WEBVTT 00:00:00.020 --> 00:00:04.094 >> We'll do that same example now only with 00:00:04.094 --> 00:00:08.370 the actual values for the circuit elements and the source here. 00:00:08.370 --> 00:00:10.560 So again, the first thing we do is convert 00:00:10.560 --> 00:00:13.800 the time-domain source to its phasor representation 00:00:13.800 --> 00:00:19.020 and then calculate the impedances associated with the capacitor, 00:00:19.020 --> 00:00:20.625 resistance and inductor here. 00:00:20.625 --> 00:00:24.800 So this the phasor representation V we'll call it 00:00:24.800 --> 00:00:31.110 cap V or phasor V is equal to 10E to the J30. 00:00:31.360 --> 00:00:33.920 Now, let's be explicit here. 00:00:33.920 --> 00:00:37.710 Omega is 1,000 radians per second. 00:00:38.200 --> 00:00:41.345 Radians per second. 00:00:41.345 --> 00:00:46.790 Z_C is equal to one over J Omega C which is equal to one over 00:00:46.790 --> 00:00:49.190 J times Omega which is 00:00:49.190 --> 00:00:53.915 1,000 times C which is 10 times 10 to the minus six to six micro, 00:00:53.915 --> 00:01:02.550 10 microfarads and that then equals negative J 100 Ohms. 00:01:02.550 --> 00:01:06.940 That's a negative J 100 Ohms. 00:01:06.940 --> 00:01:10.670 Z_R is just a value of the resistor which is 00:01:10.670 --> 00:01:16.610 50 Ohms and Z_L is equal to J Omega L which is equal to J times 00:01:16.610 --> 00:01:19.670 Omega which is 1,000 times L which is 00:01:19.670 --> 00:01:27.075 75 millihenries and that gives us a positive J 75. 00:01:27.075 --> 00:01:30.515 So this is a J 75 there. 00:01:30.515 --> 00:01:36.455 This is R which is equal to the impedance that is equal to 50 ohms and this, 00:01:36.455 --> 00:01:42.395 the impedance of the capacitor is a negative J 100. 00:01:42.395 --> 00:01:45.350 We'll go ahead and define the current 00:01:45.350 --> 00:01:47.870 flowing through here or the phasor representation of the current going 00:01:47.870 --> 00:01:51.319 through there call it phasor I and let's 00:01:51.319 --> 00:01:56.465 write a Kirchhoff's voltage law equation going around that loop. 00:01:56.465 --> 00:01:59.695 Once again, adding up the voltage drops. 00:01:59.695 --> 00:02:02.150 So, a voltage increase will be a negative, 00:02:02.150 --> 00:02:10.250 that would be a negative 10 E to the J 30 plus going across this capacitor, 00:02:10.250 --> 00:02:12.290 the voltage drop across that capacitor will 00:02:12.290 --> 00:02:14.780 be the current I times the impedance. 00:02:14.780 --> 00:02:17.660 So, that will be I times the impedance which is 00:02:17.660 --> 00:02:24.050 a negative J 100 plus the voltage drop across the resistor which is going 00:02:24.050 --> 00:02:30.575 to be I times 50 plus the voltage drop across that inductor is going to be 00:02:30.575 --> 00:02:33.320 I times the impedance of that inductor which is 00:02:33.320 --> 00:02:38.835 J 75 and the sum of those terms then equals zero. 00:02:38.835 --> 00:02:42.075 We want to solve for I the current in this circuit. 00:02:42.075 --> 00:02:45.450 So we'll factor out the I from these common terms 00:02:45.450 --> 00:02:50.300 here and bring this term here on over other side and we get then 00:02:50.300 --> 00:02:57.135 that I times a negative J 100 plus 50 00:02:57.135 --> 00:03:06.870 plus J 75 is equal to a positive 10 E to the J 30. 00:03:06.870 --> 00:03:12.845 We have a negative J 100 plus J 75 that gives us a negative J 25 plus 50. 00:03:12.845 --> 00:03:17.570 Dividing both sides by that term then gives us that I is equal 00:03:17.570 --> 00:03:23.830 to 10 E to the J 30 divided by, 00:03:23.830 --> 00:03:30.065 writing the real part first 50 minus J 25. 00:03:30.065 --> 00:03:32.270 You get out your calculator. 00:03:32.270 --> 00:03:37.850 First of all, let's convert just this denominator here to polar form. 00:03:37.850 --> 00:03:40.715 It's a little bit easier to see what's going on here if we do so. 00:03:40.715 --> 00:03:48.230 In polar form, that's equal to 55.9 E 00:03:48.230 --> 00:04:00.650 to negative J 26.57. 00:04:00.650 --> 00:04:06.170 Now, we can see there the division, 00:04:06.170 --> 00:04:13.550 it's 10 E to the J 30 divided by 55.9 E to the minus J 26.57. 00:04:13.550 --> 00:04:18.399 So 10 divided by 55.9 gives us the magnitude. 00:04:18.399 --> 00:04:20.630 The magnitude of the numerator divided by the magnitude of 00:04:20.630 --> 00:04:25.815 the denominator gives us 0.179. 00:04:25.815 --> 00:04:28.145 Now, the phase E, 00:04:28.145 --> 00:04:31.245 it will be E to the J 30 in the numerator, 00:04:31.245 --> 00:04:35.515 E to the minus J 26.57 in the denominator. 00:04:35.515 --> 00:04:41.150 We've already demonstrated that when dividing numbers involving exponents, 00:04:41.150 --> 00:04:45.830 it's the exponent of the numerator minus the exponent in the denominator. 00:04:45.830 --> 00:04:51.320 So it's J 30 minus and minus it becomes plus a 26.57 that 00:04:51.320 --> 00:05:01.815 gives us a positive J 56.57 degrees. 00:05:01.815 --> 00:05:08.455 This entire thing here is the phasor representation of the current. 00:05:08.455 --> 00:05:11.150 We can now take this back into the time domain 00:05:11.150 --> 00:05:13.400 by recognizing that the amplitude of the current 00:05:13.400 --> 00:05:17.870 is going to be the magnitude of this phasor or let's be explicit now. 00:05:17.870 --> 00:05:24.310 I of T then is equal to amplitude of 0.179. 00:05:24.670 --> 00:05:27.380 It will be a cosine term. 00:05:27.380 --> 00:05:32.075 We came in as a cosine will come back to the time domain as a cosine term. 00:05:32.075 --> 00:05:35.480 Cosine, the frequency doesn't change that's still 00:05:35.480 --> 00:05:42.335 1,000 T. But now the phase of the current is equal to 56.57 degrees. 00:05:42.335 --> 00:05:49.560 So plus 56.57 degrees. 00:05:49.560 --> 00:05:56.315 That then is the time domain expression for the current. 00:05:56.315 --> 00:05:59.765 The voltage, the source driving this circuit is oscillating at 00:05:59.765 --> 00:06:03.980 1,000 radians per second and the current is oscillating 1,000 times per second. 00:06:03.980 --> 00:06:05.630 It's going back and forth changing at 00:06:05.630 --> 00:06:08.905 the same frequency that the source is going into it. 00:06:08.905 --> 00:06:11.585 The source has an amplitude of 10 volts. 00:06:11.585 --> 00:06:17.900 The current flowing has an amplitude of 0.179 Amps. 00:06:17.900 --> 00:06:20.630 So, the amplitude of the current is smaller than the amplitude of 00:06:20.630 --> 00:06:24.440 the voltage and finally let's look at the phase. 00:06:24.440 --> 00:06:30.170 The phase angle of the voltage source is 30 degrees, positive 30 degrees. 00:06:30.170 --> 00:06:35.920 The angle of the current is 56.57 degrees. 00:06:35.920 --> 00:06:40.910 So, we'd say then that the current has been shifted ahead 00:06:40.910 --> 00:06:47.640 of the source by the 26.57 degrees. 00:06:49.550 --> 00:06:51.990 Here are plots of those, 00:06:51.990 --> 00:06:53.895 the voltage in the red, 00:06:53.895 --> 00:07:04.050 V of T is equal to 10 cosine of 1,000 T plus 30 degrees. 00:07:04.050 --> 00:07:09.725 I of T we have now found to be equal to 0.179, 00:07:09.725 --> 00:07:20.265 cosine of 1,000 T plus 56.57 degrees. 00:07:20.265 --> 00:07:22.640 So here's the voltage. It's got an amplitude of 00:07:22.640 --> 00:07:28.670 10 volts and it has been shifted to the left by 30 degrees. 00:07:28.670 --> 00:07:31.940 The current has an amplitude of 00:07:31.940 --> 00:07:39.080 0.179 and it has been shifted to the left 56.57 degrees. 00:07:39.080 --> 00:07:42.740 So the difference, I can draw this very well. 00:07:42.740 --> 00:07:44.770 I don't know if I can or not. 00:07:44.770 --> 00:07:47.870 That distance right there is 00:07:47.870 --> 00:07:52.430 a phase shift of the current ahead of the voltage. 00:07:52.430 --> 00:07:55.429 We say that the current is leading the voltage, 00:07:55.429 --> 00:07:59.140 the current peaks out before the voltage peaks. 00:07:59.140 --> 00:08:02.690 The current crosses the zero before the voltage crosses the zero. 00:08:02.690 --> 00:08:06.710 The current is leading the voltage by that 00:08:06.710 --> 00:08:11.285 much which corresponds to again that 26.57 degrees. 00:08:11.285 --> 00:08:17.600 Just one note about what appears to be an inconsistency of units. 00:08:17.600 --> 00:08:19.900 Not only does it appear to be an inconsistency of units, 00:08:19.900 --> 00:08:21.560 it isn't inconsistency of units. 00:08:21.560 --> 00:08:25.850 What are the units of omega, the radial frequency? 00:08:25.850 --> 00:08:31.220 That's 1,000 radians per second. 00:08:31.220 --> 00:08:36.200 Yet we're specifying the phase shift of 30 degrees. 00:08:36.200 --> 00:08:38.485 We're specifying it in degrees. 00:08:38.485 --> 00:08:41.669 To be consistent and frankly, 00:08:41.669 --> 00:08:44.450 we're going to leave it in this form I think simply because it's 00:08:44.450 --> 00:08:47.270 a lot easier for us to visualize. 00:08:47.270 --> 00:08:49.835 We have a better intuitive feel for 00:08:49.835 --> 00:08:53.490 phase shifts in terms of degrees than we do radians but if wherever we're 00:08:53.490 --> 00:08:57.110 doing calculations we'll need to convert E to the this into 00:08:57.110 --> 00:09:01.310 degrees per second or change this to radians. 00:09:01.310 --> 00:09:04.115 Each course the change of radius is going to just be simply 00:09:04.115 --> 00:09:10.520 the 30 degrees times Pi divided by 180. 00:09:10.520 --> 00:09:12.560 So that phase shift there, 00:09:12.560 --> 00:09:14.870 the 30 degree phase shift represents 00:09:14.870 --> 00:09:18.680 Pi sixth phase there 00:09:18.680 --> 00:09:21.650 and somebody could do the same for the phase shift and the current. 00:09:21.650 --> 00:09:23.735 But the point is let's just drive this home. 00:09:23.735 --> 00:09:26.660 This circuit here, the circuit that we just 00:09:26.660 --> 00:09:31.100 analyzed did not change the frequency. 00:09:31.100 --> 00:09:33.200 The frequency of the current flowing in 00:09:33.200 --> 00:09:35.720 here is the same as the frequency of the voltage. 00:09:35.720 --> 00:09:39.085 What did change was the amplitude and the phase. 00:09:39.085 --> 00:09:42.710 Using phasor analysis we were able to determine the phase 00:09:42.710 --> 00:09:47.330 on the amplitude and phase of the current.