L7 2 3 Introduction of Impedance of Resistor

>> In this video, we're going to introduce
the concept of complex impedance,

and then also derive the
impedance of a resistor.

Earlier in the semester, we learned
how to analyze circuits that

involved resistors only that
were driven by DC sources.

To do that analysis,

we used Ohm's law which said that V,

the voltage across the resistor

is related to the current
through that resistor.

By Ohm's law which says
V is equal to I times R,

where I is defined flowing from

the positive voltage terminal to
the negatively referenced terminal.

In fact, we define the R,

the resistance of the device as being
the ratio of the voltage to the current.

When we did this, we saw that the nature of

the voltage and current was the
same as the nature of the source.

The source was DC or constant,

and the voltages and currents

associated with each of
these devices were also constant,

meaning that they weren't varying in time.

Our job in that analysis
was to then determine

what the currents and voltages
were in each of those devices.

To do so, we used Ohm's law and
we also used Kirchhoff's Laws.

Summing the voltage drops around a loop,

and the currents leaving a node.

We were able to write equations that

establish the relationships
between the different currents,

and voltages flowing through the circuit.

We're now going to turn our attention
to circuits that are driven not by DC

constant sources rather driven
by sinusoidally varying sources.

We're also going to be analyzing circuits
that have in addition two resistors,

they will also have
capacitors and inductors.

The analysis that we're
doing as I mentioned before,

is going to be valid for the steady-state.

What that means is that the source
will have been applied to

the circuit for a long enough time that
any transients will have died out,

and we have simply the circuit operating
in its sinusoidal steady-state mode.

To analyze these circuits,

we're going to find it useful to define

a quantity that is analogous to resistance.

We're going to call
that quantity impedance,

and generally refer to it
with a variable name Z.

We're going to define.

We are now defining impedance as being

the ratio of the phasor voltage to
the phasor current of a device.

So once again, we'll define voltages
and currents in this circuit,

and we're going to recognize or
realize that these voltages and

currents are oscillating at the same
frequency that the source is oscillating.

What that means is that you have
currents going back and forth at

the same frequency that the source
is driving them back and forth,

and the voltages will also be

oscillating plus to minus
at that same frequency.

So the nature of each of these voltages and

currents will be the same as
the nature of the source, sinusoidal.

Thus each one of these voltages
and currents can be

represented in terms of its phasor.

Impedance then is the ratio
of the phasor representation

of the voltage of the device divided by

the phasor representation of the current.

We will find that all of our circuit
analysis techniques that we developed for

resistive networks driven by DC sources

using a resistance to represent the device.

All of those techniques are going to

apply to our analysis, the
sinusoidal steady-state.

So node analysis, mesh
analysis, voltage division,

current division, Thevenin equivalency,

parallel and series equivalence.

All of those tools that we
developed for this type of

a sinusoidal are also going
to apply for the RLC circuit.

Now let's start by developing or
deriving the impedance for a resistor.

To do so, we'll reference
the voltage V with a current

defined flowing from higher voltage
to lower voltage reference,

and we know from Ohm's law that
V is equal to I times R. Now,

let's just assume that I,
we know its sinusoidal.

So let's say that I is
of the form, I sub m,

cosine of Omega t plus some Theta sub I.

Once again, Omega is

the frequency at which the source
is driving this circuit.

From Ohm's law then we
can say that V of t is

equal to R times I or R times I sub m,

cosine of Omega t plus Theta sub I.

Now let's represent both of
these in their phasor form.

This then would be just I sub m,

e to the j, Theta sub I,

and this would be then R times I sub m,

e to the j Theta sub I.

With those two phaser representations,

we then can say that,

Z for a resistor is equal
to phasor V over phasor I,

which is equal to RI sub M, E to the J,

theta sub I, divided by
phasor I which is I sub m,

e to the j Theta sub I.

We notice that these terms cancel,

and we're left with the impedance of

a resistor being simply equal
to the value of the resistance.

In other words, in this phasor domain
or this impedance that

represent a ratio of phasors,

that ratio for resistors is just

R. The impedance of
a 10 Ohm resistor is 10 Ohms.

We'll see that it's not quite that
simple for inductors and capacitors.