PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
AWARDED THE ORDER OF THE OCTOBER REVOLUTION AND THE ORDER OF THE RED BANNER OF LABOR, named after S. M. Kirov

Vol. 294
1976

STUDY OF DC VOLTAGE CONVERTERS USING AN ANALOG COMPUTER

V. I. STEPANOV, I. A. GONCHAR

(Submitted by the scientific and technical seminar of the Department of Automation and Telemechanics)

Stabilized DC voltage converters with pulse-width modulation are widely used to power electronic equipment. In designing such converters it is necessary to know the stability region. To determine the boundary of the stability region, it is expedient to use analog computing machines.

Figure 1 shows the electrical circuit of a thyristor converter of DC voltage into AC voltage.

The master oscillator (MO) produces a rectangular voltage. The modulator is implemented on transistor T1 [1] with output transformer Tr4 and two switching transformers Tr2 and Tr3. The duration of the output pulses is proportional to the difference between the currents of resistor R2 and the collector of transistor T2. The voltages from the master oscillator and the modulator are transmitted through capacitors C1 and C2, respectively, to the control electrodes of thyristors T1–T4 of the controlled parallel current inverter [2]. Capacitors C1, C2, together with the winding resistances and the input resistances of the thyristors, form differentiating circuits. The inverter is started after the master oscillator has been started. The delay time is set by the parameters of resistor R3 and capacitor C3 [3].

From the secondary winding of inverter transformer Tr5 an alternating voltage, close to sinusoidal in form, is taken. From the other secondary winding of the inverter transformer, the feedback voltage is taken and, through the rectifier and the L-shaped filter Dr2, C5, is fed to the measuring bridge.

The problem of modeling the converter can be reduced to constructing a model of the pulse-width modulator, the inverter, the rectifier, and the model of the continuous part.

The block diagram of the converter model is shown in Fig. 2, where the models of the inverter, rectifier, and filter are shown as a single IRF block. The block diagram of the pulse-width modulation element is shown in Fig. 3 [4]. It includes a sweep generator 1 (sawtooth-voltage generator), a comparison device 2, and a shaping device 3. The input signal \(u_m\) and the signal from the sweep generator are applied to the comparison device with opposite signs. The shaping device produces rectangular pulses whose duration is proportional to \(u_m\). All component parts

Fig. 1. Circuit diagram of a thyristor converter of DC voltage into AC voltage.

block diagrams are constructed on the basis of a set of standard computing elements of any analog computer.

The literature [4, 5, 6] describes a large number of sawtooth-voltage generator circuits; however, the circuit using rapid discharge of a capacitor through closed relay contacts [7] most fully meets the requirements of accuracy and simplicity. For this purpose, use is made of

Fig. 2. Structural diagram of a model of a dc-to-ac voltage converter

Fig. 3. Block diagram of a pulse-width modulation element

a computing amplifier in the integrating mode (amplifier 1, Fig. 2). Short rectangular pulses are applied to relay P1. The contacts of this relay periodically close for a short time and cause the capacitance C1 to discharge through resistor R1. At the output of the integrator, sawtooth pulses are formed; their repetition frequency is equal to the frequency \(f_{\mathrm{r}}\) of the pulses applied to relay P1, and their amplitude is determined by the relation:

\[ u_m=\frac{U}{R_{11}C_1 f_{\mathrm{r}}}, \tag{1} \]

where \(U\) is the input voltage of the integrator;

\(R_{11}\) and C1 are the input resistance and feedback capacitance of integrator 1.

The functions of the comparing and shaping device are performed by a nonlinearities block (amplifier 2 with a silicon Zener diode). Such a limiter circuit reliably maintains a zero level at the amplifier output when a voltage of negative polarity is present at the input.

When modeling a high-speed magnetic-transistor modulator, an element (amplifier 3, capacitor C2, relay P3 contacts) was introduced into the structural diagram that stores the discrete values of the input signal. Relay P3 is periodically actuated for a short time, and at the output of amplifier 3, during a half-period, a voltage is maintained that is equal to the input voltage at the moment when the modulator pulse ends. Thus, the processes in the model will be analogous to the processes in a high-speed magnetic-transistor modulator [8]. To prevent relay P2 from actuating at the moment when the P3 contacts close, the circuit uses additional contacts of relay P2, which connect, through an additional resistor, the output of integrator 1 to the input of summator 2.

To describe the processes in the inverter, we shall adopt the following assumptions:

1) leakage inductances and losses in the transformer are absent;

2) the commutating choke Dr1 has inductance \(L_{\mathrm{k}}\);

3) the resistances of the thyristors in the open states are equal to zero, and in the closed state to infinity;

4) the converter load \(R_{\mathrm{n}}\) is referred to half of the transformer primary winding.

During operation of the inverter, a number of intervals can be distinguished, each of which corresponds either to the open state of one of the thyristors T1--T4, or to the closed state of all thyristors (intermittent-current mode). The thyristor turn-on sequence is as follows: T4, T1, T3, T2. The inverter equivalent circuits corresponding to this sequence are shown in Fig. 4,a--4,d. For the interval when all thyristors are closed and the choke current \(i=0\), the inverter equivalent circuit is shown in Fig. 4,e.

The processes in the inverter for the first four intervals are described by the equations

\[ \left. \begin{aligned} L_{\mathrm{k}}\frac{di}{dt} &= xE + yu_c,\\ 4R_{\mathrm{n}} C_{\mathrm{k}}\frac{du_c}{dt} + u_c &= yiR_{\mathrm{n}} \end{aligned} \right\} \quad \text{for } i>0, \tag{2} \]

where

For the interval when \(i=0\), the processes are described by the equation

\[ 4R_{\mathrm{n}}C_{\mathrm{k}}\frac{du_c}{dt}+u_c=0. \tag{3} \]

Fig. 4. Equivalent circuits of the regulated inverter

On the basis of differential equations (2), (3), a block diagram of the inverter model was drawn up (Fig. 5, amplifiers 4, 5, 6, 7). Rectangular pulses specifying the value \(x\) are applied to the input of the inverter model. To determine the value \(y\), trigger T with a counting input and relay P4 are used. The transition from equations (2) to equation (3) and back is carried out by a diode in amplifier 4.

The functions of the rectifier are performed by amplifier 8 with a symmetrical Zener diode and relay P5.

The continuous part of the converter, in this case a T-shaped smoothing filter, is described by the differential equation

\[ a_2\frac{d^2u_{\mathrm{n}}}{d^2t}+a_1\frac{du_{\mathrm{n}}}{dt}+u_{\mathrm{n}}=U_{\mathrm{in}}(t), \tag{4} \]

where \(U_{\mathrm{in}}(t)\), \(u_{\mathrm{n}}\) are the voltages at the input and output, respectively, of the continuous part, \(U_{\mathrm{in}}(t)=u_c\);

\[ a_2=L_{\mathrm{f}}C_{\mathrm{f}}; \]

\[ a_1=\frac{L_{\mathrm{f}}}{R_{\mathrm{e}}}; \]

$L_{\phi}, C_{\phi}$ are the inductance and capacitance, respectively, of choke Dr2 and capacitor C5 (Fig. 1);

$R_{e}$ is the load resistance at the output of the dc section.

Expressing equation (4) with respect to the first derivative, we obtain:

\[ \begin{aligned} R_{e} C_{\phi}\frac{d u_{n}}{dt} &= R_{e} i_{\mathrm{dr}} - u_{n},\\ L_{\phi}\frac{d i_{\mathrm{dr}}}{dt} &= u_{c} - u_{n}, \end{aligned} \tag{5} \]

where

\[ i_{\mathrm{dr}}=\frac{1}{L_{\phi}}\int (u_{c}-u_{n})\,dt . \]

To solve equations (5), amplifiers 9, 10, 11 are used (Fig. 5).

Fig. 5. Block diagram of the model of the IVF unit of a dc-to-ac voltage converter

The model of a voltage converter into a series of dc voltages differs from the model of the converter into ac by the circuit of the IVF unit. The circuit diagram of the inverter with the filter of the voltage converter into a series of dc voltages is shown in Fig. 6,a, and Fig. 6,b shows the block diagram of the IVF unit.

In this inverter, a common smoothing choke is used in the ac circuit, $Dr_{\phi}$, and the smoothing capacitors $C_{\phi}$ are connected directly to the outputs of the rectifiers [9]. Relay P5 operates when the polarity of the output voltage of amplifier 9 changes; this voltage is an analog of the current of the smoothing choke $Dr_{\phi}$. For $u_{6}<u_{10}$, where $u_{6}, u_{10}$ are the voltages at the outputs of the corresponding amplifiers, self-oscillations will be present in the closed loop composed of amplifiers 9, 8 and the contacts of relay P6. If the switching time of relay P6 is less than the operating time of relay P5, then the contacts of relay P5 will be in the neutral position, which is analogous, in the real circuit, to blocking of the rectifier diodes when $u_{c}<u_{f}$.

Modeling of both types of converters was carried out on the MN-7 machine. In this process, stability regions were found in the plane of the parameters $L_{\phi}C_{\phi}$ and the feedback coefficient $K_{\mathrm{os}}$ for the dc-to-ac voltage converter (Fig. 7,a), and in the plane of the parameters $\Sigma C_{\phi}$ (the total capacitance of the filter capacitors) and $K_{\mathrm{os}}$ for the converter of voltage into a series of dc voltages (Fig. 7,b). From analysis of the obtained dependences $K_{\mathrm{os}}=f(L_{\phi}C_{\phi})$ and $K_{\mathrm{os}}=f(\Sigma C_{\phi})$, it may be concluded that, in order to increase the stability margin, it is necessary in the dc-to-ac voltage converter to increase the filter time constant $L_{\phi}C_{\phi}$, and in the converter of voltage into a series of dc-

Fig. 6. Inverter block of a DC-voltage converter
to a DC voltage of another magnitude:
a — circuit diagram;
b — structural diagram of the model

Fig. 7. Stability regions of converters:
a — to AC voltage; b — to DC voltage

voltages to increase \(\sum C_{\mathrm{f}}\), since the inductance of the choke \(D_{\mathrm{f}}\), according to [9], is not varied. The dependences obtained make it possible to select filter parameters when designing converters.

The waveforms of currents and voltages obtained in the simulation agree well with the current and voltage curves of actual converters.

REFERENCES

1. V. I. Stepanov. Semiconductor converter. USSR Author’s Certificate No. 163262, class HO 2m, published 22.7.64.

2. V. I. Stepanov. Parallel inverter. USSR Author’s Certificate No. 379965, published in Bulletin of Inventions, 1973, No. 20.

3. V. I. Stepanov. Device for controlling a single-phase parallel inverter. USSR Author’s Certificate No. 379965, class HO2r, published 20.4.73.

4. M. A. Shneidman. A pulse-width modulation element for studying pulse-control systems on an electronic model. Avtomatika i telemekhanika, vol. 17, 1956, No. 10.

5. M. A. Shneidman. Modeling of pulse automatic-control systems. In: Theory and Application of Discrete Automatic Systems. Publishing House of the Academy of Sciences of the USSR, 1960.

6. Bedford B. D., Hoft R. G. Principles of Inverter Circuits. I. Willey and Sons, New York, 1964.

7. I. A. Gonchar. Problems in the investigation and design of a voltage regulator with pulse-width modulation. Dissertation for the degree of Candidate of Technical Sciences. Tomsk, 1967.

8. V. I. Stepanov, I. A. Gonchar, Yu. S. Melnikov. Two-stroke magnetic-transistor modulator with positive feedback. Izvestiya vuzov. Elektromekhanika, 1974, No. 7.

9. V. I. Stepanov, Yu. S. Melnikov, I. A. Gonchar. Smoothing choke in the alternating-current circuit of a multichannel DC-voltage converter. Izvestiya vuzov. Elektromekhanika, 1975, No. 6.