yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

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


2m read
·Nov 11, 2024

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.

More Articles

View All
Finding Michigan’s Wild Side: A Journey through the Upper Peninsula | National Geographic
For years, I’ve heard from friends how the Upper Peninsula of Michigan is this mythical place that I needed to see at some point in my life. I’m very grateful as a National Geographic photographer to travel all around the world to see magnificent landscap…
BULLET TIME FAIL (Slowmo) - Smarter Every Day 101
Surprised you had it running at full speed. Hey, it’s me, Destin. Welcome back to Smarter Every Day. This is Mark Rober. Mark is using me. Is this true or false? - [laugh] - Mark is using me to access your subscribers. I’m not really happy about this, Mar…
Counting faces and edges of 3D shapes
How many faces does the following shape have? Pause this video and see if you can figure that out. All right, I’m assuming you paused it, and I’ll see if we can work through it together. I’m going to actually try to color the faces. So, we have this face…
See What It Takes to Hide a Secret Tracker in a Rhino Horn | Short Film Showcase
[Music] Africa’s got the greatest number and diversity of large mammals. It’s the continent that’s been blessed with the most wildlife. Many of these animals, like the black rhino, are down to a few thousand. This is it; in the next hundred years, years m…
How To Make The Greatest Comeback Of Your Life (In 6-12 months)
You hear your alarm clock. Snooze it multiple times, and when you finally wake up, you think to yourself, “Ah, here we go again.” You get out of your bed feeling tired and unmotivated. You grab your phone, start scrolling, and see other people living thei…
Theorem for limits of composite functions | Limits and contiuity | AP Calculus | Khan Academy
In this video, we’re going to try to understand limits of composite functions, or at least a way of thinking about limits of composite functions. In particular, we’re going to think about the case where we’re trying to find the limit as x approaches a of …