Parallel resistors (part 3) | Circuit analysis | Electrical engineering | Khan Academy

In this video, we're going to talk even some more about parallel resistors. Parallel resistors are resistors that are connected end to end and share the same nodes. Here's R1 and R2; they share the same nodes, that one and that one, and that means they share the same voltage.

We worked out an expression for how to replace that with a single resistor, R parallel, and we found that ( \frac{1}{R_{\text{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} ). So in this video, I'm going to actually start working with this expression a little bit more, and we'll just change it around a little bit to an easier way to remember it. Then, we're going to do a special case where R1 and R2 are the same value, and we're going to see what happens.

Right now, I just want to do a little bit of algebra on that expression. ( \frac{1}{R_P} ) is the same as ( \frac{1}{R_1} + \frac{1}{R_2} ). What I want to end up with here is an expression on this side that says ( R_P ) equals something, and on this side, I just want one expression, not two fractions.

So we're going to go about that by combining these two fractions first. The least common denominator here, L.C.D., equals ( R_1 \times R_2 ). I'm going to convert both of these to that proper denominator. I have to multiply by ( \frac{R_2}{R_2} ) on this expression, and I have to multiply it by ( \frac{R_1}{R_1} ).

This equals, of course, ( \frac{1}{R_P} ). Continuing on, ( \frac{1}{R_P} = \frac{R_2}{R_1 R_2} + \frac{R_1}{R_1 R_2} ). Now I can combine them together. Let's move up here:

( \frac{1}{R_P} = \frac{R_1 + R_2}{R_1 R_2} ).

Now I'm going to take the reciprocal of both sides of the expression, so I get an expression in ( R_P ):

( R_P = \frac{R_1 R_2}{R_1 + R_2} ).

That is a way you can remember how to combine parallel resistors. The parallel equivalent resistor ( R_P ) is the product of the two resistors over the sum. So, it's a product over the sum; that's how I remember it.

Now, this arithmetic, this expression is exactly the same as the original one that we had. These are the same, and it's just a question of which one do you want to remember, which one's easier to remember, and which one's easiest to calculate.

I like to remember this one here. Let's do a quick example using it. I move the screen up a little bit; let's leave that there so we can see it.

Let's say we have two parallel resistors, and we'll say the first one is 1,000 ohms, and the second one is 4,000 ohms. The question is, what is the parallel combination of those things? How can I replace these with one resistor so that the same current flows?

We'll use this expression here:

( R_P = \frac{1,000 \times 4,000}{1,000 + 4,000} ).

That equals, oh my goodness, there's a lot of zeros here:

4 and 6 zeros, ( \frac{4,000,000}{5,000} ).

Alright, let's say let's take out three zeros out of this one, let's knock off three zeros here and three zeros here, and I have ( \frac{4,000}{5} ), and that equals 800 ohms. So that is this, and that's how we use this expression.

Now just something to notice here: notice that our parallel equivalent resistor is smaller than both of these, and that happens every time. The parallel resistance is smaller than the smallest resistor. So here it was 1,000; it's going to be smaller than that, and that's a property of parallel resistors.

Because you have two current paths that allow current to go two different ways, the effect of resistance is always smaller than the smallest original path because there's a way for current to go around another way. So this is now the expression for two parallel resistors, and that's a good one to remember.

Now, I'm going to show you one special case, and we'll do this in this color. What if ( R_1 = R_2 )? What is ( R_P )?

For this special case, we use the same expression; we say ( R_P ) is the product, we'll just use ( R ) because ( R ) is the same value:

( R_P = \frac{R \times R}{R + R} ).

That's multiplied, and so that is ( R^2 \over 2R ), and one of these Rs cancels. So let's cancel, let's cancel that R and that squared, and we end up with ( \frac{R}{2} ).

So, for the special case, if ( R_1 = R_2 ), then ( R_P = \frac{R}{2} ); it's just half. And that should make sense. If we do something like this, if we draw two resistors in parallel like this, and we say this is 300 ohms, and this is 300 ohms, that means the effective parallel resistor is 150 ohms.

That's got a very pleasing symmetry to it because these resistors are the same. They have the same voltage, and they're going to have the same current. Basically, twice as much current is going to flow in this circuit as would flow if there was just one of these guys. So that's where the divide by two comes from.

So, for two resistors in parallel, if the resistors are the same value, the effective parallel resistance is just half.