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>> Hi my name's Lee Brinton and I am

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an electrical engineering instructor
at Salt Lake Community College.

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In this video and actually going
to turn our attention now to

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analyzing circuits that are driven by
sinusoidally varying or AC sources.

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This is what we're going to refer
to as the sinusoidal steady state.

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By this, we're going to assume
that the system has been

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operating that the sinusoidally
varying sources have been applied long

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enough that any transients that
might have occurred at startup have

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all died out and we're now operating
in the sinusoidal steady state.

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To do this and to analyze
these kinds of circuits.

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We're going to use
something known as phasors.

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So to begin with we're going to
introduce this concept of a phasor.

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Phasor is a complex exponential way of
representing trigonometric functions.

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Will then turn our attention to and

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define what we mean by a complex impedance.

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Will then analyze or demonstrate ways
of analyzing these RLC circuits.

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Resistors inductors
capacitors are circuits that

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have resistors inductors
and capacitors in them.

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In the sinusoidal steady-state including

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looking at series connections
and voltage division,

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parallel connections and current division.

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We will see how node
mesh analysis applies in

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this sinusoidal steady state
involving phasors and impedances.

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And then we'll look at
the Thevenin equivalency and

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maximum power transfer concepts
in this sinusoidal steady state.

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So here we are again in the complex plane.

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Where we have a real axis
and an imaginary axis.

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And we know that we can
represent points in this plane.

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Now we're going to be using
a complex number to represent a voltage.

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So let's now just start
out with a complex number.

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We will call it V and say that
it has a length which we will

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call V sub M and it has some angle theta.

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So there's theta.

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Again the length of this is V
sub M. So it's a complex number.

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The magnitude of
the complex number is V sub M

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and the angle associated with it is theta.

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Of course that represents
the polar representation of

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this complex number V. We know
that another way of writing

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that is to say V sub M E to

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the J theta and we also know that we
can express it using Euler's formula or

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simply pointing out that
the projection of this point on

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the real axis is equal to
V sub M times the cosine of theta.

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So we can write this as

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V sub M, times the cosine of
theta gives us the real part.

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And the imaginary part
is just V sub M times

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the sine of theta or
this can be rewritten as

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V sub M times cosine theta
plus J sine of theta.

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Just a little bit of nomenclature here.

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We can say than that
the real part of V is equal to

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V sub M cosine of theta
and the imaginary part of

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V is equal to V sub M sine theta.

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Alright here's where it
starts to get interesting.

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Up until now theta has
been a constant value,

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what if we start allowing theta to be

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some function of time and in
fact let's begin with by letting

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theta of T equal some constant omega times

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T. Omega is going to have
the units of radians per seconds.

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So that when we have radians per
second times seconds that gives us

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radians or measure of an angle.

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So sure enough theta is
an angle or has the units of

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radians or equivalently degrees.

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By the way, omega is referred to
as the radial frequency term.

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So now, let's rewrite our V. V then
is equal to V sub M E to the theta,

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but theta is equal to
omega T. So it will be E to

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the J theta or E to the J.
Omega T. Which again

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using Euler's formula can
be written as V sub M times

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the cosine of omega T plus J sine of

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omega T. Given that the real part
of V is the projection of

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this point down onto
the real axis and given now

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that theta is starting at
T equals 0 theta equals 0.

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As T increases, we now see that

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theta is increasing linearly
as a function of time.

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Which means that the real part of

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this complex number as
theta increases with time.

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The real part of it, which is
the projection onto the real axis,

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is going to be changing as
a function of time also.

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The real part of this now becomes

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a function of time and if
we stop and think about it,

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we know that the projection onto
the real axis is just the magnitude of

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V sub M times the cosine of this angle.

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The angle being omega T. We have
then that the real part of V,

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writing it the real part
of V then is equal to

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V sub M cosine of

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omega T. Hopefully now you'll start
to see where we're headed with this.

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This V sub M cosine omega T is the way
of numerically or mathematically

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representing a waveform this oscillating

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this cosine is solely at
a frequency of omega T seconds.

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We can see graphically what's happening.

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Let's now overlay a time-domain axis
coming down like this.

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And let's plot as a function
of time the projection on

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the real axis of this point as it goes
around at omega radians per second.

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Just draw a couple of lines
down here to help me graph is.

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So at T equals 0, theta equals 0,

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the cosine of 0 is one and we
start out here at a maximum.

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Now we allow theta to increase and as we
do so the projection on the real axis

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decreases until we get to theta
equaling pie halves or 90 degrees,

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at which time the real projection equals 0.

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Continuing on through until theta gets
around here two pie or a 180 degrees.

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Once again we have a minimum here.

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It comes back crosses as theta
gets down here to 270 degrees.

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Back to where we started.

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As Theta goes around and around
this graph and continues on and on,

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and you'll see, hopefully,
you recognize then.

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Though what we have here is a plot of
the function V_M cosine of Omega T. Now,

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this analysis assumed that Theta
started at zero radians per second.

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Let's now take a look at what happens
if we start our angle at not zero,

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but at some angle Phi.

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In other words, Theta at
t equals zero equals Phi.

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Now, when we start
this projection coming down here,

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or first of all,

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we then would say that Theta
as a function of time and

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is equal to Omega T plus Phi.

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Our complex expression V
is equal to then V_M,

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E to the J Theta,

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but Theta is now Omega T plus Phi,

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which again using Euler's formula,

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we can write as V_M times
the cosine of Omega T plus Phi,

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plus J sine of Omega T plus Phi.

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We can save in that
the real part of V is equal to

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V sub m cosine of Omega T plus Phi,

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a completely arbitrary cosine term,

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specified by its amplitude V_M.

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The rate at which it's oscillating
Omega and this arbitrary term,

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or arbitrary offset of Phi radians.

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Let's plot that projection
onto our time domain.

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As we did before, only now instead of

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starting at T equals zero
with Theta equaling zero,

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we start at T equals zero at Phi,

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and again dropping these down
to just help me draw this.

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We're starting here at this point.

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It goes to zero when Theta
gets to zero or to 90 degrees.

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By the time Theta is around here to Pi,

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we're down here to minor,

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or the negative coming back this way.

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Once again, we have a cosine waveform,

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but it's a cosine waveform with
a Phi degree or Phi radian shift.

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Let's take this term,

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and rewrite it using
properties of exponents

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to take our analysis just
another step further.

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Thus, we can say then
that V is equal to V_M,

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E to the J Omega T times E to the J Phi.

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Just this the multiple
property of exponents.

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Now, let's rewrite this,

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as V_M E to the J Phi times E to
the J Omega T. As we do that,

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we see that this term here
is the time-dependent.

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The time-dependent term, and it varies as

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Omega T. But we also see

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that we have these two terms
here which are constants.

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In fact, we have a V_M E to the J Phi.

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We have another complex number
involving the amplitude of

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the cosine wave and
this arbitrary phase shift.

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It turns out in RLC circuits,

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linear circuits, in general,

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that are driven by
sinusoidally varying wave forms,

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that the frequency Omega doesn't change.

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In other words, every current
and voltage in a circuit

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that's driven will be oscillating
at the same frequency.

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The only thing that changes,

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the only differences between

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various currents and voltages
within this waveform are

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the amplitude and the phase of
those currents and voltages.

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This term here is so important.

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We give it a where we create
a new complex number called phasor V,

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and phasor V then is simply equal
to this V_M E to the J Phi.

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I'll say it one final time.

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Phasor V is a complex number.

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Its magnitude V_M is the amplitude

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of the cosine wave that we're wanting to

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represent in terms of complex exponential,

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and its angle Phi is the arbitrary offset
or the phase shift of that function.

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Lets just look graphically at
what we're talking about then.

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Here is a cosine wave form that goes
through its maximum at T equals zero,

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in other words, it has no phase shift.

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We would arrive in the phasor
representation of this would

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be V_M E to the J zero.

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Well, E to the zero is one.

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So, this is simply equal to V_M.

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For a wave form, it's got some
finite or some shift to it.

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We would write it phasor V
then is equal to V_M,

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E to the J Phi.

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So, this thing right here,

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this complex exponential term is going

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to represent a waveform
in this case that is

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shifted Phi degrees away from the
original or the non shifted waveform.

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This blue one is cosine Omega T.
The red one is cosine Omega T plus Phi.

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In this case, Phi is a positive 90 degrees.

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Alright, let's just close up
here by giving three examples.

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Here we have the time
domain representations

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and over here let's go ahead
and write the phasor transform,

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or the complex exponential way of
representing these trigonometric functions.

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So, on this first one,

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phasor V1 would be equal
to five E to the J zero.

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But again, E to the zero is one,

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so that would simply be five.

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Phasor V two would equal
four E to the J 60,

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and phasor V three would equal

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three E to the J minus 90.

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We typically put the minus
sign out here in front,

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E to the minus J 90 degrees.

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I have represented here
the time domain functions

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V one V two and V three.
V one is the blue one.

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It goes through a maximum at T
equals zero and has an amplitude

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of five, no phase shift.

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The blue one is v two.

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It has an amp I'm sorry the blue
and we're just talking about.

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The green one has an amplitude of

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four volts and it has a
phase shift of plus 60 degrees.

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So, it's been shifted to
the left by 60 degrees.

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Under these circumstances, we
would say that the green one

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because it peaks out
before the blue one does,

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we'd say that the green function V
two leads the blue function,

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in this case, by 60 degrees.

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Finally, the red function shown here,

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corresponds to V three.

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It has an amplitude of three volts and
it is shifted to the right 90 degrees,

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So, the red one lags V one by 90 degrees.

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All three of these are oscillating
at the same frequency,

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Omega is the same for all three of them.

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They differ only in amplitude,

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and in phase shift.

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We're going to use
these complex exponentials,

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instead of trigonometric functions
as we analyzed these RLC circuits,

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because it's a whole lot easier to do

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exponential mathematics than
it is trigonometry with all of

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its identities and complicating
trigonometric properties.