[BLANK_AUDIO] In the last lecture, we found that the
quantities T over 1 plus T, and 1 over 1 plus T, are important functions that we
need to, to find, to compute the closed-loop
transfer functions. In this lecture we're going to talk about
how to, to construct those quantities in a
very easy and simple way with no algebra, but rather
by using the algebra on the graph, or
graphical construction approach. Which is really a generalization of a lot
of these graphical construction techniques we
talked about in the last chapter. So, I'm going to illustrate this with,
respect to an example. And so here is a loop gain, T of s. And just an arbitrary one that I've
chosen. It has a DC gain, T 0, which here is some fairly large value, 50-something
dB. It's got two poles with a Q factor. Then there's a 0 and another pole, and so
it looks like this. This is a fairly complex function, with a
fair number of poles and 0's. And so if you try to add 1 to it, and then
refactor and compute 1 over 1 plus T, or T over 1
plus T, it's a pretty tedious chore. You'll get cubic type denominator
polynomials that will be hard to factor. And so it's, it's not a very good
candidate for, for algebra. And here's the approach.
To approximate 1 over 1 plus T, or T over 1 plus T, we use some of the same tricks
for approximations that we used in Chapter Eight. Namely, whenever we have a sum of terms,
we approximate this using the largest. So, here in the denominator, we have 1
plus T. If you look back at this typical T, I've identified what's called a crossover
frequency here. Where T passes through 0 dB. And so above the crossover frequency, the magnitude of T is less than 1.
And in fact, the farther up in frequency you go, the smaller T gets, and so it
quickly becomes much less than 1. Likewise, at low frequencies, the
magnitude of T is bigger than 1. And if you get very far down in frequency
below the crossover frequency, this magnitude gets very large.
What 60 dB is a gain of 1000, 40 is a gain of 100, so we're talking two
orders of magnitude or more. So the quantity T over 1 plus T. We can approximate then, for low
frequency, where T is large in magnitude, we can
approximate the denominator, by ignoring the 1 and just
taking the T. And in that case we get T over T, which is
1. On the other hand, at high frequency where
T is small at magnitude, we'll approximate the denominator by
throwing out the T and just use the 1. And so, T over 1 then leads to T. So, this is what we get at high frequency
or above the crossover. Similar process for 1 over 1 plus T. At low frequency where T is large, we
throw out the 1 and we get 1 over T. Or at high frequency, we have the opposite
case, so we throw out the T and we get 1 over 1, which is
just 1. Okay? These are very easy to construct.
Here I've drawn the asymptotes of T again. And to find T over 1 plus T, we have in our two cases bounded by the crossover frequency at low frequency where
T is large. This just becomes 1, and at high frequency
where T is small becomes T. So here's our low frequency asymptote. And here are our high frequency
asymptotes. So you can see that compared to T, the
quantity T over 1 plus T, doesn't have the low frequency poles and 0's.
These go away. And in fact, what happens is they get
moved up in frequency effectively. and we get instead a pull, or perhaps more
things, at the crossover frequency. Okay, the quantity T over 1 plus T is
important in the transfer function from the
reference to the output. And what we found previously was that the
output, over the reference, transfer function was 1 over H times T
over 1 plus T. What we would like is for this to simply
be 1 over H. So that the output follows the reference
with a simple gain of 1 over H. So that's, indeed, what happens, where T
over 1 plus T is 1. So here, v over v ref, is just approximately 1 over H.
But at high frequency that's not the case. So really, within the bandwidth of the
feedback loop which means below the crossover frequency, we we
have large loop gain. Then, the feedback loop works well, and
makes the output follow the reference. But at higher frequencies, the feedback
loop runs out of gain. It no longer works. And then, what actually happens, is, the gain reduces back to the,
the, open loop case. So, in this case, where T over 1 plus T
goes to T. The overall transfer function, v over v
ref, becomes T over H. Which is in fact the gains in the forward
path of the loop. For the same example, let's construct now
1 over 1 plus T. This is the quantity that multiplies the transfer functions from disturbances to
the output. Here's our same T again, and to construct
1 over 1 plus T, we again have two cases. Above the crossover frequency where T is
small, this quantity just goes to 1. And below the crossover frequency where T
is large, this quantity goes to 1 over T. Okay, so, 1 is easy enough to draw.
It's up here at high frequency. And then at low frequency, 1 over T is
found by finding the mirror image, or flipping
this T upside down. Basically, we reflect T about zero dB
axis. The 0's of T become poles and 1 over T. The poles at T become 0's, and 1 over T,
and this DC gain becomes a DC gain of 1 over the DC gain, or in
decibels minus the, the dB. So then here is what our 1 over 1 plus T
quantity looks like. What is the interpretation of this? Well, a transfer function from a
disturbance to output, say from vg to the output, is equal to the original open loop transfer function, Gvg, divided by 1 plus
T. So, we take the original transfer function
with no feedback loop and we multiply it by this
quantity. So, what happens is that, first of all, at
high frequency where we don't have much loop gain; the feedback loop doesn't change the disturbance
transfer function. So, it has no effect and you get the same
thing that you get in the open loop case. So, that's what happens here. But at low frequency, where we have a lot
of loop gain, then our transfer function from the disturbance to the
output is reduced by the amount of the loop gain. So, v over vg then becomes the open gain divided by T.
So if we have a, a loop gain, say of 60 db at
some frequency, then this transfer function
from vg to the output is reduced by 60 dB by the feedback loop.
Which is, therefore, will have minus 60 dB of
gain, relative to the open loop case, or 1/1000th of the
output variation that we would have if there was
no loop. So, this is a measure then of how much
loop gain we need to make the loop reduce the
disturbances. We'll do several examples next week using
these criteria. So, we will construct these T over 1, plus
T and 1 over 1, plus T asymptotes. And we will use them, then, to shape our
loop gain to design the system to meet requirements regarding rejection
of disturbances, such as load current variations and vg
variations. One last short note about terminology
here. Sometimes students are confused by this. We, we call things open-loop or
closed-loop. So when we talk about an open-loop
transfer function, what we mean is the transfer function of the original power stage before introducing a feedback
loop. On the other hand, the closed-loop
transfer functions, are what we get with the feedback loop
present. So, they're the transfer functions like
these. And then finally, we have a quantity just
called the loop gain. Neither open-loop gain or closed-loop
gain, but just the loop gain, which is the T of S gain around the
loop.