In this video, we're going to continue 
our discussion of general circuits. 
Circuits that contain resistors, 
capacitors, and inductors. 
And we're going to interpret the results 
we've gained already. 
We're going to, show, that simple 
circuits function like what we call 
filters. 
We'll define those in just a second. 
we'll extend our ideas of the equivalent 
circuits to the case where we have 
impedances and it's very straightforward 
and we'll also use impedances to talk 
about what's called complex power, 
makes it very easy to do power 
calculations when you use complex power. 
Okay. So, big question is what is this 
circuit doing? We have, we can figure out 
its output, but let's review how we got 
that output first and the solar input, 
and this will lead us to an understanding 
of what the circuit is really doing. 
So, what we did is that we assume that 
the source was a sinusoid of some sort. 
We assumed a times a cosine 2 pi f naught 
t and we wrote that as the real part. 
This is the critical thing, the real part 
of a complex exponential, 
because once you assume things are 
complex exponential, 
the calculation of the transfer function 
is very, very easy. 
So, we assume it was a conflict 
exponential. 
We derive a relationship between a 
complex input to the input and the 
complex input to the output by using 
voltage divider. 
Very simple calculations, 
and this, we're going to concentrate on, 
this is the transfer function. 
But in any case, now that we knew what 
the complex amplitude of the output was, 
it was very easy to find out what the 
actual waveform in time is, because we 
simply put it into the real part for a 
minute, 
because we used real part here. 
We're going to use real part calculations 
down there, 
and we discovered that the output was 
given by this rather complicated looking 
expression. 
But we're going to, 
figure out what it's really telling us in 
just a second. 
one critical thing we had to do in order 
to arrive this result, was we had to 
convert this expression for the transfer 
function into polar form. 
And it turns out that's the way we think 
about transfer functions, 
not in Cartesian form, but in polar form. 
So, here it is. 
I've written H(f) in terms of its 
magnitude, which is the quantity here and 
in terms of a phase, which was given by 
this, 
and then, those are the pieces that fit 
in in here. 
A little subtle detail here is that here, 
we assume the frequency is at zero and if 
zero occurs down in the expressions for 
the final output. 
And now, I want to think about the 
transfer function as a general function 
of f, as a function of frequency, 
that's going to be the key insight into 
what the circuit is doing. 
And so, you'll notice the notation of a 
change. 
So, here's that formula for the transfer 
function and the form for this circuit 
and I'm going to plot it. 
So, this is what it looks like. 
We plot the magnitude and phase, usually, 
they're plotted one on top of the other. 
There is the magnitude, there is the 
angle. 
So you can compare what's happening to 
the phase and the magnitude at some given 
frequency. 
Okay. So lets look at this. 
the general rule of exploring these 
transfer functions to look at what 
happens at very low frequencies, as f is 
going to zero, and at very high 
frequencies, as f is going to infinity. 
So, at zero frequency this is an easy 
calculation, 
mideterms, it has a value of one. 
Okay? As f gets very big, when this term 
is a whole lot bigger than that, 
because the square root, the magnitude is 
falling down like 1/f, going down like 
one over frequency. 
And in between, it's a nice smooth curve 
getting between those two extremes, the 
constant and behaving like 1/f. 
The phase, it is simply an arctangent 
function with a minus sign. 
So this is the arctangent function, very 
simple. 
It's heading toward minus pi over two as 
frequency gets big, and it's zero at the 
origin. 
Okay. 
There's something to notice here. 
I'm plotting this as a function of both 
positive frequency and negative 
frequency. 
As we saw in previous videos turns out 
sines and cosines actually consist of two 
complex exponential terms among the + and 
one with a -. 
So, negative frequency certainly exist, 
and this, and you should plot your 
magnitude and phase as a, realize that 
they're both positive and negative 
frequency components. 
However, it turns out there's a pretty 
easy thing to see in this example and 
it's going to happen in general. 
And that is, for the magnitude, it's a 
symmetric function about the origin 
having what we call even symmetry. 
And what that means is that a magnitude 
of H(f) is equal to the magnitude of 
H(f), of (-f), 
that's called even symmetry. And all 
magnitudes of transfer functions of 
circuits and general systems have this 
property. 
So, you can just tell me what the 
positive axis looks like and I can plot 
for you what happens if negative 
frequency very easily. 
It's a mirror image, so it's a very easy 
thing to do. 
The phase has very similar kind of 
symmetries and it has what's called odd 
symmetry, 
and that is that the angle of H(f) is 
equal to the negative of the angle of 
H(-f), and that's always true. 
So, again, if I tell you what happens at 
over the positive frequency axis, it has 
negative symmetry and that the odd 
symmetry, negative symmetry in that, this 
part is going to be the negative of what 
it is over here. 
So again, I gave, I give you the positive 
frequency part for both the magnitude and 
the phase. 
You can fill in what happens at negative 
frequency easily. 
I should mention that this is a, the e, 
the magnitude being even and the phase 
being odd, is a way of talking about 
what's called conjugate symmetry and that 
H(f) is equal to H conjugate of -f. 
Turns out, another way of thinking that 
even in, even symmetry for the magnitude, 
odd symmetry for the phase is that H(f) 
equals H conjugate of -f. 
And I writed that, again, to make it very 
clear. 
Conjugate symmetry means that H(f) is 
equal to the conjugate of what is 
happening at negative frequency. 
So all transfer functions have this 
property, 
so it makes it very easy to do your 
plotting and everything else. 
Well, okay, 
but what is this curve telling us? 
Particularly, the magnitude? So, as 
frequency increases, the, the output 
voltage, V out, will be getting smaller 
and smaller and smaller, so I'm assuming 
is that the input voltage, the amplitude 
of our sign wave is a constant across all 
frequency. 
And as I change the frequency of that 
input, the output, just because of the 
action of the circuit is going to get 
smaller and smaller and smaller. 
Okay? 
That's what we call a lowpass filter. 
And, I'm going to, give you the reason 
for the word filter here in just a 
second, 
for the lowpass I think is pretty 
straightforward. It's passing through, 
letting through, things in the low 
frequency range, and things at higher 
frequency ranges tend to be suppressed, 
and when we see that, how that happens, 
through a very simple example. 
Suppose our input voltage consisted of a 
sum of two sinusoids, f1 and f2, and you 
can see I've allowed them to have general 
amplitudes. 
What's the output going to be when the 
input consists of a sum of two 
sinusoid's? Well, we're going to use the 
principle superposition of course, 
because the circuits are linear. 
So, when you have the input is equal to 
the first sinusoid, we know what the 
output is. 
And the input is equal to just the second 
sinusoid, we know what the output is, and 
then the output input rather consists of 
the sum of those two superposition 
applies you just add those answers up. 
That is the beauty of having a linear 
circuit, 
this is the very definition of being 
linear. 
The output for sum is equal to the sum of 
the outputs. 
Okay. 
So, what's happening here? Let's assume 
that f1 is down here somewhere and lets 
assume that f2 is out here somewhere. 
So, let's assume for the sake of 
argument, that the amplitudes of these 
are about the same. 
Now, what's going to happen on the output 
side is that, the transfer function at f2 
is much smaller than it is at frequency 
f1. 
So, this term, is going to be attenuated. 
Filter is the term. 
It's going to tend to filter out the high 
frequency term. 
So the terminology is that the low 
frequency area, the place where you get 
frequency region over which things get 
through is called the passband, 
the band of frequencies over signals pass 
through. 
And for frequencies that are much higher, 
herein, was called the stopband, 
the circuit, the filter tends to stop 
that from coming out. 
So this again is a low pass, because it's 
in the low frequency regions that get 
passed through and the high frequencies 
tend to be attenuated. 
So, let's get, so what's, high and what's 
low? Well, you can see this curve is very 
gradual and it's not as if there's, 
there's was, a clear delineation between 
the two. 
an ideal low pass would have a frequency 
characteristic that looks like this. 
So, let's say it has a gain of one at low 
frequencies in and in it, instantly jumps 
down, and at some frequency, goes to 
zero. 
That would be what we call an ideal low 
pass. 
Well, the RC filter, we have, certainly 
doesn't have that characteristic, but we 
can think about it by defining what's 
called a cutoff frequency. 
So, 
the cutoff frequency is generally defined 
with frequency at which the magnitude at 
that frequency is one over the square 
root of two times the maximum value of 
the magnitude, wherever that occurred in 
the frequency. 
So for a lowpass, this maximum is 
occuring and the orgin has a maximum of 
one. 
And you change frequency till it's, the 
output is equal to one over the square 
root of two of the value at the origin, 
which here is one. 
And the, what, why one over the square 
root of two, you might be asking? If you 
think about that for a second, 
what could be so important about the one 
over square root of two? And the answer 
is power, so, at this frequency, because 
power is proportional to signal squared. 
The power of this, of the sinusoid at 
this cutoff frequency, will be 1/2, of 
the value it is at the maximum. 
So this cutoff frequency, the power at 
the, in the output will be 1/2 of what it 
was at the origin. 
If you go to higher frequencies, the 
power output gets less, and less, and 
less, and less at those frequencies. 
So, for our RC lowpass, we need to look 
at the formula to figure out where the 
cutoff frequency is. And it's pretty easy 
in this case, 
because we know at f = 0 the value is 
one. And the frequency at which it's, 
this square root is going to be two is 
when this term equals one. 
And so solving for that, we get that the 
cutoff frequency is equal to one over two 
pi times R times C. 
So, when it comes to designing a circuit, 
suppose you wanted to design a filter 
that would pass yeah, it have a cutoff 
frequency of 1 kilohertz. 
Well, what you do is set this equal to, 
to a thousand, and what you'd find is 
that the only thing that you need is for 
the product of R and C to be over two pi. 
So the important point here is that it's 
the product of the component names. 
We've seen this before. 
You have, the only thing you need is the 
product to have certain property, so you 
can choose the resistor and capacitor 
separately according to whatever criteria 
you want. 
But the, the product of the two is what 
determines the quality frequency, and so 
not the individual component you use. 
It gives you some latitude in doing the 
design. 
Okay. 
So it turns out there are more kind of 
filters out there of lowpass. 
So the lowpass we've talked about. 
So, and generically, I'm only going to 
draw the positive frequency axis, because 
we know its going to happen through the 
magnitude. 
generally look something like that. 
So generic lowpass which has a cutoff 
frequency somewhere there, 
where this value here is one over square 
root of two times whatever the value is 
at the, at the, at zero frequency. 
You can also have a pass go through. 
Highpass, what it looks like is pretty 
easy to see where the name comes from. It 
tends to suppress low frequencies, let 
through the higher frequencies and its 
cutoff frequency will be similar here. 
And that value is one over the square 
root of two times whatever the value is 
at infinite frequency. 
And then finally there's the bandpass 
filter which, what this means is that it 
passes a range of frequencies and 
suppresses the other ones. 
So, if you plotted it, it tends to look 
something like that and this means it has 
two cutoff frequencies, 
a lower cutoff frequency and an upper 
cutoff frequency. 
We refer to the width, 
the distance between the two cutoff 
frequencies as being the bandwidth of the 
filter. 
And that's an important parameter when 
you start talking about band bass 
filters. 
sometimes we refer to the upper and lower 
cutoff frequencies. 
Other times, we might refer to the center 
of the band, and the bandwith specifying 
the bandpass forward whichever way seems 
most convenient. 
We're going to discover, as we go through 
this course, that all these folders have 
very important uses and occur very 
frequently in real world application. 
Okay, I want to explore what really 
happens when you build a RC lowpass when, 
because you can see, I want to use 
specific elements that I use for, I 
really constructed this circuit. 
The input is going to be a sine wave. 
And we know that, from all we've done 
that the output for a general filter is 
going to be the transfer function, 
magnitude of the transfer function times 
the sine plus the sine wave with a given, 
that has a phase shift given by the volt. 
And we have calculated this for the RC 
lowpass several times, and we know that 
what it's going to do, is the magnitude 
is given by this expression in the phase, 
is given by this. 
Okay. 
So, first thing we want to do is 
calculate exactly where the cutoff 
frequency is going to be. 
To do that, we need to find out where the 
magnitude is it's maximum value and it's 
pretty clear from this expression that 
the maximum value occurs at a frequency 
of zero, 
this is after all a lowpass filter. 
And then, we want to see where the 
magnitude is one over the square root of 
two times that maximum value, which in 
this case is one. 
So, it's pretty clear that it's going to 
be equal to one over the square root of 
two and this expression equals one, 
because we had the square root of 1 + 1, 
will give us what we want. 
So, we've calculated this several times 
and we know that the answer is with the 
cutoff frequencies given by one over two 
pi RC. 
And when you stick in the explicit 
values, for our circuit element values, 
we have 10^4 for the resistor, and this 
is 220 nanofarads. 
Like I said, you don't usually have very 
large value capacitors in circuits, so, 
and we get 72.34 Hertz. 
We'll have to see if that really works, 
really happens when you look at the out, 
output of the circuit that has these 
circuit elements. 
When you plot this, some things that I 
want to note before we go to the 
simulation we predict the cutoff 
frequency is going to be 72.34 which is a 
frequency somewhere like this. 
Well, I want to point out something, it's 
a special thing for this RC lowpass. 
Where is the phase, what is the phase at 
the cut off frequency? 
Well, then, 
the fc in the frequency is equal to one 
pi RC. 
This expression is one, the arctangent of 
one is pi over four, 35 degrees, so the 
phase will be - 45 at that point. 
Okay. 
So here is what we should expect. 
When you have frequencies less than fc, 
the amplitude of the output should be 
going up until it gets to very low 
frequencies, 
it should be about the same as the input. 
The phase shift will become smaller and 
smaller and smaller, 
and then again, as you go toward zero in 
frequency, the output in terms of its 
phase shift will also look like the 
input. 
On the other hand, if you go way beyond 
the cutoff frequency, the amplitude of 
the output will be going down and down 
and down and down and the phase shift is 
headed toward minus 90 degrees, minus pi 
over two. 
So, what is sine of two pi ft minus pi 
over two? What is that equal? And what I 
get is that that is minus cosine. 
Okay. So that's what we do, should 
expect. 
As the frequency gets very high, 
amptitude gets smaller and smaller and 
smaller, and the phase is going, the 
output is going to look more and more 
like a minus cosine when all is said and 
done. 
Well, let's see if that really happens. 
So here is my input, the frequency is, 
here is the frequency generator, is 
setting up a sine. 
This output number here is the peak to 
peak amplitude of the output, 
while 80 Hertz is a little bit above the 
cutoff frequency, so the amplitude should 
be 1.414. 
If we're at the cutoff frequency, 
it's a little lower than that, because 
we're too high in frequency. 
I've lowered it to a 74 1/2, 
getting closer to 1.414, but it's not 
quite correct. 
I've put in 73.2, it's getting closer. 
And, you, for this oscilloscope, we could 
have typed it in all the time. 
So we point out that the white line is 
the input to our circuit, the red line is 
the output. 
So you can see the amplitude is down by 
some amount and this is telling exactly 
what it is. 
Right? Well, let's explore what happens 
as I go to the lower frequency. 
So I gotta change the what's called the 
time base of my oscilloscope. 
I'm going to make it go down by a factor 
of ten in frequency. 
Right now, it's 7.3 Hertz. 
And as you can see, the red line, the 
output is almost the same as the white 
line, the input to the filter, and it's 
got very little phase shift and that's 
exactly what we should predict. 
Let's go back to 70, 73 Hertz and we can 
see how the amplitude's going down like 
it should. 
Now, I'm going to set it up to go up to a 
very high frequency and you can see the 
output is greatly attenuated. And if you 
look carefully at that waveform, it looks 
to me like it looks like a minus cosine 
to me, so all the theoretical predictions 
are correct. 
Now, what I'm going to do is go back to 
our original setting, 
original frequency, and I'm going to 
variate continuously now, so we can see 
the go up, in frequency, so the amplitude 
will be going down as the frequency goes 
up just as predicted. 
The phase shift keeps increasing, getting 
more negative. 
And then, if we go back the other way, we 
see the amp, amplitude go up and keep 
going down, down, down in frequency, 
below the cutoff frequency now, well 
inside, looks more and more like the 
input. 
Okay. 
So, let's talk about the big picture. 
We're going to make sure we really 
understand what's been happening here. 
We started with a circuit. 
We assumed complex exponential input, 
which meant that every voltage and every 
current in the circuit was also a complex 
exponential. 
Based on that notion, we defined 
impedance and converted the circuit to 
being an equivalent circuit, so far as 
complex amplitudes are concerned so that 
we had impedances representing the 
circuit elements. 
Because of this use, assuming complex 
exponential inputs, we can now use all 
the shortcuts that we developed for 
resistor circuits and this makes the 
calculations quite easy. 
and, I think it'd be pretty obvious that 
the Mayer-Norton and Thevenin equivalents 
follow just as well. 
So, since they were based simply on using 
that series imparallel rules and, and 
even a bit more generally. 
The Thevenin equivalent, when you have 
impedances, is exactly the same, except 
this is now an impedance and this is a 
complex amplitude. So the frequency of 
the source is implicit. 
All we care about in representing that 
circuit with a Thevenin equivalent are 
the equivalent source and the equivalent 
impedance, and the Mayer-Norton has 
exactly the same properties we decide, 
drew out the four. 
The two impedances are the same and 
that's the amplitude of the current 
source. 
And that same formula applies for that we 
derive, with four impedances, 
that the equivalent. 
amplitude, the amplitude is of the 
equivalent circuit, 
obey this relationship and that's the 
equivalent impedance, so there's no real 
surprise there. 
And you can now use these equivalents in 
RLC circuits just like we did for 
resistor circuits. 
Okay. Well, there's something else that 
becomes much very easy to talk about, 
when we have complex exponential or 
sinusoidal sources. So, again I'm 
going to think about all my circuit 
elements as being an impedance with some 
complex amplitude for the voltage and 
complex amplitude for the current coming 
in. 
And now what I'm going to do, going to do 
is a power calculation, so I'm going to 
do this in general. 
I'm going to assume that the complex 
amplitudes has a magnitude and a phase 
for both the current and the voltage. 
Okay? 
Which means that speaking about them back 
in, its time-domain quantities the, here 
is the magnitude, there, 
there is the magnitude for the voltage 
and there is that phase shift. So I'm 
thinking about my v, v(t) as being the 
real part of this. And for i(t) you take 
the real part of, of that. 
There you go. 
Well, we know what power is. 
Power is always voltage times current. 
Well, I could plug in these two formulas 
into here and try to simplify it. 
Turns out there is a much easier way, 
which I'm going to show you, which 
actually brings up a very interesting 
form. 
So, I'm going to do that calculation in a 
slightly different way. What I want to do 
is use Euler's formula. 
So, in addition to, we can explicitly 
write out the real part of the rule, 
formula, as being just Euler's formula. 
We've seen how this works before. 
Now, is what I want to do is I want to 
plug that in. I want to plug these 
expressions in to that product. 
Well, it's very easy to multiply this out 
and it turns that become, it really 
simplifies the calculations a lot. 
So, the way to think about this is that 
the voltage is now a sum, 
the current is now a sum, so when you 
multiply the two together, you get all 
four possible cross-products. 
This times this, this times that, that 
times that, and that times that. 
So if we multiply them out, you can 
notice that some terms, the complex 
exponential part cancels, 
and for other terms, the complex 
exponential adds. 
And when everything is said and done, 
this is the expression that you get. 
Well, this is quite interesting. 
Now, if we look at the, these two for 
example, this is the complex conjugate of 
that. 
Right? Because the conjugate of a product 
is the product of the conjugate. 
V conjugate, V, the conjugate of V is V 
conjugate, and the conjugate of I 
conjugate is just I, we get it back 
again. 
Same thing happens out here. 
This is the conjugate of that. 
So, those are real parts. 
And noice, because we multiply by a 
quarter, we would have to put a half 
factor in to get the real part in the 
final answer. 
So, what does this say about the power 
when you this is the actual power. 
There's nothing complex here at all. 
Those are actual power values, P(t) is 
equal to some number and then you get a 
term that turns out to have a frequency, 
which is ft, twice the frequecy of the 
source. 
So if you plot this, this term is a 
constant, it's a function of time now and 
rippling around it, somehow is the time 
varying part. 
Well, we know that energy is the integral 
of power, so the time varying part really 
doesn't contribute over the long term to 
energy consumption. 
Sometimes, it goes to zero, and this 
thing, and this term goes to, this pulses 
goes to zero and sometimes goes positive. 
The only thing that contributes long term 
is that term, and that's called the 
average power. 
this is a general result for all RLC 
circuits. 
And if the source is a exponential, 
I'm sorry, is a sinusoid, which we think 
of as a complex exponential. 
So, average power that is consumed is 
given by the real part, half the real 
part of the I conjugate, for these are 
the complex amplitudes of the circuit 
element where, where you met the power 
consumption flow. 
So, this is what is known as complex 
power. 
so in general the only thing that's 
really important is the P ave term. 
I'm sorry, is the P average is the real 
part of complex power. 
And, whenever we have an impedance, it 
turns out this formula for complex power 
can be rewritten in terms of impedance. 
So, just plugging in the relationship for 
impedance in the VI conjugate formula, we 
can write it either as z times magnitude 
of the current squared or one over the 
impedance times the magnitude of the 
voltage squared. 
I'm going to stick with this one because 
it is a little bit simpler. 
And let's see what it looks like for all 
the three common circuit elements. 
And certainly, we've seen this before. 
For the resistor, the power is just 1/2 R 
times the magnitude of the current 
squared and there is no imaginary part, 
so the real part of that is just what 
you've got. 
Again, stating its resistors that 
dissipate power. 
For the capacitor though, the impedence 
is complex. 
So, it is imaginary value, right here. 
So that means when you take the real 
part, you get zero. 
Again, this is saying that the capicators 
as well as the inductors do not dissipate 
power. 
And, what they contribute to, is that 
time-varying part, 
that part where the, the power 
consumption goes up and down. 
So, sometimes they consume power, 
sometimes they give it back in certain 
times, 
but in the long term is they, they don't 
consume or produce power when you have a 
sinusoidal input. 
Okay. 
So, in summary, my thinking of each 
element as a complex valued resistor, 
what we call an impedance. We get a 
general picture of what circuits do and 
how they behave. 
We can use to solve those circuits, we 
can use the series of parallel rules 
voltage divider, current divider. We can 
use the equivalent circuits, we can talk 
about transfer functions and filters, 
these are going to be very important 
concepts coming up, transfer functions 
are going to come all the time filtering 
thinking about transfer functions is what 
kind of categorize them according to what 
kind of filter they are. We have to plot 
the transfer function and see how it 
works out. 
Weou can also talk about power using just 
the complex amplitudes, we don't have to 
really go back solve for what the v and i 
are, and then multiply now and get it, we 
have a very general formula just from 
using the complex amplitudes. 
Now, I want, do want to point out, it's 
very important, that we, these results 
only apply when the source is a sinusoid. 
Remember, what we did was assume that the 
source was the real part of something 
that looked like a complex exponential, 
and so, 
all these formulas apply only when we 
have a sinusoidal input. 
When it's a sum of sinusoids, 
superposition applies and we know how to 
solve that. 
Well, you might be wondering, what 
happens if it's not a sinusoid? For 
example, suppose it's a pulse or a square 
wave? There's other signals we've been 
talking about. 
Have we gone down a blind alley? Is this 
going to go anywhere? as you might think, 
this is not the end of the story. 
We can talk about very general source, 
source wave forms other than sinusoids. 
It turns out these are the building 
blocks that we're going to use later in 
the course.