>> In this video, we're going to show how phasors can be used to perform
a certain subset of trigonometric calculations.
The phasors don't always lend themselves,
we can't always replace trig with phasors.
But under certain circumstances we can,
and it turns out that those circumstances are very common
in dealing with sinusoidal steady state circuits.
So I have defined two different voltages,
V1 is equal to 5 times the cosine of 300t plus 50 degrees.
A second one V2, three cosine of 300t plus 25 degrees.
The first thing we note is that the frequencies of
the sinusoids have got to be the same in order for
the phasor approach to work.
But given in, that in RLC circuits,
driven by a single frequency,
all the voltages and currents in
that circuit will be oscillating at that same frequency.
So this limitation that the frequencies have to be the same,
isn't a big deal for us in what we're going to be doing.
Each of these is oscillating at different frequencies,
at the same frequency but with different phase offsets.
What I wanted to do here, is show you how we can
use phasors to add and subtract,
and multiply and divide these trigonometric functions.
To do so, we're going to be using the phasor representations of these.
So I've gone ahead and define those first, define those here.
The phasor representation of V1 is in polar form,
is equal to 5e to the j50,
five is the amplitude,
50 is the angle.
To get it to rectangular coordinates,
it's just 5 times the cosine of 50 plus 5 times j sine of 50.
Or 3.21 plus j3.82.
Similarly, the phasor representation of this time function.
The amplitude is three,
the phase angle is 25,
so we have 3e to the j25 in its polar form,
or in rectangular coordinates,
this phasor would be 2.72 plus j1.27.
So phasors are the complex representations of the trigonometric functions.
Phasors can be written in either their polar form,
or the rectangular form.
It turns out that when we're adding two sinusoids together,
it's easier because we're going to be adding their phasors.
It's easier to add with rectangular coordinates.
Because when you're adding, you're adding two complex numbers,
it's just the real part plus the real part,
and the imaginary part plus imaginary part.
So then we can say phasor V1 plus phasor V2,
we'll add the phasors and then switch the result back to the time domain.
Phasor V1 plus phasor V2 is equal to 3.21 plus j3.83.
That's phasor V1 in rectangular coordinates,
plus phasor V2 in rectangular coordinates is 2.72 plus j1.27.
You add those up and you
get 5.93 plus j5.10.
Now, let's just convert those back to
polar coordinates because in polar coordinates,
it is easiest to see what the magnitude and
the phasors that we use then to rewrite it in its time domain form.
So in polar coordinates,
this is then equal to 7.82e
to the j40.67 degrees.
I'd suggest that you stop the video now.
Just go ahead make that conversion from
the rectangle coordinates back to this.
Now, once we've got it in this form,
it's very easy to write the time-domain version.
V1 of t plus V2 of t is equal to,
the amplitude will be 7.82 times the cosine.
It's going to be oscillating at the same frequency,
300t plus 40.67 degrees.
I'll leave it to you to attempt to add these two sinusoids in the time domain.
It involves a number of trigonometric identities.
It's not undoable, but it is computationally intensive.
So when we're adding,
we add the phasors in rectangular coordinates.
Now, let's look at multiplying.
We want to multiply V1 of t times V2 of
t. We're going to do that using the phasors.
So we'll multiply phasor V1 times phasor V2,
which is 5e to the j50 times 3e to the j25.
Well, when multiplying things like this is just the coefficients,
5 times 3 is 15.
Then we add the exponents e to the J75,
that in it's already in its polar form,
we can then see that the product of
V1 times V2 is going to have an amplitude of 15.
So touching this back to V1 of t times V2 of t,
that then is equal to 15 Cosine of
300 t plus the angle of 75 degrees.
You'll notice that in both these examples,
both of the time domain functions were in terms of cosines.
When we're using phasors,
because we're talking about the real part,
or using the real part of the complex exponential,
the real part is related to the cosine.
So underlying all this is the assumption that
the time domain functions that we're going to be representing as phasors,
are going to be written in their cosine form.
Here I've got V3, it's written in its sine form,
so before we can use it or convert it to phasors,
we need to translate this,
the sine term into a cosine term.
To do that, we're going to just look at,
and remind ourselves the relationship between the sine and cosine.
The cosine waveform looks something like this,
and the sine waveform looks something like this.
If it was drawn better you'd be able to see,
but hopefully this is good enough to observe that the sine of theta here,
is simply the cosine wave form,
delayed by 90 degrees.
Or the sine of theta,
is equal to the cosine of theta minus 90 degrees.
So to change this or to transform this into a cosine term,
we can simply say then that V3 of t is equal to 5 times the cosine
of 300t minus 10, minus 90 degrees.
We take the original sine term,
delay it by 90 degrees,
and that gives us the cosine term.
We can now convert it to phasor V3,
phasor V3 would equal 5e to the j -10 -90 is -100.
Its polar form would then be 5e to the minus j100.
In rectangular coordinates, that then would be just to, for completeness.
That would be negative 0.868 plus j4.92.