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

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and then also derive the
impedance of a resistor.

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

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involved resistors only that
were driven by DC sources.

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To do that analysis,

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we used Ohm's law which said that V,

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the voltage across the resistor

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is related to the current
through that resistor.

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

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where I is defined flowing from

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the positive voltage terminal to
the negatively referenced terminal.

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In fact, we define the R,

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

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When we did this, we saw that the nature of

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

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The source was DC or constant,

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and the voltages and currents

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associated with each of
these devices were also constant,

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meaning that they weren't varying in time.

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Our job in that analysis
was to then determine

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

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

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Summing the voltage drops around a loop,

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and the currents leaving a node.

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We were able to write equations that

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establish the relationships
between the different currents,

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and voltages flowing through the circuit.

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

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constant sources rather driven
by sinusoidally varying sources.

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

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they will also have
capacitors and inductors.

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

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is going to be valid for the steady-state.

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

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

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

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To analyze these circuits,

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we're going to find it useful to define

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a quantity that is analogous to resistance.

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We're going to call
that quantity impedance,

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and generally refer to it
with a variable name Z.

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We're going to define.

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We are now defining impedance as being

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

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

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

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

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

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

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and the voltages will also be

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oscillating plus to minus
at that same frequency.

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So the nature of each of these voltages and

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

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Thus each one of these voltages
and currents can be

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represented in terms of its phasor.

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Impedance then is the ratio
of the phasor representation

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of the voltage of the device divided by

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the phasor representation of the current.

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

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resistive networks driven by DC sources

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using a resistance to represent the device.

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All of those techniques are going to

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apply to our analysis, the
sinusoidal steady-state.

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So node analysis, mesh
analysis, voltage division,

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current division, Thevenin equivalency,

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parallel and series equivalence.

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All of those tools that we
developed for this type of

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

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

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

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defined flowing from higher voltage
to lower voltage reference,

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

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

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

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cosine of Omega t plus some Theta sub I.

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Once again, Omega is

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the frequency at which the source
is driving this circuit.

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

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

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cosine of Omega t plus Theta sub I.

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

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This then would be just I sub m,

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e to the j, Theta sub I,

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and this would be then R times I sub m,

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e to the j Theta sub I.

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With those two phaser representations,

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we then can say that,

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

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

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

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e to the j Theta sub I.

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We notice that these terms cancel,

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and we're left with the impedance of

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

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

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represent a ratio of phasors,

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that ratio for resistors is just

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

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