0:00:00.000,0:00:04.905
>> In this video, we're going to introduce[br]the concept of complex impedance,

0:00:04.905,0:00:08.370
and then also derive the[br]impedance of a resistor.

0:00:08.370,0:00:11.880
Earlier in the semester, we learned[br]how to analyze circuits that

0:00:11.880,0:00:16.364
involved resistors only that[br]were driven by DC sources.

0:00:16.364,0:00:18.045
To do that analysis,

0:00:18.045,0:00:20.745
we used Ohm's law which said that V,

0:00:20.745,0:00:23.940
the voltage across the resistor

0:00:23.940,0:00:28.440
is related to the current[br]through that resistor.

0:00:28.440,0:00:38.399
By Ohm's law which says[br]V is equal to I times R,

0:00:38.399,0:00:42.230
where I is defined flowing from

0:00:42.230,0:00:46.145
the positive voltage terminal to[br]the negatively referenced terminal.

0:00:46.145,0:00:48.230
In fact, we define the R,

0:00:48.230,0:00:54.400
the resistance of the device as being[br]the ratio of the voltage to the current.

0:00:54.400,0:00:56.630
When we did this, we saw that the nature of

0:00:56.630,0:00:59.180
the voltage and current was the[br]same as the nature of the source.

0:00:59.180,0:01:01.370
The source was DC or constant,

0:01:01.370,0:01:03.005
and the voltages and currents

0:01:03.005,0:01:06.335
associated with each of[br]these devices were also constant,

0:01:06.335,0:01:09.095
meaning that they weren't varying in time.

0:01:09.095,0:01:11.750
Our job in that analysis[br]was to then determine

0:01:11.750,0:01:14.060
what the currents and voltages[br]were in each of those devices.

0:01:14.060,0:01:18.815
To do so, we used Ohm's law and[br]we also used Kirchhoff's Laws.

0:01:18.815,0:01:22.630
Summing the voltage drops around a loop,

0:01:22.630,0:01:25.100
and the currents leaving a node.

0:01:25.100,0:01:26.810
We were able to write equations that

0:01:26.810,0:01:29.360
establish the relationships[br]between the different currents,

0:01:29.360,0:01:31.910
and voltages flowing through the circuit.

0:01:31.910,0:01:36.260
We're now going to turn our attention[br]to circuits that are driven not by DC

0:01:36.260,0:01:41.000
constant sources rather driven[br]by sinusoidally varying sources.

0:01:41.000,0:01:45.080
We're also going to be analyzing circuits[br]that have in addition two resistors,

0:01:45.080,0:01:48.245
they will also have[br]capacitors and inductors.

0:01:48.245,0:01:50.780
The analysis that we're[br]doing as I mentioned before,

0:01:50.780,0:01:54.290
is going to be valid for the steady-state.

0:01:54.290,0:01:58.850
What that means is that the source[br]will have been applied to

0:01:58.850,0:02:03.830
the circuit for a long enough time that[br]any transients will have died out,

0:02:03.830,0:02:11.645
and we have simply the circuit operating[br]in its sinusoidal steady-state mode.

0:02:11.645,0:02:13.310
To analyze these circuits,

0:02:13.310,0:02:15.115
we're going to find it useful to define

0:02:15.115,0:02:18.620
a quantity that is analogous to resistance.

0:02:18.620,0:02:21.390
We're going to call[br]that quantity impedance,

0:02:21.390,0:02:24.500
and generally refer to it[br]with a variable name Z.

0:02:24.500,0:02:26.285
We're going to define.

0:02:26.285,0:02:29.825
We are now defining impedance as being

0:02:29.825,0:02:36.755
the ratio of the phasor voltage to[br]the phasor current of a device.

0:02:36.755,0:02:44.805
So once again, we'll define voltages[br]and currents in this circuit,

0:02:44.805,0:02:49.580
and we're going to recognize or[br]realize that these voltages and

0:02:49.580,0:02:54.650
currents are oscillating at the same[br]frequency that the source is oscillating.

0:02:54.650,0:02:57.620
What that means is that you have[br]currents going back and forth at

0:02:57.620,0:03:00.665
the same frequency that the source[br]is driving them back and forth,

0:03:00.665,0:03:03.125
and the voltages will also be

0:03:03.125,0:03:08.420
oscillating plus to minus[br]at that same frequency.

0:03:08.420,0:03:11.960
So the nature of each of these voltages and

0:03:11.960,0:03:17.675
currents will be the same as[br]the nature of the source, sinusoidal.

0:03:17.675,0:03:20.510
Thus each one of these voltages[br]and currents can be

0:03:20.510,0:03:24.275
represented in terms of its phasor.

0:03:24.275,0:03:28.610
Impedance then is the ratio[br]of the phasor representation

0:03:28.610,0:03:30.260
of the voltage of the device divided by

0:03:30.260,0:03:33.480
the phasor representation of the current.

0:03:34.300,0:03:40.280
We will find that all of our circuit[br]analysis techniques that we developed for

0:03:40.280,0:03:43.205
resistive networks driven by DC sources

0:03:43.205,0:03:47.375
using a resistance to represent the device.

0:03:47.375,0:03:49.160
All of those techniques are going to

0:03:49.160,0:03:52.910
apply to our analysis, the[br]sinusoidal steady-state.

0:03:52.910,0:03:55.280
So node analysis, mesh[br]analysis, voltage division,

0:03:55.280,0:03:58.840
current division, Thevenin equivalency,

0:03:58.840,0:04:01.430
parallel and series equivalence.

0:04:01.430,0:04:04.610
All of those tools that we[br]developed for this type of

0:04:04.610,0:04:08.255
a sinusoidal are also going[br]to apply for the RLC circuit.

0:04:08.255,0:04:18.390
Now let's start by developing or[br]deriving the impedance for a resistor.

0:04:18.390,0:04:22.490
To do so, we'll reference[br]the voltage V with a current

0:04:22.490,0:04:26.465
defined flowing from higher voltage[br]to lower voltage reference,

0:04:26.465,0:04:34.460
and we know from Ohm's law that[br]V is equal to I times R. Now,

0:04:34.460,0:04:38.620
let's just assume that I,[br]we know its sinusoidal.

0:04:38.620,0:04:42.105
So let's say that I is[br]of the form, I sub m,

0:04:42.105,0:04:47.355
cosine of Omega t plus some Theta sub I.

0:04:47.355,0:04:48.850
Once again, Omega is

0:04:48.850,0:04:52.835
the frequency at which the source[br]is driving this circuit.

0:04:52.835,0:04:57.420
From Ohm's law then we[br]can say that V of t is

0:04:57.420,0:05:02.415
equal to R times I or R times I sub m,

0:05:02.415,0:05:07.490
cosine of Omega t plus Theta sub I.

0:05:07.490,0:05:11.605
Now let's represent both of[br]these in their phasor form.

0:05:11.605,0:05:14.905
This then would be just I sub m,

0:05:14.905,0:05:18.265
e to the j, Theta sub I,

0:05:18.265,0:05:24.900
and this would be then R times I sub m,

0:05:24.900,0:05:27.855
e to the j Theta sub I.

0:05:27.855,0:05:30.300
With those two phaser representations,

0:05:30.300,0:05:31.845
we then can say that,

0:05:31.845,0:05:41.210
Z for a resistor is equal[br]to phasor V over phasor I,

0:05:41.210,0:05:45.980
which is equal to RI sub M, E to the J,

0:05:45.980,0:05:51.555
theta sub I, divided by[br]phasor I which is I sub m,

0:05:51.555,0:05:54.300
e to the j Theta sub I.

0:05:54.300,0:05:57.510
We notice that these terms cancel,

0:05:57.510,0:06:00.980
and we're left with the impedance of

0:06:00.980,0:06:05.900
a resistor being simply equal[br]to the value of the resistance.

0:06:05.900,0:06:10.190
In other words, in this phasor domain[br]or this impedance that

0:06:10.190,0:06:15.535
represent a ratio of phasors,

0:06:15.535,0:06:19.215
that ratio for resistors is just

0:06:19.215,0:06:24.800
R. The impedance of[br]a 10 Ohm resistor is 10 Ohms.

0:06:24.800,0:06:29.700
We'll see that it's not quite that[br]simple for inductors and capacitors.