In the
previous
lecture, we saw how to handle the case of complex poles.
So we can use this standard form for the quadratic case.
And how we approach plotting it, then, is to first work out what
Q is, and if Q is greater than a half, then we have complex poles,
and we can plot the magnitude and
phase responses as described in the previous lecture.
However, in the case where Q is less than a half, then we have real poles, and
what we would still like to do is
factor the denominator into two real poles, like this.
And then plot it, using the rules for real poles.
So, we're going to consider today, or in this lecture, the details and
mechanics of how to factor this and plot its two real poles.
So here's an example, back to our R-L-C network.
Where this was the transfer function, and
this was the example in the previous lecture.
let's factor the 2 roots, and what we find is omega 1 and
omega 2 are given by this expression using the quadratic
formula with a radical and a plus and minus sign.
Now, I think that this expression for
the two corner frequencies is not very illuminating.
It's pretty hard to just look at this expression and
tell how the corner frequencies depend on the different element values.
If you change, say, L, the, what does that do
to the two corner frequencies?
It's actually interesting to know that, in fact, when the key factor is much
less than a half, that the two quarter frequencies can be approximated like this,
where omega 1 is R over L, and omega 2 is 1 over RC.
So, for example, omega 1 doesn't even depend
on C, and omega 2 doesn't depend on L.
But you'd never know that from looking at this formula.
So the simpler in approximate equations are actually more useful.
They gain more insight into how the corner frequencies depend
on the element values, and they're easy to, to solve.
You can see how to change an element value to,
if you want to adjust the value of the frequency.
Which you would never get out of this formula.
Okay, we're going to define omega
1 as the low frequency pole, which is found by taking the the minus
sign and omega 2 is the high frequency pole, found by taking the plus sign.
And what we're going to do is derive a simple approximation to
get expressions like this without having to use the ugly quadratic formula.
So, the way I'm go, going to state it
is if we have, start with this standard function.
The first thing we will do is find Q.
And in the case where Q is less than a half.
Then, we know we have real pulls and we will factor them.
Whereas if Q is greater than a half, then we
will use the quadratic form as in the previous lecture.
So when Q is less than a half, we factor the poles, and the
quadratic formula tells us that these are the expressions then for the two poles.
and you can note that when Q is less than a half then 1
minus 4Q squared is positive, so we get a real result for the roots.
Okay, let's look at whether we can
approximate these functions in a much simpler way.
So I'm going to look first at omega 2, the high frequency pole.
So here, here was the expression we had for omega 2, and
it has a term out in front that is omega 0 over Q,
and then it has this more complicated term that has the radical
in it, and I'm going to call this more complicated term f of Q.
if we look at what f of Q does as a function of Q, first
of all, when Q is small, so much less then a half, the 4Q squared
is much less then 1.
And what we can do to approximate f of Q is just throw out the 4Q
squared term and we get 1 plus root 1 over 2 which is just 1.
And so, for low Q, F of Q actually goes to 1, which is what's plotted here.
as Q approaches a half, then this function approaches
just 1 over 2, and the, the discriminant goes
to 0.
So the, F of Q goes to a half, at Q of a half.
And in fact, it changes very quickly for Q in the vicinity of a half.
if you, in fact, look at the function for Q less than
0.3, f of Q is between 0.9 and 1. So if we
just called f of Q equal to 1, we'd be within 10% of the right answer for
Q less than 0.3, and it's only this last little part here
for Q between 0.3 and 0.5, where we, f of Q changes much at all.
So the low-Q approximation says, then, for omega
2, let's just take the f of Q function and call it equal to 1, and approximate
omega 2 as being omega 0 over Q, this part of the expression.
Okay, how about the omega 1 term? Here's omega 1, it has the minus sign.
So if you try the same thing, say, okay, let's let
four Q squared be much less than 1 and ignore it.
What you get then is 1 minus 1 over 2, or 0.
So then we are approximating omega 1 as 0. Which is not a very useful approximation.
So I need to do a little more work.
And here's a trick, a way to do it.
What we do is we, you can think of this as rationalizing the numerator.
We're going to multiply by 1 plus the radical.
In the numerator and the denominator.
And if you multiply this term by this
term, it removes the radical from the numerator.
And you can do the algebra.
I'll let you do that on your own.
And what you can show then is that omega one is given by Q omega 0 over F of
Q where F of Q is the same function we defined on the last slide.
Which we've already plotted. So for omega 1 then, we get Q times omega
0 divided by the f of Q.
And again, f of Q is close to 1, particularly when Q is less than 0.3.
So the low Q approximation, then, approximates the low
frequency pull as omega 1 is Q omega 0.
So then here's a summary.
For the two-pole case when we have low-Q,
the two poles split into a low frequency pole,
omega 1, and a high-frequency pole, omega 2, and they're in fact spaced
evenly about this resonant frequency, f 0. It's no longer resonant.
but so, f1 is f0 times Q, where Q is a small number, and f2 is f0 divided by Q.
If we increase Q, the two poles move together towards
f0 and then in fact, when Q is greater than a half, they turn into a resonant
response with a resonance that f0. So, what we do then is we calculate Q.
If we find Q is less than a half then we can use
the low Q approximation and quickly get the real poles from these expressions.
Back to our R-L-C example. Here was
our original transfer function, and here are the expressions that we had for
f0 and Q. So f1 we said would be Q times
f0. So Q is R root C
over L, and f0 is 1 over 2 pi root LC.
So the C's cancel, and what we get is R over 2 pi L.
It's a nice simple expression.
For f2, we said that was f0 divided by Q, with
f0 again is 1 over 2 pi with LC, and 1 over Q is
1 over R times root L over C. So let's see, the root L's
in this case is cancel, and we get 1 over 2 pi R times C.
So again, nice, simple expression.
So the low-Q approximation gives us a nice way to keep control
of the algebra and get simpler expressions that are easy to use and design.