Here is the diagram of a basic control system, feedback loop. that we or that I introduced at the
beginning of this week. so the goal now or for this week has been
to model the elements around this feedback loop that we can use in our upcoming week
to design the control system. So, what we've done so far this week is
derive an AC equivalent circuit for the switching
converter that models its dynamics. So now we have that.
Let's look at the rest of the loop. The feedback connection, summing node and
compensator are generally built with analog circuits such as off amp
circuits. And I'm going to presume that you already
know how to solve those and design them from
other classes. But the thing that's missing is the pulse
width modulator. And so we need to take a few minutes now
and discuss how the pulse width modulator
works, and how to model it. So the pulse width modulator is a circuit
rhat has an analog control signal at its input, we're calling VC, is
some smooth signal. And it's output is the gate drive signal
that switches high and low. when it's high the mosfet is on, when it's
low the mosfet is off. And the duty cycle of this waveform is
supposed to be proportional to the control signal VC. So let's talk about what is that
relationship between the duty cycle and the analog
input VC. Here is a common way to build a pulse
width modulator. We have a saw-tooth wave generator which
is an oscillator, and it produces a signal such
as this one. This is also sometimes called a ramp
generator. so, it makes a periodic signal, whose
period is the switching frequency here. So this is the switching period, t sub s.
And, in fact, this saw-tooth wave generator is what sets the switching
frequency of the switching converter. Here I've drawn the, the saw-tooth
waveform, V saw T as being this function that starts
at zero and it increases linearly with a
constant slope up to some maximum value v sub m. at the to slip into the switching period,
then it jumps back down to zero and starts over
again. The saw-tooth wave is put into an analog
comparitor circuit where it is compared with this control input, the
analog input VC of T. VC of T is this wave form right here. The function of the analog compartor is
that it compares the two analog input signals at it's plus
and minus input terminals, and it produces an output logic
signal that is high when the plus input is greater than
the minus input. And otherwise, it's low.
So, this pwm output then will be high, here until at this point the
saw-tooth and the control input are equal. After that point, the saw-tooth will be
greater than VC and the logic output will switch low for the rest
of the switching period. So the output of the comparator is then this logic signal having a duty cycle of
d. What is the equation between the input and
output of the pulse-width modulator? Well if the saw-tooth waveform is linear,
so it has constant slope, then we can write that the duty cycle is actually linearly proportional to, to the control
input VC. And you can compare or look at just a few
different cases suppose VC is zero then we'll switch at the very beginning and our
duty cycle will be zero and on the other hand if VC is
equal to the peak value of VM, then we won't
switch until the end of the period and our duty cycle
will be one. And with a straight line, or linear
sawtooth, the duty cycle will vary linearly in between
those two extremes. So we can write then, that the duty cycle
is the control input divided by the, the peak amplitude of the ramp.
V sub M. Sometimes commercial saw-tooth or pulse
with modulators have saw-tooth waves that start at some dc bias, or some value
greater than zero. And in that case, we need to subtract that
bias from VC in this formula. So here is that equation again. If you like, we can perturb and linearize
this equation. So we let VC equal capital VC plus VC hat,
but d equal capital D plus d hat. If you plug those into here, we get this
expression. And from these we can see that the quesen
or steady state value capital D is equal to
capital VC over VM this is in fact a practice how the
quiescent duty cycle is set by the DC component of the control
voltage we've seen. And likewise D hat is equal to VC hat over
VM so the block diagram of the pulse with
modulator then is a gain. Equal to one over VM.
and in a small signal sense then, DC hat goes in and D hat comes out.
I should note one other thing; the pulse width modulator effectively
samples, it inherently samples. which means that we have one duty cycle
per switching period. If we go back and look at these wave
forms, the duty cycle is determined only at this, at this
point right here, and there is simply one value of duty cycle here
over a switching period that is proportional to the, the voltage,
VC, sampled at that instant. It doesn't matter what VC does at other
times. So we can have things going on here at, at other times and they won't effect the
duty cycle. Now you might say, well suppose we
actually had variations that did this, and made it switch many
times. That's actually an undesirable thing. It causes a lot of switching loss. It can happen when there is noise in our
control signal, but we do everything we can to eliminate it.
One thing that's often done in commercial pulse width modulator chips is that
there's a latch driven by this comparator. The latch gets set by the saw-tooth wave generator, at
the beginning of the switching period, and it gets reset
by this comparator signal and then the output
of the latch is this signal that goes to the gate
driver. What this does, is it sets the latch here
to turn the fet on, or mosfet on, or whatever our
switch is. At this point, the transistor's turned off, and once it's off, no amount of noise
can turn it back on again. Okay. So we have sampling then, there is one value of duty cycle for every switching
period. Regardless of, of VC, and we have a
sampler built in. sampling is actually beyond the scope of
what we're going to talk about in this course, but what I'll say is that we must restrict
our, the bandwidth of our control system. To be sufficiently less than the nyquist
trait which is half the switching frequency or half
the sampling rate. models of the modulator that don't account
for sampling therefore are not going to be accurate except at
frequencies that are sufficiently less then this nyquist rate. One quick summary slide something sampling
does is it repeats the spectrum of the signal at the switching frequency
in its multiples and so we get something that is known as alias,
aliasing where the original signal hs frequency content at dc and low frequencies that frequency content
gets folded by aliasing up to higher frequencies at
about the switching frequency. And its harmonics. Though we're not predicting that in our
models, and in fact when there is aliasing generally bad things happen to our control
system and so we don't want that to happen. Here is a plot of the effective transfer
function of the zero order hold circuit which is what the
host width modulator behaves as that I've
described so far. and so, it has behavior at the Nyquist
rate and higher and generally what we want to do is restrict our bandwidth to
be sufficiently less than that. And at these lower frequencies the
modulator essentially has a constant gain that is the one we already discussed, the one over V sub
M. So for frequencies sufficiently less than
the Nyquist rate, we can model the pulse-width modulator with a
simple gain block. With a gain of one divided by the peak to
peak amplitude of, of the saw-tooth. And this is the model that we're going to
use then in our control system designs that are coming
up in the future weeks.