Proceedings
of the Tomsk Polytechnic Institute named after S. M. Kirov,
holder of the Order of the October Revolution and the Order of the Red Banner of Labor

Volume 294 1976

STEADY-STATE PROCESSES IN A PULSE DC VOLTAGE STABILIZER WITH PULSE-WIDTH MODULATION

O. S. Vadutov

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

A pulse DC voltage stabilizer (Fig. 1,a), in which the pulse-width modulator consists of a sawtooth-voltage generator (SVG), a comparison circuit, and a shaping device, may be regarded as a relay system operating in the forced-oscillation regime [1]. In this relay system (Fig. 1,b), the relay element combines the shaping element of the modulator itself and the regulating element (RE) of the stabilizer, operating in switching mode. The external periodic signal for the system is the output signal of the SVG, \(f_n(t)\).

In [1], stability conditions were found for the steady-state regime in which the RE switches with the period \(T\) of the SVG output signal and with a constant duty ratio of its closed state \(\psi_0\). In the present article, the conditions for existence of this steady-state regime will be determined.

Fig. 1.

For convenience, as in [1], we shall write all variables in relative units:

\[ \begin{aligned} z &= U_{\mathrm{n}}/U_{\mathrm{in}} &&\text{— output voltage of the stabilizer;}\\ y &= U/U_{\mathrm{in}} &&\text{— output signal of the modulator;}\\ x &= (U_0 - U_{\mathrm{n}})/U_{\mathrm{in}} &&\text{— input signal of the modulator;}\\ f_0 &= U_0/U_{\mathrm{in}} &&\text{— reference signal (setpoint).} \end{aligned} \]

Assuming that the resistances of the RE and diode \(D\) in the nonconducting state are equal to infinity, and in the conducting state satisfy the condition \(R_{\mathrm{re}} \simeq R_{\mathrm{d}}\), the continuous part of the stabilizer in the continuous-current regime is described by the equation

\[ T_0^2 \frac{d^2 z}{dt^2} + 2 \xi T_0 \frac{dz}{dt} + z = k_{\mathrm{n}} y, \tag{1} \]

where

\[ k_{\mathrm{n}} = R_{\mathrm{n}}/(R + R_{\mathrm{n}}); \]

\[ T_0=\sqrt{LC\kappa_{\mathrm{n}}}; \]

\[ \xi=(L+RR_{\mathrm{n}}C)/2\sqrt{LCR_{\mathrm{n}}(R+R_{\mathrm{n}})}; \]

\[ R=R_L+R_{\mathrm{e}}\simeq R_L+R_{\mathrm{d}}; \]

\(R_L\) is the active resistance of the choke.

The transfer function of the continuous part, written in terms of the dimensionless Laplace-transform parameter \(q=pT\), is equal to

\[ W_{\mathrm{n}}(q)=\frac{\kappa_{\mathrm{n}}P_{\mathrm{n}}(q)}{Q_{\mathrm{n}}(q)} =\frac{\kappa_{\mathrm{n}}\beta^2}{q^2+2\xi\beta q+\beta^2}, \tag{2} \]

where \(\beta=T/T_0\).

The output signal of the modulator in the circuit of Fig. 1b is determined by the expression

\[ y= \begin{cases} 1 & \text{for } x_1-\delta>0,\quad \dot{x}_1>0 \text{ and}\\ & \quad x_1+\delta>0,\quad \dot{x}_1<0;\\ 0 & \text{for } x_1-\delta<0,\quad \dot{x}_1>0 \text{ and}\\ & \quad x_1+\delta<0,\quad \dot{x}_1<0, \end{cases} \tag{3} \]

where \(x_1=f_0-z\div f_{\mathrm{p}}\).

For the investigation of steady-state processes in stabilizers of the class considered, the methods developed in the theory of relay systems [2] may be used.

In the presence of an external periodic action in a relay system, the following modes are possible:

1) a mode of forced oscillations with the frequency of the external action (synchronous mode);

2) a mode of subharmonic oscillations with a frequency an integral number of times lower than the frequency of the external action;

3) a mode of beats between self-oscillations and forced oscillations.

The operating mode for the stabilizer is the first mode, characterized by minimum energy losses and minimum ripples of the output voltage. Let us determine the conditions for the existence of this mode.

Assume that a periodic mode takes place in the system with the frequency of the external action (Fig. 2). In this case, a sequence of rectangular pulses with period \(T\) and relative duration \(\gamma_0 T\) is applied to the input of the continuous part, and all quantities vary periodically with period \(T\). We shall denote the periodically varying quantities by \(\tilde{x}(t)\), \(\tilde{y}(t)\), and \(\tilde{z}(t)\).

Fig. 2.

The external periodic signal, under the condition that time is reckoned from the beginning of the pulse arriving at the input of the continuous part, is determined by the expression:

\[ f_{\mathrm{p}}(t)= \begin{cases} f_2(t)=\dfrac{A}{aT}(\varphi-t), & \text{for } 0\leq t\leq \varphi;\\[6pt] f_1(t)=-\dfrac{A}{(1-a)T}(t-\varphi), & \text{for } \varphi\leq t\leq \varphi+(1-a)T;\\[6pt] f_2(t)=\dfrac{A}{aT}(T+\varphi-t), & \text{for } \varphi+(1-a)T\leq t\leq T, \end{cases} \tag{4} \]

where \(\varphi\) is the synchronization phase.

According to [2], a synchronous mode will exist in the system when the following conditions are satisfied:

1) switching instants

\[ \tilde{x}_1(0)=f_0-f_{\mathrm{p}}(0)-\tilde{z}(0)=\delta, \]

\[ \tilde{x}_1(\gamma_0T)=f_0-f_{\mathrm{p}}(\gamma_0T)-\tilde{z}(\gamma_0T)=-\delta; \tag{5} \]

2) switching directions

\[ \dot{\tilde{x}}_1(0)=-\dot{f}_{\mathrm{p}}(0)-\dot{\tilde{z}}(0)>0, \]

\[ \dot{\tilde{x}}_1(\gamma_0T)=-\dot{f}_{\mathrm{p}}(\gamma_0T)-\dot{\tilde{z}}(\gamma_0T)<0; \tag{6} \]

3) absence of additional switchings within the period

\[ \tilde{x}_1(t)=f_0-f_{\mathrm{p}}(t)-\tilde{z}(t)>-\delta \quad \text{for } 0\leq t\leq \gamma_0T, \]

\[ \tilde{x}_1(t)=f_0-f_{\mathrm{p}}(t)-\tilde{z}(t)<\delta \quad \text{for } \gamma_0T\leq t\leq T. \tag{7} \]

The values of the output coordinate of the system \(\tilde{z}(t)\) and its derivative, which enter into these conditions, can be found by summing the response of the continuous part to the infinite sequence of pulses preceding the time instant \(t=0\):

\[ \tilde{z}(0)=\kappa_{\mathrm{n}}\sum_{\nu=1}^{l} C_{\nu0}\, \frac{e^{q_\nu}-e^{(1-\gamma_0)q_\nu}}{1-e^{q_\nu}}; \tag{8} \]

\[ \dot{\tilde{z}}(0)=\frac{\kappa_{\mathrm{n}}}{T}\sum_{\nu=1}^{l} C_\nu\, \frac{e^{q_\nu}-e^{(1-\gamma_0)q_\nu}}{1-e^{q_\nu}}; \tag{9} \]

\[ \tilde{z}(\gamma_0T)=\kappa_{\mathrm{n}}\sum_{\nu=1}^{l} C_{\nu0}\, \frac{e^{\gamma_0 q_\nu}-1}{1-e^{q_\nu}}; \tag{10} \]

\[ \dot{\tilde{z}}(\gamma_0T)=\frac{\kappa_{\mathrm{n}}}{T}\sum_{\nu=1}^{l} C_\nu\, \frac{e^{\gamma_0 q_\nu}-1}{1-e^{q_\nu}}. \tag{11} \]

Here

\[ C_{\nu0}=\frac{P_{\mathrm{n}}(q_\nu)}{Q'_{\mathrm{n}}(q_\nu)q_\nu}; \qquad C_\nu=\frac{P_{\mathrm{n}}(q_\nu)}{Q'_{\mathrm{n}}(q_\nu)}; \]

\(q_\nu\) are the poles of the transfer function of the continuous part.

Substituting into (5) the values \(f_{\mathrm{p}}(0)\) and \(f_{\mathrm{p}}(\gamma_0T)\) according to (4), we obtain expressions determining the switching instants of the relay element in the following form:

\[ f_0-\frac{A\varphi}{aT}-\tilde z(0)=\delta, \tag{12} \]

\[ f_0-\frac{A}{(1-a)T}(\gamma_0T-\varphi)-\tilde z(\gamma_0T)=-\delta_* . \]

Eliminating \(f_0\) from these, we find

\[ \varphi=a\gamma_0T-\frac{a(1-a)}{A}(D+\delta), \tag{13} \]

where

\[ D=\tilde z(0)-\tilde z(\gamma_0T). \tag{14} \]

A synchronous mode can exist in the system if the amplitude of the signal \(f_n(t)\) exceeds a certain critical value \(A_{\mathrm{cr}}\) [3, 4]. We determine the values of \(A_{\mathrm{cr}}\) from the obvious necessary conditions:

\[ \begin{aligned} 0 \leq f_1(\gamma_0T) \leq A;\\ 0 \leq f_2(0) \leq A, \end{aligned} \tag{15} \]

These conditions, taking into account (4) and (13), give the conditions for existence of a synchronous mode:

\[ A \geq A_{1\mathrm{cr}}=-\frac{a(D+\delta)}{\gamma_0}; \qquad A \geq A_{2\mathrm{cr}}=\frac{a(D+\delta)}{1-\gamma_0}; \]

\[ A \geq A_{3\mathrm{cr}}=\frac{(1-a)(D+\delta)}{\gamma_0}; \qquad A \geq A_{4\mathrm{cr}}=-\frac{(1-a)(D+\delta)}{1-\gamma_0}. \tag{16} \]

In the most frequently encountered special cases, formulas (16) are written as:

1) for \(a=0\) (modulation of the trailing edge of the pulse)

\[ A \geq A_{3\mathrm{cr}}=\frac{D+\delta}{\gamma_0}; \qquad A \geq A_{4\mathrm{cr}}=-\frac{D+\delta}{1-\gamma_0}; \tag{17} \]

2) for \(a=1\) (modulation of the leading edge of the pulse)

\[ A \geq A_{1\mathrm{cr}}=-\frac{D+\delta}{\gamma_0}; \qquad A \geq A_{2\mathrm{cr}}=\frac{D+\delta}{1-\gamma_0}. \tag{18} \]

It is not difficult to see that the smaller the value of the amplitude of the sawtooth voltage \(A\), the greater the steepness of the characteristic of the pulse-width modulator and, consequently, the higher the relative stabilization coefficient of the regulator. It is therefore desirable to obtain values of \(A_{\mathrm{cr}}\) as small as possible.

Let us consider the influence of the principal parameters of the regulator on the value of \(A_{\mathrm{cr}}\). Figures 3 and 4 present the dependences of the critical amplitude values, calculated from expressions (17) for \(a=0\) and for various values of \(\beta\) and \(\delta\). The analogous dependences for \(a=1\), as is easy to note by comparing expressions (17) and (18), are mirror images, with respect to the line \(\gamma_0=0.5\), of the dependences constructed for \(a=0\).

In practice, most often \(\gamma_0 \simeq f_0 > 0.5\). In these cases, when a hysteresis loop is present in the shaping element of the pulse-width modulator, it is expedient to use trailing-edge modulation (\(a=0\)), for which the conditions for existence of the synchronous mode give

less than for other values of \(a\), the critical value of the amplitude of the sawtooth voltage.

The formulas obtained make it possible to calculate the exact static characteristics of regulators of the type under consideration. In particular, eliminating the quantity \(\varphi\) from equations (12), we find the dependence of the established relative pulse duration \(\gamma_0\) on the set point \(f_0\) and the system parameters:

Fig. 3.

Fig. 4.

\[ f_0 = (2a - 1)\delta + a\widetilde{z}(0) + (1-a)\widetilde{z}(\gamma_0 T) + A\gamma_0 . \tag{19} \]

Since the relationship between \(\gamma_0\) and the constant component of the output voltage \(z_0\) is easily established, formula (19) makes it possible to obtain exact static characteristics of pulse-width dc voltage regulators.

Conclusions

1. For a pulse regulator with pulse-width modulation, which can be regarded as a relay system operating in a forced-oscillation mode, the conditions for the existence of a synchronous mode and a formula for calculating exact static characteristics have been found.

2. It has been shown that in regulators of this type it is more expedient to use modulation of the trailing edge of the pulses.

References

1. O. S. Vadutov. Stability of the steady-state mode in a pulse dc voltage regulator with pulse-width modulation. In: Electrical Engineering Apparatus. Vol. I. Control and monitoring devices. Moscow, “Energiya,” 1971.

2. Ya. Z. Tsypkin. Theory of Relay Systems of Automatic Control. Moscow, Gostekhizdat, 1955.

3. G. S. Pospelov. Dynamics of relay systems of automatic control. In: Modern Methods for Designing Automatic Control Systems. Moscow, “Mashinostroenie,” 1967.

4. E. G. Orlov, I. V. Bulin-Sokolov. Steady-state processes in a pulse control system for electric machines. Elektrichestvo, 1970, No. 4.