>> Hi my name's Lee Brinton and I am
an electrical engineering instructor at Salt Lake Community College.
In this video and actually going to turn our attention now to
analyzing circuits that are driven by sinusoidally varying or AC sources.
This is what we're going to refer to as the sinusoidal steady state.
By this, we're going to assume that the system has been
operating that the sinusoidally varying sources have been applied long
enough that any transients that might have occurred at startup have
all died out and we're now operating in the sinusoidal steady state.
To do this and to analyze these kinds of circuits.
We're going to use something known as phasors.
So to begin with we're going to introduce this concept of a phasor.
Phasor is a complex exponential way of representing trigonometric functions.
Will then turn our attention to and
define what we mean by a complex impedance.
Will then analyze or demonstrate ways of analyzing these RLC circuits.
Resistors inductors capacitors are circuits that
have resistors inductors and capacitors in them.
In the sinusoidal steady-state including
looking at series connections and voltage division,
parallel connections and current division.
We will see how node mesh analysis applies in
this sinusoidal steady state involving phasors and impedances.
And then we'll look at the Thevenin equivalency and
maximum power transfer concepts in this sinusoidal steady state.
So here we are again in the complex plane.
Where we have a real axis and an imaginary axis.
And we know that we can represent points in this plane.
Now we're going to be using a complex number to represent a voltage.
So let's now just start out with a complex number.
We will call it V and say that it has a length which we will
call V sub M and it has some angle theta.
So there's theta.
Again the length of this is V sub M. So it's a complex number.
The magnitude of the complex number is V sub M
and the angle associated with it is theta.
Of course that represents the polar representation of
this complex number V. We know that another way of writing
that is to say V sub M E to
the J theta and we also know that we can express it using Euler's formula or
simply pointing out that the projection of this point on
the real axis is equal to V sub M times the cosine of theta.
So we can write this as
V sub M, times the cosine of theta gives us the real part.
And the imaginary part is just V sub M times
the sine of theta or this can be rewritten as
V sub M times cosine theta plus J sine of theta.
Just a little bit of nomenclature here.
We can say than that the real part of V is equal to
V sub M cosine of theta and the imaginary part of
V is equal to V sub M sine theta.
Alright here's where it starts to get interesting.
Up until now theta has been a constant value,
what if we start allowing theta to be
some function of time and in fact let's begin with by letting
theta of T equal some constant omega times
T. Omega is going to have the units of radians per seconds.
So that when we have radians per second times seconds that gives us
radians or measure of an angle.
So sure enough theta is an angle or has the units of
radians or equivalently degrees.
By the way, omega is referred to as the radial frequency term.
So now, let's rewrite our V. V then is equal to V sub M E to the theta,
but theta is equal to omega T. So it will be E to
the J theta or E to the J. Omega T. Which again
using Euler's formula can be written as V sub M times
the cosine of omega T plus J sine of
omega T. Given that the real part of V is the projection of
this point down onto the real axis and given now
that theta is starting at T equals 0 theta equals 0.
As T increases, we now see that
theta is increasing linearly as a function of time.
Which means that the real part of
this complex number as theta increases with time.
The real part of it, which is the projection onto the real axis,
is going to be changing as a function of time also.
The real part of this now becomes
a function of time and if we stop and think about it,
we know that the projection onto the real axis is just the magnitude of
V sub M times the cosine of this angle.
The angle being omega T. We have then that the real part of V,
writing it the real part of V then is equal to
V sub M cosine of
omega T. Hopefully now you'll start to see where we're headed with this.
This V sub M cosine omega T is the way of numerically or mathematically
representing a waveform this oscillating
this cosine is solely at a frequency of omega T seconds.
We can see graphically what's happening.
Let's now overlay a time-domain axis coming down like this.
And let's plot as a function of time the projection on
the real axis of this point as it goes around at omega radians per second.
Just draw a couple of lines down here to help me graph is.
So at T equals 0, theta equals 0,
the cosine of 0 is one and we start out here at a maximum.
Now we allow theta to increase and as we do so the projection on the real axis
decreases until we get to theta equaling pie halves or 90 degrees,
at which time the real projection equals 0.
Continuing on through until theta gets around here two pie or a 180 degrees.
Once again we have a minimum here.
It comes back crosses as theta gets down here to 270 degrees.
Back to where we started.
As Theta goes around and around this graph and continues on and on,
and you'll see, hopefully, you recognize then.
Though what we have here is a plot of the function V_M cosine of Omega T. Now,
this analysis assumed that Theta started at zero radians per second.
Let's now take a look at what happens if we start our angle at not zero,
but at some angle Phi.
In other words, Theta at t equals zero equals Phi.
Now, when we start this projection coming down here,
or first of all,
we then would say that Theta as a function of time and
is equal to Omega T plus Phi.
Our complex expression V is equal to then V_M,
E to the J Theta,
but Theta is now Omega T plus Phi,
which again using Euler's formula,
we can write as V_M times the cosine of Omega T plus Phi,
plus J sine of Omega T plus Phi.
We can save in that the real part of V is equal to
V sub m cosine of Omega T plus Phi,
a completely arbitrary cosine term,
specified by its amplitude V_M.
The rate at which it's oscillating Omega and this arbitrary term,
or arbitrary offset of Phi radians.
Let's plot that projection onto our time domain.
As we did before, only now instead of
starting at T equals zero with Theta equaling zero,
we start at T equals zero at Phi,
and again dropping these down to just help me draw this.
We're starting here at this point.
It goes to zero when Theta gets to zero or to 90 degrees.
By the time Theta is around here to Pi,
we're down here to minor,
or the negative coming back this way.
Once again, we have a cosine waveform,
but it's a cosine waveform with a Phi degree or Phi radian shift.
Let's take this term,
and rewrite it using properties of exponents
to take our analysis just another step further.
Thus, we can say then that V is equal to V_M,
E to the J Omega T times E to the J Phi.
Just this the multiple property of exponents.
Now, let's rewrite this,
as V_M E to the J Phi times E to the J Omega T. As we do that,
we see that this term here is the time-dependent.
The time-dependent term, and it varies as
Omega T. But we also see
that we have these two terms here which are constants.
In fact, we have a V_M E to the J Phi.
We have another complex number involving the amplitude of
the cosine wave and this arbitrary phase shift.
It turns out in RLC circuits,
linear circuits, in general,
that are driven by sinusoidally varying wave forms,
that the frequency Omega doesn't change.
In other words, every current and voltage in a circuit
that's driven will be oscillating at the same frequency.
The only thing that changes,
the only differences between
various currents and voltages within this waveform are
the amplitude and the phase of those currents and voltages.
This term here is so important.
We give it a where we create a new complex number called phasor V,
and phasor V then is simply equal to this V_M E to the J Phi.
I'll say it one final time.
Phasor V is a complex number.
Its magnitude V_M is the amplitude
of the cosine wave that we're wanting to
represent in terms of complex exponential,
and its angle Phi is the arbitrary offset or the phase shift of that function.
Lets just look graphically at what we're talking about then.
Here is a cosine wave form that goes through its maximum at T equals zero,
in other words, it has no phase shift.
We would arrive in the phasor representation of this would
be V_M E to the J zero.
Well, E to the zero is one.
So, this is simply equal to V_M.
For a wave form, it's got some finite or some shift to it.
We would write it phasor V then is equal to V_M,
E to the J Phi.
So, this thing right here,
this complex exponential term is going
to represent a waveform in this case that is
shifted Phi degrees away from the original or the non shifted waveform.
This blue one is cosine Omega T. The red one is cosine Omega T plus Phi.
In this case, Phi is a positive 90 degrees.
Alright, let's just close up here by giving three examples.
Here we have the time domain representations
and over here let's go ahead and write the phasor transform,
or the complex exponential way of representing these trigonometric functions.
So, on this first one,
phasor V1 would be equal to five E to the J zero.
But again, E to the zero is one,
so that would simply be five.
Phasor V two would equal four E to the J 60,
and phasor V three would equal
three E to the J minus 90.
We typically put the minus sign out here in front,
E to the minus J 90 degrees.
I have represented here the time domain functions
V one V two and V three. V one is the blue one.
It goes through a maximum at T equals zero and has an amplitude
of five, no phase shift.
The blue one is v two.
It has an amp I'm sorry the blue and we're just talking about.
The green one has an amplitude of
four volts and it has a phase shift of plus 60 degrees.
So, it's been shifted to the left by 60 degrees.
Under these circumstances, we would say that the green one
because it peaks out before the blue one does,
we'd say that the green function V two leads the blue function,
in this case, by 60 degrees.
Finally, the red function shown here,
corresponds to V three.
It has an amplitude of three volts and it is shifted to the right 90 degrees,
So, the red one lags V one by 90 degrees.
All three of these are oscillating at the same frequency,
Omega is the same for all three of them.
They differ only in amplitude,
and in phase shift.
We're going to use these complex exponentials,
instead of trigonometric functions as we analyzed these RLC circuits,
because it's a whole lot easier to do
exponential mathematics than it is trigonometry with all of
its identities and complicating trigonometric properties.