Welcome back to our course on Linear Circuits.
At this point, you've finished the first three modules.
So now, we've covered most of the basic circuit elements that we'll be looking at
and learned a little bit about some analysis techniques.
Now, we're going to start moving on to some more advanced topics.
This module's going to be covering frequency analysis.
This was mentioned before. We will start doing our math a little bit
differently, but a lot of the material that was covered in the first three
modules will turn out to be very useful going forward and we'll see how it can
continue to be applied in the new material.
This particular lesson talks about sinusoids and phasors.
And so, we will be reviewing the properties of sinusoids We'll be
describing phasors and what they are, as well as how to use them.
This particular lesson uses a fair bit of information from complex numbers.
So if you don't remember complex numbers, it might be good to go and review.
or if, as you're going through the lesson, if there's something you don't
quite understand, go back and review that and you should be able to, to follow
along pretty easily. In this module, we'll start with
sinusoids and phasors and understanding those topics will allow us to start
talking about impedance which is somewhat analogous to resistance but for circuits
that has sinusoidal inputs. And then, we'll look at doing circuit
analysis in AC systems, so systems with this Is a frequency, the sinusoidal
frequency behaviors. The objectives for this lesson are to
identify the properties of sinusoids like amplitude, frequency period, and so on.
Describe the properties of sinusoids for capacitors and inductors.
Find phasors of sinusoidal functions and then seeing how we can use phasors To add
two sinusoid functions today. First well looking at the sinusoidal
curve. Lets go through some definitions and some
important things that we should note. First of all,we have a function v of t
which is equal to Vm cosine of omega t plus theta.
So this Vm is our peak value and it identifies the highest passive point as
well as the lowest negative point of this centered sinusoid.
We also have this theta which is a phase angle and we'll kind of see how that
relates to the time or this plot here soon.
This T here is our period. And basically the period is just the
amount of time between two corresponding points on our sinusoid.
So, in this case, I picked the two maximal points.
You could use the two minimal points or even if you look at these two points at
the zero axis that correspond to the function going down.
Any two points that correspond to one another.
You mention, you measured the distance and time between them and that gives you
your period. And from that you can calculate your
frequency where the frequency is equal to 1 over that value.
Frequency could also be thought of as the number of cycles, number of times you go
through one sin-, one complete sinusoid. Every second, measured in Hertz, where 1
hertz is equal to 1 over 1 second. Now, all of these things are kind of just
looking at the function itself. We actually have an angular frequency
where we kind of think of this, distance, as corresponding to 2 pi radians.
Or 360 degrees. And so the angular frequency tells us how
many degrees, or how many radians we go through every second.
And so we can calculate that by just multiplying our frequency f by 2 pi to
get radians per second. You could also do this in, in degrees per
second, I suppose. But typically, it'll be radians per
second, another phase angle. Phase angle is a little hard to wrap your
head around, but if you look at this function here, you can see that if were a
real cosign function, basic cosign function it would have maximal value for
time t equals 0. But it's been shifted to the right by
some small amount. The phase angle measures how much of a
phase, how many degrees it has been shifted over.
So we can calculate that as negative 2 pi times tp over t, where that t is the
period, and tp is the point where that first p occurs, and so, that's how we
find a phase angle. Our phase angles are going to be negative
if we're shifting to the right and it will be positive if we're shifting to the
left. So, let's see how sinusoids behave if we
put them through a real element. So, we'll use a capacitor.
If our input is for example voltage going through this capacitor and we know that
the relationship between voltage and current in the capacitors i equals Cc dv
dt. So if I have this sinusoidal input V of
C, I can calculate my i of c by taking the derivative and scaling it by C, which
gives us this. So let's think about what's kind of going
on. As my current goes up, that corresponds
to putting more charge onto these plates. But anywhere my current is positive, I'm
putting charge onto my capacitive plates, which means that my voltage is
increasing. But at a certain point, when my current
starts to go negative, now, I'm removing the charge from my plates.
So now, my voltage is going to drop. So what do we observe here?
Well, we observe that my current is going to go up before my voltage does, which is
to say that we're, well, we're going to say that the current leads the voltage.
So there's a little bit of phase shift between these two curves.
We also notice that the voltage reach, reaches its peak value where the current
reaches its 0 value. Well, that makes sense because of this
derivative property, the slope of this l curve right here is going to be 0.
So the currents going to be 0. So it's kind of the behavior that we're
going to observe when we're working with sinusoids in capacitors and it's also
important notice this omega here. As my frequency increases, my current
increases, so what does that means? Well, that means if I make this frequency
really really fast. If I want to get that to the same voltage
values, I need bigger currents. Because I have less time to put charge
onto the plates. And so, as we increase our frequency, the
relative size of this current is going to be smaller.
Or sorry, the relative size of the current's going to be larger, with
respect to the voltage as the dot omega increases.
Now, we're going to talk about this thing called phasors.
A phasor is where we take a sinusoid and we re-encode it as a complex number, and
that allows us to do some math in a very simple way.
As opposed to having to use all of the sinusoidal functions and a bunch of
trigonometric identities. We can simply just do complex number
arithmetic, and we can do some really neat things.
So, in this example, we have this cosine function here, described by the function
Vm cosine 2000 pi minus pi 3rds. So here we see the frequency is 1000
hertz. So our angular frequency this would
actually be 2000 pi t. [UNKNOWN] the angular frequency here is
2000 pi. We also notice that there's been a phase
shift that's going to be negative because we've shifted it to the right.
And how much have we shifted it? Well, it's pi thirds radians.
What I'm going to do is, I'm going to take this.
And I'm going to take its amplitude. Put it right here.
Then take its phase. Which is minus pi thirds.
And put it here. And this becomes a complex number.
Where Vm is the amplitude, and minus pi 3rds is the phase, so if we plot that on
a complex plane, it looks like this. We could also describe this using
rectangular coordinates. so here, because this is 60 degrees, this
distance is going to be half of Vm, and this distance is going to be minus the
square root of 3 over 2vm. We should also notice that here there's a
j in the rectangular, so 1 half Vm minus j squared 3 over 2.
Here, j is just equal to the square root of negative 1.
We're going to use the square root of negative 1 as j instead of i, which you
might be more familiar with because we're already using i is current.
And it turns out that in a lot of engineering, people use J instead of I.
So, just don't let that startle you. Anytime you see a j, just think it's an
i. Now, it's pretty simple to go from here,
and if we wanted to go back, the same thing applies.
We find the amplitude of this as well as the phase here, and then you just go
back. Vm cosine of whatever your frequency is,
and then your phase. Notice that the phasor has an implied
frequency. There is a frequency there.
But nowhere here in this complex number do we say what the frequency is.
But it's very important because if the frequencies don't match, you can't do the
basic arithmetic. Now let's see how these can be useful.
Suppose that I wanted to add 7 cosine of 120 pi t plus 30 degrees plus 3 cosine
120 pi t minus 60 degrees. You wanted to add these two functions
together, you'd probably have to use a lot of trigonometric identities trying to
combine these can be a real pain. But it turns out that you can do it very
simply by using phasors. So as we already saw going from this time
function tho this phasor is simply take in this amplitude and this phase angle,
and going like this. The same thing for V2, which gives us
these two vectors. Again, in our complex plane.
If I want to add these two vectors together, it's kind of the same thing as
concatenating this vector on the end of this one or this one on the end of this
one. So we're going to get some vector that
kind of goes out like this. It turns out that when you want to add
phasors together, add complex numbers together, it's easier to put them into
rectangular format, and then convert back to the phasor format.
So just quickly so see how you would do that.
for 7 arc 30, because this is a 30 60 90 triangle.
V1 is just going to be so the square root of 3 divided by 2 times 7 plus j times 1
half, times 7. And if you're kind of confused as where
this come, came from, just go back to your trigonometry.
And look at 30, 60, 90 triangles and it'll be pretty simple to see where it
comes from. So it's just a 30, 60, 90 triangle.
So you just convert them into rectangular.
You add them together. And then you convert them back into this
polar form, kind of this amplitude and face type of a thing.
So let's see what we get. Adding them together gives us a phases
format of 7.62 arc 6.8, right here. And since I already have this phasor
format, all I have to do is take the amplitude pop it down there.
The angular frequency here is going to be the same as my 2 inputs.
And this is one of the reasons that phasors have this implied frequency and
is so important. So that just goes back where it was, and
then, our phase comes back to 6.8. So we see that we've been able to add
these two cosines functions together which would be really difficult to do
using our normal functions. We put them into phasors and math is a
lot easier to do. Couple of comments about phasors.
First of all, you cannot compare phasors with different frequencies.
They just don't mesh together. and there's a lot of ways you can go and
practice that, try taking two functions together, you can try and add them as
phases and see what you get. Also, you cannot multiply phasors
together and expect it to give you the same thing as multiplying the two
functions together. For example, if I have the cosine of x
and I wanted to multiply by x, it's going to be cosine squared of x.
And going to trigonometric identity turns out that that's 1 half plus 1 half of the
cosine of 2x. But if you did this in phasor format,
this would be 1 arc 0 degrees times 1 arc 0 degrees and those that are just 1 times
1 complex numbers are just 1. So the result would be 1 over 0 degrees,
which would say that multiplying these two things together just gives you the
cosine of x. So clearly, they're not the same.
But doing this multiplication of phasors has its own meaning.
You just can't kind of interpret it that way.
And especially, taking the ratios of phasors will be very useful.
And so, we'll see in the next section where that's used for analysis, and ways
that that's actually applied. So to summarize, we reviewed sinusoidal
properties, and identified the sinusoid behavior in linear devices and found
phasors and used them to add sinusoids together, which is much easier than just
adding the functions together themselves. There's probably going to be some
questions about this material, it's a lot of things to cover very quickly.
So if there are any questions that you have go to forums, post in there, we'll
be sure to try and make sure everything is clear to you.
In our next lesson we will be defining a, this thing called impedance, which is a
property that describes how currents and voltages relate to each other when we
have sinusoidal systems. So, until then.