Parallel resistors (part 2) | Circuit analysis | Electrical engineering | Khan Academy

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 a parallel resistor.

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 = \frac{1}{\frac{1}{R1} + \frac{1}{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 are more resistors? What if there are more resistors in parallel here? What if I have R3 and R4, R and RN all connected up here? What happens to this expression?

Like we did before, we had a current here, and we know that current comes back here. The first current splits; some current goes down through R1, some goes through R2, and if we add more resistors, some goes down through R3, as some goes down through RN. So, the current basically is coming down here and splitting amongst all the resistors.

Now, all the resistors share the same voltage. So, 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 we did before, which was we know that I here has to be the sum. There's the summation symbol of all the I's: ( I1 + I2 + I3 + ... + IN ). That's as many as we have, so we know that's true.

We also know that the current in each individual resistor ( I_N ) 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. We get the big I. The overall I is equal to voltage times it's going to be a big expression:

[
I = V \left( \frac{1}{R1} + \frac{1}{R2} + \frac{1}{R3} + ... + \frac{1}{R_N} \right)
]

And we do the same thing as we did before, which was we say this expression here is equivalent to one parallel resistor. We're going to make that equal to one parallel resistor.

So, this whole guy here is going to become:

[
\frac{1}{RP}
]

That gives us a way to simplify any number of resistors down to a single parallel resistor.

I'll write that over here. So for ( n ) resistors, multiple resistors:

[
\frac{1}{RP} = \frac{1}{R1} + \frac{1}{R2} + ... + \frac{1}{R_N}
]

So, this tells you how to simplify any number of parallel resistors down to one equivalent parallel resistor.