STUDY ON AN ANALOG COMPUTER OF A CLOSED-LOOP AUTOMATIC CONTROL SYSTEM CONTAINING AN IONIC CONVERTER

V. A. KOCHEGUROV, A. A. TERESHCHENKO

(Presented by the scientific seminar of the Scientific Research Institute of Nuclear Physics)

In nuclear-research technology, pulse-current generators with capacitive energy storage are widely used. The operation of such a generator is based on the relatively slow charging of storage capacitors from a controlled rectifier, followed by a rapid discharge into the load. From the standpoint of the experiment, requirements are imposed on the stability of the pre-discharge voltage on the storage capacitors.

Fig. 1.

We have developed a voltage-stabilization system with automatic regulation of the charging current. The block diagram of such a regulator is shown in Fig. 1. The storage capacitor bank is charged from a powerful controlled thyratron rectifier assembled according to a three-phase bridge circuit. To limit the current in the secondary windings of the anode transformer, inductances \(L_g\) are connected.

The rectifier is controlled by a fast-acting grid-control system. The controlling element is amplifier 8, the signals

from whose output they are fed to the grid-control system 10 through the cathode follower 9. The current and voltage measuring elements are sensors 1 and 2, connected into the load circuit. The voltages from the sensors, proportional respectively to the charging current and to the voltage on the storage unit, are compared with reference voltages 3, 4, and their differences pass through the correcting elements 5, 6 to the input of the discriminator 7. From the output of the discriminator the error signal is fed to the input of the amplifier 8.

The stabilizer with the charging-current regulator operates as follows. Upon complete or partial discharge of the capacitor bank, a voltage error signal arises, the rectifier opens, and current appears in the charging circuit. At this time the automatic current regulator comes into operation, seeking to keep the current at the prescribed level. When the voltage error signal becomes equal to the reference value, the rectifier is blocked, thereby ensuring stabilization of the voltage on the storage unit.

The analytical calculation of the optimum adjustment of this system proves rather complicated unless certain simplifying assumptions are adopted concerning the influence of various factors on the operation of the system. To carry out this stage of the work we used analog computing techniques. In studying such a closed control system on an analog computer, it is easy to introduce additional connections and correcting links into the circuit, to take into account the influence of various harmful disturbances on the system, to change the values of the feedback parameters, etc.

Let us write the equations of the elements of the block diagram shown in Fig. 1. The equations of the three-phase bridge rectifier and the load are written [1]:

\[ \begin{aligned} e_1 &= Lp(i_1+i_6)+r_{\mathrm{tr}}(i_1+i_6)+r_{\mathrm{v}1}i_1+u_{\mathrm{v}1};\\ e_2 &= Lp(i_2+i_5)+r_{\mathrm{tr}}(i_2+i_5)+r_{\mathrm{v}2}i_2+u_{\mathrm{v}1};\\ e_3 &= Lp(i_4+i_3)+r_{\mathrm{tr}}(i_4+i_3)+r_{\mathrm{v}3}i_3+u_{\mathrm{v}1};\\ e_1 &= Lp(i_1+i_6)+r_{\mathrm{tr}}(i_1+i_6)+r_{\mathrm{v}6}i_6+u_{\mathrm{v}2};\\ e_2 &= Lp(i_2+i_5)+r_{\mathrm{tr}}(i_2+i_5)+r_{\mathrm{v}5}i_5+u_{\mathrm{v}2};\\ e_3 &= Lp(i_3+i_4)+r_{\mathrm{tr}}(i_3+i_4)+r_{\mathrm{v}4}i_4+u_{\mathrm{v}2};\\ i_1+i_2+i_3 &= i_4+i_5+i_6=i_{\mathrm{n}};\\ u_c &= \frac{1}{C}\int i_{\mathrm{n}}\,dt;\\ u_{\mathrm{v}1}+u_{\mathrm{v}2} &= u_{\mathrm{v}}=f(\alpha,i_{\mathrm{n}}), \end{aligned} \]

where \(L\) is the inductance of one conversion branch, including the leakage inductance \(L\) of the transformer phase and the additional \(L_g\);

\(r_{\mathrm{tr}}\) is the active resistance of the transformer;

\(r_{\mathrm{v}}\) is the valve resistance;

\(i_j\) are the phase currents;

\(i_{\mathrm{n}}\) is the load current;

\(u_{\mathrm{v}1}\) and \(u_{\mathrm{v}2}\) are the voltages of the anode and cathode groups;

\[ T=\frac{L}{r_{\mathrm{tr}}} \]

is the electromagnetic time constant;

\(\alpha\) is the valve control angle;

\(C\) is the capacitance of the capacitor storage unit;

\(u_c\) is the voltage on the storage unit.

Let us write the equations describing the operation of the regulator. The voltages

the feedbacks with respect to load current and voltage on the storage device has the form:

\[ u_i = \kappa_1 i_{\mathrm{n}}; \]

\[ u_u = \kappa_2 u_c . \]

where \(\kappa_1\) and \(\kappa_2\) are proportionality coefficients between the load current and voltage and the voltages at the outputs of sensors 1 and 2.
The current and voltage error signals are written as:

\[ \Delta u_i = u_{0i} - u_i; \]

\[ \Delta u_u = u_{0u} - u_u; \]

where \(u_{0u}\) and \(u_{0i}\) are reference voltages.
Correction is carried out by passive phase-advancing \(RC\) circuits, whose equations are written

for the current regulator:

\[ \Delta u'_i = \kappa_3 \left( \Delta u_i + \frac{1}{T_1}\int \Delta u_i dt \right)^* \]

for the voltage regulator:

\[ \Delta u'_u = \kappa_4 \left( \Delta u_u + \frac{1}{T_2}\int \Delta u_u dt \right). \]

where \(\kappa_3\) and \(\kappa_4\) are the gain coefficients of the correction circuits,
\(T_1\) and \(T_2\) are the time constants of the correction circuits.
The amplifier output voltage is expressed as:

\[ u_y = \kappa(\Delta u'_i + \Delta u'_u), \]

where \(\kappa\) is the gain coefficient.
The equation of the grid-control system has the form:

\[ \vartheta = f[u_y(t-\tau)], \]

where

\[ \vartheta = \frac{\pi}{2} - \alpha \]

is the angle measured from the beginning of the sinusoid;

\[ \tau \]

is the delay constant of the grid-control circuit.

From the equations given, the block diagram of the model, shown in Fig. 2*), was compiled.
The parameters of the model of the closed-loop system should be calculated on the basis of the dependencies:

\[ \kappa_{12} = \frac{M_t M_e}{M_i L}; \qquad \kappa_1 = \frac{M_t r_{\mathrm{tr}}}{L}; \qquad \kappa_{\mathrm{v}} = \frac{M_t R_{\mathrm{v}}}{L}; \]

\[ \kappa_{71} = \frac{M_t}{L\cdot \kappa_{11}}; \qquad \alpha = \frac{M_i^2}{L\cdot 2\cdot C\cdot \kappa_{91}\cdot \kappa_{12}\cdot \kappa_{71}}. \]

\[ T_1 = M_t R_{14}C_{14}; \qquad T_2 = M_t R_{13}C_{13}; \qquad \tau = M_t R_{15}C_{15} \]

where \(M_t\) is the time scale;

\[ M_e = \frac{u_{\mathrm{a.f}}}{e'_1} \]

is the voltage scale;

\(u_{\mathrm{a.f}}\) is the amplitude of the phase voltage of the secondary winding of the anode transformer;

\(e'_1\) is the amplitude of the phase voltage of the generator of low-frequency sinusoidal oscillations;

\[ M_i = \frac{M_u a2C}{M_t} \]

is the current scale;

\(\kappa_{jl}\) are the transfer coefficients of the operational amplifiers.

*) The solution of the problem on the model was carried out by TPI student A. A. Shatov.

Fig. 2

An analysis of the dynamic modes of the stabilization system was carried out on the model. The parameters of the correcting elements were selected so as to ensure stable operation of the regulator under various disturbances that may arise during operation. The influence of individual parameters and of the circuit structure on the nature of the transient processes was established.

Figure 3 shows an oscillogram of the transient process for

\[ T_1 = 0.012\ \text{s}, \qquad T_2 = 0.0135\ \text{s}, \qquad \tau = 0.015\ \text{s}, \]

and Fig. 4 for

\[ T_1 = 0.038\ \text{s}, \qquad T_2 = 0.01\ \text{s}, \qquad \tau = 0.004\ \text{s}. \]

Fig. 3

The quality of the automatic system satisfying the stability condition was evaluated by the method proposed in the literature [3, 4].

Fig. 4

The developed electronic model of the automatic control system containing an ionic converter makes it possible to study the quality of the system with accuracy sufficient for engineering calculations.

REFERENCES

1. V. A. Kochegurov, A. A. Tereshchenko. Application of analog computing machines for the analysis of electromagnetic processes in three-phase rectifiers. Proceedings of TPI, 1966.

2. B. M. Gutkin. Ionic DC Drive. Energiya Publishing House, 1965.

3. A. L. Goflev, V. A. Karabanov. Dynamics of the Control System of Unmanned Aerial Vehicles. Mashinostroenie Publishing House, 1965.

4. I. N. Pechorin. Calculation of Automatic Control Systems. Mashgiz Publishing House, 1962.