ON SYSTEMS WITH PULSE-WIDTH MODULATION OF THE FIRST AND SECOND KINDS

O. S. Vadutov

(Presented by the Scientific-Technical Seminar on Automation and Telemechanics)

As is known [1], pulse-width modulation (PWM) may be of two kinds: of the first kind—PWM I, and of the second kind—PWM II. In PWM I the duration of the modulator output pulses changes depending on the values of the input quantity at equidistant discrete instants of time. In PWM II the duration of the modulator output pulses changes in accordance with the current value of the input quantity.

Pulse systems with PWM II, which have found wide application in practice, may be regarded as insufficiently studied. This is explained by the fact that the difference equations describing the behavior of these systems are nonlinear and transcendental. For this reason, the methods proposed for the study of systems with PWM I cannot be extended to systems with PWM II. As far as we know, only one work [2] is devoted to the study of closed-loop systems with PWM II; in it a method is proposed for constructing transient processes in such systems.

In the present work a graphical method is proposed for studying first-order closed-loop systems with PWM II, and a comparison is made between systems with a modulator of the first and second kinds. We note that, for the sake of simplification, in this work only systems are considered in which the output pulses of the modulator can have only one polarity, and all coordinates of the system are written in relative units (the amplitude of the modulator output pulses is taken as the unit of measurement).

The structural diagram of the system under investigation is shown in Fig. 1. The equation of the pulse-width modulator has the form

Fig. 1

\[ y(\bar t)= \begin{cases} 1, & \text{for } n < \bar t \le n+\gamma_n,\\ 0, & \text{for } n+\gamma_n < \bar t \le n+1, \end{cases} \tag{1} \]

where the dependence of the relative pulse duration \(\gamma_n\) on the input quantity of the modulator can be written as

\[ \gamma_n = \begin{cases} \chi x[n], & \text{for } 0 < x[n] < \dfrac{1}{\chi},\\[6pt] 1, & \text{for } x[n] > \dfrac{1}{\chi},\\[6pt] 0, & \text{for } x[n] < 0, \end{cases} \tag{2} \]

for modulation of the first kind, and

\[ \gamma_n = \begin{cases} \chi x[n+\gamma_n], & \text{for } 0 < x[n+\gamma_n] < \dfrac{1}{\chi},\\[6pt] 1, & \text{for } x[n+1] > \dfrac{1}{\chi},\\[6pt] 0, & \text{for } x[n] < 0, \end{cases} \tag{3} \]

for modulation of the second kind. Here

\[ x(\bar t)=f_0-z(\bar t). \tag{4} \]

It is obvious that the processes in the system are described by the equations

\[ \frac{dz(\bar t)}{dt}+\beta z(\bar t)=\beta \quad \text{for } n<\bar t<n+\gamma_n, \tag{5} \]

\[ \frac{dz(\bar t)}{dt}+\beta z(\bar t)=0 \quad \text{for } n+\gamma_n<\bar t<n+1, \tag{6} \]

where \(\beta=\alpha T\), and \(T\) is the pulse repetition period.

The solution of equation (5) can be found by applying the ordinary Laplace transform under the initial conditions \(z[n]\). As a result we obtain

\[ z(\bar t)=z[n]e^{-\beta(\bar t-n)}+1-e^{-\beta(\bar t-n)}. \tag{7} \]

For \(\bar t=n+\gamma_n\) we shall have

\[ z[n+\gamma_n]=z[n]e^{-\beta\gamma_n}+1-e^{-\beta\gamma_n}. \tag{8} \]

The solution of equation (6), under the initial conditions \(z[n+\gamma_n]\), is written as

\[ z(\bar t)=z[n+\gamma_n]e^{-\beta(\bar t-n-\gamma_n)}. \tag{9} \]

Hence, for \(\bar t=n+1\),

\[ z[n+1]=z[n+\gamma_n]e^{-\beta(1-\gamma_n)}. \tag{10} \]

From equations (8) and (10) one clearly sees the difference between systems with PWM-I and PWM-II. If, in a system with PWM-I, in which the law of variation of \(\gamma_n\) is determined by expression (2), it is possible to eliminate \(z[n+\gamma_n]\) and obtain a recurrence relation connecting the values of the variable \(z(t)\) at discrete instants of time (the method proposed in [3] is based on the graphical solution of this recurrence relation), then in a system with PWM-II this cannot be done.

However, the following way of solving equations (8) and (10) may be proposed, making it possible to avoid eliminating \(z[n+\gamma_n]\). Let us construct, in a plane along whose abscissa axis we shall plot the values \(z[n]\) and \(z[n+1]\), and along whose ordinate axis \(z[n+\gamma_n]\), according to equations (8) and

(10), taking into account the modulation laws (2) and (3) and the closing equation of the system (1), curves that would make it possible to find successively the values of the output coordinate of the system \(z[n]\), \(z[n+\gamma_n]\), \(z[n+1]\), \(z[n+1+\gamma_{n+1}]\), etc.

For a system with PWM-II, using equations (8), (3), and (4), and specifying the values \(z[n+\gamma_n]\), one can construct a curve which, from the known value of the output coordinate of the system at the instant of the beginning of the pulse \(z[n]\), makes it possible to find its value at the instant of the end of the pulse \(z[n+\gamma_n]\) (curves 2 in Figs. 2, 3, and 4). From equations (10), (3), and (4) one can construct a curve determining the dependence between \(z[n+1]\) and \(z[n+\gamma_n]\) (curves \(2'\) in Figs. 2, 3, and 4).

In order to be able to compare systems with PWM-I and PWM-II, we shall construct analogous curves also for the system with PWM-I. From equations (8), (2), and (4), specifying the values \(z[n]\), we find the values \(z[n+\gamma_n]\) (curves 1 in Figs. 2, 3, and 4). The curve that makes it possible to find \(z[n+1]\) from the known value \(z[n+\gamma_n]\) is constructed from equations (10), (2), and (4), specifying the same values \(z[n]\) and taking into account the values \(z[n+\gamma_n]\) obtained above (curves \(1'\) in Figs. 2, 3, and 4).

For the system having the parameters \(\beta=0.5\), \(\chi=2\), and \(f_0=0.8\), the curves are shown in Fig. 2. As can be seen from the figure, the steady-state values

Fig. 2

of the output coordinate of the systems with PWM-I and PWM-II, determined respectively by the coordinates of the points of intersection of curves 1 and \(1'\), 2 and \(2'\), differ. It can also be shown that the steady-state values of the relative duration of the pulses at the modulator output in the system with PWM-II are smaller than in the system with PWM-I.

Figure 2 shows an example of finding the transient process for the initial condition \(z[0]=0\) in the system with PWM-I (solid line)

Fig. 3

Fig. 4

and in the system with PWM-I (dash-dot line). It can be seen that the transient processes in the systems under consideration have different characters: aperiodic in the system with PWM-II and oscillatory in the system with PWM-I.

Figure 3 gives analogous curves for systems with the parameters: $\beta = 0.5$, $\chi = 5$, and $f_0 = 0.8$, from which it is clear that in the system with PWM-II the transient process becomes oscillatory, while in the system with PWM-I self-oscillations are established with a period equal to $2T$.

Let us consider how the parameter $\beta$ affects the process. If $\beta = 0.3$, $\chi = 2$, and $f_0 = 0.8$, the curves will have the form shown in Fig. 4, whence it follows that as $\beta$ decreases, the difference between the systems with PWM-I and PWM-II decreases. However, even in this case, if $\chi$ is increased, overshoot, and then self-oscillations, will appear in the system with PWM-I at smaller values of $\chi$ than in the system with PWM-II.

Thus, comparison of systems with PWM-I and PWM-II by means of the proposed method shows that systems with PWM-II have better transient-process quality indices, and the limiting gain coefficient in them is higher. Simulation of systems with PWM on an analog computer confirmed the validity of this conclusion also for systems of higher order.

REFERENCES

1. Ya. Z. Tsypkin. Theory of Linear Pulse Systems. Moscow, Fizmatgiz, 1963.
2. I. A. Gonchar. Calculation of transient processes in pulse-width systems of the second kind. Izv. TTI, vol. 139, 1965.
3. E. Polak. Stability and Graphical Analysis of First-Order Pulse-Width-Modulated Sampled-Data Regulator Systems. IRE Transactions on Automatic Control, vol. AC—6, No. 3, 1961.