ENERGY STORAGE IN AN INDUCTIVE LOAD FROM A PULSE GENERATOR

V. V. Ivashin, A. V. Loos, G. A. Sipailov

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

Energy storage in rotating flywheel masses is the most economical in comparison with capacitive and even inductive energy storage. The conversion of the kinetic energy stored by flywheel masses into electromagnetic energy is usually carried out by means of electrical machines. From the point of view of its commutation, the transfer of electromagnetic energy from an electrical machine into an inductive load is a less complicated problem than the transfer of energy from a capacitor bank or from an inductive storage unit.

A disadvantage of the pulsed production of electromagnetic energy by means of electrical machines is that, during one half-wave of alternating voltage, only a small fraction of the energy stored by the flywheel masses can be transferred to the load. In order to increase the electromagnetic energy obtained during a single pulse, special electrical machines are created, one of which is a single-phase generator of pulse power—the “pulse generator” of the Kapitsa–Kostenko generator type [1].

The connection of a pulse generator to a load is usually carried out at the moment when the e.m.f. passes through zero. The current of the first positive half-wave, due to the aperiodic component, then has the maximum amplitude and duration. The current increases throughout the entire positive half-period of the voltage, and its maximum is reached at the moment when the sign of the e.m.f. changes, at $\omega t = \pi$. However, even with the aid of a pulse generator, no more than 5–10% of the kinetic energy stored in the rotating rotor can be transferred to the load during one first pulse.

The fraction of energy transferred by a pulse generator to the load can be substantially increased if the transfer is carried out not during one, but during several positive half-periods of the voltage. Taking into account that, when a pulse generator is switched onto a load, the current in the circuit may reach tens and even hundreds of thousands of amperes, it is practically impossible to commutate such currents by means of ordinary valves, for example thyratrons or ignitrons. This, in particular, explains the fact that pulse-power generators have not yet attained the necessary widespread use.

The use of arcless ion-mechanical switching devices [2, 3] solves the problem of commutating the current of a pulse-power generator and thereby makes it possible to transfer energy into the de-

tens of megajoules from a pulse generator into an inductive load in a time measured in tens of milliseconds.

Principle of Operation

The basic circuit for storing energy in an inductive load and for forming in it current pulses, or, correspondingly, magnetic-field pulses of quasi-trapezoidal shape from a pulsed-power generator is shown in Fig. 1, where UG is a single-phase pulsed-power generator,

Fig. 1. Circuit for storing energy in an inductance from a single-phase pulse generator.

a source of energy; $L_{\mathrm{g}}$, $r_{\mathrm{g}}$ are the leakage inductance and active resistance of the generator; $L_{\mathrm{n}}$, $r_{\mathrm{n}}$ are the same for the load; $r_{\mathrm{z}}$ is the active resistance of the short circuit; $K$-1, $K$-2 are arcless ion-mechanical switching devices.

In the initial state the switching devices $K$-1 and $K$-2 are open; the generator, excited to its rated voltage, operates at no load. The generator is connected to the load by switch $K$-1 at the instant when the emf passes through zero. During the first, positive, half-wave of voltage, the current in the load increases from zero to the value $I_{1m}$ (Fig. 2). When the current in the generator–load circuit reaches its maximum value, the load is closed (short-circuited) by switch $K$-2. During the time when the load is short-circuited by switch $K$-2, the energy in it, apart from losses in the active resistances of the circuit, is retained. The decay of the current during this time interval is determined by the time constant of the load–short-circuit loop.

At the instant of the current maximum $I_{1m}$, the generator voltage passes through zero $(\omega t = \pi, e_{\mathrm{g}} = 0)$. When the sign of the voltage changes, the current in the generator decreases rapidly; when the current passes through zero, the generator is disconnected by switch $K$-1 (Fig. 2).

For the duration of the next positive half-wave of the emf, at the instant when it passes through zero at $\omega t = 2\pi$, the generator is again switched on by means of switch $K$-1, but now already to the load short-circuited by switch $K$-2. In this case the current in the generator increases rapidly, while in switch $K$-2 it decreases. When the current in switch $K$-2 passes through zero—that is, at the instant when the generator and load currents are equal—switch $K$-2 opens, and thereafter the current in the load increases to the value $I_{2m}$.

At the instant of the current maximum $I_{2m}$, switch $K$-2 closes again; subsequently the process proceeds in the same way as after the closure of switch $K$-2 at the first current maximum, equal to $I_{1m}$. Thus, it is possible to ensure the input of energy from a pulse generator into an inductive load.

over several positive half-periods of the voltage. In this case the rising part of the current curve becomes stepped. The duration of the rising part of the current pulse, at a generator emf frequency of 50 Hz, can in this case be made equal to 10, 30, 50, 70, etc., milliseconds. It should be noted that the increment of the load current at each subsequent step will be smaller than at the preceding one. The opti-

Fig. 2. Shape of the curves of the generator emf and of the currents in the different elements of the circuit: \(i_{\text{н}}\)—in the load, \(i_{\text{г}}\)—in the generator, \(i_{\text{з}}\)—in the short circuit.

mal number of steps in the load-current curve is determined mainly by the ratio between the parameters of the generator and of the load. If necessary, however, the circuit makes it possible to limit oneself to any preassigned number of steps.

If all the energy delivered by the shock generator is expended in the load, then after the maximum is reached the current in the load, short-circuited by switch \(K\)-2, will decrease exponentially with a time constant determined by the parameters of the short-circuited circuit. If necessary, the energy accumulated in the load, less losses, can be returned to the generator by successive switching on and off of switches \(K\)-1 and \(K\)-2. In doing so, the generator should be connected for the time, or part of the time, of the negative half-wave of the voltage (inverter mode).

In the circuit of operation of the shock generator under consideration, the inverter mode can be carried out at any of the angles that are multiples of \(\pi(2n+1)\). If the inverter mode is carried out at the angle \(\omega t \simeq 3\pi\), then after switch \(K\)-2 is opened at \(i_{\text{з}}=0\), the current in the load will rapidly decrease to zero, and when the current passes through zero value

the load can be disconnected by switch \(K\text{-}1\). This instant corresponds to point \(t_4\) in Fig. 3. As a result of such commutation, a current pulse, or correspondingly a magnetic-field pulse, of quasi-trapezoidal form is produced in the load, with an almost flat top, a “plateau,” of duration \(T\), where \(T\) is the period of the generator emf (Fig. 3).

Fig. 3. Quasi-trapezoidal form of the current in the inductive load.

The limiting value of the energy that can be accumulated in the inductance \(L_{\mathrm{n}}\) according to the circuit under consideration is equal to

\[ Q_m=\frac{1}{2}L_{\mathrm{n}} I_{\mathrm{ud}}^{2}, \]

where \(I_{\mathrm{ud}}\) is the maximum value of the short-circuit current, taking into account the aperiodic component.

Investigation of the Generator Operation Using an Analog Computer

An analytical investigation of the operation of the described circuit by analyzing the differential equations written for each stage of operation is a complex and very labor-intensive task. Solving the equations describing the individual stages on the analog computer MNB-1 presents no difficulty. However, separate solution of the equations for each stage does not give an overall picture of the operation of the circuit under consideration, and requires, for each solution, determination and input into the computer of the initial conditions from the preceding solution.

To obtain an integral picture of the operation of the surge generator with an inductive load, a mathematical model was developed in which commutation and the introduction of initial conditions during the transition from one stage to another were carried out by a special programming device. In the modeling, the following assumptions were adopted: 1) the active resistances of the windings are constant; 2) losses in the steel are absent; 3) the inductance of the windings does not depend on the magnitude of the current; 4) the rotational speed of the generator is constant. Although the last two assumptions are crude, they do not distort the qualitative picture of the operation of the circuit and have practically little effect on the number of stages at which a current of the specified magnitude is reached.

A decrease in the rotational speed of the generator affects mainly the increase in the duration of each subsequent stage and, in connection with this, the increase in losses in the circuit. In further investigations of the circuit it is proposed to take into account both steel saturation and the change in the rotational speed of the generator.

The transient process in the first stage of operation of the circuit, i.e., when switch K-1 is closed at the instant when the e.m.f. passes through zero with K-2 open, is described by the equation

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

In the first stage the generator current is equal to the load current.

The second stage begins from the instant when switch K-2 closes the inductance \(L_{\mathrm{n}}\), in which the current by this time has reached the maximum value \(I_m\); in this case the transient process is described by the equations

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

\[ 0 = i_{\mathrm{n}}r_{\mathrm{n}} + L_{\mathrm{n}}\frac{di_{\mathrm{n}}}{dt} + i_{\mathrm{s}}r_{\mathrm{s}}, \tag{3} \]

\[ 0 = i_{\mathrm{g}} + i_{\mathrm{s}} - i_{\mathrm{n}}. \tag{4} \]

At the instant when the generator current passes through zero, K-1 is opened. From this time the operation of the circuit is described by equation (3). The load current in this stage is equal to the short-circuit current. During the next positive half-wave of the e.m.f., at the instant when it passes through zero, K-1 is closed. The transient processes in the fourth stage are described by equations (2), (3), and (4).

At the instant when the current in the short circuit passes through zero, K-2 is opened. The transient process in the fifth stage is described by equation (1). Thereafter all stages of operation are repeated.

The mathematical model of the operation of a surge generator with the inductance \(L_{\mathrm{n}}\) is shown in Fig. 4. Switching in the mathematical model is carried out by means of relays \(P_1\), \(P_2\), \(P_3\), and \(P_4\), which operate at definite instants of time. These relays are connected in series with tubes \(L_1\), \(L_2\), \(L_3\), and \(L_4\), to whose grids the control voltage is supplied from blocks 8, 9, and 10. Diode limiters are included in the feedback circuits of blocks 8, 9, and 10; therefore, when the input quantity passes through zero, the voltage at the output of these blocks changes sign abruptly. This makes it possible to control the operation of the circuit as a function of the generator e.m.f., the generator current, and the short-circuit current. Relay \(P_5\) serves to prepare the control circuit for the beginning of the next accumulation cycle.

A sinusoidal e.m.f. is obtained from a special generator (blocks 5, 6, 7). Block 1 serves to reproduce the load current. The generator e.m.f. is applied to the second input of the block with the transmission coefficient

\[ k_{12} = \frac{1}{L_{\mathrm{g}}}\cdot \frac{M_E M_I}{M_t}, \]

where
\(M_E\) is the voltage scale,
\(M_I\) is the current scale,
\(M_t\) is the time scale.

The magnitude of the active resistance of the load is set by means of the coefficient

\[ k_{11} = M_t \frac{r_{\mathrm{n}}}{L_{\mathrm{n}}}. \]

Block 2 serves to reproduce the generator current. The value of the generator’s active resistance is set by means of the coefficient

\[ k_{21}=M_t\frac{r_{\mathrm{g}}}{L_{\mathrm{g}}}. \]

Fig. 4. Mathematical model of the operating circuit of a pulse generator for an inductive storage device.

The transfer coefficients of the remaining inputs are calculated from the expressions:

\[ k_{22}=\frac{1}{L_{\mathrm{g}}}\cdot\frac{M_E M_I}{M_t}, \]

\[ k_{23}=\frac{1}{L_{\mathrm{g}}+L_{\mathrm{n}}}\cdot\frac{M_F M_I}{M_t}. \]

From the output of summing block 4 a voltage proportional to \(i_3\) is taken. The influence of various parameters on the operation of the energy-storage circuit is investigated by changing the coefficients \(\kappa_{11}\), \(\kappa_{12}\), \(\kappa_{21}\), \(\kappa_2\), \(\kappa_{23}\).

The transient processes occurring in the mathematical model are described by the system of equations:

\[ pU_1 = -(\kappa_{11}U_1+\kappa_{12}U_7), \tag{5} \]

\[ pU_2 = -(\kappa_{21}U_2+\kappa_{22}U_7+\kappa_{23}U_7), \tag{6} \]

\[ U_4=-U_1+U_2, \tag{7} \]

where \(U_1, U_2,\ldots\) are the voltages at the outputs of the operational amplifiers. The mathematical model considered reproduces the entire process of energy storage in the inductance.

Investigations of the operation of a pulse generator with an inductive storage element were carried out for the following ratios between the inductances of the generator and the load:

\[ L_{\mathrm{n}}=L_{\mathrm{g}},\ 2.5L_{\mathrm{g}},\ 5L_{\mathrm{g}},\ 7.5L_{\mathrm{g}},\ 10L_{\mathrm{g}} \quad \text{at} \quad \frac{r_{\mathrm{g}}}{L_{\mathrm{g}}}=0.05. \]

The maximum value of the current in the storage element is equal to the current of the sudden short circuit of the pulse generator. As calculations have shown, for an inductance \(L_{\mathrm{n}}=10L_{\mathrm{g}}\) such a current is reached in 40 cycles, where each cycle corresponds to one current step. In practice, however, it is not expedient to strive to reach the maximum current value, since at the last steps the current increment is insignificant, while the losses are maximal.

Table 1 gives the results of an investigation of the circuit operation, showing the number of periods of the generator emf, \(n\), during which the current in the load reaches a value equal to \(0.9 I_{\mathrm{d}}\), for various ratios of the load and generator inductances.

Table 1

One of the oscillograms, showing the character of the change of current in the load and in other elements of the circuit for \(L_{\mathrm{n}}/L_{\mathrm{g}}=2.5\), is shown in Fig. 5, where \(i_{\mathrm{n}}\) is the load current, \(i_{\mathrm{z}}\) is the shorting current, \(i_{\mathrm{g}}\) is the generator current, and \(e_{\mathrm{g}}\) is the generator emf.

The circuit considered can be used both when the pulse generator operates directly into an inductive load and for storing energy in an inductance. If, during one half-wave of emf from a pulse generator of limiting dimensions, about 20 mJ of electromagnetic energy can be delivered to a load whose inductance is equal to the generator inductance, then over many half-waves of emf of one sign this energy can be increased severalfold.

Thus, energy storage in an inductive load is carried out in two stages. In the first stage, a prime mover of comparatively low power accelerates the rotor of the pulse generator to its rated speed; the energy losses are then insignificant and are determined mainly by the no-load losses of the generator. In the se-

... of the second stage, the pulse generator converts the stored kinetic energy into electromagnetic energy and transfers it to the inductive load. Since the rate of energy transfer from the pulse generator to the load is high, the energy losses in this process will also be small in comparison with the energy being transferred. Thus, the overall efficiency of the charging installation when operating according to the scheme considered will be high, while the power of the energy source—the primary motor—will be relatively small.

Fig. 5. Oscillogram.

Step-shaped current pulses, like flat-topped current pulses obtained from a pulse generator, may find application in installations for thermonuclear research and in a number of other cases where energies measured in tens of megajoules are required, and where obtaining them from capacitor banks or from any other energy sources presents major technical and economic difficulties.

REFERENCES

1. P. L. Kapitsa. Proc. Roy. Soc. A115, No. 772, 1927.

2. V. V. Ivashin, G. A. Sipailov. Commutation of the current of a pulse generator. Problems in the Theory and Design of Electrical Machines, Interuniversity Collection of Papers, issue 3, Novosibirsk, 1963.

3. V. V. Ivashin, G. A. Sipailov. Arcless interruption of large currents. Elektrotekhnika, No. 9, 1964.