a process of elastic compression of the magnetic flux captured by the armature winding in the narrow space between the copper-plated surface of the rotor and the armature. The magnetic energy of the field of the armature circuit increases, and the current rises. The rotor, losing kinetic energy in doing work against the ponderomotive forces, is braked. During the subsequent half-revolution, the energy of the magnetic field, less the losses dissipated in the active resistances, is converted into mechanical energy, accelerating the rotation of the rotor.

If the transient inductance of the generator, together with the load inductance, is equal to the critical value \(L=L_{\mathrm{cr}}\), we have a very unstable critical mode, in which all the kinetic energy initially stored in the rotor is spent on producing electromagnetic fields. The rotor of a two-pole machine comes to a stop after half a revolution. The shock current, having reached its maximum by this moment, subsequently remains constant for an indefinitely long time (Fig. 1b). In a real system \(R\ne 0\), the process decays.

When the total inductance of the generator and load is less than the critical value, \(L<L_{\mathrm{cr}}\), the current reaches its maximum not after half a revolution, but earlier. The rotor reduces its speed to zero and then reverses (Fig. 1c). On the basis of the physical picture of the operation, this phenomenon can be interpreted as follows. Here the initial kinetic energy of the rotor is insufficient to compress the magnetic flux to the end (to the angle \(\gamma=\pi\)); therefore the rotor, losing speed as its kinetic energy is converted into electromagnetic energy, stops at angles \(\gamma<\pi\) and begins to rotate in the opposite direction, and then again in the forward direction, etc., like a pendulum.

Active resistances stabilize the instabilities of the critical mode, and when their magnitude is sufficiently large a stable aperiodic discharge without speed reversal occurs. In this case all the kinetic energy of the rotor will be transferred to the active resistances.

Operation of a generator with an inductance close to the critical value is, on the one hand, of interest from the standpoint of the possibility of obtaining long current pulses and a flat top; on the other hand, however, it is uneconomical, since stretching the pulse increases the losses, while the current amplitude and, accordingly, the transmitted energy decrease. Oscillograms of the solution (Fig. 2), obtained at constant quality factor of the discharge circuit \(\left(\frac{R}{L}=\mathrm{const}\right)\), demonstrate this clearly. Of course, in a mode close to the critical one, the use of kinetic energy is more complete. However, increasing the degree of utilization of the rotor energy is associated with difficulties in achieving a small inductance and with an increase in the level of energy losses in the generator or load during the current pulse. It is precisely these limitations that will be decisive in choosing the depth of discharge.

Figure 3 shows the variation of the duration of the shock-current pulse \(t_i\) as a function of the magnitude \(L\), expressed in fractions of the critical inductance \(L_{\mathrm{cr}}\) (for \(R=0\)). As the time constant \(T\), specified in periods

\[ T_0=\frac{2\pi}{\omega_0} \]

of the initial frequency, decreases, the resonance-type curves characterizing the pulse duration shift to the left. Practically, the maximum duration of the shock-current pulse at

Fig. 2. Change in current for different ratios \(\dfrac{L}{L_{\mathrm{cr}}}\).

Fig. 3. Change in pulse duration as a function of the parameters of the generator and the load circuit

\(L = L_{\mathrm{cr}} R\) will be a finite quantity determined by the time of dissipation of the energy \(\dfrac{1}{2} I \omega_0^2\) in the active resistances of the generator and the load.

In experiments on models of shock generators with pulsed power of 0.5 MVA and 7.5 MVA, the discharge regime corresponding to the critical or ordinary supercritical case was selected by appropriately changing the rotational speed and forcing the excitation flux. Oscillograms of currents and speeds in transient processes, obtained experimentally and on an AVM (Fig. 4), agree satisfactorily. The discrepancies are caused mainly by inaccurate modeling.

Fig. 4. Comparative oscillograms recorded on a model of a shock generator and on an analog computer

losses and the nonsinusoidal character of the field and current as a result of the considerable saturation of the magnetic system of the model.

Thus, the fundamental possibility has been demonstrated of realizing discharge modes of a shock generator into a load with a high coefficient of conversion of the stored energy into the energy of the stray fields and the load, which indicates the great potential of inertial storage devices.

REFERENCES

1. G. A. Sipaylov, A. V. Loos, É. I. Sobko. Mathematical modeling of transient processes of a single-phase synchronous generator. Elektrotekhnika, 1972, No. 10.