>> 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.