0:58
Now, after we know the current, we can then find the voltage across this
capacitor as a function of frequency. Now, we're going to use our phasor
notation and write Kirchhoff's voltage law here, just like we did before.
And we'll start at the ground and then go up through the AC source of voltage.
So, you pick up a voltage V, and then when we pass through the resistor, we
drop a voltage equal to I times R. And then when we go through the
capacitor, there's a voltage drop, which is the impedance of the capacitor times
I. So, we're treating the impedance here
just like resistance. And so instead of this equation being,
having derivatives in it like it did before, actually the derivative was over
here. instead of having derivatives, these are
just algebraic equations now. Now that is the great simplification in
AC circuit analysis, is that these differential equations where we had the a
function, like the current and derivatives of the current, and voltage
and derivative of the voltage in one equation.
Instead of having those differential equations in the frequency domain here.
We have simple algebraic equation, and we can just solve these using the rules of
algebra, but we have to keep the track of the complex numbers.
2:51
And then the voltage across the capacitor is the impedance of the capacitor times
the current. Now, this current here, this is a current
going through both elements. It's the same current going through R and
C. If I want the voltage across the C, I
just multiply this current times 1 over j omega, C.
So I plug this in here and I get this expression, and now I'm going to take and
just multiply this j omega C through the denominator.
And so it cancels out this term and I get a 1, and then I get j omega C times R.
So here's the voltage across the capacitor, is equal to this factor times
the voltage driving the circuit. Now, what I want to do is compute the
magnitude of VC. So, we introduced back when we were
talking about complex numbers, we introduced the magnitude of a complex
number. So, I don't care about the phase, I just
want to know the magnitude of the voltage.
And so, the magnitude of VC over V is just the magnitude of this factor here.
And I just rewrote it as one plus j omega CR.
So, I can write this magnitude of 1 over 1 plus j omega CR is, the factor times
its complex conjugate. And I get the complex conjugate just by
changing the sign in front of the j, then I take the square root of that entire
expression. And so, if I multiply these two, then I
get 1 plus omega squared, C squared, R squared.
And the cross terms, here's plus j omega CR, and this is minus J omega CR, they
cancel. So here is the expression for the
voltage, magnitude of the voltage across the capacitor divided by the voltage
driving the circuit. So if I plot that, VC over V.
And now we're plotting this as a function of frequency.
So remember it's very, it's very important to make a distinction between
this and when we did the transient analysis of the circuit, we plotted the
response of the circuit as a function of time.
Now, we're plotting the response of the circuit as a function of frequency.
Now, I'm taking and there's I'm expressing the frequency as the frequency
divided by 1 over RC. Now RC, R times C is the so called time
constant of the circuit, it has units of time.
So if I multiply a frequency times a time, this is dimension less.
so what I'm really doing is I'm taking my frequency and I'm normalizing it by 1
over that time, that characteristic time of the circuit.
Now, this point when omega is 1 over RC, then that makes omega times RC equal to
1. And at that point, this factor, 1 over 1,
becomes 1 over 1 plus 1. So that's 1 over the square root of 2 and
that's 0.707, and that's this point right here.
So this is the frequency when omega equals 1 over RC.
The response of this circuit is down to 1 over the square root of two, times the
response at DC. So this is an example of a low pass
filter. What happens here, and it's much easier
to see here in the frequency domain than it was when we were talking about this in
the time domain. The low frequencies pass through with
very little attenuation. The voltage across the capacitor is
almost equal to the voltage driving this entire circuit.
As I, the frequency goes up, the size of the voltage appearing here across the
capacitor gets smaller. And when omega is 1 over RC, it's 1 over
the square root of 2 times this driving voltage.
when omega is much bigger than 1 over RC, the response goes down considerably.
So high frequencies do not pass through this circuit very efficiently.
Low frequencies are passed through this circuit very efficiently.
And there is a very solid connection between this and the transient analysis
we were talking about. The, at low frequencies, there's plenty
of time for this capacitor to charge up and this voltage to become quite large.
At high frequencies, there isn't enough time for the capacitor to charge up and
for this voltage to become very significant.
And so, high frequencies are attenuated by the circuit, low frequencies pass
through this circuit.
Robert ClarkProvost and Senior Vice President for Research and Professor
Mark BockoDistinguished Professor and Chair
