Combining Resistors in Series & Parallel

by Patrick Ford
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Hey, guys. So in this video, we're gonna talk about combining resistors, which is a key skill that you have to master for this chapter. Let's go. All right, so in circuit problems, you're often gonna be asked to collapse or combine or merge multiple resistors into a single equivalent resistor single equivalent. Resist store just one. And resistance could be connected to each other in two ways. It could be connected in series or in parallel. And this looks like this in Syria's you're gonna have resistors sort of side by side like this. And parallel, you're gonna have resistors like this. All right? And what you want to do is you want to go from a number of multiple resistors? Let's say here, three into a single equivalent resistor. Same thing here. You wanna turn these three resistors into just a single resistor. Now, what makes this a serious connection is that you have a direct connection between the resisters without any splits in the wire. So imagine if you are a new electron traveling through this wire, you go straight through with no forks on the road. All right, so and then here the wires splits and because the wire splits, you get some loops. So let's say we connect this to let's connect this to a battery toe. A voltage source. Kind of like a battery, right? Remember, charge charge moves out of the current gonna flow out of the positive. The larger terminal here. It's gonna go this way when he gets here, it has the choice of moving this way or this way. So some of the charge will go one way some of the charges go the other way. Eso the wire splits and forms a loop. So one other thing that's important to note is you have a parallel connection whenever you have two resistors that are alone on opposite page on opposite sides on opposite branches. So let me write this alone on opposite branches where opposite sides You can think of it that way as well. So, for example, this guy is alone in this branch. This guy is alone on this branch. So they are alone on opposite sides are on opposite branches. Therefore they are in parallel. All right now, the way you combine them is by using the equivalent resistance equation, which is different for Siris in parallel and you have to memorize this one. The equivalent resistance if you're in Siris is just the addition of the individual resistances R one plus r two plus r three. For example, if this is a one own resistor and this one is too and this one's three, this equivalent resistor here is going to be six. You just add up the numbers Now, if you are in parallel, the equation is a little bit more complicated. It's one over. Our equivalent equals one over R one plus one over r two and you keep adding one of these fractions for each resistor. Here I have three someone, right? One over are three. Okay. And what it means to be an equivalent resistor is that you behave the same as the original group of resistors. So let me illustrate this. Let's connect the battery here positive side of the battery So the current gonna flow out this way. The idea is that this group of resistors, this group of resistors, behaves just as this single resistor would Here, as far as the batteries concern, it is the same exact thing. The battery cannot distinguish these three resistors from a single equipment resistor, which is why they're called equivalents. Same thing here. If you combine these three resistors into a single equivalent resistor, as far as the battery is concerned, the battery sees the same exact amount of resistance. Okay, one last point I want to make is that if you combine resistors in Syria's like we did hear the equivalent resistance we're always going is always going to be higher than the individual resistance is. And this should make sense. Since you're adding the numbers right one plus two plus three, the total number is obviously going to be higher. Now. If you have parallel connection, the number is always going to be lower. And that's because, in this case, remember I mentioned that the the current splits. So now, because the electrons have an option of going one way or the other, there is effectively less resistance because they have a choice co. So let's do two examples here. What is the equipment resistance of the following resistors? What we want to do is get from four resistors into a single resistor, and there are sometimes they're multiple ways of doing this multiple paths that you can take to get to the answer. Sometimes there's only one path, and what you have to sort of do is map about how are you gonna go from 4 to 1? And what you want to do is look for places where you can easily combine these resistors. So, for example, one thing you might do is look at one and start comparing it and start seeing how it's connected to every single one of these. So is one in series or in parallel with two? Well, remember, Siri's means that they have a direct connection. Is there a direct connection from 1 to 2? So if you're if you're a charge, you're going through here and then right here the wire splits. There is no direct connection. So one and two are not in Siris with each other. Okay? They're not in serious. What about in parallel? Well, parallel means that there alone and opposite sides of a branch, they're not even in the same loop. This is a loop. This is a loop. There are different loops, so they're not in parallel either. So you cannot combine one and two right now. What you can do is you can combine one and four because one and four are alone on opposite sides on opposite branches off the loop so you can combine these two in parallel so you can combine these two in parallel. And you can also combine these two in parallel. And if you do that, you go from having 2 to 1, and then you go from having 2 to 1. So now you have a simpler circuit. So let's do that. Let's show that here, I'm gonna go and redraw this where this entire thing is gonna become a single resistor. Let's call this single resistor R one. And over here I'm gonna have another resistor R two. Okay, now how can I get these two guys to be just a single resisted? Well, these guys are in Siris with each other. Hopefully, you see that right away because they're just sitting next to each other. There's no, um, splits on the wire, right? One flows directly to the other. So they are in Siris, which means I'm going to be able to easily combine them into a single resistor. Let's call that are three. So the idea is that everything that's inside of green here becomes just this single resistor right here. So what I like to do is I like to draw the paths without doing the math, get all the way to the end. So I know my path, and then we're now actually going to calculate. Okay, so the first thing I did is I emerged these two into our one. So let's calculate R one r one is in parallel are one is the parallel connection of one and four. So to find our one, I'm gonna have to write the parallel equation, which is this one here. Cool. So it's gonna be one over R one, which is what I'm looking for. And then one over plus one over. I like to make room to plug in variables just so it's a little bit more organized. And then we just gotta plug in the numbers one and +41 And for now, this is a fraction. So you have to get a common denominator, remember, And to get a common denominator between one and four, you could just multiply this side by four. And if you do that in the bottom, have to do at the top and then you can multiply, decided by one, which is effectively not doing anything. And you do the same up here. Now you have four. You have 4/1. I'm sorry. 4/4 plus 1/4. Because the denominators the same. I can combine them into five at the top ads. And the bottom combines 5/4. Are we done? Is that the answer? No. There's one last step which is noticed that the R one isn't the denominators on the bottom of the left side. And I have to solve for R one one way to do this that I like is you can just flip these two. But then if you flip that on the left side, you have to flip on the right side as well. So I'm gonna get one over r one and flip, and I'm gonna get the 5/4 and flip, and it's gonna become 4/5. So our one is 4/5, which is 0.8. Oh, cool. We're gonna do the same thing for eso. We got this guy boom gonna do the same thing to find are 21 over r two equals one over apprentices plus one over apprentices. And this is the This is the parallel connection between the two and the two over here in blue. So we're gonna put a to here and the two year the denominator is already the same. So I'm just gonna merge and say one plus one is to the and then this is one. Now remember, I have to flip this, but here because you have a one. And remember, there's always an implicit one here. If you flip 1/1, you still get a one long story short are too is simply one. Oh, okay. We're almost done. I have war. One I have are two. Now we're just combine them in Siris. Whenever you add something, whatever you combined resistors in Syria's you just add their resistance is so finally are. Three is R one plus R two, which is one plus 10.8, which is 1.8 OEMs. Cool. That's it. That's the final answer for this one. Let's do one more. And what you might want to do here is maybe a positive video and at least lay out the steps off. How would you combine these guys? Okay, I'm gonna keep rolling here. Um, if you if you hopefully you saw right away that these Aaron Siri's with each other and then that these air in series with each other as well, let's color code. You can't merge anything else yet. Okay, So this is going to give me if you follow the wire here carefully, you're gonna go up, and then this entire red thing over here is gonna be replaced by Let's call This guy are one. So that's gonna be our one. And I'm actually gonna make this black. So that's our one. And then down here, the two resistors and blue are gonna become are too. And then once I have these two resistors, hopefully so this isn't seriously is in serious, hopefully see that these guys are in parallel because they're alone on opposite sides on opposite branches of the loop. And I could go one step further and say that these two will just combine into our three. Okay, Now let's calculate r one r two and then our three are one is the Siri's connection. So it's just one plus 21 plus three is four, so I can actually just put it over here. This is four homes. This is two and four, which is six homes. So to find our three, I just have to merge. The four owns with the six homes. So let's do it. Over Here are three one over R three is one over the first resistor, one over the second resistor. In this case, we have a four and a six, So I have a four and a six. The best way to get the easiest way to get a common denominator here is just to multiply this four by six and then put a six on top. And then this six by four, and then put this on top is well, six times four times six is 24 on both of them and on top. I have six plus four, which is 10. So I got 10/24. Are we done? Is that the answer? No. Be careful. You gotta flip the r three. So I'm gonna flip the left and right. So I'm gonna get our 3/1 or just r three equals the flip of these guys. 24 the body by 10 which is just 24 homes and that's the end of this one. Hopefully makes sense. Hopefully you got it. Hope you think it's easy. Let's keep rolling.