1 00:00:08,600 --> 00:00:15,844 Let's take a look at our circuit. We want to apply mesh analysis to solve 2 00:00:15,844 --> 00:00:21,383 for this current right here. 3 00:00:21,383 --> 00:00:23,462 [NOISE] 4 00:00:23,462 --> 00:00:34,441 For I sub 0. In mesh analysis, we have to identify our 5 00:00:34,441 --> 00:00:40,480 mesh currents. I've got three loops here that are 6 00:00:40,480 --> 00:00:43,850 non-inclusive, so they don't include any other loop. 7 00:00:43,850 --> 00:00:56,576 And I've got to define my mesh currents, so I'll call this one I1, this one I2, 8 00:00:56,576 --> 00:01:05,976 and this one I3. So, when I solve for this problem I've 9 00:01:05,976 --> 00:01:11,136 got to set up three simultaneous equations and they're going to be in the 10 00:01:11,136 --> 00:01:18,510 form of the unknowns being I1, I2 and I3, a mesh currents. 11 00:01:18,510 --> 00:01:21,860 So, I should have three equations and three unknowns. 12 00:01:21,860 --> 00:01:24,566 Now, sometimes when you set this up, you might find that you can reduce out one of 13 00:01:24,566 --> 00:01:27,940 these pretty easily. But I'll go head and set it up with the 14 00:01:27,940 --> 00:01:33,641 three equations to begin with. Now notice that I got a dependent source 15 00:01:33,641 --> 00:01:38,063 in here this is the dependent current source,, and you often times find circuit 16 00:01:38,063 --> 00:01:43,095 sources in certain examples like amplifier examples. 17 00:01:43,095 --> 00:01:47,442 Amplifiers might become from non-linear elements, and some, sometimes we redraw 18 00:01:47,442 --> 00:01:51,285 those and model those, and when we model them to an equivalent circuit, we 19 00:01:51,285 --> 00:01:56,100 oftentimes find dependent sources like this. 20 00:01:56,100 --> 00:01:58,430 So I just want you to be able to see this. 21 00:01:58,430 --> 00:02:01,724 It does complicate it a little bit, but if you can understand how to handle 22 00:02:01,724 --> 00:02:07,320 dependent sources it'll help you under to be able to solve other problems. 23 00:02:07,320 --> 00:02:14,030 So, I got 3 equations that I need to, to find and I'll find them by doing a KVL 24 00:02:14,030 --> 00:02:20,410 around each of these loops so let me start out with loop 1, that's the I1 25 00:02:20,410 --> 00:02:27,250 mesh, okay. So, this one is pretty straightforward 26 00:02:27,250 --> 00:02:31,340 because I've got this source right here. It's a dependent source. 27 00:02:31,340 --> 00:02:35,160 But the current has to match up with what this mesh current is. 28 00:02:35,160 --> 00:02:41,019 So, I1 It's going to be equal to capital I over 2, alright? 29 00:02:41,019 --> 00:02:44,739 Capital I is a variable I don't want to have in my final equations, because I 30 00:02:44,739 --> 00:02:49,930 only want I1, I2, and I3, so I've got to try to get rid of capital I. 31 00:02:49,930 --> 00:02:55,880 So let me see where else I've got an I. I've got an I going through this loop 32 00:02:55,880 --> 00:03:03,267 right here, this branch. And I also know in terms of my mesh 33 00:03:03,267 --> 00:03:13,360 currents, that I would be equal to capital I sub 2 minus I sub 1. 34 00:03:13,360 --> 00:03:17,540 Because going in this direction, I two is going in this direction. 35 00:03:17,540 --> 00:03:20,320 I one is going in the opposite direction so I have to subtract it off. 36 00:03:21,440 --> 00:03:25,210 So I can now solve for I in terms of I one and I two. 37 00:03:25,210 --> 00:03:31,950 That means plugging back into this equation [SOUND] I had that let's see. 38 00:03:31,950 --> 00:03:55,190 I can reduce this out by looking at yeah, I can plug this back into here. 39 00:03:55,190 --> 00:04:00,034 It's actually easier if I say, 2 times I1 is equal to I. 40 00:04:00,034 --> 00:04:04,914 If I have 2 times I1 is equal to I and then just equate these, I end up with the 41 00:04:04,914 --> 00:04:11,004 expression 3I1 is equal to I2. So now I have an equation that only 42 00:04:11,004 --> 00:04:16,132 involves my mesh currents. Let me look at loop 2. 43 00:04:19,970 --> 00:04:24,326 Around this loops let me start out with the voltage source going around the loop 44 00:04:24,326 --> 00:04:28,616 in a clockwise direction, I'll get to the plus sign first, so I'm going to have a 45 00:04:28,616 --> 00:04:32,642 plus 2 volts, so coming up to here remember that I want to have equations in 46 00:04:32,642 --> 00:04:36,800 terms of my mesh currents so I don't want to write in terms of I, I want to write 47 00:04:36,800 --> 00:04:46,550 in terms of I1 and I2 So, it's plus 5, I 2 minus I 1, going in this direction. 48 00:04:46,550 --> 00:04:50,744 Plus a voltage drop across 2 ohm resistor. 49 00:04:50,744 --> 00:04:58,094 And going into this direction it would be I 2 minus I 3. 50 00:05:00,540 --> 00:05:03,556 Plus the voltage drop across this resistor and the only current going 51 00:05:03,556 --> 00:05:10,792 through that is a mesh current of I 2. Plus I 2 times 1 ohm, that has to equal 52 00:05:10,792 --> 00:05:14,349 0. If I want to clean this up. 53 00:05:14,349 --> 00:05:23,376 Let me re-group my terms. I will have a minus 5I1 plus 8 I2 minus 2 54 00:05:23,376 --> 00:05:35,187 I3 is equal to minus 2 loop 3. So I now have two equations and I need 55 00:05:35,187 --> 00:05:43,666 one more equation, I go to loop 3 and looking at this loop I'm going to have 4 56 00:05:43,666 --> 00:05:52,562 ohms times 3, which is this voltage drop plus this voltage drop which is 2 ohms 57 00:05:52,562 --> 00:06:01,736 times this current which is I3 minus I2 plus this voltage drop, which is going to 58 00:06:01,736 --> 00:06:18,797 be, I 3 minus I 1, is going to equal 0. And if I clean this up by regrouping my 59 00:06:18,797 --> 00:06:26,260 terms, I'll get a minus 10, I1 minus 2I2 plus 16I3 equals 0. 60 00:06:26,260 --> 00:06:38,852 And this is, these are my equations. In this case, I've got three equations 61 00:06:38,852 --> 00:06:46,296 and three unknowns. Now, one of the equations is really 62 00:06:46,296 --> 00:06:49,148 simple. I could substitute this back into here 63 00:06:49,148 --> 00:06:53,970 and solve, reduce this down to two equations and two unknowns. 64 00:06:53,970 --> 00:06:57,330 I'm going to go ahead and keep it as three unknowns right now, because I want 65 00:06:57,330 --> 00:07:01,620 to show you how to do matrix analysis of this. 66 00:07:01,620 --> 00:07:12,139 So, I've got three unknowns I want I2 and I3, that's sum matrix times that is equal 67 00:07:12,139 --> 00:07:20,192 to something. So, taking my first equation this is a 68 00:07:20,192 --> 00:07:25,380 pretty simple one, I've got 3I1 equaled to I2. 69 00:07:25,380 --> 00:07:33,950 Or in other words, I can write this as 3 I 1, minus I 2, equals 0. 70 00:07:33,950 --> 00:07:38,198 So, putting that into the first row of this matrix, I'm going to have a 0, 71 00:07:38,198 --> 00:07:42,662 because it equals 0 and then I've got a 3 times I 1, minus 1 times I 2, and then a 72 00:07:42,662 --> 00:07:49,520 0 plus 0 times I3, that's my first equation. 73 00:07:49,520 --> 00:07:55,951 My second equation, I put in the coefficients here, a minus 5 times I1, 8 74 00:07:55,951 --> 00:08:02,709 times I2, minus 2 times I3 and that equals minus 2, and my last equation I've 75 00:08:02,709 --> 00:08:14,270 got a minus 10 times our 1 minus 2 times I 2 and a 16 times I 3 equals 0. 76 00:08:14,270 --> 00:08:17,381 If I solve this, again, if your calculator does this, use your 77 00:08:17,381 --> 00:08:21,414 calculator. Otherwise, you can go online and find a 78 00:08:21,414 --> 00:08:29,100 linear equation solver. If I solve this, then I would get I 1, I 79 00:08:29,100 --> 00:08:42,582 2, and I 3 equals minus 0 point 118 minus 0 point 353 and minus 0.118. 80 00:08:42,582 --> 00:08:43,939 And of these mesh currents I was interested in I zero to begin with well I 81 00:08:43,939 --> 00:08:45,870 zero is equal to I three minus 0.118 and that's my final answer. 82 00:09:09,230 --> 00:09:13,360 So, to summarize what we did, we were looking for I0, we defined our mesh 83 00:09:13,360 --> 00:09:18,050 currents, which are currents around each of my non inclusive loops, and I want to 84 00:09:18,050 --> 00:09:22,320 make sure you remember I only want equations that only involve these mesh 85 00:09:22,320 --> 00:09:28,488 currents, those are going to be my unknowns. 86 00:09:28,488 --> 00:09:33,233 So I solve to find those equations, I do a [UNKNOWN] around each of my loops and I 87 00:09:33,233 --> 00:09:38,343 make whatever substitutions I make to get rid of any extraneous variables because I 88 00:09:38,343 --> 00:09:45,781 only want the the mesh currents to be in, my only variables in there. 89 00:09:45,781 --> 00:09:49,502 So I do a [UNKNOWN] around each of the loops and in my case, and I came up with 90 00:09:49,502 --> 00:09:53,589 three equations, and three unknowns, and I wrote it into a matrix form, and then I 91 00:09:53,589 --> 00:10:01,225 used a matrix inversion solver. because I invert this matrix to solve for 92 00:10:01,225 --> 00:10:07,520 the I's. Thank you.