 We're going to continue working through this handout on circuits and circuit analysis, but we're going to scroll down to the third page where we're going to get some practice with these total resistance formulas for resistors in parallel and in series, and I'm going to show you how to work through some of these calculations here. The first one is really easy. We've got just a parallel, pardon me, a series circuit and you can see we have three resistors here and the resistance of each of them is listed along with the current of the whole circuit. And all we're going to do is find the total resistance. So it's as simple as just adding together the resistance of each individual resistor here. So you can show your work, although I think for this honestly it's probably easy enough. You could just add them up in your head or on your calculator to see what we get in total. And so obviously this partly has to do with you being able to work out whether it's parallel or series. I know this one is series because the current coming out of the battery really doesn't have any options in terms of where it's going. It has to go through every one of those resistors. That's what makes this one a series circuit. All right, let's take a look at our next example. We're going to do the same thing, but in parallel. Now this is a parallel circuit. The reason is parallel is because the electrons as they come out of the battery, and they don't show the battery in this example, which is sort of annoying, but it should be somewhere in this area here. The electrons have choices. They can go down one loop or the other. And when you see that sort of design, you know that you're dealing with a parallel circuit. All right, so how do I work this out? Well, the formula is a little different. It says the one over total resistance, so this isn't actually my answer. It says one over the total resistance is equal to one over the first resistance. So I'll put maybe 60 ohms in there, plus one over the second resistance. Maybe I'll put in 30 ohms. Now let me show you how to type this here calculator really, really easily so that we can come up with the total resistance automatically. So I'm going to use my x to the negative one button on one of these TI-83 calculators here. So I'm going to type in 60, and I'm going to hit this power of negative one button. All right, then I'm going to go through and add to that 30 to the power of negative one. Putting something, the power of negative one means the same as putting one over that amount. Hit enter, and then hit that power of negative one button one last time. So our total resistance in this question is going to be 20 ohms. So that's a really nice way of being able to go through and do the parallel series total resistances. Now that we've seen how those formulas work, we can get into some more complicated sorts of questions. So let's try number three. It says we're going to work out the total resistance of the circuit, and then we're going to work out the total voltage of the circuit by adding each individual voltage and work out the resistance, pardon me, the current at each resistor using ohm's law. And we have to remember that current is the same everywhere when we're in a series circuit. All right, so it says first thing here, find the total resistance. So I'm not going to really show them all of my work on this one here. I'm just going to say that my total resistance is going to be, what do we got here? 24 plus 20 plus 4. So I guess that's 48 ohms. Just total it all up because it's a series circuit, straightforward, easy to do. Second thing it says, determine the total voltage of the circuit. Okay, so there's a couple of ways I can do that. The easiest way I think though is just to go and look at what I have for the voltage at each individual resistor. All right, so I have 12, 10, and 2. So the total voltage is going to be the sum of those numbers. So that's going to be 12, 22, 24 volts. All right, again if you're kind of wondering why is that, imagine an electron going through this little circuit and as it goes through each of those resistors it's going to go and drop a certain amount of its energy per unit of charge which is its voltage. It drops two volts at the first one and then 10 and then 12 and as it does that it has none left over when it's done. So it had to start off with in that case 24 volts. All right, last but not least we're going to work out the current each resistor and to do that I'm going to use Ohm's law which is V equals IR. So I'm going to go through and for the first resistor here I'll do V1. I'll put in 12 volts. The current there is what I'm going to work out and the resistance at that first resistor was 24 Ohms. So if I was to divide both sides by 24 to solve for this I'd get 0.50 amps. All right, so that's the current going through the first resistor. Now that's not going to be the same as the current at every other resistor as well. You could have done that series of steps at any one of the three resistors and gotten 0.50 amps. That's because the current is the same everywhere in a series circuit. Let's try another one here, a different variation. We're going to find the total resistance of the circuit, number four, then determine the current each resistor and then we're going to work out the voltage using Ohm's law. All right, total resistance again, so very similar. I'm going to add together 2.2 Ohms, 1.6 and 1.2. Let's grab the old calculator out for that one. Works out to a total of 5.0 Ohms. So you could kind of think of that as the value going right in here. All of those resistors together would be the same as having a single circuit with just a single 5 Ohm resistor. Then we're going to work with the current at each resistor. Now to do that, I need to figure out what, you know, find one place to kind of get started here, one place where I can find the current. And the only spot I can really do that is here at the total because I already have two pieces of information. I have the voltage of 12 volts and I have the resistance now of 5 Ohms. So here I can work out using Ohm's law, the current in total is V equals IR, 12 volts in total. Let's figure out what the total current is, 5 Ohms in total. So that's going to work out to whatever 12 divided by 5 is. And that's going to be a current of 2.4 Ohms. So now I know what the total current going through that circuit is, which actually means that I know the current at each individual location as well, because every resistor, if we've got a series circuit, is going to have the same current. So a lot of times these problems might seem like a lot of work, but they're not bad once you get going, if you can remember those shortcuts. I'm going to jump over number five and I'm going to post the key so you can see the numerical answers for that. We're going to try out number six now. So let's find the total resistance. I'm going to use that reciprocal calculation again. So one over the total resistance is one over 60 Ohms plus one over 60 Ohms plus one over 60 Ohms. I'll show you how I type that into the calculator again. It's not too bad to do. So I just put 60 to the negative one, plus 60 to the negative one, plus 60 to the negative one. And don't forget this last step to hit the power of negative one one more time. That's a total resistance of 20 Ohms. You can go and check out the key to see the answer to number seven. I'll post that on the website. Let's take a look here at number eight. We're going to first figure out the total resistance of the circuit, then we're going to determine the voltage of each resistor. And then finally, we're going to get the current at each resistor using Ohm's law. The thing to remember here is that the voltage is the same across each resistor in parallel. So let's see, I can start off here by finding the total resistance again. It's going to be a very similar calculation to what we've done a few times here already. Since it's a parallel circuit, I've got to use my reciprocal formula. 1 over 20 Ohms plus 1 over 40 Ohms plus 1 over 80 Ohms. Let's see what that is as our total resistance here. Works out to, I'm going to round it to 11.4 Ohms in total. Now that I know what the total resistance is, I can go in there and I can figure out, since I know the voltage, my current. So I have 11.4 Ohms and I have a voltage of 120 volts. It's listed as an E in this diagram that I found online, but that's a voltage. Same here. Those are all voltages. And we're going to go through and use Ohm's law to see if we can go in and find the current. Alright, so let's do that. So current and voltage are related through Ohm's law, V equals IR, 120 volts equals the unknown current times that resistance of 11.4 Ohms. So if I divide both sides by 11.4, let's see here that's 120 over 11.4. It's about 10.5 amps. So now we have the total current in the circuit and we need to work out each individual little current. So this is going to be a little repetitive. The current going through the first resistor, I'll call that current 1, doesn't look much like a 1, is just going to be what we get when we take the voltage and divide it by the resistance at resistor one. So the voltage is going to be exactly the same as the total voltage is 120 volts in each of those situations for each of those resistors because the voltage stays the same when you've got resistors in parallel. So kind of like in the last situation where we took the voltage of 120 volts and we divided it by 11.4 Ohms. Now I'm just going to divide it by 20 Ohms since that's the resistance of the first resistor. That'll give me 6.0 amps. And if I go through and do the same thing for the other two resistors, I'll get a current for the second one of 3.0 amps and for the third one of 1.5 amps. I hope this helps if you have more questions about these circuit analysis situations, check out some of the other resources on the website at ldindustries.ca.