0:00:01.270,0:00:04.930 Hello, this is Dr.[br]Cynthia Furse from the University of Utah. 0:00:04.930,0:00:07.430 Today we're going to be talking[br]about Equivalent Circuits. 0:00:09.960,0:00:12.510 First we're going to talk about[br]what an equivalent circuit is. 0:00:12.510,0:00:14.900 It's basically a circuit that[br]gives you the same voltage and 0:00:14.900,0:00:17.650 current that another circuit[br]would have given you. 0:00:17.650,0:00:21.730 We'll talk about what it means to be[br]series and parallel for resistance and 0:00:21.730,0:00:24.960 conductance, and also for[br]voltage and current sources. 0:00:24.960,0:00:28.500 We'll talk about voltage and current[br]dividers and then equivalent sources. 0:00:30.620,0:00:34.360 Equivalent circuits are two circuits[br]that are the same if the voltage and 0:00:34.360,0:00:37.300 the current characteristics[br]at the nodes are identical. 0:00:37.300,0:00:40.516 Then the circuits are considered[br]equivalent, what does that mean? 0:00:42.560,0:00:46.730 So right here are two nodes,[br]here's V1 and here's is V2. 0:00:46.730,0:00:50.090 If the currents i1, i2 and v1, 0:00:50.090,0:00:54.020 v2 are the same, then it means that[br]we have an equivalent circuit. 0:00:54.020,0:00:57.640 Basically we're going to be changing[br]the part that's right here, and 0:00:57.640,0:00:59.140 the rest of the circuit. 0:00:59.140,0:01:04.450 So that we have two circuits that give us[br]the same voltage and current situation. 0:01:04.450,0:01:07.120 So let's say here's a very simple example. 0:01:07.120,0:01:11.600 If I had a circuit right here,[br]notice this is the part on the left and 0:01:11.600,0:01:13.691 it has two nodes, v1 and v2. 0:01:13.691,0:01:18.328 And there are five resistors,[br]R1, R2, R3, R4, and R5. 0:01:18.328,0:01:22.738 If we combine all of these resistors and[br]series right there, 0:01:22.738,0:01:25.880 we'd have an equivalent resistance. 0:01:25.880,0:01:31.770 The equivalent resistance is basically[br]just the sum of all five of the resistors. 0:01:31.770,0:01:34.210 And if we did that, this circuit and 0:01:34.210,0:01:38.120 this circuit would have[br]exactly the same front-end. 0:01:38.120,0:01:41.060 The voltage and the current at[br]the front would be exactly the same. 0:01:41.060,0:01:42.450 These two circuits would be equivalent. 0:01:45.110,0:01:48.600 So if we have circuits that are in series, 0:01:48.600,0:01:52.050 it means they have the same[br]current as shown here at the top. 0:01:52.050,0:01:55.960 If we have circuits that are unparallel,[br]it means they have the same voltage, 0:01:55.960,0:01:57.119 as shown here at the bottom. 0:01:59.320,0:02:02.390 Resistors that are in series add,[br]that's what we just did. 0:02:02.390,0:02:06.660 So if we had four resistors in series, and 0:02:06.660,0:02:09.580 we want to find their equivalent[br]resistance, we just add them up. 0:02:11.090,0:02:14.280 The equivalent resistance is just[br]the sum of all the four resistors. 0:02:16.200,0:02:20.930 A voltage divider uses resistors in series[br]to be able to divide up the voltage. 0:02:20.930,0:02:25.270 We take the original voltage, VS, and[br]we run it across a couple of resistors. 0:02:25.270,0:02:30.556 And you can see right here that one of the[br]resistors is used to divide the voltage so 0:02:30.556,0:02:31.896 that we can get v2. 0:02:31.896,0:02:39.210 V2 is going to be equal to R2[br]over R1 + R2 as shown here. 0:02:39.210,0:02:44.021 If we had many resistors, the voltage[br]would be the resistor that the voltage 0:02:44.021,0:02:48.098 has taken across divided by all[br]of the other resistor in series. 0:02:49.927,0:02:54.445 Here's an example of the voltage divider[br]cards that you have in your package and 0:02:54.445,0:02:58.828 you can see that we can take one voltage[br]such as the voltage across this battery 0:02:58.828,0:03:02.492 and divide it into two parts by[br]running it across two resistors. 0:03:02.492,0:03:06.180 Now, the other thing that we can do[br]with the voltage divider is we can take 0:03:06.180,0:03:10.510 two voltages and[br]put them together in series, so 0:03:10.510,0:03:14.540 basically stack up two batteries and[br]we'll get the sum of the two voltages. 0:03:14.540,0:03:17.910 A voltage divider works as[br]both a voltage divider and 0:03:17.910,0:03:20.827 in the other direction[br]as a voltage summer. 0:03:20.827,0:03:23.000 So voltages in series add. 0:03:23.000,0:03:25.260 Here we have three voltages in a circuit. 0:03:25.260,0:03:28.840 And the way they're going to add is[br]we'll just go in this direction. 0:03:28.840,0:03:33.861 Here is -v1 + v2- v3 and 0:03:33.861,0:03:38.255 so right here the voltage 0:03:38.255,0:03:43.914 equivalent is v1- v2 + v3. 0:03:43.914,0:03:49.226 The equivalent resistance is found[br]by summing up the two resistors that 0:03:49.226,0:03:54.546 are in series, so the bottom circuit[br]is an equivalent of a top circuit. 0:03:56.588,0:04:00.190 Currents in series have to be the same or[br]else something blows up. 0:04:00.190,0:04:04.770 So this circuit, this right here, you[br]we have a 6 amp in series to the 4 amp. 0:04:04.770,0:04:07.370 That's unrealizable impossible circuit. 0:04:07.370,0:04:09.360 This can't happen or[br]else the circuit will blow up. 0:04:11.200,0:04:14.110 Resistors that are in parallel[br]add in a different way. 0:04:14.110,0:04:18.640 The equivalent resistance is taken by,[br]take the inverse of each resistor, 0:04:18.640,0:04:22.245 1 divided by R1 + 1 divided[br]by R2 + 1 divided by R3, and 0:04:22.245,0:04:26.840 invert all of that and that's going[br]to be the equivalent resistance. 0:04:26.840,0:04:31.580 So no matter how many resistors we have,[br]we can find the equivalent resistance for 0:04:31.580,0:04:32.490 this circuit right here. 0:04:34.440,0:04:37.678 There's another way to think of that,[br]and that's in terms of conductance. 0:04:37.678,0:04:42.260 Conductance is 1 divided by the[br]resistance, so here is the conductance, 0:04:42.260,0:04:45.700 1 divided by R1,[br]that you can see right here. 0:04:47.090,0:04:51.150 Adding up the resistors in parallel is the[br]same thing as adding the conductance in 0:04:51.150,0:04:53.050 parallel except it' very simple. 0:04:53.050,0:04:59.636 The conductances in parallel add,[br]G equivalent = G1 + G2 + G3. 0:04:59.636,0:05:02.967 Remember the R equivalent[br]1 divided by G equivalent. 0:05:02.967,0:05:07.890 So this is often used in electromagnetics[br]as well as other aspects of electrical 0:05:07.890,0:05:08.850 engineering. 0:05:10.650,0:05:14.690 The current divider is what happens[br]when we have resistors in parallel. 0:05:14.690,0:05:22.130 We bring in a single current, is, and[br]it divides into two currents, i1 and i2. 0:05:22.130,0:05:25.581 According to this equation here, 0:05:25.581,0:05:30.936 you can see that i1 is going[br]to be dependent upon R2, 0:05:30.936,0:05:34.039 and i2 is dependent upon R1. 0:05:36.141,0:05:38.170 Here's your current divider card. 0:05:38.170,0:05:41.360 Now a current divider can also[br]be used as a current adder. 0:05:41.360,0:05:46.530 If we had one current,[br]we could divide it into two or 0:05:46.530,0:05:50.960 if we had two currents coming in,[br]they could be combined into one current. 0:05:52.360,0:05:55.960 Now currents can be added up in[br]parallel but voltages can't. 0:05:55.960,0:05:59.810 So voltages and parallel have to be[br]the same or else the socket blows up. 0:05:59.810,0:06:04.540 Here, for example, are three different[br]batteries, this is a 1.5 volt battery, 0:06:04.540,0:06:08.280 this is a 9 volt battery, and[br]this is a 12 volt battery. 0:06:08.280,0:06:11.710 We can't put those in parallel[br]unless they were equal, or 0:06:11.710,0:06:13.340 else we blow up our circuit. 0:06:13.340,0:06:15.405 So this is also an unrealizable circuit. 0:06:18.090,0:06:22.350 In order to transform a source, we often[br]do this in order to simplify our circuit 0:06:22.350,0:06:25.020 or better understand what's[br]going on in the circuit. 0:06:25.020,0:06:29.260 So if, for example, we had had a voltage[br]source, remember how we had the voltage 0:06:29.260,0:06:35.140 source card for a realistic voltage[br]that had a resistor in series with it? 0:06:35.140,0:06:40.730 If we want to convert that instead[br]to a realistic current source, 0:06:40.730,0:06:43.560 not an ideal current source,[br]a realistic current source, 0:06:43.560,0:06:46.670 here's the transformation[br]that we would do. 0:06:46.670,0:06:54.910 Is would be vs divided by R1, and R1 and[br]R2 in these pictures would be equal. 0:06:54.910,0:07:00.160 We can go back and forth between[br]these two equivalent circuits and 0:07:00.160,0:07:01.890 have different source transformations. 0:07:02.990,0:07:05.040 Now, how might we use this. 0:07:05.040,0:07:09.030 I want you to take a look in your[br]text book of example 2-10 and 0:07:09.030,0:07:11.770 go through this example[br]in some level of detail. 0:07:11.770,0:07:14.940 Basically, transforming back and[br]forth between current and 0:07:14.940,0:07:20.000 voltage sources allows us to more[br]easily analyze this particular circuit. 0:07:20.000,0:07:24.580 So we started out with[br]a current source and 0:07:24.580,0:07:27.540 several resistors in series and[br]in parallel. 0:07:27.540,0:07:32.624 Then, if we convert the current[br]source to a voltage source, 0:07:32.624,0:07:37.425 this allows us to combine[br]these two resistors in series. 0:07:37.425,0:07:42.204 Then, if we convert this new voltage[br]source back to a current source, 0:07:42.204,0:07:47.631 that's going to allow us to more easily[br]include these resistors in parallel and 0:07:47.631,0:07:50.010 so on until we're finished. 0:07:50.010,0:07:53.700 This analysis is often done when we're[br]doing filter design and development. 0:07:55.810,0:07:58.890 So in short,[br]we've talked about equivalent circuits. 0:07:58.890,0:08:02.620 Basically, an equivalent circuit[br]is if we have the same voltage and 0:08:02.620,0:08:06.310 current at the front end we know[br]that two circuits are equivalent. 0:08:06.310,0:08:08.390 We talked about series and parallel. 0:08:08.390,0:08:13.520 Remember that resistances in series[br]add and conductances in parallel add. 0:08:13.520,0:08:15.480 Voltages can be added in series, and 0:08:15.480,0:08:19.570 current sources can be added in parallel,[br]but not the other way around. 0:08:19.570,0:08:22.140 We also talked about voltage and[br]current dividers. 0:08:22.140,0:08:24.840 Remember that those can[br]also be used as summers and 0:08:24.840,0:08:27.960 then we talked about equivalent sources. 0:08:27.960,0:08:32.265 The picture from the front is from the rim[br]of Snow Canyon in Saint George, Utah.