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