STUDY ON AN ANALOG COMPUTER OF TRANSIENT PROCESSES IN THE SWITCHING DEVICE OF A SYNCHROTRON POWER-SUPPLY CIRCUIT

V. A. KOCHEGUROV, A. A. TERESHCHENKO

The power-supply circuit of the synchrotron for an energy of 1500 MeV at the Research Institute of Nuclear Physics, EA, shown in Fig. 1, contains an energy source—a transformer with a rectifier, a storage capacitor bank, and a switching device. The precharged capacitor bank, through the switching device, made with controlled mercury thyratrons connected in parallel, is discharged at a specified instant onto the electromagnet. Stable parallel operation of the thyratrons is ensured by anode current dividers (AD) [1]. To provide de-forcing of the magnetic field of the synchrotron, saturable reactors (ND) are included [2]. A mathematical study of the transient processes presents certain difficulties because of the nonlinear and non-single-valued relationship between induction and field strength in the magnetic cores of the AD and ND, and also because of their mutual influence during operation. For the study of various transient operating modes of the switching-device circuit, we used an analog computer.

Below is the method for solving, on an electronic model, the equations of the transient processes in the circuit shown in Fig. 1. The calculations were carried out under the following assumptions:

a) losses in the iron of the magnetic cores of the AD and ND were not taken into account,

b) the inductance of the synchrotron electromagnet was considered as a linear quantity.

Taking the noted assumptions into account and in accordance with the designations given in Fig. 1, the operation of the circuit is described by nine differential equations, compiled on the basis of Kirchhoff’s laws:

\[ i_1 = -\frac{R}{L}\int i_1\,dt -\frac{1}{LC}\iint i_1\,dt^2 -\frac{\omega S}{L}(B_1-B_7) -\frac{\omega S}{L}(B_2-B_{17}) -\frac{\omega_g s_g}{L}B_{g1} -\frac{(U_{c0}-U_{c1})}{R_0} +\frac{(U_{c1}-U_{c2})}{R_1}, \]

\[ i_2 = -\frac{R}{L}\int i_2\,dt -\frac{1}{LC}\iint i_2\,dt^2 -\frac{\omega S}{L}(B_3-B_6) -\frac{\omega S}{L}(B_4-B_1) -\frac{\omega_g s_g}{L}B_{g2} -\frac{(U_{c1}-U_{c2})}{R_1} +\frac{(U_{c2}-U_{c3})}{R_2}, \]

Fig. 1.

Fig. 2.

\[ i_9=-\frac{R}{L}\int i_9 dt-\frac{1}{LC}\iint i_9dt^2-\frac{ws}{L}(B_{17}-B_2)-\frac{ws}{L}(B_{18}-B_{15})- \]

\[ -\frac{w_gs_g}{L}B_{g_9}-\frac{(U_{C_8}-U_{C_9})}{R_8}+\frac{(U_{C_9}-U_{C_1})}{R_9}, \]

\[ Hl=wi;\qquad U_C=\frac{1}{C}\int i_Cdt; \]

where \(R\) is the total ohmic resistance of the circuit, taking into account active losses in the windings of the AD, ND, and electromagnet;

\(L\) is the total inductance of the circuit, taking into account the leakage inductances of the AD and ND and the inductance of the electromagnet;

\(w, w_g\) are the numbers of turns of the AD and ND windings;

\(s, s_g\) are the cross-sectional areas of the magnetic cores of the AD and ND;

\(R_j\) are the resistances equalizing the voltages on the capacitor cells;

\(U_{C_j}\) are the voltages on the capacitor cells;

\(C_j\) are the capacitances of the capacitor cells;

\(B_j, H_j\) are the instantaneous values of induction and field strength in the magnetic cores of the AD and ND;

\(i_j\) are the currents in the discharge circuits;

\(l, l_g\) are the lengths of the magnetic cores of the AD and ND.

Fig. 3

The block diagram of the model for calculating transient processes in the circuit under consideration is shown in Fig. 2. The operational amplifiers with diodes \(6\times 2\text{T}\) in the feedback loops simulate the discharge and recharge valves. The principal magnetization curves of the steel for the AD and ND are set on the nonlinearity blocks (NB). Hysteresis losses were not taken into account in this case.

When it is necessary to take hysteresis losses into account, we have developed a device that makes it possible to simulate the hysteresis loop by means of ordinary elements of analog machines, as shown in Fig. 3.

Modeling of the hysteresis loop in this case is based on the principle of inverting one half of the limiting symmetric cycle of the loop, set up on the nonlinearity block. The input signal, through the normally closed contact of relay \(P_1\), is fed to the input of the BN, on which one half of the limiting symmetric cycle of the loop, approximated by linear segments, is set up. The output of the BN in this case is connected through the normally closed contact of relay \(P_3\). When the maximum value of the amplitude of the input signal is reached, the inverting blocks 1 and 2 are switched on through the contacts of relays \(P_2\) and \(P_4\), thereby ensuring reproduction of the second half of the hysteresis loop.

Fig. 4

The magnitude of the coercive force can be regulated by means of an additional \(RC\)-circuit connected to an additional input of the BN through the contact of the machine starting relay, as shown in Fig. 3. The \(RC\)-circuit provides a reduction of the voltage equivalent to the residual induction in a prescribed time. The relay connection circuit is shown in Fig. 3. Inversion of one branch of the hysteresis loop can also be provided by means of the electronic machine, i.e., by one integrating amplifier and the contacts of the block of special nonlinearities.

The relation of the elements of the model (Fig. 2) to the parameters of the real circuit (Fig. 1), taking into account the equations for transforming the variables, is expressed by the following relations [3]:

\[ \frac{ws}{L}=\frac{\kappa_{14}\cdot M_i}{M_B}, \qquad \alpha_1=\alpha_2=\frac{wM_i}{M_H\cdot l}, \qquad \alpha_3=\frac{w_g M_i}{M_{H_g}\cdot l_g}, \]

\[ \frac{1}{LC}=\kappa_{11}\cdot\kappa_{21}\cdot\kappa_{31}\frac{1}{M_t^2}; \qquad \frac{w_g s_g}{L}=\frac{\kappa_{15}\cdot M_i}{M_B}, \qquad \frac{R}{L}=\kappa_{21}\cdot\kappa_{41}\cdot\kappa_{16}\frac{1}{M_t}, \]

where \(\kappa_{ij}\) are the transfer coefficients of the operational amplifiers;
\(\alpha_1,\alpha_2,\alpha_3\) are constant coefficients;
\(M_t, M_H, M_B, M_i\) are scale factors.

On the electronic model, normal operation and a number of possible emergency modes of the switching device were investigated. In Fig. 4 oscillograms are given for several operating modes of the switching device:

a) normal operation;
b) the pre-discharge voltage on one section is less than on the others (corresponds to breakdown of one of the series-connected capacitor sections at the moment of discharge);
c) one of the discharge valves fired later;
d) one of the recharge valves did not fire.

A rational notation for the equations describing the system made it possible to use the minimum number of operational amplifiers in solving the problem.

The method presented for modeling circuits with nonlinear elements can also be applied to the modeling of transient processes and steady-state conditions of other electromagnetic devices containing elements with steel and rectifiers.

REFERENCES

1. I. P. Chuchalin, V. A. Kochegurov. Application of two-core anode current dividers with parallel connection of rectifiers. Izv. vuzov. “Elektromekhanika”, 7, 1960.

2. V. A. Kochegurov. Derating of the magnetic field in synchrotrons by means of a saturating choke. Izv. vuzov, “Fizika”, 1, 1957.

3. B. Ya. Kogan. Electronic modeling devices and their application to the study of automatic control systems. GIFML. Moscow, 1959.