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Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:04.90,Default,,0000,0000,0000,,>> In this video, we're going to introduce\Nthe concept of complex impedance,
Dialogue: 0,0:00:04.90,0:00:08.37,Default,,0000,0000,0000,,and then also derive the\Nimpedance of a resistor.
Dialogue: 0,0:00:08.37,0:00:11.88,Default,,0000,0000,0000,,Earlier in the semester, we learned\Nhow to analyze circuits that
Dialogue: 0,0:00:11.88,0:00:16.36,Default,,0000,0000,0000,,involved resistors only that\Nwere driven by DC sources.
Dialogue: 0,0:00:16.36,0:00:18.04,Default,,0000,0000,0000,,To do that analysis,
Dialogue: 0,0:00:18.04,0:00:20.74,Default,,0000,0000,0000,,we used Ohm's law which said that V,
Dialogue: 0,0:00:20.74,0:00:23.94,Default,,0000,0000,0000,,the voltage across the resistor
Dialogue: 0,0:00:23.94,0:00:28.44,Default,,0000,0000,0000,,is related to the current\Nthrough that resistor.
Dialogue: 0,0:00:28.44,0:00:38.40,Default,,0000,0000,0000,,By Ohm's law which says\NV is equal to I times R,
Dialogue: 0,0:00:38.40,0:00:42.23,Default,,0000,0000,0000,,where I is defined flowing from
Dialogue: 0,0:00:42.23,0:00:46.14,Default,,0000,0000,0000,,the positive voltage terminal to\Nthe negatively referenced terminal.
Dialogue: 0,0:00:46.14,0:00:48.23,Default,,0000,0000,0000,,In fact, we define the R,
Dialogue: 0,0:00:48.23,0:00:54.40,Default,,0000,0000,0000,,the resistance of the device as being\Nthe ratio of the voltage to the current.
Dialogue: 0,0:00:54.40,0:00:56.63,Default,,0000,0000,0000,,When we did this, we saw that the nature of
Dialogue: 0,0:00:56.63,0:00:59.18,Default,,0000,0000,0000,,the voltage and current was the\Nsame as the nature of the source.
Dialogue: 0,0:00:59.18,0:01:01.37,Default,,0000,0000,0000,,The source was DC or constant,
Dialogue: 0,0:01:01.37,0:01:03.00,Default,,0000,0000,0000,,and the voltages and currents
Dialogue: 0,0:01:03.00,0:01:06.34,Default,,0000,0000,0000,,associated with each of\Nthese devices were also constant,
Dialogue: 0,0:01:06.34,0:01:09.10,Default,,0000,0000,0000,,meaning that they weren't varying in time.
Dialogue: 0,0:01:09.10,0:01:11.75,Default,,0000,0000,0000,,Our job in that analysis\Nwas to then determine
Dialogue: 0,0:01:11.75,0:01:14.06,Default,,0000,0000,0000,,what the currents and voltages\Nwere in each of those devices.
Dialogue: 0,0:01:14.06,0:01:18.82,Default,,0000,0000,0000,,To do so, we used Ohm's law and\Nwe also used Kirchhoff's Laws.
Dialogue: 0,0:01:18.82,0:01:22.63,Default,,0000,0000,0000,,Summing the voltage drops around a loop,
Dialogue: 0,0:01:22.63,0:01:25.10,Default,,0000,0000,0000,,and the currents leaving a node.
Dialogue: 0,0:01:25.10,0:01:26.81,Default,,0000,0000,0000,,We were able to write equations that
Dialogue: 0,0:01:26.81,0:01:29.36,Default,,0000,0000,0000,,establish the relationships\Nbetween the different currents,
Dialogue: 0,0:01:29.36,0:01:31.91,Default,,0000,0000,0000,,and voltages flowing through the circuit.
Dialogue: 0,0:01:31.91,0:01:36.26,Default,,0000,0000,0000,,We're now going to turn our attention\Nto circuits that are driven not by DC
Dialogue: 0,0:01:36.26,0:01:41.00,Default,,0000,0000,0000,,constant sources rather driven\Nby sinusoidally varying sources.
Dialogue: 0,0:01:41.00,0:01:45.08,Default,,0000,0000,0000,,We're also going to be analyzing circuits\Nthat have in addition two resistors,
Dialogue: 0,0:01:45.08,0:01:48.24,Default,,0000,0000,0000,,they will also have\Ncapacitors and inductors.
Dialogue: 0,0:01:48.24,0:01:50.78,Default,,0000,0000,0000,,The analysis that we're\Ndoing as I mentioned before,
Dialogue: 0,0:01:50.78,0:01:54.29,Default,,0000,0000,0000,,is going to be valid for the steady-state.
Dialogue: 0,0:01:54.29,0:01:58.85,Default,,0000,0000,0000,,What that means is that the source\Nwill have been applied to
Dialogue: 0,0:01:58.85,0:02:03.83,Default,,0000,0000,0000,,the circuit for a long enough time that\Nany transients will have died out,
Dialogue: 0,0:02:03.83,0:02:11.64,Default,,0000,0000,0000,,and we have simply the circuit operating\Nin its sinusoidal steady-state mode.
Dialogue: 0,0:02:11.64,0:02:13.31,Default,,0000,0000,0000,,To analyze these circuits,
Dialogue: 0,0:02:13.31,0:02:15.12,Default,,0000,0000,0000,,we're going to find it useful to define
Dialogue: 0,0:02:15.12,0:02:18.62,Default,,0000,0000,0000,,a quantity that is analogous to resistance.
Dialogue: 0,0:02:18.62,0:02:21.39,Default,,0000,0000,0000,,We're going to call\Nthat quantity impedance,
Dialogue: 0,0:02:21.39,0:02:24.50,Default,,0000,0000,0000,,and generally refer to it\Nwith a variable name Z.
Dialogue: 0,0:02:24.50,0:02:26.28,Default,,0000,0000,0000,,We're going to define.
Dialogue: 0,0:02:26.28,0:02:29.82,Default,,0000,0000,0000,,We are now defining impedance as being
Dialogue: 0,0:02:29.82,0:02:36.76,Default,,0000,0000,0000,,the ratio of the phasor voltage to\Nthe phasor current of a device.
Dialogue: 0,0:02:36.76,0:02:44.80,Default,,0000,0000,0000,,So once again, we'll define voltages\Nand currents in this circuit,
Dialogue: 0,0:02:44.80,0:02:49.58,Default,,0000,0000,0000,,and we're going to recognize or\Nrealize that these voltages and
Dialogue: 0,0:02:49.58,0:02:54.65,Default,,0000,0000,0000,,currents are oscillating at the same\Nfrequency that the source is oscillating.
Dialogue: 0,0:02:54.65,0:02:57.62,Default,,0000,0000,0000,,What that means is that you have\Ncurrents going back and forth at
Dialogue: 0,0:02:57.62,0:03:00.66,Default,,0000,0000,0000,,the same frequency that the source\Nis driving them back and forth,
Dialogue: 0,0:03:00.66,0:03:03.12,Default,,0000,0000,0000,,and the voltages will also be
Dialogue: 0,0:03:03.12,0:03:08.42,Default,,0000,0000,0000,,oscillating plus to minus\Nat that same frequency.
Dialogue: 0,0:03:08.42,0:03:11.96,Default,,0000,0000,0000,,So the nature of each of these voltages and
Dialogue: 0,0:03:11.96,0:03:17.68,Default,,0000,0000,0000,,currents will be the same as\Nthe nature of the source, sinusoidal.
Dialogue: 0,0:03:17.68,0:03:20.51,Default,,0000,0000,0000,,Thus each one of these voltages\Nand currents can be
Dialogue: 0,0:03:20.51,0:03:24.28,Default,,0000,0000,0000,,represented in terms of its phasor.
Dialogue: 0,0:03:24.28,0:03:28.61,Default,,0000,0000,0000,,Impedance then is the ratio\Nof the phasor representation
Dialogue: 0,0:03:28.61,0:03:30.26,Default,,0000,0000,0000,,of the voltage of the device divided by
Dialogue: 0,0:03:30.26,0:03:33.48,Default,,0000,0000,0000,,the phasor representation of the current.
Dialogue: 0,0:03:34.30,0:03:40.28,Default,,0000,0000,0000,,We will find that all of our circuit\Nanalysis techniques that we developed for
Dialogue: 0,0:03:40.28,0:03:43.20,Default,,0000,0000,0000,,resistive networks driven by DC sources
Dialogue: 0,0:03:43.20,0:03:47.38,Default,,0000,0000,0000,,using a resistance to represent the device.
Dialogue: 0,0:03:47.38,0:03:49.16,Default,,0000,0000,0000,,All of those techniques are going to
Dialogue: 0,0:03:49.16,0:03:52.91,Default,,0000,0000,0000,,apply to our analysis, the\Nsinusoidal steady-state.
Dialogue: 0,0:03:52.91,0:03:55.28,Default,,0000,0000,0000,,So node analysis, mesh\Nanalysis, voltage division,
Dialogue: 0,0:03:55.28,0:03:58.84,Default,,0000,0000,0000,,current division, Thevenin equivalency,
Dialogue: 0,0:03:58.84,0:04:01.43,Default,,0000,0000,0000,,parallel and series equivalence.
Dialogue: 0,0:04:01.43,0:04:04.61,Default,,0000,0000,0000,,All of those tools that we\Ndeveloped for this type of
Dialogue: 0,0:04:04.61,0:04:08.26,Default,,0000,0000,0000,,a sinusoidal are also going\Nto apply for the RLC circuit.
Dialogue: 0,0:04:08.26,0:04:18.39,Default,,0000,0000,0000,,Now let's start by developing or\Nderiving the impedance for a resistor.
Dialogue: 0,0:04:18.39,0:04:22.49,Default,,0000,0000,0000,,To do so, we'll reference\Nthe voltage V with a current
Dialogue: 0,0:04:22.49,0:04:26.46,Default,,0000,0000,0000,,defined flowing from higher voltage\Nto lower voltage reference,
Dialogue: 0,0:04:26.46,0:04:34.46,Default,,0000,0000,0000,,and we know from Ohm's law that\NV is equal to I times R. Now,
Dialogue: 0,0:04:34.46,0:04:38.62,Default,,0000,0000,0000,,let's just assume that I,\Nwe know its sinusoidal.
Dialogue: 0,0:04:38.62,0:04:42.10,Default,,0000,0000,0000,,So let's say that I is\Nof the form, I sub m,
Dialogue: 0,0:04:42.10,0:04:47.36,Default,,0000,0000,0000,,cosine of Omega t plus some Theta sub I.
Dialogue: 0,0:04:47.36,0:04:48.85,Default,,0000,0000,0000,,Once again, Omega is
Dialogue: 0,0:04:48.85,0:04:52.84,Default,,0000,0000,0000,,the frequency at which the source\Nis driving this circuit.
Dialogue: 0,0:04:52.84,0:04:57.42,Default,,0000,0000,0000,,From Ohm's law then we\Ncan say that V of t is
Dialogue: 0,0:04:57.42,0:05:02.42,Default,,0000,0000,0000,,equal to R times I or R times I sub m,
Dialogue: 0,0:05:02.42,0:05:07.49,Default,,0000,0000,0000,,cosine of Omega t plus Theta sub I.
Dialogue: 0,0:05:07.49,0:05:11.60,Default,,0000,0000,0000,,Now let's represent both of\Nthese in their phasor form.
Dialogue: 0,0:05:11.60,0:05:14.90,Default,,0000,0000,0000,,This then would be just I sub m,
Dialogue: 0,0:05:14.90,0:05:18.26,Default,,0000,0000,0000,,e to the j, Theta sub I,
Dialogue: 0,0:05:18.26,0:05:24.90,Default,,0000,0000,0000,,and this would be then R times I sub m,
Dialogue: 0,0:05:24.90,0:05:27.86,Default,,0000,0000,0000,,e to the j Theta sub I.
Dialogue: 0,0:05:27.86,0:05:30.30,Default,,0000,0000,0000,,With those two phaser representations,
Dialogue: 0,0:05:30.30,0:05:31.84,Default,,0000,0000,0000,,we then can say that,
Dialogue: 0,0:05:31.84,0:05:41.21,Default,,0000,0000,0000,,Z for a resistor is equal\Nto phasor V over phasor I,
Dialogue: 0,0:05:41.21,0:05:45.98,Default,,0000,0000,0000,,which is equal to RI sub M, E to the J,
Dialogue: 0,0:05:45.98,0:05:51.56,Default,,0000,0000,0000,,theta sub I, divided by\Nphasor I which is I sub m,
Dialogue: 0,0:05:51.56,0:05:54.30,Default,,0000,0000,0000,,e to the j Theta sub I.
Dialogue: 0,0:05:54.30,0:05:57.51,Default,,0000,0000,0000,,We notice that these terms cancel,
Dialogue: 0,0:05:57.51,0:06:00.98,Default,,0000,0000,0000,,and we're left with the impedance of
Dialogue: 0,0:06:00.98,0:06:05.90,Default,,0000,0000,0000,,a resistor being simply equal\Nto the value of the resistance.
Dialogue: 0,0:06:05.90,0:06:10.19,Default,,0000,0000,0000,,In other words, in this phasor domain\Nor this impedance that
Dialogue: 0,0:06:10.19,0:06:15.54,Default,,0000,0000,0000,,represent a ratio of phasors,
Dialogue: 0,0:06:15.54,0:06:19.22,Default,,0000,0000,0000,,that ratio for resistors is just
Dialogue: 0,0:06:19.22,0:06:24.80,Default,,0000,0000,0000,,R. The impedance of\Na 10 Ohm resistor is 10 Ohms.
Dialogue: 0,0:06:24.80,0:06:29.70,Default,,0000,0000,0000,,We'll see that it's not quite that\Nsimple for inductors and capacitors.