The first part of the circuit in figure 34 consists of a resistor, , and capacitor, connected in series. The circuit is shown in figure 35. The circuit is driven by a time-varying voltage and we're interested in the voltage that develops over the capacitor.

We now assume that is a -periodic pwm signal.
Over a
single period, , the input voltage to the circuit
has two distinct parts. There is the "forced" part from
during which the applied voltage is . In this
interval, the capacitor is being charged by the external
voltage source. After this charging phase, the applied
voltage drops to zero and the capacitor is *discharging* through its resistor.

During the "charging" phase, we can think of the RC circuit
as being driven by a step function of magnitude volts.
If we assume that the capacitor has an initial voltage of
at time (the start of the period), then the
circuit's response is simply the total response of the
system. So the capacitor's voltage is given by the
equation

for .

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 . As a result the capacitor's
voltage over the time interval is captured by the
RC circuit's natural response. So the capacitor's voltage
is given by the equation

for . Note that equals the voltage on the capacitor at time (i.e., ) and also note that we've used the term (rather than ) in the exponent. This last substitution was made because equation 3 represents the response of the system starting at time rather than time . As a result we had to shift the time origin from to . The substitution of by accomplishes this time-shift.

The top drawing in figure 36 illustrates the output signal we expect from a PWM driven RC circuit over an interval from . For times beyond this interval, we expect to see the waveform shown in the bottom drawing in figure 36. 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 and volts. The exact value of these steady state voltages is dependent on the period and the duty cycle . What we now want to do is characterize these voltages in terms of and .

To determine and at steady state, we note that
is the capacitor's charge after the discharge phase.
Since we expect the steady state behavior to be
-periodic, we can conclude that
which means that

In a similar way we know that is the cap's voltage after charging up from for seconds. This means that

Equations 4 and 5 are
two nonlinear equations in to unknowns. We now try to
solve these equations. To start we insert equation
4 into equation 5 to
obtain

Equation 6 only has the unknown in it, so we can re-arrange this equation to see that

We now take equation 7 and back
substitute it into equation 4 to obtain

Equations 8 and 7 completely characterize the steady-state "ripple" of the RC circuit in response to a PWM voltage whose period is and whose duty cycle is .

Figure 37 shows two graphs
characterizing the voltage, , over the capacitor as
a function of the duty cycle. In particular, these
graphs assume that volt, k-ohm, F, and
the period milliseconds. We've plotted duty
cycles at to in increments of . The top
graph plots and . Note that these two values
have a symmetry around a steady state value. The steady
state value is simply the average of and .
We've plotted this value
in the
second graph. We've also plotted the ripple
in this graph. The important
thing to note here is that the average value of the
capacitor voltage varies in a **linear** manner with
duty cycle.

Note that the ripple shown in figure 37 is about volts. Is it possible to reduce this. Intuitively we might suppose that if the period is much smaller, then there is less time to discharge to cap and this might result in a smaller ripple. To test this hypothesis, we plot the ripple as a function of PWM period, . In this case, we assume that k-ohm, F, volt, and we fix the duty cycle at 50 percent. The plots are shown in figure 38. In these plots, ranges over two decades from milliseconds to milliseconds. What should be apparent here is that as we decrease the period beneath milliseconds that the ripple in our voltage becomes negligible. This time, of course, is comparable to the RC time constant of seconds for this circuit, which is entirely reasonable.

What should be apparent from the preceding discussion that the RC circuit essentially turns the pwm input signal, , into a constant analog voltage with a small ripple on it. We can make this ripple arbitrarily small through proper selection of the period. Moreover, we can see that the value of is proportional to the pwm's duty cycle.

As we'll see below, it is relatively easy for the MicroStamp11 to generate a pwm signal through the use of output compare interrupts. This means that with the help of the RC circuit in figure 35, we can use the MicroStamp11 to generate an analog voltage that lies between zero and 5 volts. Moreover, we can tune this output voltage by simply tuning the duty cycle of the pwm signal. Unfortunately, there is one small problem with the circuit shown in figure 35. This problem and its solution are discussed in the next subsection.