1 00:00:00,012 --> 00:00:04,742 Well, in this video, we're going to break out of circuits consisting only of 2 00:00:04,742 --> 00:00:08,788 resistors and sources. We're going to talk about more generally 3 00:00:08,788 --> 00:00:14,469 circuits having capacitors and inductors. this will open the door so we can handle 4 00:00:14,469 --> 00:00:18,275 any circuit that we want. Now, if we go down the path we went 5 00:00:18,275 --> 00:00:23,095 through with resistive circuits, when we use the VR relations, KVL and 6 00:00:23,095 --> 00:00:28,290 KCL, it turns out, what you're going to wind up with, is, a differential equation 7 00:00:28,290 --> 00:00:30,242 to solve. I'll show you that. 8 00:00:30,242 --> 00:00:35,389 And that turns out out does not give us much insight into how the circuit is 9 00:00:35,389 --> 00:00:40,486 working, what they're doing, why were they designed this way, how do I change 10 00:00:40,486 --> 00:00:45,237 designs. We're going to introduce a new idea, something called impedance, 11 00:00:45,237 --> 00:00:49,894 which is going to allow us to think about,every element in the circuit as 12 00:00:49,894 --> 00:00:54,211 being a kind of resistor. That means we can use current divider, 13 00:00:54,211 --> 00:00:59,354 voltage divider all the techniques we've already learned with resistors. 14 00:00:59,354 --> 00:01:02,372 That's the point. Let's see how that works. 15 00:01:02,372 --> 00:01:07,832 So, here's a simple circuit. It's what's known in the trade as an RC 16 00:01:07,832 --> 00:01:11,645 circuit. And that terminology means it has one 17 00:01:11,645 --> 00:01:17,766 resistor and one capacitor usually. And here, they're connected in series. 18 00:01:17,766 --> 00:01:22,766 So, the source here is voltage of whatever it may be. 19 00:01:22,766 --> 00:01:28,466 And I've decided to take the output to be the voltage across the capacitor. 20 00:01:28,466 --> 00:01:31,496 Okay. So, what we want to do is find V out for 21 00:01:31,496 --> 00:01:36,055 any V in, that's the thing we have to do to solve the circuit. 22 00:01:36,055 --> 00:01:40,642 Well, let me remind you what the VR relation is for the capacitor. 23 00:01:40,642 --> 00:01:45,062 The current is proportional to the derivative of the voltage. 24 00:01:45,062 --> 00:01:50,790 So, we need to see how that comes in to solve the circuit. 25 00:01:50,790 --> 00:01:58,308 So remember, that current i has to be, for this VR relation, has to be going in 26 00:01:58,308 --> 00:02:05,882 the positive side of voltage, and V is now V out for this circuit, that's what 27 00:02:05,882 --> 00:02:10,863 we're playing with. So, what I'm going to do is write KVL. 28 00:02:10,863 --> 00:02:14,960 I'm going to start here and go around like this. 29 00:02:14,960 --> 00:02:20,823 And we know from KCL that the current through the resistor is also i. 30 00:02:20,823 --> 00:02:25,082 So, when I write the KVL equation, V out is equal to, 31 00:02:25,082 --> 00:02:31,387 but I want to go plus to minus this way. And that's voltage in that direction is 32 00:02:31,387 --> 00:02:34,447 -Ri, okay? From the VR relation. 33 00:02:34,447 --> 00:02:40,442 The current's going in the negative side not the positive side, 34 00:02:40,442 --> 00:02:51,470 we have to flip the sign. Plus the, well now, I'm going to substitute for i 35 00:02:51,470 --> 00:02:56,055 in there and you get -RC dV out, dt + V in. 36 00:02:59,114 --> 00:03:06,056 Okay. So, if I bring the derivative over to the other side, 37 00:03:06,056 --> 00:03:11,235 we wind up with the input/output relationship for this circuit. 38 00:03:11,235 --> 00:03:14,943 And it's what's known as an implicit relationship. 39 00:03:14,943 --> 00:03:18,944 This is the relationship you have to solve something. 40 00:03:18,944 --> 00:03:24,544 It isn't directly given to you like it was for the resistor circuits. 41 00:03:24,544 --> 00:03:28,950 So, RC times the derivative V out, plus V out has got to be V in. 42 00:03:28,950 --> 00:03:32,876 So now, what you have to do is you tell me what V in is. 43 00:03:32,876 --> 00:03:38,117 It could be a sinusoid, pulse, whatever. And now, I have to solve this 44 00:03:38,117 --> 00:03:43,627 differential equation for each case. Well, it turns out that's a little 45 00:03:43,627 --> 00:03:47,850 tedious and doesn't really tell us what's going on. 46 00:03:47,850 --> 00:03:50,727 So, I'm going to take a different approach. 47 00:03:50,727 --> 00:03:55,732 We're, so we're not going to solve differential equations in this course. 48 00:03:55,732 --> 00:04:00,709 We're going to sneak around them and figure out a better way to solve them. 49 00:04:00,709 --> 00:04:07,913 So, this is a very different approach which is based on a somewhat surprising 50 00:04:07,913 --> 00:04:13,242 assumption. Just suppose, just suppose the voltage 51 00:04:13,242 --> 00:04:20,188 source is a complex exponential. So, V in of t is equal to capital V in, 52 00:04:20,188 --> 00:04:27,980 this is known as the complex amplitude. Could be a complex number, that is just a 53 00:04:27,980 --> 00:04:32,202 number. The only pa, par, part it varies with 54 00:04:32,202 --> 00:04:37,453 time is the complex exponential. You do the j 2 pi ft. 55 00:04:37,453 --> 00:04:41,307 Okay. So, we're just going to assume that for 56 00:04:41,307 --> 00:04:44,687 right now. Well, here's the claim. 57 00:04:44,687 --> 00:04:51,412 That every voltage in the circuit, once you solve it for, for the complex 58 00:04:51,412 --> 00:04:56,503 exponential source, also turning out to be complex exponentials. 59 00:04:56,503 --> 00:05:00,279 Every voltage and every currents at that form. 60 00:05:00,279 --> 00:05:05,975 So, what I want to do now, is check to see if such a solution is consistent with 61 00:05:05,975 --> 00:05:09,182 the VI relations, and with KVL and KCL. 62 00:05:09,182 --> 00:05:14,737 So, let's do the VI relations. So let's start with our resistor with a 63 00:05:14,737 --> 00:05:20,487 VI relation whose voltage is equal to the resistance times the current. 64 00:05:20,487 --> 00:05:25,722 And I'm just going to plug in the voltage and the current in here. 65 00:05:25,722 --> 00:05:31,370 And it turns out it is consistent as long as the camp, complex samplitude of the 66 00:05:31,370 --> 00:05:36,144 voltage is relate, is R times the complex amplitude of the current. 67 00:05:36,144 --> 00:05:41,837 The other thing to notice is, I can cancel these complex exponentials. And 68 00:05:41,837 --> 00:05:46,486 there is explicitly relationship between the current and voltage complex 69 00:05:46,486 --> 00:05:52,023 amplitudes. This is an interesting point about this 70 00:05:52,023 --> 00:05:57,139 cancellation. So, let's talk about a capacitor, i = C 71 00:05:57,139 --> 00:06:00,301 dv / dt. So, here's the current i. 72 00:06:00,301 --> 00:06:07,695 And the derivative of this is just j 2 pi f times the same thing because of the 73 00:06:07,695 --> 00:06:13,787 properties of the exponential. And we need to plug in the capacitor 74 00:06:13,787 --> 00:06:21,012 values because that's what's over here. So again, with cancellation, we now find 75 00:06:21,012 --> 00:06:28,397 that this assumption is consistent as long as the amplitudes are related this 76 00:06:28,397 --> 00:06:32,347 way. The amplitude is current = c * j2pif 77 00:06:32,347 --> 00:06:37,762 times the amplitude of the voltage. And I'm pretty sure you can believe that 78 00:06:37,762 --> 00:06:41,761 for the inductors, some other kind of thing is going to happen. 79 00:06:41,761 --> 00:06:46,697 Amplitude is related by a different formula, but still they're related. 80 00:06:46,697 --> 00:06:51,709 Okay. This cancellation's really important. Because let's think about 81 00:06:51,709 --> 00:06:57,647 defining a quantity Z for each element that is the ratio of the complex 82 00:06:57,647 --> 00:07:04,048 amplitude of the voltage divided by the complex amplitude of the current. 83 00:07:04,048 --> 00:07:09,690 This is known as impedance. So, we're assuming that the source is a 84 00:07:09,690 --> 00:07:15,589 complex exponential at some frequency. And we're going to, we now know that 85 00:07:15,589 --> 00:07:20,774 every voltage and current in the circuit is also a complex exponential. 86 00:07:20,774 --> 00:07:26,171 Well, let's sort of hide the complex exponential for a while and just look at 87 00:07:26,171 --> 00:07:30,764 the complex amplitudes. We're going to define Z, the impedance to 88 00:07:30,764 --> 00:07:35,997 be V/I. I want to point out that Volt, V as in 89 00:07:35,997 --> 00:07:40,578 Volts, and I as in Ampere, so the ratio is Ohms, 90 00:07:40,578 --> 00:07:44,636 okay? So, impedence has got units of ohms. 91 00:07:44,636 --> 00:07:51,096 However, for the capacity and the inductor, they're complex valued. 92 00:07:51,096 --> 00:07:56,907 So, the impedience of a resistor is just the resistance, 93 00:07:56,907 --> 00:08:05,202 but the impedance of the capacitor is 1/j2pifc, impedance of the inductor is 94 00:08:05,202 --> 00:08:10,037 j2pifl. Alright? So, what this means is that we 95 00:08:10,037 --> 00:08:16,364 can think about every element in the circuit, resistors, capacitors, 96 00:08:16,364 --> 00:08:20,244 inductors, as being complex valued resistors. 97 00:08:20,244 --> 00:08:25,729 Which is going to be the key thing. But we've now just checked the VI 98 00:08:25,729 --> 00:08:29,843 relations. We've got to check KVL and KCL before we 99 00:08:29,843 --> 00:08:33,563 get to figure out the consequences of that. 100 00:08:33,563 --> 00:08:39,011 So, it looks, it's looking good. But when you look at KVL and KCL, you see 101 00:08:39,011 --> 00:08:44,500 that everything hangs together again. Because if every voltage in the circuit 102 00:08:44,500 --> 00:08:48,432 is a complex exponential always at the same frequency, 103 00:08:48,432 --> 00:08:54,641 but they have different amplitudes. you can cancel that common complex 104 00:08:54,641 --> 00:09:01,197 exponential to find the result that the complex amplitude of the voltage around 105 00:09:01,197 --> 00:09:04,355 any loop equal zero, which means it obeys KVL. 106 00:09:04,355 --> 00:09:10,761 And same thing for the currents, the complex amplitudes of the currents 107 00:09:10,761 --> 00:09:15,515 obey KCL. So, we now have a very consistent picture. 108 00:09:15,515 --> 00:09:21,608 When the sources are complex exponentials, so they look like the V in 109 00:09:21,608 --> 00:09:26,895 we've just had, it turns out all the that we had just discovered that all currents 110 00:09:26,895 --> 00:09:31,483 and voltages are complex exponentials. If you had solved that differential 111 00:09:31,483 --> 00:09:36,029 equation for the input voltage being a complex exponential, you would have 112 00:09:36,029 --> 00:09:39,763 discovered that if you assumed V out was complex exponential, 113 00:09:39,763 --> 00:09:43,862 yeah, you get, you get that answer. It's a consistent solution. 114 00:09:43,862 --> 00:09:47,148 Once you have a solution to a differential equation. 115 00:09:47,148 --> 00:09:50,741 No matter how you got it, you know it's the only solution. 116 00:09:50,741 --> 00:09:53,538 That's the way differential equations work. 117 00:09:53,538 --> 00:09:56,178 So, all sources and elements, and element 118 00:09:56,178 --> 00:10:01,314 voltages and currents, can be considered just to be complex amplitudes because 119 00:10:01,314 --> 00:10:05,207 we're going to assume the source frequency is understood. 120 00:10:05,207 --> 00:10:09,532 We're going to assume the complex exponential is setting there, 121 00:10:09,532 --> 00:10:14,832 that's the cancellation I mentioned. And when we do that, all elements can be 122 00:10:14,832 --> 00:10:21,302 considered as complex value resistors the technical word is they are impedance. 123 00:10:21,302 --> 00:10:28,050 That is really makes our life a whole lot easier as I can, let's go back to our RC 124 00:10:28,050 --> 00:10:31,258 circuit. So, we're going to talk about our RC 125 00:10:31,258 --> 00:10:34,892 circuit. And, instead of it being out of real 126 00:10:34,892 --> 00:10:41,202 world, we're going to consider it as being built out of impedances. 127 00:10:41,202 --> 00:10:47,487 So, be sure you notice something in the 128 00:10:47,487 --> 00:10:54,120 notation. Over here, this is a lowercase v in of t. 129 00:10:54,120 --> 00:11:04,380 We're assuming that's v in e^j 2pift. This is a lowercase b which is going to 130 00:11:04,380 --> 00:11:11,973 be something which we know is going to be an uppercase V out e^j 2pift. 131 00:11:11,973 --> 00:11:16,897 So these are uppercase, and these are lowercase. 132 00:11:16,897 --> 00:11:23,817 So, what you do is you think about everything as being a impedance of the 133 00:11:23,817 --> 00:11:28,412 resistors, capacitors, inductors, are impedance's. 134 00:11:28,412 --> 00:11:34,367 The source is a constant given by the complex amplitude of the complex 135 00:11:34,367 --> 00:11:37,727 exponential back here. So, time is gone. 136 00:11:37,727 --> 00:11:41,277 There is no time variation in this circuit, 137 00:11:41,277 --> 00:11:46,980 and we have a V out. And now, we want to figure out what the amplitude of the 138 00:11:46,980 --> 00:11:50,193 output is given the amplitude of the input. 139 00:11:50,193 --> 00:11:54,925 Well, now there are things resistor we can use voltage dividers. 140 00:11:54,925 --> 00:12:00,554 What could be simpler? You're going to use the, instead of resistor values, 141 00:12:00,554 --> 00:12:04,782 we're going to use impedence values in voltage divider. 142 00:12:04,782 --> 00:12:10,572 So, the voltage divider says that the output voltage is equal to the impedance 143 00:12:10,572 --> 00:12:16,302 of the capacitor divided by the sum of the impedances of the resistor and the 144 00:12:16,302 --> 00:12:19,757 capacitor. Just like we had before with the 145 00:12:19,757 --> 00:12:23,619 resistors. Plug in what your impedances are and 146 00:12:23,619 --> 00:12:29,924 simplify a little bit, we now know what V out is. 147 00:12:29,924 --> 00:12:38,866 It is equal to the amplitude of source times a complex number, 1/j2pifRC + 1. 148 00:12:38,866 --> 00:12:46,522 So, what could be simplier? Now to complete the story, we now know that this 149 00:12:46,522 --> 00:12:52,987 is true. We put back in the complex exponentials, we will uncancel the 150 00:12:52,987 --> 00:13:00,267 complex exponential that was lurking in the setup from the very beginning. And 151 00:13:00,267 --> 00:13:07,129 now, we know that when the input is a complex exponential, the output is also a 152 00:13:07,129 --> 00:13:11,851 complex exponential where V out is equal to this quantity. 153 00:13:11,851 --> 00:13:18,441 We have solved the circuit for the case of a complex exponential source. 154 00:13:18,441 --> 00:13:22,971 That's the way the impedance gain works if you will. 155 00:13:22,971 --> 00:13:29,788 So, when the circuit consists of sources and any number of resistors, capacitors, 156 00:13:29,788 --> 00:13:36,512 inductors, it doesn't really matter. you can pretend the sources are complex 157 00:13:36,512 --> 00:13:43,211 exponentials having some frequency, f. Many consider each element to be an 158 00:13:43,211 --> 00:13:48,777 impedance, a complex value resistor. And now you can use all the tools we've 159 00:13:48,777 --> 00:13:52,364 already had, voltage divider, current divider, 160 00:13:52,364 --> 00:13:58,511 series/parallel rules, to figure out how the output's complex amplitude is related 161 00:13:58,511 --> 00:14:03,876 to the complex amplitude of the source. Okay, so I'm going to show you the 162 00:14:03,876 --> 00:14:09,240 consequences of this. You may be a little concerned that what about this pretend 163 00:14:09,240 --> 00:14:14,531 part? suppose my source isn't a complex exponential, which is usually the case. 164 00:14:14,531 --> 00:14:19,017 You're not going to, you're going to have a hard time generating a complex 165 00:14:19,017 --> 00:14:25,374 exponential with any real world source. Well, it turns out we're not really going 166 00:14:25,374 --> 00:14:29,726 down a blind alley. This is going to open the door to a whole 167 00:14:29,726 --> 00:14:35,371 new way of thinking about things, and we'll get that in succeeding videos.