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added lecture 2 subs

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fischerl
2012-01-30 14:16:42 -05:00
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subs/38euKuG1cUs.srt.sjson Normal file
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"text": [
"SPEAKER 1: All right.",
"So v1 here is simply the difference between v0 and e1.",
"So v1 is simply v0 minus e1.",
"That's it.",
"What about i1?",
"That is pretty simple, too.",
"i1 is simply v1 divide by r1.",
"And because v1 is v0 minus e1, I can replace v1 by v0 minus",
"e1 and divide by r1.",
"That gives me i1.",
"So in this manner, I can go ahead and find all the vi's",
"and ii's very straightforwardly from the",
"known voltages.",
"So most often when we analyze circuits, we just go ahead and",
"find the e's-- the known voltages--",
"because we know we can very quickly find all the branch",
"voltages and currents.",
"And so, we often stop at the known voltage step."
]
}

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"SPEAKER 1: Well, it turns out that I don't need a variable",
"there because of the trick that I used in picking my",
"ground node.",
"By picking my ground node where it was, notice that the",
"voltage at this node is simply v0, because this node is",
"connected to ground by a voltage source.",
"And so the voltage source would hold the voltage of that",
"node at v0.",
"So I directly have the voltage of that node.",
"So right there I've done my second step.",
"So that's my step two of node analysis.",
"So let's move on and start off with step",
"three of node analysis.",
"All right?",
"Step three would be to write KCL for the nodes-- on each of",
"the nodes-- e1 and e2 here, which are my unknowns.",
"I'm going to write KCL for these nodes, and I am going to",
"substitute device laws and KVL in dividing the current that",
"leaves those nodes.",
"But also remember that as I work with currents, and sum",
"the currents for KCL, I'll simply use my usual",
"convention, which is sum the currents leaving a node always",
"so I don't make mistakes.",
"Now again, for another piece of convenience, I'm going to",
"use conductances.",
"So the variable I have is resistance.",
"1 by that resistance is the conductance, since I would",
"just use that to make my algebra a little simpler.",
"So let's get started.",
"Let's start with node e1.",
"And I'm going to write KCL at node e1.",
"Now, with KCL, notice that I'm going to follow my usual",
"convention to avoid mistakes.",
"I'm going to add up the currents that leave a node so",
"that I don't make mistakes.",
"So let's go ahead and start.",
"Let me start with KCL at e1.",
"So if you look at e1, what is the current that is going up",
"in that direction?",
"I know the node voltage here is e1.",
"I know the voltage of this node is v0.",
"So can you tell me what the current is going up?",
"Let me pause for a second so that you can think about it.",
"And then I'll go back and do it for you."
]
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"SPEAKER 1: Our third method is going to be the node method.",
"This method is going to be the most important method that we",
"learn, and you can apply it to all the circuits that you're",
"going to see.",
"This method is going to be a very specific application of",
"the KVL/KCL method.",
"This method has five steps.",
"The first step would be you go and select a reference node--",
"we call that a ground node--",
"represented by an inverted T symbol.",
"And you're going to measure all our voltages with",
"reference to this ground node.",
"The second step.",
"We label all the voltages of the remaining nodes with",
"respect to ground.",
"And these will be our primary unknowns.",
"As a third step, we will write KCL for all the nodes, leading",
"out the ground node.",
"And while writing KCL, we will apply a cool trick.",
"So KCL involves currents.",
"But notice that your primary variables",
"are your node voltages.",
"But KCL involves current, so what we will do is in step",
"three, we will derive the current, in terms of the node",
"voltages, and directly apply the element relationship",
"without actually having to write down",
"any specific current.",
"Then we go and solve for the node",
"voltages as our step four.",
"And finally, we go and back solve for the branch voltages",
"and branch currents.",
"Notice that for circuit analysis, our goal is to find",
"branch voltages and currents, and so to the extent that you",
"need to know all the branch voltages and currents, you can",
"go ahead and find them.",
"The node analysis method is the workhorse of our industry",
"and the workhorse of 6.002x.",
"And so you will be using it all over the place.",
"It's a very, very important method.",
"Let's get started.",
"And for this method, I'm going to use our",
"old faithful circuit.",
"For fun, I'm going to add a extra current source, I1, to",
"our circuit.",
"Notice that we now have seven elements in the circuit.",
"And with seven elements, the number of equations is going",
"to be completely unwieldy.",
"And so you will notice that with the node method, you will",
"be amazed that at the end, we will have to",
"solve only two equations.",
"Watch and see.",
"So as the first step of the node method, I have to select",
"the ground node.",
"Now, in general, you can select any node as the ground,",
"because voltages are always measured with",
"respect to a point.",
"There are differences in potentials between a pair of",
"points, so you can select any node as the ground.",
"But in general, to simplify our lives, I'd recommend using",
"a couple of rules to pick a ground node to",
"simplify the analysis.",
"The first thing to do is to find a node which has the",
"largest number of edges coming to it.",
"And in our circuit here, this node here, which connects R to",
"R5, and the current and voltage sources has the",
"largest number of edges.",
"So that seemed like a good candidate.",
"A second thing you can look at is look for a node that has",
"many of the sources connected to it.",
"And in this case, it does happen that the same node has",
"the voltage source also connected",
"to it with one terminal.",
"So let me go ahead and pick that as my ground node.",
"So let me go ahead and mark that as my ground node.",
"This will be my ground, and I mark it",
"with my ground terminal.",
"So this is step one.",
"Now, let's apply step two.",
"Step two will be label node voltages with respect to the",
"ground node.",
"So what are my nodes?",
"So I have a node here.",
"That's one node.",
"Let me label the node voltage with respect to ground as E1.",
"So if I take this as my ground voltage and",
"take that as a zero--",
"so the voltage at this node here is going to be E1 with",
"respect to the ground.",
"Let me pick this node here as E2 with respect to ground.",
"Finally, I have a third node here.",
"And what can you say about the voltage at that node?",
"Let me pause for a second while you think about it, and",
"then I will continue.",
"So what do you think the voltage is at that node?",
"Do I need a variable there?"
]
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"text": [
"SPEAKER 1: All right, let's begin by building a tool chest",
"of our analysis techniques.",
"But before we begin, it would worth while asking the",
"question, what does it mean to analyze a circuit?",
"Well analyzing a circuit simply means find all the",
"element v's and i's.",
"In other words, find all the branch variables.",
"And so what are these branch variables?",
"Well if you look at element R1, I can write its branch",
"variables as plus minus v1 here, so the",
"voltage across R1.",
"And I can assign a current i.",
"And for R4, as an example, I can say there's",
"v4 and there's i4.",
"And in general I'm pretty free to assign voltage and currents",
"in any direction.",
"But you will notice that I'm going to follow a discipline",
"where I'm going to draw the current entering into the",
"positive terminal.",
"And I will show you why in a quick second.",
"So as another example, I can define the element value for v",
"for the voltage source.",
"The branch variables here would be v0 and a current i0",
"going here.",
"And so on and so forth and I can do the",
"same for all the elements.",
"So analyzing a circuit really means go find all the element",
"v's and i's for the circuit.",
"Now we're ready to start with our analysis techniques that",
"will go into our tool chest. The first method that we will",
"look at is called a very basic KVL, KCL",
"method of circuit analysis.",
"And as I mentioned earlier, our goal with circuit analysis",
"is to find all the element v's and i's.",
"The KVL, KCL method is extremely simple.",
"In the first step in this method, you take your circuit",
"and you write down for each element their v-i",
"relationships.",
"Then, the second step, you pick all the nodes and for",
"each node you write KCL.",
"Then as a third step, you go through the loops one by one",
"and you write KVL for each of the nodes.",
"Notice that each of these equations is a very simple",
"equation containing simple algebraic expressions of the",
"currents and the voltages.",
"Now you'll end up with lots and lots of equations and you",
"will have lots of fun solving them.",
"Let's take a look at an example.",
"So our goal, as I said earlier, was to find all",
"elements v's and i's.",
"Now before we go in and do an example of using the KVL, KCL",
"method, I want to spend 30 seconds telling you about a",
"couple of ideas on how to label element v's and i's.",
"So for example, here's an element, the",
"Element e in some circuit.",
"And notice it has a v and an i.",
"In general, what we're going to do is we are going to",
"follow a little discipline or a little convention in how we",
"assign the voltage and currents.",
"Specifically, we are going to take the current to be",
"positive going into the positive voltage terminal",
"So in this case, here is the positive voltage terminal of",
"the Element e.",
"And so we're going to be assigning currents to be going",
"into the positive voltage terminal.",
"Now you can do whatever you want, it",
"really doesn't matter.",
"You can go and assign currents and voltages any which way and",
"reverse them and KVL, KCL will simply work.",
"It just turns out that in a lot of what we do in circuits",
"and electronics, it is useful to follow conventions so that",
"you don't make silly algebraic mistakes.",
"So whenever you have a choice of being able to do something",
"10 different ways, it is just good to be disciplined and use",
"one technique all the time so that you",
"just don't make mistakes.",
"Now one benefit of taking the current to be going into the",
"positive terminal of the element is that the power",
"consumed by the element, which is given by vi, is positive.",
"So if you choose your convention of the current",
"going into the positive terminal, then the power",
"consumed by the element, vi, is going to be positive.",
"So this way, or this convention, of assigning",
"voltage and currents as the branch variables is called the",
"associated variables discipline.",
"And again, as with all disciplines, it's something we",
"impose on ourselves, it's a convention.",
"We just follow this convention to simplify our lives and to",
"reduce the chance that we're going to make mistakes.",
"And if you don't like this, don't worry, go ahead, make my",
"day, pick different convention for v and i, just be careful.",
"If you follow this convention, it is less likely you will",
"make mistakes.",
"Now before we actually begin the KVL method, there is one",
"more thing that I need to quickly cover which is a few",
"elements relationships.",
"So you have seen resistors before.",
"So for a resistor with the resistance value R, the v-i",
"relationship is simply given by v equals iR.",
"Where if this is my voltage v, and the",
"current i, v equals iR.",
"Next, for our voltage source, let me draw out a voltage",
"source with voltage V0 in this case.",
"And let me assign branch variable, v plus v and v, and",
"the current i.",
"Then the element relationship, v is given by V0.",
"One of the places where I find students getting confused is",
"the difference between element values and the",
"component value itself.",
"So if you look at a voltage source of this sort, V0 is a",
"property or this voltage source.",
"So if it's a battery, it might supply a",
"voltage V0 of 1.5 volts.",
"However, when I take this voltage source and I put that",
"into a circuit, with the circuit it's going to draw",
"some current i and there will be some voltage across it.",
"If there's a resistor, it will be some of the voltage.",
"But if I put an ideal voltage source with internal voltage",
"V0 and I stick that into a circuit, then the branch",
"variable v associated with that will also be equal to V0.",
"But in general, that doesn't have to be true.",
"Next, let me go to a current source.",
"In this case, I would define my current source like so.",
"I can also apply branch variables v and i.",
"And the equation, if the current source supplies a",
"current I0, then the element relationship for this is given",
"by i equals I0.",
"So these are three examples of element relationships for",
"resistors, voltage sources, and current sources.",
"OK, we are done with the preamble.",
"We are now all set to dive into method one of circuit",
"analysis, the KVL, KCL method.",
"So let's apply this method to the demo circuit",
"that we used earlier.",
"And let's see how we can find out all the",
"element v's and i's.",
"Now recall, the KVL, KCL method said, in fact circuit",
"analysis said, that solving a circuit involves finding all",
"the element v's and i's.",
"So why don't we start this example with labeling all the",
"v's and i's for our demo circuit.",
"OK, so let's get started and very quickly label all the",
"branch variables.",
"And I'm going to to follow the associated variable discipline",
"in doing that.",
"So let me start by labelling all the branch variables.",
"Let me call this v0, current is i0 coming in here.",
"Let me call this v1 and let me say this is current i1.",
"Let me say this is v4, current i4.",
"Let me say this is v3 and current i3.",
"Let me then say that this guy here is v5 and current is i5.",
"In this case, it's v2 and the current is i2.",
"So I have a bunch of branch variables assigned.",
"And I also have these branch variables involved with the",
"currents and voltages.",
"And notice here that in this example I have used the",
"associated variable discipline, that I have",
"assigned all my currents flowing into the positive",
"terminal of each of the assigned branch voltages.",
"So here I get 12 unknowns in all.",
"If we count v0, v1, v2, v3, v4, and v5, that's 6.",
"And i0 through i5, that's another 6, so I have a total",
"of 12 unknowns.",
"Also notice that is a is a node, b is a node, c is a",
"node, and d is a node.",
"And so those are my nodes.",
"I will be writing KCL for those nodes.",
"And then I have a bunch of loops.",
"So what are my loops here?",
"So I have a loop L1, I have a loop L2, I have a loop L3.",
"Now I claim there is one more loop.",
"Like let me pause here for a second and have you think",
"about where that fourth loop is."
]
}

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"SPEAKER 1: All right, so to do KCL at e2, let's start with",
"the current leaving this node.",
"Remember, my usual habit is to sum the",
"nodes leaving the node.",
"So the current heading in this direction will be e2 minus e1",
"divided by R3.",
"So current going left from that node will be e2 minus e1",
"divided by R3 or multiplied by G3.",
"Then the current going up will be e2 minus V0 times G4.",
"And the current leaving the node and going down",
"will be e2 times G5.",
"Have I missed anything?",
"Aha, there is a current here that's coming into the node.",
"And so therefore, the current leaving the mode is minus I1.",
"And all of that sums up to zero.",
"There you have it.",
"So I've written KCL for the two nodes.",
"And I've substituted the device laws as I've been",
"writing it so I don't have this",
"intermediate current variable.",
"So just stare at the beauty of this.",
"Notice that I have two equations, one and two, and I",
"have two variables, e1 and e2, two unknowns and two equations",
"and that's it.",
"OK, so let me go on to step four of the node method.",
"The step four of the node method is simply to go ahead",
"and solve for the node voltages, e1 and e2.",
"So I have my two equations which reflect the KCL at node",
"e1 and KCL at e2.",
"So now it's simply juggling it on with some math and trying",
"to collect all the coefficients of e1 and those",
"of e2, collect them all and just write",
"them a little cleanly.",
"So let me move all the constant terms to the",
"right-hand side.",
"And I will move the variables, or the unknowns, to the",
"left-hand hand side.",
"So let me start with e1.",
"So notice that e1, I multiplied by G1, so",
"I write that down.",
"Multiply by G3, so I write that down.",
"And multiply by G2, so I write that down too.",
"So I get e1 times G1 plus G3 plus G2.",
"And then I collect the terms in e2.",
"And there's only one here, which is minus G3.",
"And the constant terms are here, V0 G1.",
"And I'm going to move that to the right-hand side and write",
"that down as V0 G1.",
"For the next set of steps, I go ahead and do e1 for the",
"next equation.",
"I get minus G3.",
"And then for e2, I go ahead and get a G5, and I get a G4,",
"and I get a G3.",
"So it's G3, G4, and a G5.",
"And I can move the constant terms, which is V0 G4, and I",
"want the right-hand side.",
"And in this case, they both become positive as they move",
"to the right.",
"And I get a I1 here.",
"Good, there you go.",
"So I end up with two equations and two unknowns.",
"And notice I have by e1 times something plus e2 times",
"something equals a constant.",
"And then again, that e1 times something plus e2 times",
"something is another constant.",
"So I two equations and two unknowns.",
"And see how remarkably simple this is.",
"This is basic high school math to solve a pair of",
"simultaneous equations in order to get the answer.",
"And just to be a complete here, let me go ahead and",
"complete it for you.",
"So supposing in step four number four you've gone and",
"solved for the node method, and by solving the node method",
"you get the values of e1 and e2.",
"So let's say you've gone and solved the simultaneous",
"equations for step four and you have e1 and e2.",
"Then, to complete your circuit analysis you need to find the",
"branch voltages.",
"And so you can go ahead and find all the v-i's and i-i's.",
"So in this case, one of my branch voltages is v1 i1.",
"So I just do a couple of quick examples just to show you that",
"it's pretty simple.",
"So what is v1 here?",
"Well let me pause and have you do it."
]
}

5
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"SPEAKER 1: OK, let's now work on the second method of",
"circuit analysis.",
"This is actually a very simple method.",
"And oftentimes when you're applying some of the more",
"sophisticated methods, it is helpful to apply these methods",
"as some of the partial steps.",
"You can find more about this method and the KVL KCL method",
"in chapter two of the textbook.",
"So this method is very simple.",
"If you have a set of resistors in series, R1 through RN, then",
"you can write up an equivalent resistor whose resistance is",
"equal to the sum of the individual resistances.",
"Now you can also do a complimentary circuit.",
"You can make a set of conductances.",
"Notice that G is our nomenclature for conductance.",
"So G1, for example, is 1 divided by R1 and so on, so",
"G's a conductance.",
"So when I have a set of conductances in parallel, then",
"I can add them all up and replace them with a resulting",
"conductance given by G1 plus G2 plus G3 dot dot dot GN,",
"where G1 equals 1 by RI is a conductance.",
"Another simple thing to remember is when you have two",
"resistances in parallel, let's say R1 and R2, the equivalent",
"resistance is given by R1 R2 divide by R1 plus R2.",
"This is just a special case for two resistors.",
"Let's continue one with element combinations rules.",
"In case C is when I have two voltage sources in parallel.",
"Let's call them V1 and V2.",
"Then I can replace them with a single voltage source whose",
"voltage is given by V1 plus V2.",
"I can get the current based circuit in a similar manner.",
"So let's say I have two current sources in parallel,",
"I1 in parallel with I2.",
"Then I can replace them with a single current source whose",
"current is I1 plus I2.",
"In this example, I would like to use element combination",
"rules to solve for the following situation.",
"So I have a little circuit here with a voltage source of",
"three resistors.",
"And let's say I care about finding the current I through",
"that branch.",
"So how do I go about doing it?",
"So let's start applying element competition rules.",
"As a first step, let me combine these two resistances.",
"And so I get the same voltage source, my resistance R1 here.",
"But in place of R2 and R3, I get the combination of",
"resistors given by R2 R3 3 divide by R2 plus R3.",
"Now as soon as I get that, I now have two resistances in",
"series, so I can sum them up very quickly like so.",
"And I can get a single resistance here, R, which is",
"the sum of R1 plus R2 R3 divide by R2 plus R3.",
"So now notice that this was my current that I needed to find.",
"And I was the current I needed to find.",
"So I brought this down to a very simple pattern.",
"And so I can very quickly write I is simply given by V",
"divided by R.",
"Now the textbook in this particular section gives you a",
"lot more interesting examples of how you can use intuitive",
"analysis to analyze many, many circuits like this.",
"And I would encourage you to go and read that section",
"that's part of section two."
]
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"SPEAKER 1: All right.",
"So I can apply the KVL rule.",
"The KVL rule simply says that I can take the difference in",
"the node voltages.",
"And then applying the element rule, I can divide by R1 to",
"get the current going up.",
"So the current going up is simply E1 minus V0, which is",
"the voltage difference between these two points.",
"And I can divide that by R1 to get the current going up.",
"And dividing by R1 is the same as multiplying by the",
"conductance, G1.",
"OK.",
"Next.",
"What is the current going in this direction?",
"The current going in that direction is simply E1 minus",
"E3 divided by R3, or multiplied by the",
"conductance, G3.",
"And finally, I can figure out the current going down, which",
"is simply E1 minus 0, which is the ground",
"potential, times G2.",
"And that sums to 0 by KCL.",
"OK.",
"Next let me write down KCL at the node two.",
"So let me give you some practice, and let me pause",
"while you think about doing it yourself.",
"And then I'll go ahead and do it."
]
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"SPEAKER 1: So the fourth loop really is this outer loop here",
"that goes like so.",
"That is four.",
"So notice that I have four nodes, and I have four loops,",
"and I have 12 unknowns.",
"And the goal of the method will be to find out the values",
"of each of these 12 unknowns.",
"If you're ready, now that you've labeled the circuit,",
"let's go to step one of the KVL, KCL method.",
"And our overall goal will be to find out the values of",
"these 12 unknowns.",
"You see the circuit on the right-hand side, and then",
"notice that all the branch variables have been assigned",
"as we did in the previous step.",
"So the first step of the KVL, KCL method is to write down",
"all the element v, i relationships so",
"there are six elements--",
"one, two, the R2, R3, R4, R5, so six elements.",
"So let's write down six v, i relationships.",
"So for the volted source, I get V0 is capital V0.",
"And this is given because I know the value",
"of the voltage here.",
"I can do V1 is i1R1.",
"V2 is i2R2.",
"And then for resistor three, I can write V3 is i3R3.",
"V4 is i4R4.",
"And V5 is i5R5.",
"So notice that there are six equations here.",
"So we're done with the element relationships.",
"And let me now go on to the next step of the KVL, KCL",
"method, which is to write KCL at each of the nodes.",
"So I have four nodes, A, B, C, and D. So let's start by",
"writing KCL for node eight.",
"So let's write KCL for node eight.",
"So at node eight, notice that I have three currents deal",
"with, i0, i1, and i4.",
"So here, again, I'm going to use a little bit of a",
"convention in how I sum up currents.",
"And as a rule, I can sum currents going in or",
"currents going out.",
"But again, I'm just going to use a convention.",
"I am going to sum currents leaving the node.",
"So for every node, I'm just going to sum currents leaving",
"the node if they should add up to 0.",
"So for node eight, notice that current i0 is leaving the",
"node, i1 is leaving the node, and i4 is leaving the node.",
"So I can sum them all up to be i0 plus i4 plus i1 equals 0.",
"Similarly, for node b, I have currents--",
"let's see, we have i2 leaving the node, so i2.",
"Then we have i3 leaving the node.",
"So I write i3.",
"And then, notice that i1 is coming into the node.",
"So there, it's not a plus, but it's a minus i1",
"coming into the node.",
"And that equals 0.",
"Then, similarly, at the next node, b, I can write down i5",
"leaving the node minus i4 leaving the node.",
"And then I have i3 coming into the node.",
"So it is minus i3 leaving the node.",
"Finally, for c, I have minus i0 minus i2 minus i5 equals 0.",
"So in this case, I have four equations.",
"But notice that even though there's four equations,",
"because of the circuit topology, there is one",
"dependent equation.",
"So I really end up getting three independent equations.",
"So as you're doing the analysis, you will find that",
"there's only three independent equations, given these four",
"nodes in a circuit.",
"OK, let's go to step three of the KVL, KCL method.",
"And the step three says write down the Kirchhoff's Voltage",
"Law, KVL, for each of the nodes.",
"So here, again, let me start by writing KVL for loop one.",
"So what I'm going to do is I'm going to follow loop one in",
"this direction.",
"I'm going to go around clockwise for loop one.",
"So here, again, I have a convention to follow.",
"As I go down the loop, how do I write down the voltages?",
"There's many conventions people follow.",
"But the convention that I follow and I recommend, which",
"is very simple, is as you go around the loop, you simply",
"assign to each voltage the first encountered sign.",
"So as you go around this loop here, I get the v0 voltage.",
"And the sign encountered is minus.",
"So I'm going to assign that a minus sign.",
"So let's do that.",
"So for the loop one, I'm going to go around the",
"loop in this direction.",
"And the first encountered voltage is v0.",
"And the first encountered sign is a minus.",
"So I write minus v0.",
"And I continue down this way.",
"And I hit v1.",
"In this case, I hit the plus sign.",
"So it is plus v1.",
"And I go down.",
"And then I hit plus v2.",
"And that sums to 0.",
"Similarly, for loop two, I'm going to go around the loop.",
"Why don't I say just for fun, for this loop, why don't I go",
"around in a counterclockwise manner.",
"So in this case, I'm going to hit the positive terminal of",
"v1 first. So I write plus v1.",
"Then I'm going to hit the positive terminal of v3.",
"So I write plus v3.",
"Then the minus terminal of v4, so minus v4.",
"So similarly, for loop three, I get v3 plus v5",
"minus v2 equals 0.",
"And for loop four, I get minus v0 plus v4 plus v5 equals 0.",
"And again, because of the circuit topologies, there's",
"three independent loops.",
"And so I get three independent equations.",
"And so this last equation is redundant.",
"OK, so putting it all together, I'm in",
"the KVL, KCL method.",
"You saw that we developed six equations for the element",
"relationships.",
"And then we had three independent equations from the",
"KCL at the nodes and three more KVL at the nodes.",
"So in all, notice that we ended up with 12 equations.",
"And recall we had 12 unknowns, v0 through v5",
"and i0 through i5.",
"So now you can take these 12 equations and put them into a",
"hopper and turn the crank.",
"And you can go and get the answer.",
"However, this is very grubby.",
"But even a simple circuit like this, now, with just a small",
"number of elements, three inner loops and one outer",
"loop, I had a horrible mess, 12 equations, 12 unknown.",
"And so this works, but this is a real pain.",
"We will see that the node method, which builds on the",
"KVL, KCL method, will be much, much better",
"and will be way simpler.",
"And we'll have far fewer equations than",
"the KVL, KCL method.",
"But it's good to know about the KVL, KCL method, because",
"it's the simplest method that there is.",
"Just that it ends up with a whole boatload of equations",
"that you have to solve."
]
}