L7 2 1 The Phasor

>> 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.