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Use nodal analysis to determine the node voltages defined in the circuit in Fig. P3.64....

Question

Use nodal analysis to determine the node voltages defined in the circuit in Fig. P3.64.

Use nodal analysis to determine the node voltages defined in the circuit in Fig. P3.64.



Answers

Use nodal analysis to determine the node voltages defined in the circuit in Fig. P3.64.

Okay, so this is a very lengthy problem. Um, first, So we have this circuit here, and Ah, As you can see, I have marked all the currents that's passing through our each resistors. Now, I marked the current with the, uh, value of the resistor correspondent. I'm among the current corresponding to the value of the resistor. So, for example, I six is the one that's passing to six homes. Idol is passing through twelve homes. I six point eight passing to six point agent. So on, so forth on DH. Now we see we have one, two, three, four, five unknowns. So we need five questions Now we can use Junction rule at A and E, and then we can use lope rule at left, middle and right, Luke. So let's first supply junction rule at point and be so that point a we see that top is going away. I six is going away and I five is going into the junction. So that means I five should be called Ice Express. I adopt similarly in this junction. I top is going coming in, and I six hundred it is coming in and then twelve is going over it. So that's what we want here. And if we act, it's all for it. If we actually substitute items from both of the questions, we get a relation between I five, six point eight, twelve and six, right? So that's That's a little attention now for the loop rule. What we're doing here is first. We're starting with the seven bowls here, so these two are in serious. So that's why we add them. And we are taking our direction of current as clock ways, and similarly, we're taking for the middle one. We're taking the direction as club plays again. And for the right Luke, we're taking the direction as anti clockwise, right? So here, as you can see, we're adding twelve and five so way have folded gain on both Ah, marriage. So that's where we have seven holes on. There's a plus sign. Then the kind is in the same direction as the look. So that's why you have a negative sign and then we have I five times five homes, then who have negative sixteen six jobs because of this current. So that's our left. Look for the right look. We have to various connected in Siri's. And then we have a voltage again. So we have ten bowls, then twelve times twelve, and there's a negative because of the same direction. Then again, six fine date is in same direction with the low obstructions, or it's going to be six point eight times Isaac's pointy and is a native. And for the center look, we see that ice sixes in opposite direction with the obstructions or that. So sorry. Ah, it's going to be the opposite. But it doesn't matter, actually, because since there are only two currents, we'LL have the same expression. But anyway, so we see that Ice twelve is in opposite direction with the low traction. So that's why it's positive. And I six is in the same direction of the looks, tricks and direction. So so it's native, but we're not concerning. This ight off current here because there's no resistors. So that's what it is, a double digit drop, so we don't have to worry about that right? So now we have, ah, technically, four questions. So we have marched this equation into one, so that's where four questions Let's try to solve them. So we start with the fourth question. So you see, it's it's the shortest off all. So we get a relation between six and twelve, and then we can use this and substitute I six on the other equation. So that's what we did. So when we substituted all the I six, we got expression like this. So these are our new ah equations. So these are the three new questions that we have. So we go to the bar for decoration. Now, what we can do here is from there we used question one or the first one toe eliminate six point eight. So in our forest ignition is this so we are eliminating six find? Ah, yeah, we're eliminating six point eight from the first equation. So we're writing in terms ofthe tighter than I fight. So write if we substitute back Toa I too and I we get these to falling expressions on da. Okay, so this thing is going to be zero, right? So then this is our third equation. This is our second question. Now we see that we only have five and I dwell on both of the questions. So what we did here is we now found an expression for if I from the second equation s o a cz we see here that we got rid of our first equations. Now, from the second creation, we write I five in terms ofthe twelve, and we substitute that Any question three And if we do so we'll have on ly I dwelt, um, present in this third equation and then we can solve for it. Well, so that's what we did here. And then once we put all that on all the numbers together, we got I twelve as point six seven nine, eight amps. Now, since we know I'd well, we can figure out if I buy this expression. So And we got a fight as one point seven six eight EMS again, we can go back. And since we know I five, we can know. We know I six point eight by this expression over here. So that gave us this this point two seven one for EMS, And then finally, if we use this expression like here, that gives us the expression for our value for six, which is one point two six APS. So, yeah, that's how we're going tow. We got our occurrence, but make sure you check the calculation so that you can do it by yourself. Thank you.

We have a 2.7 kilo ohm resistor on DDE. We want to find the peak current when the army's voltage is to 20 votes. So to do this we find our peak voltage, which is basically just the r. M s times square root of two. And so I peek is equal to re peak over our, which is equal to the R. M s times grew too divided by our and this is just to 20 times rue tune divided by 2700. That's our resistance and calculating this we get 0.12 m's, so that's our peak current.

So as you can see in this good, I gotta Ah, there are two different currents. Sorry. There are six different. Currently, currents are labeled. So in order to find these six currents, we should have six unknown. Sorry. 60 e questions to solve. So we applied the junction roll toe junctions A, B and C. So these are the three questions that we get by doing so. So now me apply the loop rule toe loops. 123 So this is applying to Luke one. Uh, this is you I do on then we have no tree. So now we have six months All those currents on dis currents and six aggressions. So trust, uh, we let's Victor simplistic question on eliminate Joe. One of the baby of this from them. So let's eliminate I by using these two questions so far, Slits named those questions. So now let's use the question One country in order to eliminate I. So why don't so we get I want blast by two equal door. I treat less, I force. So this is a new a new equation. And let's call this question three. So now when these two questions are gone, and we're going to use this a question instead of this too. So now that one off the variables is are also eliminated. And we have five questions now to this 3456 So no, we eliminate. I won using a question took So we can we have I won expressed in terms off three and five. So we will plug this expression in other questions where we have Ivan. So question, fool. So we will use this equation and a good job with you. So we will have another question where we will have I t and my five in place off. I want So this is our moral question for Aunt have cancelled the question toe s so that we don't have to worry about these three. Now, this tree are already are sorted out. So now we have for questions on dhe for unknowns because I want is already has already been eliminated. So now let's use ah question lot. Mal, it's eliminate a question four. Bye. Using a question, she gets it this tree. So from here we will express I for that will be that big Black only so we will have I for equal toe. I want this. I do minus I tree. And we will use this expression and these two questions. Well, we have I for so doing that, we get negative. I three all three Plus, I want less. I too, minus I three times. Ah, yeah, are for less. I five dimes are five. He quoted a Lucille, but we already have. I won expressed in terms off I triplets I for So we're going to use that also over here. So if you if we blood this over here, we will get. So if we plug this a question over here, we will get I fought equal toe. I won. Yes, I do. Minus I five holes. So we will get this Toby I five plus I too. So instead of this began just right, I five less I So now we have, uh, three unknowns. Wait. We also need toe put eye for at this place substitute. I fall over here, so we will get Linus. I do art too. Minus I five plus two dimes. I forced. You got dizzy. So now if you see these two equations, we have todo three on loans off these three questions, and we have three unknowns. I three, I five and I to on we have three questions. So let's call this as our question five. This other questions, it's not. They're basically the same. I stood before a cushion, but the variables have been Oh, we have substituted for some off the Peebles. So now the eliminate. I do, uh, using these two questions on, then we will eliminate I three some. What's so Just like we have eliminated the ah, rest of the variables I won, I, for one, will eliminate I tow and I, Pfeiffer as well, by going similarly on. Finally, we will find a tree. So that will come out, Toby negative. By five times I do our force. Plus after what? Five bless all one are full. Plus, our five are for overs. So now that we have I three expressed in terms off I five so we can plug this expression back. And so maybe this question for, uh, remember that Do you need to eliminate I, too, as it so basically, we're going by the method off elimination and substitution. So I'm just gonna write or the final answer after doing so, because it's the same method that, well, you'll be repeating again and again until you finally arrive at Bonnie. Question containing just one unknown so far. My case. The first variable that I could find is I fight and health care's depending on how you Oh, Jules, your elimination and substitution We might find some ever occurring first. So yeah, so I five we got that to be negative. 1.14 Siegel on the I five, as you can see, is here taken down one well, at this point. But since they found it to be negative, therefore the actual direction off I five is upwards on the magnitude is does this positive part. So then I go backwards and like these values again, back in tow, the rest off the questions and identify and I find I treat of is zero point 1771 Basically, this was my expression off fight three rounds off. I fight so I can just plug I five back into their on. This is positives on my election. Off a tree is correct. It's going this way. The southwest, the southeast direction, then similarly, I can find. I told by may be using this equation and substituting I tree and I five I just found so I don't will be 0.154 to Anfield. So now it's going to be easy because these are three questions that we're gonna mostly use now. So I three is I five. Plus I do. So I'll just do that on DDE. So that's I fought, which is I five less. I do. So that will be subtle. 0.1531 and beers. And we have I won. It will be I treat less I five according to the question to this one. So we have I'm gonna call It was 0.176 serial champion. So now we have five parents. Onda, We can also find this. I was just a total current if you want, So that is going to be I want us. I too So I won Here I do here. So really add them up. This is I on Duh. Now that we have all the currents, let's write them down in a practice. So Ivan is the current that is flowing through printed it warms. Don't you do. A woman is a star, then I Ford, Mr Curren, that is flowing through 25 formula. Sister, let me just cross check. Yes, Andi, I three is so the eye for is flowing through. 14 homeless, The stars. Then this is the garden that's going through 15 armed. Is it stirred? And then I three is the one that's room to do a little mist, and I use the current. That's the total current and that's flowing from the battery.

So here I have labeled the resistance on the in Memphis with the drop appropriate the the value or corresponding toe that variable. So, for example, are five is the resistance. Ah ah, Off this resistor on has a value five homes. Similarly, the steep in which is that Yemen has an E m f a o off 10 bowls on for the currents. Uh, we have but the Sox trip. It's a start. That's the current corresponding toe, that particular variable like that particular resistant or enough. So I 6.8 is the current flowing through the six point assistance Mr Onda. Awesome. Really, I to always the current flowing through the 12 room resistor and I five is the current flowing through the five or Bold or E M. If you can call this, I fight for my train, whichever one you want to pick. So yeah, so now we apply the junction. Jewels are two points A and B onda. We apply the loop. You'll toe loops 12 tree that have already labeled here on this will give us our five questions on this will make sense because we have five unknown current that we need to solve for and therefore we need five questions for that. So let's start off like the junction rule out O A N B at point c nd So we have my five equal toe. I felt less by six or junction eight and then I don't Yes, I 6.8 still be equal to duel. I dress because they have the opposite direction. So I thought this this Ray and ice explained it is like this sword point readings to combine to make twins and therefore the system relation. So from here, we can solve for Oh, I thought so because I don't this common in both of these equations. So doing that, we get I five minus I six, which is high top equal Dole I do of minus I 618 on dhe begin. Oh, arranged this a question like this having although positive signs, less signs My six. So this is AJ question from those true junction Who's that's called that a question one. So we got the situation by solving two equations on dhe now. Therefore, we have eliminated the variable I top, so we will have 40 questions now because we have reduced to these doing one. So now we applied the new proof. So this is large. No one. Remember, if you're going from Wasit if two negative Ah, inside a back alley, there should be a negative sign off before that. And if you're going negative to positive in a battery, there should be a positive sign before the M f on for current. If you're going along the direction off current, there should be a negative sign before that on. If you're going opposite to the direction off current Ah, then there should be a Was it the same for that? So those are the simple shores on, I have applied them on DDE. You can inject them if there got it. Also, it doesn't really matter how you go how your Luke looks like. For example, you can choose this loop one toe be counterclockwise, and you should be able to arrive are disa question. So this is for loop one. That's all. That a question. So the sense for Luke I do. And this is for Luke Tree. So let me destroy that so that you don't get confused with the question number. Yeah, and this is the question that we got from the junction. Now we will solve these four questions by the method of substitution elimination, my demented off elimination and substitution. So let's use the question for because I find this a question the simplest off this fall. So let's use this to solve for I six. So we will get I six in terms off 12 Here from the secretion on dhe are 12 over our state. So since these toe ah, basically present the value that sickle tow the sub script. So we can like that. You can cross that that the resistance have activities values. So we have our sex equal to twice off. Sorry, we have ice six equal to twice off. I do have. So now that we have ice six in terms off are you to have began eliminate Wherever they have only six on DDE That baby will be able to eliminate Ah, one more valuable. So thank you. Let's come buying these two immense by eye the adding the voltages. So we have I five less ice x 0.8 equal toe. So this is a question once I have other days too. So this will be equal to go three. I too. Let's call this a question One again, because these are practically the same. So now well, we look that tow the loop. One question a question to plug this value off. I six over here. So we will get I 15 minus I five. Odd. Five minus. I will. I drove you quit the seal. So I have added these two because in five has value five. Fold it and it's Turnbull's of Fife. Less dryness, too. And that's why I have returned here to be 50. So again we have 12. Which is why, for the four plus e eight on this to add up to 12 mutual physical toe minus, I do it outweighs minus 6.8, not a 6.8. But this is a question duel. This is a question she so you might get confused as to how this became. I, too, have afterwards basically have substituted our 12 as well in terms off our six. So basically are six is articles over too. So if you combine these two, you'll see that I six times are sixes impractical dry, too, as my massage. Because these two. Randy was canceled. So that's what I have done in order to make Ari questions look nice on dhe. You congee lately substitute all the values of that, uh, my ability, but simple. So, yeah, you can do that. Still now way have three questions on three young girls. I five by 6.8 and I do is we have already eliminated. Eliminated two off the really of this. So among this tree, a z can see question one is the simplest. So we will use this toe. Let's say eliminate I 6.8. So I six point it will be I three idol minus five, and we will block this. Ah, until this a question, This has ice Expo Nadiya question toe doesn't have ah, ice explain dates, So yeah, so lugging this over here, you know that I 12 minus sorry. It well minus I our minds it is. Ice explained to each times AJ 6.8. We called Rosie. And this is actually question three, Except we have a plug this over here. So now we have to It questions the question to one. The question three. So we will ah, eliminate. Let's say we will out of these two, This one is the simplest. So we will use this equation. You know that. Eliminate one more variable. So let's say V eliminate I five. So we express I five using the question does so from that we have this and we will blood this into a question three. Still, and then, ah, as you must have already guessed the question tree, they'll have just one unknown i 12 because everything else is now expressed in terms off. I drive only so then we can solve for I so doing that. So this will be, though, finally, question three. So now we solved this aggression for I two l's. It's with the equal to trends. I fight less, and then we need to plug the values off the image and assistance. And, uh, this should not be confusing a call because the values off these two variables are to present it by their sub scripts. So during that, so eat will this Basically 12 are five hits, five rooms on and so on. So, solving them we get I tried to be subtle 0.66502 and this is approximately 0166 five years. So this is I do have or the current flowing through. Let's see, I tow the current flowing through the 12 armed assistance. It's right that in records. So now we can, uh, go oh, backwards on dhe soul for, um, that is stuff, though. Parents. So because we have solved for 12. So we need an equation that that just has 12 minutes of the system. So this has just warm unknown by five. Rest of the things are no now. So they find i five by using this a question. So doing that, we find i five to the equal dough one point fold Syria am Peel nearly called one point 40 amperes of the sister resistance going through the fife gold E m f or the 10 body Enough. Okay, just a straight toward the end. So then you have high five on DS. I fight and I too have now. So let's say we can find ice 6.8 because these two are already known. So you reuse that a question and I 6.8 comes out to be 0.591 ampule. That is the current flowing through and 6.8 or Melissa start on dhe then began find I six. That should be easy, because I six is just twice off 12. We could have done that just after finding I do it. What? Anyway? So I six is 1.33 amperes, and that is the current flowing through, though. Six older sister. So yeah, on dhe, there's one were current to find, which is I twelves. Look just right that over here so that I don't need to go to the next page and all the currency will be here. So I talk is basically that's used execution. So I talked this guy five minus I six. So I five is 1.4 and pubes on I six is one point 33 ambulance. So this should be querido. I see your point CEO seven AM This is the current flowing through this region from A to B


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