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Parallel resistors (derivation continued)

Learn how to calculate the equivalent resistance for multiple parallel resistors. This method simplifies complex resistor networks by replacing them with a single equivalent parallel resistor, streamlining circuit analysis and improving efficiency. Grasp the concept of current splitting among resistors and understand that all parallel resistors share the same voltage, making this technique applicable to various electrical engineering scenarios. Created by Willy McAllister.

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Video transcript

- [Voiceover] In the last video, we introduced the idea of parallel resistors. These two resistors are in parallel with each other because they share nodes and they have the same voltage across them. So that configuration is called the parallel resistor. And we also showed that these two resistors could be replaced by a single resistor we labeled this one R1, this is R2. We showed that we can replace R1 and R2 by an equivalent parallel resistor with this expression here for two resistors. RP ..one over RP equals one over R1 plus one over R2. So that's how you calculate the equivalent resistance for two parallel resistors. Now you can ask and it's a good thing to ask what if there's more resistors? What if there's more resistors in parallel here? What if I have R3 and R4 and RN all connected up here. What happens in this expression? So like we did before we had a current here and we know that current comes back here. The first current split, some current goes down through R1, some current goes through R2 and if we had more resistors some goes down through R3, and some goes through RN. So the current basically is coming down here and splitting amongst all the resistors. Now all the resistors share the same all the resistors share the same voltage. So that's just V, let's label V. That's just V, they all share the same V and they all have a different current. Assuming they all have a different resistance value. So we do exactly the same analysis that we did before. Which was, we know that I here here has to be the sum, there's the summation symbol of all the I's. I1, plus I2, plus I3 plus IN, that's as many as we have. So we know that's true. And we also know that the current, we also know that the current in each individual resistor IN is equal to one over that resistor times V, and V is the same for every one of them. So now we substitute this equation into here for I For I. We get the big I, the overall I is equal to voltage times, it's gotta be a big expression, one over R1 plus one over R2 plus one over R3, plus as many resistors as we have One over RN, like that. And we do the same thing as we did before which was we say that this expression here is equivalent to one parallel resistor we're going to make the equal to one parallel resistor. So this whole guy here is gonna become one over RP That gives us a way to simplify any number of resistors down to a single parallel resistor and I'll write that over here. So for N resistors, multiple resistors one over RP the equivalent parallel resistor is equal to the same thing. One over R1 plus one over R2 plus dot, dot, dot, one over RN. So this tells you how to simplify any number of parallel resistors down to one equivalent parallel resistor.