WEBVTT 00:00:00.410 --> 00:00:02.460 >> Hi my name's Lee Brinton and I am 00:00:02.460 --> 00:00:05.120 an electrical engineering instructor at Salt Lake Community College. 00:00:05.120 --> 00:00:10.620 In this video and actually going to turn our attention now to 00:00:10.620 --> 00:00:16.920 analyzing circuits that are driven by sinusoidally varying or AC sources. 00:00:16.920 --> 00:00:20.520 This is what we're going to refer to as the sinusoidal steady state. 00:00:20.520 --> 00:00:23.910 By this, we're going to assume that the system has been 00:00:23.910 --> 00:00:28.620 operating that the sinusoidally varying sources have been applied long 00:00:28.620 --> 00:00:32.630 enough that any transients that might have occurred at startup have 00:00:32.630 --> 00:00:37.400 all died out and we're now operating in the sinusoidal steady state. 00:00:37.400 --> 00:00:41.975 To do this and to analyze these kinds of circuits. 00:00:41.975 --> 00:00:44.045 We're going to use something known as phasors. 00:00:44.045 --> 00:00:47.105 So to begin with we're going to introduce this concept of a phasor. 00:00:47.105 --> 00:00:52.175 Phasor is a complex exponential way of representing trigonometric functions. 00:00:52.175 --> 00:00:54.620 Will then turn our attention to and 00:00:54.620 --> 00:00:57.500 define what we mean by a complex impedance. 00:00:57.500 --> 00:01:01.955 Will then analyze or demonstrate ways of analyzing these RLC circuits. 00:01:01.955 --> 00:01:04.190 Resistors inductors capacitors are circuits that 00:01:04.190 --> 00:01:06.680 have resistors inductors and capacitors in them. 00:01:06.680 --> 00:01:09.290 In the sinusoidal steady-state including 00:01:09.290 --> 00:01:12.245 looking at series connections and voltage division, 00:01:12.245 --> 00:01:14.680 parallel connections and current division. 00:01:14.680 --> 00:01:17.090 We will see how node mesh analysis applies in 00:01:17.090 --> 00:01:20.915 this sinusoidal steady state involving phasors and impedances. 00:01:20.915 --> 00:01:24.110 And then we'll look at the Thevenin equivalency and 00:01:24.110 --> 00:01:28.375 maximum power transfer concepts in this sinusoidal steady state. 00:01:28.375 --> 00:01:31.800 So here we are again in the complex plane. 00:01:32.590 --> 00:01:38.105 Where we have a real axis and an imaginary axis. 00:01:38.105 --> 00:01:40.445 And we know that we can represent points in this plane. 00:01:40.445 --> 00:01:45.755 Now we're going to be using a complex number to represent a voltage. 00:01:45.755 --> 00:01:51.110 So let's now just start out with a complex number. 00:01:51.110 --> 00:01:55.040 We will call it V and say that it has a length which we will 00:01:55.040 --> 00:01:59.555 call V sub M and it has some angle theta. 00:01:59.555 --> 00:02:01.220 So there's theta. 00:02:01.220 --> 00:02:04.670 Again the length of this is V sub M. So it's a complex number. 00:02:04.670 --> 00:02:07.490 The magnitude of the complex number is V sub M 00:02:07.490 --> 00:02:10.460 and the angle associated with it is theta. 00:02:10.460 --> 00:02:13.970 Of course that represents the polar representation of 00:02:13.970 --> 00:02:18.410 this complex number V. We know that another way of writing 00:02:18.410 --> 00:02:21.110 that is to say V sub M E to 00:02:21.110 --> 00:02:27.080 the J theta and we also know that we can express it using Euler's formula or 00:02:27.080 --> 00:02:30.920 simply pointing out that the projection of this point on 00:02:30.920 --> 00:02:36.080 the real axis is equal to V sub M times the cosine of theta. 00:02:36.080 --> 00:02:37.985 So we can write this as 00:02:37.985 --> 00:02:44.780 V sub M, times the cosine of theta gives us the real part. 00:02:44.780 --> 00:02:47.510 And the imaginary part is just V sub M times 00:02:47.510 --> 00:02:50.150 the sine of theta or this can be rewritten as 00:02:50.150 --> 00:02:58.200 V sub M times cosine theta plus J sine of theta. 00:02:58.490 --> 00:03:01.145 Just a little bit of nomenclature here. 00:03:01.145 --> 00:03:09.260 We can say than that the real part of V is equal to 00:03:09.260 --> 00:03:15.130 V sub M cosine of theta and the imaginary part of 00:03:15.130 --> 00:03:22.640 V is equal to V sub M sine theta. 00:03:22.640 --> 00:03:25.345 Alright here's where it starts to get interesting. 00:03:25.345 --> 00:03:28.205 Up until now theta has been a constant value, 00:03:28.205 --> 00:03:31.040 what if we start allowing theta to be 00:03:31.040 --> 00:03:35.390 some function of time and in fact let's begin with by letting 00:03:35.390 --> 00:03:39.470 theta of T equal some constant omega times 00:03:39.470 --> 00:03:44.060 T. Omega is going to have the units of radians per seconds. 00:03:44.060 --> 00:03:46.880 So that when we have radians per second times seconds that gives us 00:03:46.880 --> 00:03:49.730 radians or measure of an angle. 00:03:49.730 --> 00:03:52.880 So sure enough theta is an angle or has the units of 00:03:52.880 --> 00:03:57.240 radians or equivalently degrees. 00:03:57.240 --> 00:04:03.530 By the way, omega is referred to as the radial frequency term. 00:04:03.530 --> 00:04:11.460 So now, let's rewrite our V. V then is equal to V sub M E to the theta, 00:04:11.460 --> 00:04:14.690 but theta is equal to omega T. So it will be E to 00:04:14.690 --> 00:04:20.180 the J theta or E to the J. Omega T. Which again 00:04:20.180 --> 00:04:24.560 using Euler's formula can be written as V sub M times 00:04:24.560 --> 00:04:29.580 the cosine of omega T plus J sine of 00:04:29.580 --> 00:04:36.560 omega T. Given that the real part of V is the projection of 00:04:36.560 --> 00:04:40.520 this point down onto the real axis and given now 00:04:40.520 --> 00:04:45.065 that theta is starting at T equals 0 theta equals 0. 00:04:45.065 --> 00:04:48.605 As T increases, we now see that 00:04:48.605 --> 00:04:54.125 theta is increasing linearly as a function of time. 00:04:54.125 --> 00:04:57.020 Which means that the real part of 00:04:57.020 --> 00:05:01.085 this complex number as theta increases with time. 00:05:01.085 --> 00:05:04.909 The real part of it, which is the projection onto the real axis, 00:05:04.909 --> 00:05:09.845 is going to be changing as a function of time also. 00:05:09.845 --> 00:05:11.960 The real part of this now becomes 00:05:11.960 --> 00:05:14.360 a function of time and if we stop and think about it, 00:05:14.360 --> 00:05:19.100 we know that the projection onto the real axis is just the magnitude of 00:05:19.100 --> 00:05:25.520 V sub M times the cosine of this angle. 00:05:25.520 --> 00:05:31.775 The angle being omega T. We have then that the real part of V, 00:05:31.775 --> 00:05:36.815 writing it the real part of V then is equal to 00:05:36.815 --> 00:05:39.710 V sub M cosine of 00:05:39.710 --> 00:05:45.430 omega T. Hopefully now you'll start to see where we're headed with this. 00:05:45.430 --> 00:05:51.215 This V sub M cosine omega T is the way of numerically or mathematically 00:05:51.215 --> 00:05:54.905 representing a waveform this oscillating 00:05:54.905 --> 00:05:59.705 this cosine is solely at a frequency of omega T seconds. 00:05:59.705 --> 00:06:01.775 We can see graphically what's happening. 00:06:01.775 --> 00:06:07.955 Let's now overlay a time-domain axis coming down like this. 00:06:07.955 --> 00:06:12.155 And let's plot as a function of time the projection on 00:06:12.155 --> 00:06:19.955 the real axis of this point as it goes around at omega radians per second. 00:06:19.955 --> 00:06:24.005 Just draw a couple of lines down here to help me graph is. 00:06:24.005 --> 00:06:26.060 So at T equals 0, theta equals 0, 00:06:26.060 --> 00:06:29.855 the cosine of 0 is one and we start out here at a maximum. 00:06:29.855 --> 00:06:35.930 Now we allow theta to increase and as we do so the projection on the real axis 00:06:35.930 --> 00:06:41.630 decreases until we get to theta equaling pie halves or 90 degrees, 00:06:41.630 --> 00:06:45.485 at which time the real projection equals 0. 00:06:45.485 --> 00:06:53.870 Continuing on through until theta gets around here two pie or a 180 degrees. 00:06:53.870 --> 00:06:56.555 Once again we have a minimum here. 00:06:56.555 --> 00:07:03.180 It comes back crosses as theta gets down here to 270 degrees. 00:07:03.450 --> 00:07:06.700 Back to where we started. 00:07:06.700 --> 00:07:11.680 As Theta goes around and around this graph and continues on and on, 00:07:11.680 --> 00:07:14.050 and you'll see, hopefully, you recognize then. 00:07:14.050 --> 00:07:20.500 Though what we have here is a plot of the function V_M cosine of Omega T. Now, 00:07:20.500 --> 00:07:27.100 this analysis assumed that Theta started at zero radians per second. 00:07:27.100 --> 00:07:33.430 Let's now take a look at what happens if we start our angle at not zero, 00:07:33.430 --> 00:07:36.700 but at some angle Phi. 00:07:36.700 --> 00:07:42.625 In other words, Theta at t equals zero equals Phi. 00:07:42.625 --> 00:07:46.225 Now, when we start this projection coming down here, 00:07:46.225 --> 00:07:47.425 or first of all, 00:07:47.425 --> 00:07:51.340 we then would say that Theta as a function of time and 00:07:51.340 --> 00:07:56.320 is equal to Omega T plus Phi. 00:07:56.320 --> 00:08:02.830 Our complex expression V is equal to then V_M, 00:08:02.830 --> 00:08:05.230 E to the J Theta, 00:08:05.230 --> 00:08:10.480 but Theta is now Omega T plus Phi, 00:08:10.480 --> 00:08:13.420 which again using Euler's formula, 00:08:13.420 --> 00:08:23.880 we can write as V_M times the cosine of Omega T plus Phi, 00:08:23.880 --> 00:08:32.059 plus J sine of Omega T plus Phi. 00:08:32.059 --> 00:08:37.765 We can save in that the real part of V is equal to 00:08:37.765 --> 00:08:44.155 V sub m cosine of Omega T plus Phi, 00:08:44.155 --> 00:08:46.795 a completely arbitrary cosine term, 00:08:46.795 --> 00:08:50.110 specified by its amplitude V_M. 00:08:50.110 --> 00:08:55.435 The rate at which it's oscillating Omega and this arbitrary term, 00:08:55.435 --> 00:08:59.890 or arbitrary offset of Phi radians. 00:08:59.890 --> 00:09:05.935 Let's plot that projection onto our time domain. 00:09:05.935 --> 00:09:08.740 As we did before, only now instead of 00:09:08.740 --> 00:09:11.215 starting at T equals zero with Theta equaling zero, 00:09:11.215 --> 00:09:14.665 we start at T equals zero at Phi, 00:09:14.665 --> 00:09:19.585 and again dropping these down to just help me draw this. 00:09:19.585 --> 00:09:22.000 We're starting here at this point. 00:09:22.000 --> 00:09:27.250 It goes to zero when Theta gets to zero or to 90 degrees. 00:09:27.250 --> 00:09:29.830 By the time Theta is around here to Pi, 00:09:29.830 --> 00:09:30.985 we're down here to minor, 00:09:30.985 --> 00:09:34.540 or the negative coming back this way. 00:09:35.660 --> 00:09:39.000 Once again, we have a cosine waveform, 00:09:39.000 --> 00:09:46.360 but it's a cosine waveform with a Phi degree or Phi radian shift. 00:09:46.770 --> 00:09:49.120 Let's take this term, 00:09:49.120 --> 00:09:51.490 and rewrite it using properties of exponents 00:09:51.490 --> 00:09:54.970 to take our analysis just another step further. 00:09:54.970 --> 00:10:00.265 Thus, we can say then that V is equal to V_M, 00:10:00.265 --> 00:10:07.435 E to the J Omega T times E to the J Phi. 00:10:07.435 --> 00:10:11.965 Just this the multiple property of exponents. 00:10:11.965 --> 00:10:14.275 Now, let's rewrite this, 00:10:14.275 --> 00:10:22.510 as V_M E to the J Phi times E to the J Omega T. As we do that, 00:10:22.510 --> 00:10:26.750 we see that this term here is the time-dependent. 00:10:27.390 --> 00:10:31.510 The time-dependent term, and it varies as 00:10:31.510 --> 00:10:34.390 Omega T. But we also see 00:10:34.390 --> 00:10:37.810 that we have these two terms here which are constants. 00:10:37.810 --> 00:10:40.480 In fact, we have a V_M E to the J Phi. 00:10:40.480 --> 00:10:44.545 We have another complex number involving the amplitude of 00:10:44.545 --> 00:10:50.215 the cosine wave and this arbitrary phase shift. 00:10:50.215 --> 00:10:53.350 It turns out in RLC circuits, 00:10:53.350 --> 00:10:55.060 linear circuits, in general, 00:10:55.060 --> 00:10:59.530 that are driven by sinusoidally varying wave forms, 00:10:59.530 --> 00:11:03.505 that the frequency Omega doesn't change. 00:11:03.505 --> 00:11:07.300 In other words, every current and voltage in a circuit 00:11:07.300 --> 00:11:11.230 that's driven will be oscillating at the same frequency. 00:11:11.230 --> 00:11:12.820 The only thing that changes, 00:11:12.820 --> 00:11:14.200 the only differences between 00:11:14.200 --> 00:11:17.380 various currents and voltages within this waveform are 00:11:17.380 --> 00:11:23.410 the amplitude and the phase of those currents and voltages. 00:11:23.410 --> 00:11:25.540 This term here is so important. 00:11:25.540 --> 00:11:32.395 We give it a where we create a new complex number called phasor V, 00:11:32.395 --> 00:11:38.120 and phasor V then is simply equal to this V_M E to the J Phi. 00:11:42.530 --> 00:11:44.910 I'll say it one final time. 00:11:44.910 --> 00:11:47.940 Phasor V is a complex number. 00:11:47.940 --> 00:11:52.934 Its magnitude V_M is the amplitude 00:11:52.934 --> 00:11:55.080 of the cosine wave that we're wanting to 00:11:55.080 --> 00:11:57.984 represent in terms of complex exponential, 00:11:57.984 --> 00:12:05.950 and its angle Phi is the arbitrary offset or the phase shift of that function. 00:12:05.950 --> 00:12:09.100 Lets just look graphically at what we're talking about then. 00:12:09.100 --> 00:12:12.910 Here is a cosine wave form that goes through its maximum at T equals zero, 00:12:12.910 --> 00:12:15.115 in other words, it has no phase shift. 00:12:15.115 --> 00:12:19.270 We would arrive in the phasor representation of this would 00:12:19.270 --> 00:12:24.160 be V_M E to the J zero. 00:12:24.160 --> 00:12:25.720 Well, E to the zero is one. 00:12:25.720 --> 00:12:30.365 So, this is simply equal to V_M. 00:12:30.365 --> 00:12:35.200 For a wave form, it's got some finite or some shift to it. 00:12:35.200 --> 00:12:39.745 We would write it phasor V then is equal to V_M, 00:12:39.745 --> 00:12:43.285 E to the J Phi. 00:12:43.285 --> 00:12:45.400 So, this thing right here, 00:12:45.400 --> 00:12:49.060 this complex exponential term is going 00:12:49.060 --> 00:12:52.870 to represent a waveform in this case that is 00:12:52.870 --> 00:13:02.230 shifted Phi degrees away from the original or the non shifted waveform. 00:13:02.230 --> 00:13:08.575 This blue one is cosine Omega T. The red one is cosine Omega T plus Phi. 00:13:08.575 --> 00:13:12.320 In this case, Phi is a positive 90 degrees. 00:13:12.900 --> 00:13:18.040 Alright, let's just close up here by giving three examples. 00:13:18.040 --> 00:13:21.925 Here we have the time domain representations 00:13:21.925 --> 00:13:25.720 and over here let's go ahead and write the phasor transform, 00:13:25.720 --> 00:13:32.185 or the complex exponential way of representing these trigonometric functions. 00:13:32.185 --> 00:13:33.775 So, on this first one, 00:13:33.775 --> 00:13:42.070 phasor V1 would be equal to five E to the J zero. 00:13:42.070 --> 00:13:45.940 But again, E to the zero is one, 00:13:45.940 --> 00:13:48.850 so that would simply be five. 00:13:48.850 --> 00:13:56.270 Phasor V two would equal four E to the J 60, 00:13:56.610 --> 00:14:01.795 and phasor V three would equal 00:14:01.795 --> 00:14:07.720 three E to the J minus 90. 00:14:07.720 --> 00:14:09.565 We typically put the minus sign out here in front, 00:14:09.565 --> 00:14:14.845 E to the minus J 90 degrees. 00:14:14.845 --> 00:14:18.160 I have represented here the time domain functions 00:14:18.160 --> 00:14:21.025 V one V two and V three. V one is the blue one. 00:14:21.025 --> 00:14:24.280 It goes through a maximum at T equals zero and has an amplitude 00:14:24.280 --> 00:14:29.230 of five, no phase shift. 00:14:29.230 --> 00:14:32.695 The blue one is v two. 00:14:32.695 --> 00:14:35.410 It has an amp I'm sorry the blue and we're just talking about. 00:14:35.410 --> 00:14:37.630 The green one has an amplitude of 00:14:37.630 --> 00:14:41.840 four volts and it has a phase shift of plus 60 degrees. 00:14:41.840 --> 00:14:46.195 So, it's been shifted to the left by 60 degrees. 00:14:46.195 --> 00:14:48.500 Under these circumstances, we would say that the green one 00:14:48.500 --> 00:14:50.560 because it peaks out before the blue one does, 00:14:50.560 --> 00:14:55.100 we'd say that the green function V two leads the blue function, 00:14:55.100 --> 00:14:57.005 in this case, by 60 degrees. 00:14:57.005 --> 00:14:59.355 Finally, the red function shown here, 00:14:59.355 --> 00:15:01.620 corresponds to V three. 00:15:01.620 --> 00:15:08.450 It has an amplitude of three volts and it is shifted to the right 90 degrees, 00:15:08.450 --> 00:15:14.615 So, the red one lags V one by 90 degrees. 00:15:14.615 --> 00:15:17.660 All three of these are oscillating at the same frequency, 00:15:17.660 --> 00:15:19.910 Omega is the same for all three of them. 00:15:19.910 --> 00:15:23.520 They differ only in amplitude, 00:15:24.220 --> 00:15:28.260 and in phase shift. 00:15:31.560 --> 00:15:35.460 We're going to use these complex exponentials, 00:15:35.460 --> 00:15:41.695 instead of trigonometric functions as we analyzed these RLC circuits, 00:15:41.695 --> 00:15:44.015 because it's a whole lot easier to do 00:15:44.015 --> 00:15:48.170 exponential mathematics than it is trigonometry with all of 00:15:48.170 --> 00:15:58.170 its identities and complicating trigonometric properties.