Understanding Circuit Analysis Basics

Welcome to my video on circuit analysis. In this video we're going to talk about how to find the current through and the voltage across every resistor in this circuit. But before I get started let me introduce myself. My name is Chris. I'm the creator of all the videos here at MathMeeting. And if you need any extra help with your homework, send me pictures to homeworksolutionsatmathmeeting.com. My email is on the right-hand side of the screen. Send pictures, and I'll get back to you immediately with a quote. Or if you want tutoring, send me an email. live tutoring at mathmeaning.com and we can set something up. But let's get started right away with this example. Alright so in this example we need to find the current through and the voltage across each resistor. And to do this our first step is to find an equivalent circuit. So we need to simplify this circuit so there's an equivalent circuit with only one resistor. All right so if you want to simplify this circuit so that there's only one resistor, we need to use these two formulas that I wrote for you at the top of the screen. And if there's two resistors or two or more resistors that are in series, then we use this formula here on the left side of the screen. screen. And the formula is the total resistance is equal to R1 plus R2 plus R3. In other words, we just add all of the resistors together. But if two or more resistors are in parallel, then we use this formula here on the right. And that's equal to 1 over the total resistance or the equivalent resistance is equal to 1 over R1 plus 1 over R2 plus 1 over R3, etc, etc, depending on how many resistors we have. So we're going to use these two formulas to simplify this circuit so that there's only one resistor left. And what I like to do is to start with the resistors that are the farthest away from the battery source. So we have a battery source over here of 24 volts and the resistors that are farthest away from this battery source are these two resistors over here on the right hand side of the screen. These two resistors are in series. Because they're in series we just need to add them together. We're going to use this formula here on the left hand side of the screen. So if we add them together we're going to have 3 ohms plus 12 ohms. 3 plus 12 is equal to 15. So we can erase these two resistors on the right hand side of our circuit. After we erase these two resistors we can place them with one equivalent resistor of 15 ohms Alright so the 12 ohm and the 3 ohm resistor in series made this 15 ohm resistor and Now once again, we want to simplify the resistors that are farthest to the right of our battery source So I'm going to simplify these two resistors that are in parallel with each other. Because they're in parallel with each other we have to use this formula on the right-hand side of the screen. And just to give myself a little bit more space I'm going to erase these formulas here on the top. Alright so once again we're going to simplify these two resistors that are in parallel with each other. So we're going to use our parallel resistor formula and the equivalent resistance or the total resistance for two resistors that are in parallel is equal to 1 over R1. R1 is equal to 10 ohms plus 1 over R2 which is equal to R2 is equal to 15 ohms. So we have 1 over 10 plus 1 over 15 which is equal to 5 over 30. 30 simplifies to 1 over 6. So 1 over the total resistance is equal to 1 over 6. So we know that the total resistance, we'll say the total resistance we know is equal to 6 ohms. So I can erase both of these resistors on the right hand side of our circuit and replace them with one equivalent resistor of 6 ohms. Okay, so we're almost finished simplifying this. So there's only one reason. resistor left. The only thing we have to do now is simplify these three circuits that are in series with each other. All three of these circuits are in series with each other. So the only thing we need to do is add them together to get our equivalent or total resistance. All right, so the total resistance for three resistors that are in series is just R1 plus R2 plus R3. So we have two... plus 6. 2 plus 6 is equal to 8. 8 plus 4 is equal to 12. So if we add all of the resistors together, we have a total or equivalent resistance of 12 ohms. So I'm going to erase all of these resistors that are in series and replace them with one equivalent resistor of 12 ohms. Okay, so now that we've simplified this circuit so that there's only one resistor, now we can find the total current. running through the circuit. And we can use the formula the voltage is equal to the current which is written with the letter I times the resistance. Okay so the voltage we know is 24 volts and the the resistance the equivalent or total resistance is equal to 12 ohms. So we can plug that in and we know that the current is going to be equal to the the voltage which is 24 volts divided by the the resistance which is 12 ohms. 24 divided by 12 is equal to 2 so the current is going to be equal to 2 amps. Alright, so we know that the total current running through the circuit is 2 amps. So we know that the current leaving the battery is equal to 2 amps. And we can use this information to find the current through and the voltage across every resistor in the circuit. entire circuit. So what I like to do is to keep going back to each diagram that we simplified. So if we go back one more diagram that we simplified, the circuit is going to look something like this. we know once again that the total current is 2 amps so the current leaving the battery is 2 amps and we know from my last video that the current never changes for any resistor that's in series. So when this current goes through this resistor that's in series we know that's not going to change. So we know that the current going through this first resistor is also equal to 2 amps. So the current I1 is going to be equal to 2 amps. And we also know that the current going through this second resistor, which is also in series, isn't going to change. So the current going through this second resistor, I2, is also equal to 2 amps. And I think you get the idea by now. This current going through this third resistor is not going to change because it's in series. It's also going to be equal to 2 amps. All right, so now that we've found the current through each resistor, now let's find the voltage across each resistor. And we can use the same... formula as before. To find the voltage across resistor number one, the voltage across resistor number one is equal to the current of resistor number one times the resistance. All right, so we know that the voltage across one is equal to the current across through one, which is two amps. So we have two amps for the current. We know that the resistance is equal to two ohms, so we got to multiply it by two ohms. Two times two is equal to four. voltage across resistor number one is equal to four volts. Alright so now let's find the voltage across resistor number two. The voltage across two is going to be equal to the current of two amps. So we have a current of two amps multiplied times the resistance of six ohms multiplied times six ohms. Two times six is equal to twelve. So the voltage across the second resistor is going to be equal to twelve volts. All right, and now let's do the same exact thing for our third resistor. We know that the voltage across the third resistor is going to be the current, which is 2 amps. So we have 2 amps being multiplied by the resistance. We know the resistance is equal to 4 ohms. so 2 times 4 ohms is equal to 8 so the voltage across the last resistor resistor number 3 the voltage across resistor number 3 is going to be equal to 8 volts Alright, so now that we found the current through and the voltage across every one of these resistors, let's go back one more diagram when we were simplifying. Alright so let's recap what we already know. We know that the current going through resistor number 1 is equal to 2 amps. We know that the voltage across resistor number 1 is equal to 4 volts. And we also know that the current going through resistor number 3 is equal to 2 amps. And we know that the voltage across resistor number 3 is equal to 8 volts. All of this we already know from the work we did before. All right, so now let's try and solve for these two resistors here on the right side of the circuit. All right, so how do we do this? Well, we know that the voltage leaving the battery is equal to 24 volts. And we know that once we go across this resistor, we have a 4 volt drop. So 24 minus 4 is equal to 20. So the voltage after that first resistor is equal to 20 volts. volts, which means that the voltage over here is also equal to 20 volts because we haven't crossed any resistors. All right, and we can do the same thing for the end of the circuit. We know that the voltage at the end of the circuit is equal to zero volts because the potential always goes to zero. And we know that there's an eight volt drop before we crossed that last resistor. So zero plus eight is equal to eight. So we know that we have a voltage of 8 volts before we cross the third resistor. So we know that the voltage over here is also equal to 8 volts because we have not crossed any resistor. If you don't cross any resistor, the voltage is going to stay the same. All right, now look what we have. We know that before we cross this resistor, we have a voltage of 20 volts. And then once we cross that resistor, it goes from 20 to 8. So we know 20 to 8 is a difference of 12. So we know the voltage across this resistor is equal to 12. So we'll call this resistor 4. The voltage across resistor 4 is equal to 12 volts. And the same thing is going to happen for this resistor on the right. We know that we're starting with 20 volts. Once we cross the resistor, it goes to 8 volts. 20 minus 8 is equal to 12. We'll call this resistor number 5. So the voltage across resistor 5 is going to be equal to 12 volts as well. Thank you. All right, so now let's find the current going through these resistors. And we can use the same formula. If we just rearrange the formula we know that the current going through resistor number four is going to be equal to the voltage across four divided by the resistance of four. And we know that the voltage across four is equal to 12 volts. So we have 12 volts being divided by the resistance of four which we know is 10 ohms. So 12 divided by 10 ohms. 12 over 10 is equal to 1.2. So we know that the current going through resistor number four is equal to 1.2 amps. And now let's do the same thing for resistor number two. resistor number 5. Alright once again the current through resistor number 5 is equal to the voltage across 5 divided by the resistance of 5. And we know that the voltage across 5 is equal to 12 volts. So we have 12 volts being divided by the resistance of 5 which we know is equal to 15. So we have 15 ohms as the resistance. 12 divided by 15 is equal to 0.8. So we know that the resistance or sorry the current through resistor number five is equal to 0.8 amps and this totally makes sense because we started with a current of two amps and when it splits off into these two parallel resistors we have one current of 1.2 and one current of 0.8 1.2 plus 0.8 is equal to 2 so the total current hasn't changed it just splits and some of it goes down the left side and some of it goes down the right side Alright, so now let's go back one more diagram so we can completely solve the original circuit. Alright, so now the only thing we have left are these two resistors. So once we solve those two resistors, we have completely solved our entire circuit. Alright, so we know that the current going through the right side of these resistors in parallel is equal to 0.8 amps. So we'll call this resistor number 6. So the current through resistor 6 is equal to 0.8 amps. amps. Alright and once again these resistors are in series with each other so the current doesn't change so we'll call this resistor number seven. So the current through resistor number seven is also equal to 0.8 amps. Alright so now the only thing we have left to do is use our formula. So we know that the voltage across resistor number six is going to be equal to the current across six or through six multiplied times the resistance resistor number six. So this is going to be equal to the current which is 0.8 amps. So the current is equal to 0.8 amps multiplied times the resistance. The resistance we know is 3 ohms. Multiply times 3 ohms. 0.8 times 3 is equal to 2.4. So we know that the voltage across resistor 6 is equal to 2.4 volts. Alright so now let's move on. on to resistor number 7. We know that the voltage across resistor number 7 is equal to the current of 7 multiplied times the resistance of 7. This is equal to the current, which is 0.8. So we have 0.8 amps multiplied times the resistance. The resistance we know is 12 ohms multiplied times 12 ohms. 0.8 times 12 is equal to 9.6. So we know the voltage across resistor number 7 equal to 9.6 volts which makes sense because 9.6 and plus 2.4 is equal to 12. So the voltage across these two resistors are exactly the same as the voltage across this resistor. And now we've officially found the current through and the voltage across every single resistor in this original circuit. So I hope this gave you a better idea on circuit analysis and how to solve circuits. Check out my next video if you want to keep on learning. Thank you so much for watching. Don't forget to subscribe. I really appreciate all of you guys who are supporting me making these videos. And I will see you in my next one.

My name is Chris. I'm the creator of all the videos here at MathMeeting. And if you need any extra help with your homework, send me pictures to homeworksolutionsatmathmeeting.com. My email is on the right-hand side of the screen. Send pictures, and I'll get back to you immediately with a quote.

Or if you want tutoring, send me an email. live tutoring at mathmeaning.com and we can set something up. But let's get started right away with this example. Alright so in this example we need to find the current through and the voltage across each resistor. And to do this our first step is to find an equivalent circuit.

So we need to simplify this circuit so there's an equivalent circuit with only one resistor. All right so if you want to simplify this circuit so that there's only one resistor, we need to use these two formulas that I wrote for you at the top of the screen. And if there's two resistors or two or more resistors that are in series, then we use this formula here on the left side of the screen.

screen. And the formula is the total resistance is equal to R1 plus R2 plus R3. In other words, we just add all of the resistors together. But if two or more resistors are in parallel, then we use this formula here on the right. And that's equal to 1 over the total resistance or the equivalent resistance is equal to 1 over R1 plus 1 over R2 plus 1 over R3, etc, etc, depending on how many resistors we have.

So we're going to use these two formulas to simplify this circuit so that there's only one resistor left. And what I like to do is to start with the resistors that are the farthest away from the battery source. So we have a battery source over here of 24 volts and the resistors that are farthest away from this battery source are these two resistors over here on the right hand side of the screen. These two resistors are in series. Because they're in series we just need to add them together.

We're going to use this formula here on the left hand side of the screen. So if we add them together we're going to have 3 ohms plus 12 ohms. 3 plus 12 is equal to 15. So we can erase these two resistors on the right hand side of our circuit. After we erase these two resistors we can place them with one equivalent resistor of 15 ohms Alright so the 12 ohm and the 3 ohm resistor in series made this 15 ohm resistor and Now once again, we want to simplify the resistors that are farthest to the right of our battery source So I'm going to simplify these two resistors that are in parallel with each other.

Because they're in parallel with each other we have to use this formula on the right-hand side of the screen. And just to give myself a little bit more space I'm going to erase these formulas here on the top. Alright so once again we're going to simplify these two resistors that are in parallel with each other. So we're going to use our parallel resistor formula and the equivalent resistance or the total resistance for two resistors that are in parallel is equal to 1 over R1. R1 is equal to 10 ohms plus 1 over R2 which is equal to R2 is equal to 15 ohms.

So we have 1 over 10 plus 1 over 15 which is equal to 5 over 30. 30 simplifies to 1 over 6. So 1 over the total resistance is equal to 1 over 6. So we know that the total resistance, we'll say the total resistance we know is equal to 6 ohms. So I can erase both of these resistors on the right hand side of our circuit and replace them with one equivalent resistor of 6 ohms. Okay, so we're almost finished simplifying this.

So there's only one reason. resistor left. The only thing we have to do now is simplify these three circuits that are in series with each other. All three of these circuits are in series with each other. So the only thing we need to do is add them together to get our equivalent or total resistance.

All right, so the total resistance for three resistors that are in series is just R1 plus R2 plus R3. So we have two... plus 6. 2 plus 6 is equal to 8. 8 plus 4 is equal to 12. So if we add all of the resistors together, we have a total or equivalent resistance of 12 ohms. So I'm going to erase all of these resistors that are in series and replace them with one equivalent resistor of 12 ohms. Okay, so now that we've simplified this circuit so that there's only one resistor, now we can find the total current.

running through the circuit. And we can use the formula the voltage is equal to the current which is written with the letter I times the resistance. Okay so the voltage we know is 24 volts and the the resistance the equivalent or total resistance is equal to 12 ohms. So we can plug that in and we know that the current is going to be equal to the the voltage which is 24 volts divided by the the resistance which is 12 ohms. 24 divided by 12 is equal to 2 so the current is going to be equal to 2 amps.

Alright, so we know that the total current running through the circuit is 2 amps. So we know that the current leaving the battery is equal to 2 amps. And we can use this information to find the current through and the voltage across every resistor in the circuit. entire circuit. So what I like to do is to keep going back to each diagram that we simplified.

So if we go back one more diagram that we simplified, the circuit is going to look something like this. we know once again that the total current is 2 amps so the current leaving the battery is 2 amps and we know from my last video that the current never changes for any resistor that's in series. So when this current goes through this resistor that's in series we know that's not going to change.

So we know that the current going through this first resistor is also equal to 2 amps. So the current I1 is going to be equal to 2 amps. And we also know that the current going through this second resistor, which is also in series, isn't going to change. So the current going through this second resistor, I2, is also equal to 2 amps.

And I think you get the idea by now. This current going through this third resistor is not going to change because it's in series. It's also going to be equal to 2 amps.

All right, so now that we've found the current through each resistor, now let's find the voltage across each resistor. And we can use the same... formula as before.

To find the voltage across resistor number one, the voltage across resistor number one is equal to the current of resistor number one times the resistance. All right, so we know that the voltage across one is equal to the current across through one, which is two amps. So we have two amps for the current.

We know that the resistance is equal to two ohms, so we got to multiply it by two ohms. Two times two is equal to four. voltage across resistor number one is equal to four volts. Alright so now let's find the voltage across resistor number two. The voltage across two is going to be equal to the current of two amps.

So we have a current of two amps multiplied times the resistance of six ohms multiplied times six ohms. Two times six is equal to twelve. So the voltage across the second resistor is going to be equal to twelve volts.

All right, and now let's do the same exact thing for our third resistor. We know that the voltage across the third resistor is going to be the current, which is 2 amps. So we have 2 amps being multiplied by the resistance. We know the resistance is equal to 4 ohms.

so 2 times 4 ohms is equal to 8 so the voltage across the last resistor resistor number 3 the voltage across resistor number 3 is going to be equal to 8 volts Alright, so now that we found the current through and the voltage across every one of these resistors, let's go back one more diagram when we were simplifying. Alright so let's recap what we already know. We know that the current going through resistor number 1 is equal to 2 amps.

We know that the voltage across resistor number 1 is equal to 4 volts. And we also know that the current going through resistor number 3 is equal to 2 amps. And we know that the voltage across resistor number 3 is equal to 8 volts.

All of this we already know from the work we did before. All right, so now let's try and solve for these two resistors here on the right side of the circuit. All right, so how do we do this? Well, we know that the voltage leaving the battery is equal to 24 volts. And we know that once we go across this resistor, we have a 4 volt drop.

So 24 minus 4 is equal to 20. So the voltage after that first resistor is equal to 20 volts. volts, which means that the voltage over here is also equal to 20 volts because we haven't crossed any resistors. All right, and we can do the same thing for the end of the circuit.

We know that the voltage at the end of the circuit is equal to zero volts because the potential always goes to zero. And we know that there's an eight volt drop before we crossed that last resistor. So zero plus eight is equal to eight.

So we know that we have a voltage of 8 volts before we cross the third resistor. So we know that the voltage over here is also equal to 8 volts because we have not crossed any resistor. If you don't cross any resistor, the voltage is going to stay the same. All right, now look what we have. We know that before we cross this resistor, we have a voltage of 20 volts.

And then once we cross that resistor, it goes from 20 to 8. So we know 20 to 8 is a difference of 12. So we know the voltage across this resistor is equal to 12. So we'll call this resistor 4. The voltage across resistor 4 is equal to 12 volts. And the same thing is going to happen for this resistor on the right. We know that we're starting with 20 volts.

Once we cross the resistor, it goes to 8 volts. 20 minus 8 is equal to 12. We'll call this resistor number 5. So the voltage across resistor 5 is going to be equal to 12 volts as well. Thank you.

All right, so now let's find the current going through these resistors. And we can use the same formula. If we just rearrange the formula we know that the current going through resistor number four is going to be equal to the voltage across four divided by the resistance of four.

And we know that the voltage across four is equal to 12 volts. So we have 12 volts being divided by the resistance of four which we know is 10 ohms. So 12 divided by 10 ohms.

12 over 10 is equal to 1.2. So we know that the current going through resistor number four is equal to 1.2 amps. And now let's do the same thing for resistor number two.

resistor number 5. Alright once again the current through resistor number 5 is equal to the voltage across 5 divided by the resistance of 5. And we know that the voltage across 5 is equal to 12 volts. So we have 12 volts being divided by the resistance of 5 which we know is equal to 15. So we have 15 ohms as the resistance. 12 divided by 15 is equal to 0.8. So we know that the resistance or sorry the current through resistor number five is equal to 0.8 amps and this totally makes sense because we started with a current of two amps and when it splits off into these two parallel resistors we have one current of 1.2 and one current of 0.8 1.2 plus 0.8 is equal to 2 so the total current hasn't changed it just splits and some of it goes down the left side and some of it goes down the right side Alright, so now let's go back one more diagram so we can completely solve the original circuit.

Alright, so now the only thing we have left are these two resistors. So once we solve those two resistors, we have completely solved our entire circuit. Alright, so we know that the current going through the right side of these resistors in parallel is equal to 0.8 amps.

So we'll call this resistor number 6. So the current through resistor 6 is equal to 0.8 amps. amps. Alright and once again these resistors are in series with each other so the current doesn't change so we'll call this resistor number seven.

So the current through resistor number seven is also equal to 0.8 amps. Alright so now the only thing we have left to do is use our formula. So we know that the voltage across resistor number six is going to be equal to the current across six or through six multiplied times the resistance resistor number six. So this is going to be equal to the current which is 0.8 amps.

So the current is equal to 0.8 amps multiplied times the resistance. The resistance we know is 3 ohms. Multiply times 3 ohms.

0.8 times 3 is equal to 2.4. So we know that the voltage across resistor 6 is equal to 2.4 volts. Alright so now let's move on.

on to resistor number 7. We know that the voltage across resistor number 7 is equal to the current of 7 multiplied times the resistance of 7. This is equal to the current, which is 0.8. So we have 0.8 amps multiplied times the resistance. The resistance we know is 12 ohms multiplied times 12 ohms.

0.8 times 12 is equal to 9.6. So we know the voltage across resistor number 7 equal to 9.6 volts which makes sense because 9.6 and plus 2.4 is equal to 12. So the voltage across these two resistors are exactly the same as the voltage across this resistor. And now we've officially found the current through and the voltage across every single resistor in this original circuit.

So I hope this gave you a better idea on circuit analysis and how to solve circuits. Check out my next video if you want to keep on learning. Thank you so much for watching.

Don't forget to subscribe. I really appreciate all of you guys who are supporting me making these videos. And I will see you in my next one.

Transcript for:Understanding Circuit Analysis Basics

Transcript for:
Understanding Circuit Analysis Basics