What is the RC circuit's response to a PWM signal?

In this section, we discuss the RC circuit's (figure 2) response to an input signal that is a pulse width modulated signal of known period and duty cycle. The pulse width modulated input signal is shown in figure 6. Over a single period, $[0,T]$, the input voltage to the circuit has two distinct parts. There is the "charging" part from $[0,T_1]$ during which the applied voltage is $V$. In this interval, the capacitor is being charged by the external voltage source. The second part is the "discharging" part from $[T_1,T]$. During this interval the applied voltage is zero and so the capacitor is discharging through its resistor.

Figure 6: Pulse Width Modulated Signal

During the "charging" phase, we can think of the RC circuit as being driven by a step function of magnitude $V$ volts. If we assume that the capacitor has an initial voltage of $V_0$ at the beginning of the charging phase (time $t=0$), then the circuit's response is simply given by equation 2 for $t \in [0,T_1]$.

During the "discharge" phase, there is no external voltage being applied to the RC circuit. This means that the system response is due solely to the capacitor voltage that was present at time $T_1$ after the charging period. As a result, the capacitor's voltage over the time interval $[T_1,T]$ is simply the RC circuit's natural response. This means, of course, that the capacitor voltage for $t \in [T_1,T]$ is given by equation 1 of the form $V_1 e^{-(t-T_1)/RC}$ for $t \in [T_1,T]$, where the initial voltage, $V_1$, is the voltage on the capacitor at time $t=T_1$.

The top drawing in figure 7 illustrates the output signal we expect from a PWM signal driving an RC circuit over an interval from $[0,T]$. For times beyond this interval, we expect to see the waveform shown in the bottom drawing in figure 7. In this drawing we assume that the capacitor is initially uncharged. As our circuit cycles through its charge and discharge phases, the voltage over the capacitor follows a saw-tooth trajectory that eventually reaches a steady state regime. In this steady-state region, the capacitor on the voltage zigzags between $V_0$ and $V_1$ volts. The exact value of these steady state voltages is dependent on the period $T$ and the duty cycle $T_1/T$.

Figure 7: Response to PWM signal over a single period

The steady state region shown in figure 7 is usually characterized by two "figures of merit". The first "figure of merit" is the mean voltage, $V_m$, of the steady state response and it is given by the equation

$$\begin{eqnarray*} V_{\rm m} = \frac{V_1+V_0}{2} \end{eqnarray*}$$

where $V_1$ and $V_0$ are the maximum and minimum voltages over the steady state region, respectively. The other "figure of merit" is the ripple. The ripple measures the peak variation in the steady state region and it is given by the equation

$$\begin{eqnarray*} V_{\rm r} = \frac{V_1-V_0}{2} \end{eqnarray*}$$

We often specify the ripple as a percentage in which $V_{\rm r}$ is normalized by the steady state voltage $V_{\rm m}$. For example if $V_{\rm m} = 10$ volts and $V_{\rm r} = 1$ volt, then the ripple would be $10 \%$.

In this lab you will be using the output of the RC network as the analog voltage generated by a digital-to-analog converter (DAC). As you can see in figure 7, this analog voltage is not really constant, it has a mean value and a small ripple. So the performance of the RC-DAC can be characterized by these two figures of merit. If our DAC performs well, then its mean voltage $V_{\rm m}$ must vary in a linear manner with the commanded voltage and its ripple, $V_r$, should be very very small. In return for accepting a small ripple, we gain some important benefits. In the first place the DAC only needs to use a single output line and the precision of the DAC increases significantly (from 3 to 6 bits).

The reason for the "increased" precision is that we are no longer using the output lines to encode the digital number we want to convert. Instead, we are using a time-varying signal (the PWM signal) to encode the voltage we wish to convert. We don't get something for nothing. In return for this enhanced DAC, we must settle for a small ripple on the converted voltage and our DAC's response time to changes in the requested voltage will be governed by our circuit's RC time constant.