STUDY OF A CIRCUIT FOR ACCELERATED ENERGY TRANSFER FROM A SHOCK GENERATOR TO AN INDUCTIVE LOAD USING AN ANALOG COMPUTER

A. V. LOOS, K. A. KHOR'KOV, T. A. SIGAILOV, V. V. IVASHIN

(Presented by the scientific seminar of the Electromechanical Faculty)

An analytical study of a circuit for increasing the energy and the rate of its transfer from a shock generator to an inductive load is rather difficult [1]. This difficulty is explained not only by the complexity of solving the systems of differential equations describing the processes in the circuit, but also by the difficulty of determining the initial conditions for each stage of operation of the circuit, since they are determined from the results of solving the preceding stage.

Therefore, the studies were carried out on an MNB-1 analog computer. Solving the differential equations stage by stage on an analog computer presents no difficulty. However, this path of investigation does not give an overall picture of the operation of the circuit and requires large expenditures of time, since, in order to solve the equations describing the processes at each stage of operation of the circuit, it is necessary to determine the initial conditions from the results of the solution at the preceding stage. In doing so, errors introduced in determining and setting the initial conditions are inevitable.

To obtain the complete picture of the circuit operation, a mathematical model was developed in which switching was performed automatically during the transition from stage to stage by a special programming device. The use of the programming device increased the accuracy of solving the problem, since the need to determine the initial conditions for the subsequent stage from the results of the solution at the preceding stage was eliminated; it considerably shortened the investigation time and made it possible to follow visually the entire process of operation of the circuit.

The transient process in the first stage of operation of the circuit (closing of K-1) is described by the equation

\[ E_m \sin \omega t = L_{\mathrm{r}} \frac{di_{\mathrm{r}}}{dt} + i_{\mathrm{r}} r_{\mathrm{r}} + \frac{1}{C}\int i_c\,dt. \tag{1} \]

At the moment when the generator emf passes through zero, K-2 closes. The transient processes for the second stage of operation of the circuit are described by equations (1) and (2)

\[ 0 = L_{\mathrm{н}} \frac{di_{\mathrm{н}}}{dt} + i_{\mathrm{н}} r_{\mathrm{н}} + \frac{1}{C}\int i_c\,dt. \tag{2} \]

When the generator current passes through zero, K-1 opens. The transient process for the third stage \((t_1 \div t_2)\) is described by equation (2). When the voltage on the capacitor bank passes through zero, K-3 closes. The transient process in the fourth stage \((t_2 \div t_3)\) is described by equations (3) and (4):

\[ L_{\mathrm{n}}\frac{di_{\mathrm{n}}}{dt}+i_{\mathrm{n}}r_{\mathrm{n}}+i_3r_3=0, \tag{3} \]

\[ \frac{1}{C}\int i_c\,dt+i_3r_3=0. \tag{4} \]

At the beginning of the next positive half-wave of the generator emf, K-1 closes. The transient processes in the fifth stage \((t_3 \div t_4)\) of the circuit operation are described by equations (1), (2), (3), (4). When the current in the short-circuit jumper reaches zero, K-3 opens. The transient processes in this stage \((t_4 \div t_5)\) of the circuit operation are described by equations (1) and (2). The subsequent stages of circuit operation repeat.

Fig. 1. Mathematical model of the circuit.

The mathematical model of the circuit under consideration is shown in Fig. 1. The mathematical model was developed so that, by means of switching, it would be possible to simulate the operation of the real ...

of the storage circuit at each stage. Switching in the mathematical model during the transition from one stage to another is carried out by means of a special programming device consisting of relays \(P_1, P_2, P_3, P_4, P_5\), tubes \(L_1, L_2, L_3, L_4, L_5\), and dc amplifiers 8, 9, 10, 11, in whose feedback circuits diode limiters are connected. When the input signal passes through zero, the sign of the output voltage at the outputs of blocks 8, 9, 10, 11 changes instantaneously. This control voltage is applied to the grids of tubes \(L_1, L_2, L_3, L_4, L_5\). With a positive grid potential the tubes open (each at the appropriate instant), the relays operate, and the necessary switching is performed in the mathematical model. After the completion of one storage cycle it is necessary to return the circuit of the programming device to its initial position. For this purpose relay \(P_5\) is used; its contacts \(1P_5\) open the supply circuit of relay \(P_4\), which, by its contacts \(1P_4, 2P_4, 4P_4\), prepares relays \(P_1, P_2, P_3\) for operation. Thus, the programming device makes it possible to investigate the operation of the circuit for any number of cycles.

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

\[ pU_1=-k_{11}U_1-\frac{1}{p}k_{12}k_{61}k_{51}k_{41}k_{21}U_1 +\frac{1}{p}k_{12}k_{61}k_{51}k_{42}U_3+ \]

\[ +\frac{1}{p}k_{12}k_{61}k_{51}k_{13}U_{10}-k_{13}U_1; \tag{5} \]

\[ pU_3=-k_{31}U_3+\frac{1}{p}k_{21}k_{22}k_{51}k_{41}U_1 +\frac{1}{p}k_{63}k_{51}k_{32}U_{10} -\frac{1}{p}k_{32}k_{51}k_{42}U_3; \tag{6} \]

\[ pU_{10}=k_{101}k_{51}k_{11}k_{21}U_1-k_{101}k_{51}k_{43}U_{10} +k_{101}k_{51}k_{42}U_3. \tag{7} \]

Here

\(U_1, U_6, U_3, U_{10}\) are the voltages at the outputs of the blocks,

\(k_{12}, k_{13}, k_{21}, \ldots\) are the transfer coefficients of the blocks, where the first digit

Fig. 2. Oscillogram of the solution on an analog computer.

denotes the block number, and the second the input number. The initial charge on the capacitor bank corresponds to the initial condition on block 5.

From comparison of equations (1), (2), (3), (4) with equations (5), (6), (7), we obtain expressions for the transfer coefficients of the blocks in terms of the parameters of the actual circuit, the time scale \(M_t\), the current scale \(M_i\), and the emf scale \(M_E\):

\[ k_{11}=M_t\frac{r_{\mathrm{g}}}{L_{\mathrm{g}}};\qquad k_{12}=M_t^2\frac{1}{L_{\mathrm{g}}C};\qquad k_{31}=M_t\frac{r_{\mathrm{n}}}{L_{\mathrm{n}}}; \]

\[ k_{32}=M_t^2\frac{1}{L_{\mathrm{n}}C};\qquad k_{101}=\frac{M_t}{Cr_3};\qquad k_{13}=\frac{1}{L_{\mathrm{g}}}\cdot\frac{M_E M_i}{M_t}. \]

The values of all the remaining transfer coefficients were chosen equal to unity. The mathematical model developed makes it possible to study the operation of the circuit for various parameters and various initial voltages of the capacitor bank. Studies of the operation of the circuit were carried out in order to determine the influence of individual circuit parameters on the magnitude of the energy accumulated in the load and on the rate of its accumulation. Figure 2 presents an oscillogram of the solution for

\[ \frac{L_{\mathrm{n}}}{L_{\mathrm{g}}}=5;\qquad \nu^2=3.5;\qquad U_{c0}=0.5E_m;\qquad \frac{r}{x}=0.05. \]

Results of the Study of Circuit Operation

The operation of the circuit was studied for various ratios between the inductances of the load and the generator and for a constant natural frequency of oscillation of the circuit \(L_{\mathrm{g}}-C-L_{\mathrm{n}}\). To estimate the rate of rise of the current in the load up to a specified level, the number of periods of the generator emf was determined at which, for each of the circuits considered, the current in the load reached a value equal to \(0.9I_{\mathrm{ud}}\). It is known that the greatest increase in the energy transferred from the generator to the load when an auxiliary capacitor bank is used occurs at the ratio

\[ \nu^2=\frac{\omega_{\mathrm{k}}^2}{\omega^2}\simeq 3.5\ [2]. \]

Here \(\omega_{\mathrm{k}}\) is the natural frequency of oscillation of the circuit \(L_{\mathrm{g}}-C-L_{\mathrm{n}}\).

The value of the ratio \(\dfrac{L_{\mathrm{n}}}{L_{\mathrm{g}}}\) was varied within the limits \(1\div 10\), and the voltage on the capacitance was \(U_{c0}=0\) and \(U_{c0}=0.5E_m\). The results of these studies are presented in Table 1. In the last row of Table 1, for comparison, the number of positive half-periods of the emf is also shown for operation of the shock generator on the load without auxiliary capacitance.

The maximum energy of the load and of the capacitor bank was calculated from the formulas:

\[ Q_{\mathrm{n}}=0.5L_{\mathrm{n}}I_{\mathrm{ud}}^2,\qquad Q_c=0.5CU_{\max}^2, \]

\[ C=\frac{L_{\mathrm{n}}+L_{\mathrm{g}}}{\omega_{\mathrm{k}}^2L_{\mathrm{n}}L_{\mathrm{g}}}. \]

Study of the Influence of Preliminary Charging of the Capacitor Bank

These studies were carried out for the natural frequency of oscillation of the circuit

\[ \omega_{\mathrm{k}}=\sqrt{3.5}\,\omega,\qquad L_{\mathrm{n}}=5L_{\mathrm{g}},\qquad \frac{r}{x}=0.05. \]

The results of these studies are presented in Table 2.

Table 1

Note: \(n\) is the number of positive half-periods of the emf;
\(Q_c\) is the energy for which the capacitor bank must be designed;
\(Q_{\mathrm{н}}\) is the energy that can be transferred to the load in the circuit under consideration;
\(P_{\mathrm{н}}\) is the loss in the load when operating without a capacitor bank;
\(P_c\) is the loss in the load when operating with capacitance in the circuit under consideration.

Table 2

Conclusions

1. The use of an auxiliary capacitor bank designed for an energy of less than 5% of the energy transferred to the load (for \(L_{\mathrm{н}} > 2L_{\mathrm{г}}\)) leads to a reduction in the energy-transfer time by a factor of 2–2.5. At the same time, the dissipation of energy in the active resistances is reduced by a factor of 1.5–2.

2. Preliminary charging of the capacitor bank further increases the rate of energy transfer to the inductive load in comparison with a circuit without preliminary charging of the capacitor bank.

3. The use of a capacitor bank connected in parallel with the load facilitates the operating conditions of the switching device K-2 when the short circuit is removed.

Literature

1. A. V. Loos, K. A. Khorkov, G. A. Sipailov, V. V. Ivashin. A circuit for increasing the energy and the rate of its transfer from a shock generator to an inductive load. Izvestiya TPI, vol. 139, 1965.

2. G. A. Sipailov, K. A. Khorkov. On the joint operation of a shock generator and a capacitor bank into an inductive load. Izvestiya TPI, vol. 132, 1964.