[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.41,0:00:02.46,Default,,0000,0000,0000,,>> Hi my name's Lee Brinton and I am
Dialogue: 0,0:00:02.46,0:00:05.12,Default,,0000,0000,0000,,an electrical engineering instructor\Nat Salt Lake Community College.
Dialogue: 0,0:00:05.12,0:00:10.62,Default,,0000,0000,0000,,In this video and actually going\Nto turn our attention now to
Dialogue: 0,0:00:10.62,0:00:16.92,Default,,0000,0000,0000,,analyzing circuits that are driven by\Nsinusoidally varying or AC sources.
Dialogue: 0,0:00:16.92,0:00:20.52,Default,,0000,0000,0000,,This is what we're going to refer\Nto as the sinusoidal steady state.
Dialogue: 0,0:00:20.52,0:00:23.91,Default,,0000,0000,0000,,By this, we're going to assume\Nthat the system has been
Dialogue: 0,0:00:23.91,0:00:28.62,Default,,0000,0000,0000,,operating that the sinusoidally\Nvarying sources have been applied long
Dialogue: 0,0:00:28.62,0:00:32.63,Default,,0000,0000,0000,,enough that any transients that\Nmight have occurred at startup have
Dialogue: 0,0:00:32.63,0:00:37.40,Default,,0000,0000,0000,,all died out and we're now operating\Nin the sinusoidal steady state.
Dialogue: 0,0:00:37.40,0:00:41.98,Default,,0000,0000,0000,,To do this and to analyze\Nthese kinds of circuits.
Dialogue: 0,0:00:41.98,0:00:44.04,Default,,0000,0000,0000,,We're going to use\Nsomething known as phasors.
Dialogue: 0,0:00:44.04,0:00:47.10,Default,,0000,0000,0000,,So to begin with we're going to\Nintroduce this concept of a phasor.
Dialogue: 0,0:00:47.10,0:00:52.18,Default,,0000,0000,0000,,Phasor is a complex exponential way of\Nrepresenting trigonometric functions.
Dialogue: 0,0:00:52.18,0:00:54.62,Default,,0000,0000,0000,,Will then turn our attention to and
Dialogue: 0,0:00:54.62,0:00:57.50,Default,,0000,0000,0000,,define what we mean by a complex impedance.
Dialogue: 0,0:00:57.50,0:01:01.96,Default,,0000,0000,0000,,Will then analyze or demonstrate ways\Nof analyzing these RLC circuits.
Dialogue: 0,0:01:01.96,0:01:04.19,Default,,0000,0000,0000,,Resistors inductors\Ncapacitors are circuits that
Dialogue: 0,0:01:04.19,0:01:06.68,Default,,0000,0000,0000,,have resistors inductors\Nand capacitors in them.
Dialogue: 0,0:01:06.68,0:01:09.29,Default,,0000,0000,0000,,In the sinusoidal steady-state including
Dialogue: 0,0:01:09.29,0:01:12.24,Default,,0000,0000,0000,,looking at series connections\Nand voltage division,
Dialogue: 0,0:01:12.24,0:01:14.68,Default,,0000,0000,0000,,parallel connections and current division.
Dialogue: 0,0:01:14.68,0:01:17.09,Default,,0000,0000,0000,,We will see how node\Nmesh analysis applies in
Dialogue: 0,0:01:17.09,0:01:20.92,Default,,0000,0000,0000,,this sinusoidal steady state\Ninvolving phasors and impedances.
Dialogue: 0,0:01:20.92,0:01:24.11,Default,,0000,0000,0000,,And then we'll look at\Nthe Thevenin equivalency and
Dialogue: 0,0:01:24.11,0:01:28.38,Default,,0000,0000,0000,,maximum power transfer concepts\Nin this sinusoidal steady state.
Dialogue: 0,0:01:28.38,0:01:31.80,Default,,0000,0000,0000,,So here we are again in the complex plane.
Dialogue: 0,0:01:32.59,0:01:38.10,Default,,0000,0000,0000,,Where we have a real axis\Nand an imaginary axis.
Dialogue: 0,0:01:38.10,0:01:40.44,Default,,0000,0000,0000,,And we know that we can\Nrepresent points in this plane.
Dialogue: 0,0:01:40.44,0:01:45.76,Default,,0000,0000,0000,,Now we're going to be using\Na complex number to represent a voltage.
Dialogue: 0,0:01:45.76,0:01:51.11,Default,,0000,0000,0000,,So let's now just start\Nout with a complex number.
Dialogue: 0,0:01:51.11,0:01:55.04,Default,,0000,0000,0000,,We will call it V and say that\Nit has a length which we will
Dialogue: 0,0:01:55.04,0:01:59.56,Default,,0000,0000,0000,,call V sub M and it has some angle theta.
Dialogue: 0,0:01:59.56,0:02:01.22,Default,,0000,0000,0000,,So there's theta.
Dialogue: 0,0:02:01.22,0:02:04.67,Default,,0000,0000,0000,,Again the length of this is V\Nsub M. So it's a complex number.
Dialogue: 0,0:02:04.67,0:02:07.49,Default,,0000,0000,0000,,The magnitude of\Nthe complex number is V sub M
Dialogue: 0,0:02:07.49,0:02:10.46,Default,,0000,0000,0000,,and the angle associated with it is theta.
Dialogue: 0,0:02:10.46,0:02:13.97,Default,,0000,0000,0000,,Of course that represents\Nthe polar representation of
Dialogue: 0,0:02:13.97,0:02:18.41,Default,,0000,0000,0000,,this complex number V. We know\Nthat another way of writing
Dialogue: 0,0:02:18.41,0:02:21.11,Default,,0000,0000,0000,,that is to say V sub M E to
Dialogue: 0,0:02:21.11,0:02:27.08,Default,,0000,0000,0000,,the J theta and we also know that we\Ncan express it using Euler's formula or
Dialogue: 0,0:02:27.08,0:02:30.92,Default,,0000,0000,0000,,simply pointing out that\Nthe projection of this point on
Dialogue: 0,0:02:30.92,0:02:36.08,Default,,0000,0000,0000,,the real axis is equal to\NV sub M times the cosine of theta.
Dialogue: 0,0:02:36.08,0:02:37.98,Default,,0000,0000,0000,,So we can write this as
Dialogue: 0,0:02:37.98,0:02:44.78,Default,,0000,0000,0000,,V sub M, times the cosine of\Ntheta gives us the real part.
Dialogue: 0,0:02:44.78,0:02:47.51,Default,,0000,0000,0000,,And the imaginary part\Nis just V sub M times
Dialogue: 0,0:02:47.51,0:02:50.15,Default,,0000,0000,0000,,the sine of theta or\Nthis can be rewritten as
Dialogue: 0,0:02:50.15,0:02:58.20,Default,,0000,0000,0000,,V sub M times cosine theta\Nplus J sine of theta.
Dialogue: 0,0:02:58.49,0:03:01.14,Default,,0000,0000,0000,,Just a little bit of nomenclature here.
Dialogue: 0,0:03:01.14,0:03:09.26,Default,,0000,0000,0000,,We can say than that\Nthe real part of V is equal to
Dialogue: 0,0:03:09.26,0:03:15.13,Default,,0000,0000,0000,,V sub M cosine of theta\Nand the imaginary part of
Dialogue: 0,0:03:15.13,0:03:22.64,Default,,0000,0000,0000,,V is equal to V sub M sine theta.
Dialogue: 0,0:03:22.64,0:03:25.34,Default,,0000,0000,0000,,Alright here's where it\Nstarts to get interesting.
Dialogue: 0,0:03:25.34,0:03:28.20,Default,,0000,0000,0000,,Up until now theta has\Nbeen a constant value,
Dialogue: 0,0:03:28.20,0:03:31.04,Default,,0000,0000,0000,,what if we start allowing theta to be
Dialogue: 0,0:03:31.04,0:03:35.39,Default,,0000,0000,0000,,some function of time and in\Nfact let's begin with by letting
Dialogue: 0,0:03:35.39,0:03:39.47,Default,,0000,0000,0000,,theta of T equal some constant omega times
Dialogue: 0,0:03:39.47,0:03:44.06,Default,,0000,0000,0000,,T. Omega is going to have\Nthe units of radians per seconds.
Dialogue: 0,0:03:44.06,0:03:46.88,Default,,0000,0000,0000,,So that when we have radians per\Nsecond times seconds that gives us
Dialogue: 0,0:03:46.88,0:03:49.73,Default,,0000,0000,0000,,radians or measure of an angle.
Dialogue: 0,0:03:49.73,0:03:52.88,Default,,0000,0000,0000,,So sure enough theta is\Nan angle or has the units of
Dialogue: 0,0:03:52.88,0:03:57.24,Default,,0000,0000,0000,,radians or equivalently degrees.
Dialogue: 0,0:03:57.24,0:04:03.53,Default,,0000,0000,0000,,By the way, omega is referred to\Nas the radial frequency term.
Dialogue: 0,0:04:03.53,0:04:11.46,Default,,0000,0000,0000,,So now, let's rewrite our V. V then\Nis equal to V sub M E to the theta,
Dialogue: 0,0:04:11.46,0:04:14.69,Default,,0000,0000,0000,,but theta is equal to\Nomega T. So it will be E to
Dialogue: 0,0:04:14.69,0:04:20.18,Default,,0000,0000,0000,,the J theta or E to the J.\NOmega T. Which again
Dialogue: 0,0:04:20.18,0:04:24.56,Default,,0000,0000,0000,,using Euler's formula can\Nbe written as V sub M times
Dialogue: 0,0:04:24.56,0:04:29.58,Default,,0000,0000,0000,,the cosine of omega T plus J sine of
Dialogue: 0,0:04:29.58,0:04:36.56,Default,,0000,0000,0000,,omega T. Given that the real part\Nof V is the projection of
Dialogue: 0,0:04:36.56,0:04:40.52,Default,,0000,0000,0000,,this point down onto\Nthe real axis and given now
Dialogue: 0,0:04:40.52,0:04:45.06,Default,,0000,0000,0000,,that theta is starting at\NT equals 0 theta equals 0.
Dialogue: 0,0:04:45.06,0:04:48.60,Default,,0000,0000,0000,,As T increases, we now see that
Dialogue: 0,0:04:48.60,0:04:54.12,Default,,0000,0000,0000,,theta is increasing linearly\Nas a function of time.
Dialogue: 0,0:04:54.12,0:04:57.02,Default,,0000,0000,0000,,Which means that the real part of
Dialogue: 0,0:04:57.02,0:05:01.08,Default,,0000,0000,0000,,this complex number as\Ntheta increases with time.
Dialogue: 0,0:05:01.08,0:05:04.91,Default,,0000,0000,0000,,The real part of it, which is\Nthe projection onto the real axis,
Dialogue: 0,0:05:04.91,0:05:09.84,Default,,0000,0000,0000,,is going to be changing as\Na function of time also.
Dialogue: 0,0:05:09.84,0:05:11.96,Default,,0000,0000,0000,,The real part of this now becomes
Dialogue: 0,0:05:11.96,0:05:14.36,Default,,0000,0000,0000,,a function of time and if\Nwe stop and think about it,
Dialogue: 0,0:05:14.36,0:05:19.10,Default,,0000,0000,0000,,we know that the projection onto\Nthe real axis is just the magnitude of
Dialogue: 0,0:05:19.10,0:05:25.52,Default,,0000,0000,0000,,V sub M times the cosine of this angle.
Dialogue: 0,0:05:25.52,0:05:31.78,Default,,0000,0000,0000,,The angle being omega T. We have\Nthen that the real part of V,
Dialogue: 0,0:05:31.78,0:05:36.82,Default,,0000,0000,0000,,writing it the real part\Nof V then is equal to
Dialogue: 0,0:05:36.82,0:05:39.71,Default,,0000,0000,0000,,V sub M cosine of
Dialogue: 0,0:05:39.71,0:05:45.43,Default,,0000,0000,0000,,omega T. Hopefully now you'll start\Nto see where we're headed with this.
Dialogue: 0,0:05:45.43,0:05:51.22,Default,,0000,0000,0000,,This V sub M cosine omega T is the way\Nof numerically or mathematically
Dialogue: 0,0:05:51.22,0:05:54.90,Default,,0000,0000,0000,,representing a waveform this oscillating
Dialogue: 0,0:05:54.90,0:05:59.70,Default,,0000,0000,0000,,this cosine is solely at\Na frequency of omega T seconds.
Dialogue: 0,0:05:59.70,0:06:01.78,Default,,0000,0000,0000,,We can see graphically what's happening.
Dialogue: 0,0:06:01.78,0:06:07.96,Default,,0000,0000,0000,,Let's now overlay a time-domain axis\Ncoming down like this.
Dialogue: 0,0:06:07.96,0:06:12.16,Default,,0000,0000,0000,,And let's plot as a function\Nof time the projection on
Dialogue: 0,0:06:12.16,0:06:19.96,Default,,0000,0000,0000,,the real axis of this point as it goes\Naround at omega radians per second.
Dialogue: 0,0:06:19.96,0:06:24.00,Default,,0000,0000,0000,,Just draw a couple of lines\Ndown here to help me graph is.
Dialogue: 0,0:06:24.00,0:06:26.06,Default,,0000,0000,0000,,So at T equals 0, theta equals 0,
Dialogue: 0,0:06:26.06,0:06:29.86,Default,,0000,0000,0000,,the cosine of 0 is one and we\Nstart out here at a maximum.
Dialogue: 0,0:06:29.86,0:06:35.93,Default,,0000,0000,0000,,Now we allow theta to increase and as we\Ndo so the projection on the real axis
Dialogue: 0,0:06:35.93,0:06:41.63,Default,,0000,0000,0000,,decreases until we get to theta\Nequaling pie halves or 90 degrees,
Dialogue: 0,0:06:41.63,0:06:45.48,Default,,0000,0000,0000,,at which time the real projection equals 0.
Dialogue: 0,0:06:45.48,0:06:53.87,Default,,0000,0000,0000,,Continuing on through until theta gets\Naround here two pie or a 180 degrees.
Dialogue: 0,0:06:53.87,0:06:56.56,Default,,0000,0000,0000,,Once again we have a minimum here.
Dialogue: 0,0:06:56.56,0:07:03.18,Default,,0000,0000,0000,,It comes back crosses as theta\Ngets down here to 270 degrees.
Dialogue: 0,0:07:03.45,0:07:06.70,Default,,0000,0000,0000,,Back to where we started.
Dialogue: 0,0:07:06.70,0:07:11.68,Default,,0000,0000,0000,,As Theta goes around and around\Nthis graph and continues on and on,
Dialogue: 0,0:07:11.68,0:07:14.05,Default,,0000,0000,0000,,and you'll see, hopefully,\Nyou recognize then.
Dialogue: 0,0:07:14.05,0:07:20.50,Default,,0000,0000,0000,,Though what we have here is a plot of\Nthe function V_M cosine of Omega T. Now,
Dialogue: 0,0:07:20.50,0:07:27.10,Default,,0000,0000,0000,,this analysis assumed that Theta\Nstarted at zero radians per second.
Dialogue: 0,0:07:27.10,0:07:33.43,Default,,0000,0000,0000,,Let's now take a look at what happens\Nif we start our angle at not zero,
Dialogue: 0,0:07:33.43,0:07:36.70,Default,,0000,0000,0000,,but at some angle Phi.
Dialogue: 0,0:07:36.70,0:07:42.62,Default,,0000,0000,0000,,In other words, Theta at\Nt equals zero equals Phi.
Dialogue: 0,0:07:42.62,0:07:46.22,Default,,0000,0000,0000,,Now, when we start\Nthis projection coming down here,
Dialogue: 0,0:07:46.22,0:07:47.42,Default,,0000,0000,0000,,or first of all,
Dialogue: 0,0:07:47.42,0:07:51.34,Default,,0000,0000,0000,,we then would say that Theta\Nas a function of time and
Dialogue: 0,0:07:51.34,0:07:56.32,Default,,0000,0000,0000,,is equal to Omega T plus Phi.
Dialogue: 0,0:07:56.32,0:08:02.83,Default,,0000,0000,0000,,Our complex expression V\Nis equal to then V_M,
Dialogue: 0,0:08:02.83,0:08:05.23,Default,,0000,0000,0000,,E to the J Theta,
Dialogue: 0,0:08:05.23,0:08:10.48,Default,,0000,0000,0000,,but Theta is now Omega T plus Phi,
Dialogue: 0,0:08:10.48,0:08:13.42,Default,,0000,0000,0000,,which again using Euler's formula,
Dialogue: 0,0:08:13.42,0:08:23.88,Default,,0000,0000,0000,,we can write as V_M times\Nthe cosine of Omega T plus Phi,
Dialogue: 0,0:08:23.88,0:08:32.06,Default,,0000,0000,0000,,plus J sine of Omega T plus Phi.
Dialogue: 0,0:08:32.06,0:08:37.76,Default,,0000,0000,0000,,We can save in that\Nthe real part of V is equal to
Dialogue: 0,0:08:37.76,0:08:44.16,Default,,0000,0000,0000,,V sub m cosine of Omega T plus Phi,
Dialogue: 0,0:08:44.16,0:08:46.80,Default,,0000,0000,0000,,a completely arbitrary cosine term,
Dialogue: 0,0:08:46.80,0:08:50.11,Default,,0000,0000,0000,,specified by its amplitude V_M.
Dialogue: 0,0:08:50.11,0:08:55.44,Default,,0000,0000,0000,,The rate at which it's oscillating\NOmega and this arbitrary term,
Dialogue: 0,0:08:55.44,0:08:59.89,Default,,0000,0000,0000,,or arbitrary offset of Phi radians.
Dialogue: 0,0:08:59.89,0:09:05.94,Default,,0000,0000,0000,,Let's plot that projection\Nonto our time domain.
Dialogue: 0,0:09:05.94,0:09:08.74,Default,,0000,0000,0000,,As we did before, only now instead of
Dialogue: 0,0:09:08.74,0:09:11.22,Default,,0000,0000,0000,,starting at T equals zero\Nwith Theta equaling zero,
Dialogue: 0,0:09:11.22,0:09:14.66,Default,,0000,0000,0000,,we start at T equals zero at Phi,
Dialogue: 0,0:09:14.66,0:09:19.58,Default,,0000,0000,0000,,and again dropping these down\Nto just help me draw this.
Dialogue: 0,0:09:19.58,0:09:22.00,Default,,0000,0000,0000,,We're starting here at this point.
Dialogue: 0,0:09:22.00,0:09:27.25,Default,,0000,0000,0000,,It goes to zero when Theta\Ngets to zero or to 90 degrees.
Dialogue: 0,0:09:27.25,0:09:29.83,Default,,0000,0000,0000,,By the time Theta is around here to Pi,
Dialogue: 0,0:09:29.83,0:09:30.98,Default,,0000,0000,0000,,we're down here to minor,
Dialogue: 0,0:09:30.98,0:09:34.54,Default,,0000,0000,0000,,or the negative coming back this way.
Dialogue: 0,0:09:35.66,0:09:39.00,Default,,0000,0000,0000,,Once again, we have a cosine waveform,
Dialogue: 0,0:09:39.00,0:09:46.36,Default,,0000,0000,0000,,but it's a cosine waveform with\Na Phi degree or Phi radian shift.
Dialogue: 0,0:09:46.77,0:09:49.12,Default,,0000,0000,0000,,Let's take this term,
Dialogue: 0,0:09:49.12,0:09:51.49,Default,,0000,0000,0000,,and rewrite it using\Nproperties of exponents
Dialogue: 0,0:09:51.49,0:09:54.97,Default,,0000,0000,0000,,to take our analysis just\Nanother step further.
Dialogue: 0,0:09:54.97,0:10:00.26,Default,,0000,0000,0000,,Thus, we can say then\Nthat V is equal to V_M,
Dialogue: 0,0:10:00.26,0:10:07.44,Default,,0000,0000,0000,,E to the J Omega T times E to the J Phi.
Dialogue: 0,0:10:07.44,0:10:11.96,Default,,0000,0000,0000,,Just this the multiple\Nproperty of exponents.
Dialogue: 0,0:10:11.96,0:10:14.28,Default,,0000,0000,0000,,Now, let's rewrite this,
Dialogue: 0,0:10:14.28,0:10:22.51,Default,,0000,0000,0000,,as V_M E to the J Phi times E to\Nthe J Omega T. As we do that,
Dialogue: 0,0:10:22.51,0:10:26.75,Default,,0000,0000,0000,,we see that this term here\Nis the time-dependent.
Dialogue: 0,0:10:27.39,0:10:31.51,Default,,0000,0000,0000,,The time-dependent term, and it varies as
Dialogue: 0,0:10:31.51,0:10:34.39,Default,,0000,0000,0000,,Omega T. But we also see
Dialogue: 0,0:10:34.39,0:10:37.81,Default,,0000,0000,0000,,that we have these two terms\Nhere which are constants.
Dialogue: 0,0:10:37.81,0:10:40.48,Default,,0000,0000,0000,,In fact, we have a V_M E to the J Phi.
Dialogue: 0,0:10:40.48,0:10:44.54,Default,,0000,0000,0000,,We have another complex number\Ninvolving the amplitude of
Dialogue: 0,0:10:44.54,0:10:50.22,Default,,0000,0000,0000,,the cosine wave and\Nthis arbitrary phase shift.
Dialogue: 0,0:10:50.22,0:10:53.35,Default,,0000,0000,0000,,It turns out in RLC circuits,
Dialogue: 0,0:10:53.35,0:10:55.06,Default,,0000,0000,0000,,linear circuits, in general,
Dialogue: 0,0:10:55.06,0:10:59.53,Default,,0000,0000,0000,,that are driven by\Nsinusoidally varying wave forms,
Dialogue: 0,0:10:59.53,0:11:03.50,Default,,0000,0000,0000,,that the frequency Omega doesn't change.
Dialogue: 0,0:11:03.50,0:11:07.30,Default,,0000,0000,0000,,In other words, every current\Nand voltage in a circuit
Dialogue: 0,0:11:07.30,0:11:11.23,Default,,0000,0000,0000,,that's driven will be oscillating\Nat the same frequency.
Dialogue: 0,0:11:11.23,0:11:12.82,Default,,0000,0000,0000,,The only thing that changes,
Dialogue: 0,0:11:12.82,0:11:14.20,Default,,0000,0000,0000,,the only differences between
Dialogue: 0,0:11:14.20,0:11:17.38,Default,,0000,0000,0000,,various currents and voltages\Nwithin this waveform are
Dialogue: 0,0:11:17.38,0:11:23.41,Default,,0000,0000,0000,,the amplitude and the phase of\Nthose currents and voltages.
Dialogue: 0,0:11:23.41,0:11:25.54,Default,,0000,0000,0000,,This term here is so important.
Dialogue: 0,0:11:25.54,0:11:32.40,Default,,0000,0000,0000,,We give it a where we create\Na new complex number called phasor V,
Dialogue: 0,0:11:32.40,0:11:38.12,Default,,0000,0000,0000,,and phasor V then is simply equal\Nto this V_M E to the J Phi.
Dialogue: 0,0:11:42.53,0:11:44.91,Default,,0000,0000,0000,,I'll say it one final time.
Dialogue: 0,0:11:44.91,0:11:47.94,Default,,0000,0000,0000,,Phasor V is a complex number.
Dialogue: 0,0:11:47.94,0:11:52.93,Default,,0000,0000,0000,,Its magnitude V_M is the amplitude
Dialogue: 0,0:11:52.93,0:11:55.08,Default,,0000,0000,0000,,of the cosine wave that we're wanting to
Dialogue: 0,0:11:55.08,0:11:57.98,Default,,0000,0000,0000,,represent in terms of complex exponential,
Dialogue: 0,0:11:57.98,0:12:05.95,Default,,0000,0000,0000,,and its angle Phi is the arbitrary offset\Nor the phase shift of that function.
Dialogue: 0,0:12:05.95,0:12:09.10,Default,,0000,0000,0000,,Lets just look graphically at\Nwhat we're talking about then.
Dialogue: 0,0:12:09.10,0:12:12.91,Default,,0000,0000,0000,,Here is a cosine wave form that goes\Nthrough its maximum at T equals zero,
Dialogue: 0,0:12:12.91,0:12:15.12,Default,,0000,0000,0000,,in other words, it has no phase shift.
Dialogue: 0,0:12:15.12,0:12:19.27,Default,,0000,0000,0000,,We would arrive in the phasor\Nrepresentation of this would
Dialogue: 0,0:12:19.27,0:12:24.16,Default,,0000,0000,0000,,be V_M E to the J zero.
Dialogue: 0,0:12:24.16,0:12:25.72,Default,,0000,0000,0000,,Well, E to the zero is one.
Dialogue: 0,0:12:25.72,0:12:30.36,Default,,0000,0000,0000,,So, this is simply equal to V_M.
Dialogue: 0,0:12:30.36,0:12:35.20,Default,,0000,0000,0000,,For a wave form, it's got some\Nfinite or some shift to it.
Dialogue: 0,0:12:35.20,0:12:39.74,Default,,0000,0000,0000,,We would write it phasor V\Nthen is equal to V_M,
Dialogue: 0,0:12:39.74,0:12:43.28,Default,,0000,0000,0000,,E to the J Phi.
Dialogue: 0,0:12:43.28,0:12:45.40,Default,,0000,0000,0000,,So, this thing right here,
Dialogue: 0,0:12:45.40,0:12:49.06,Default,,0000,0000,0000,,this complex exponential term is going
Dialogue: 0,0:12:49.06,0:12:52.87,Default,,0000,0000,0000,,to represent a waveform\Nin this case that is
Dialogue: 0,0:12:52.87,0:13:02.23,Default,,0000,0000,0000,,shifted Phi degrees away from the\Noriginal or the non shifted waveform.
Dialogue: 0,0:13:02.23,0:13:08.58,Default,,0000,0000,0000,,This blue one is cosine Omega T.\NThe red one is cosine Omega T plus Phi.
Dialogue: 0,0:13:08.58,0:13:12.32,Default,,0000,0000,0000,,In this case, Phi is a positive 90 degrees.
Dialogue: 0,0:13:12.90,0:13:18.04,Default,,0000,0000,0000,,Alright, let's just close up\Nhere by giving three examples.
Dialogue: 0,0:13:18.04,0:13:21.92,Default,,0000,0000,0000,,Here we have the time\Ndomain representations
Dialogue: 0,0:13:21.92,0:13:25.72,Default,,0000,0000,0000,,and over here let's go ahead\Nand write the phasor transform,
Dialogue: 0,0:13:25.72,0:13:32.18,Default,,0000,0000,0000,,or the complex exponential way of\Nrepresenting these trigonometric functions.
Dialogue: 0,0:13:32.18,0:13:33.78,Default,,0000,0000,0000,,So, on this first one,
Dialogue: 0,0:13:33.78,0:13:42.07,Default,,0000,0000,0000,,phasor V1 would be equal\Nto five E to the J zero.
Dialogue: 0,0:13:42.07,0:13:45.94,Default,,0000,0000,0000,,But again, E to the zero is one,
Dialogue: 0,0:13:45.94,0:13:48.85,Default,,0000,0000,0000,,so that would simply be five.
Dialogue: 0,0:13:48.85,0:13:56.27,Default,,0000,0000,0000,,Phasor V two would equal\Nfour E to the J 60,
Dialogue: 0,0:13:56.61,0:14:01.80,Default,,0000,0000,0000,,and phasor V three would equal
Dialogue: 0,0:14:01.80,0:14:07.72,Default,,0000,0000,0000,,three E to the J minus 90.
Dialogue: 0,0:14:07.72,0:14:09.56,Default,,0000,0000,0000,,We typically put the minus\Nsign out here in front,
Dialogue: 0,0:14:09.56,0:14:14.84,Default,,0000,0000,0000,,E to the minus J 90 degrees.
Dialogue: 0,0:14:14.84,0:14:18.16,Default,,0000,0000,0000,,I have represented here\Nthe time domain functions
Dialogue: 0,0:14:18.16,0:14:21.02,Default,,0000,0000,0000,,V one V two and V three.\NV one is the blue one.
Dialogue: 0,0:14:21.02,0:14:24.28,Default,,0000,0000,0000,,It goes through a maximum at T\Nequals zero and has an amplitude
Dialogue: 0,0:14:24.28,0:14:29.23,Default,,0000,0000,0000,,of five, no phase shift.
Dialogue: 0,0:14:29.23,0:14:32.70,Default,,0000,0000,0000,,The blue one is v two.
Dialogue: 0,0:14:32.70,0:14:35.41,Default,,0000,0000,0000,,It has an amp I'm sorry the blue\Nand we're just talking about.
Dialogue: 0,0:14:35.41,0:14:37.63,Default,,0000,0000,0000,,The green one has an amplitude of
Dialogue: 0,0:14:37.63,0:14:41.84,Default,,0000,0000,0000,,four volts and it has a\Nphase shift of plus 60 degrees.
Dialogue: 0,0:14:41.84,0:14:46.20,Default,,0000,0000,0000,,So, it's been shifted to\Nthe left by 60 degrees.
Dialogue: 0,0:14:46.20,0:14:48.50,Default,,0000,0000,0000,,Under these circumstances, we\Nwould say that the green one
Dialogue: 0,0:14:48.50,0:14:50.56,Default,,0000,0000,0000,,because it peaks out\Nbefore the blue one does,
Dialogue: 0,0:14:50.56,0:14:55.10,Default,,0000,0000,0000,,we'd say that the green function V\Ntwo leads the blue function,
Dialogue: 0,0:14:55.10,0:14:57.00,Default,,0000,0000,0000,,in this case, by 60 degrees.
Dialogue: 0,0:14:57.00,0:14:59.36,Default,,0000,0000,0000,,Finally, the red function shown here,
Dialogue: 0,0:14:59.36,0:15:01.62,Default,,0000,0000,0000,,corresponds to V three.
Dialogue: 0,0:15:01.62,0:15:08.45,Default,,0000,0000,0000,,It has an amplitude of three volts and\Nit is shifted to the right 90 degrees,
Dialogue: 0,0:15:08.45,0:15:14.62,Default,,0000,0000,0000,,So, the red one lags V one by 90 degrees.
Dialogue: 0,0:15:14.62,0:15:17.66,Default,,0000,0000,0000,,All three of these are oscillating\Nat the same frequency,
Dialogue: 0,0:15:17.66,0:15:19.91,Default,,0000,0000,0000,,Omega is the same for all three of them.
Dialogue: 0,0:15:19.91,0:15:23.52,Default,,0000,0000,0000,,They differ only in amplitude,
Dialogue: 0,0:15:24.22,0:15:28.26,Default,,0000,0000,0000,,and in phase shift.
Dialogue: 0,0:15:31.56,0:15:35.46,Default,,0000,0000,0000,,We're going to use\Nthese complex exponentials,
Dialogue: 0,0:15:35.46,0:15:41.70,Default,,0000,0000,0000,,instead of trigonometric functions\Nas we analyzed these RLC circuits,
Dialogue: 0,0:15:41.70,0:15:44.02,Default,,0000,0000,0000,,because it's a whole lot easier to do
Dialogue: 0,0:15:44.02,0:15:48.17,Default,,0000,0000,0000,,exponential mathematics than\Nit is trigonometry with all of
Dialogue: 0,0:15:48.17,0:15:58.17,Default,,0000,0000,0000,,its identities and complicating\Ntrigonometric properties.