1

concept

## Impedance in AC Circuits

8m

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Hey, guys, in this video, we want to talk about this quantity of a circuit called impedance. Okay, It's gonna be very similar to reactant. So let's get to it. We know how to find the current in any a C circumvent circuit with a single element. Okay, so we saw an A C source connected to a resistor and a C source connected to a capacitor and a C source connected to an induct. Er, the maximum current in that circuit was just the maximum voltage divided by the reactant since in this case, I'm considering the reactant of a resistor to just be its resistance. Okay, because reactant and resistance are the same for resisters. Now there are two types of circuits for combining multiple elements. We know there are serious circuits and parallel circuits. There are also types of circuits that have neither serious nor parallel connections, but those who are not going to encounter when discussing a C circuits, they will either be purely Siri's or purely parallel. Whenever an A C circuit has multiple elements in Siris, the current phasers all line up. They are all in phase, okay? And that just comes from the fact that the current is the same for all elements in Siris. That's the definition of a serious connection. Whenever a D. C. Circuit has multiple elements in parallel the voltage phasers line up the voltage phasers are in phase, right? That's because the voltage for all elements in parallel is the same. That's just the definition of a parallel connection. Okay, Now let's consider one particular circuit which happens to be an A C source connected in Syria's two resistor in a capacitor. Okay, in this case, So let's draw that circuit. Here we have our A C source here we have our resistor in our capacitor connected to our A C source. I have defined in this case the voltage across the resistor and the capacitor to be V R c. Notice that that voltage the voltage across both of those elements is the same as the voltage across the source. V Max. Okay, so those have to be the same. The same maximum, at least. Okay. In this case, the maximum voltage across the resistor and capacitor v. R. C will not simply be the sum of er NBC. Okay, remember that we have equations for both of these. This is simply I r. And this is I xy. It's not just gonna be the some of those two because the maximum voltage is don't appear at the same time. Okay. Instead, the maximum voltage is actually going to be the vector sum off voltage phasers. This is one of the particular reasons why we use phasers, because you can add them like vectors. Okay, so this is a serious circuit. We're going tohave both elements current phasers to be in parallel. Now a resistor always has its voltage phaser in Siris with its current phaser. So right here we have the voltage phaser off the resistor. Now a capacitor always has its voltage phaser lagging by 90 degrees to its current phaser. So right here we have the voltage phaser of the capacitor. Okay, so what is the total voltage going to be while V. R c is just going to be Pythagorean theorem v r Square plus V C squared? Right. This is the vector sum off those two phasers. So this is going to be I Max squared r squared, which is just a maximum voltage across the resistor. Plus I max squared XY squared, which is just the maximum voltage across the capacitor. I can pull out the factor of imax that they both share and then I get this equation. Okay, We want to rewrite this like OEMs law. Like having a reactant like having a resistance. And we rewrite it with this variable Z and Z. We call the impedance off this a C circuit which acts like the effective reactant off the entire circuit with all the elements taken into account. Okay. And the maximum current output by a source is always going to be defined in terms of the reactant, since it's always going to be defined as v Max divided by Z. Okay, In this particular case, the case of a Siri's R C circuit, we saw that the reactant is just r squared. Plus, I'm gonna substitute in the capacitive reactant, since this is one over omega squared C squared. OK, this is the impedance off a Siri's R C circuit. But that's Onley for Syria's RC circuit. The impedance can be found for multiple different types of circuits, and it's all found the same way you draw the phaser diagram and you do the vector sum of something that you're looking for that will lead you to the impedance. Okay, let's do an example to illustrate that. What's the impedance oven? A C circuit with a resistor and an induct er in Siris. Okay, In this case, once again, since they're in Siris, the current is gonna be the same for both of them, right? So this is the current. Now the voltage across a resistor is always in phase with its current. So this is a voltage across the resistor and the voltage across an induct er always leads its current by 90. So this is the voltage across the induct, er it's leading by 90. So what is the maximum voltage in this circuit? It's just gonna be the vector sum of those two voltage phasers. So I'm gonna use Pythagorean theorem right now. I'm gonna substitute in. I Max squared r squared for the maximum voltage across the resistor and imax squared X l squared for the maximum voltage across a sorry across an induct er this factor of imax they both share. So, Aiken, factor that out and the inductive reactant. Since I can substitute in in terms of the angular frequency And don't forget this term right here is the react. Sorry is the impedance. So the impedance of this circuit is the square root of R squared, plus Omega squared elsewhere. Okay. And this is absolutely different than the impedance off a Siri's R C circuit that we saw before this. Alright, guys, that wraps up our discussion on impudence. Thanks for watching.

2

example

## Impedance of a Parallel LR AC Circuit

2m

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Hey, guys, let's do an example using impedance. Okay. What's the impedance of a parallel l R circuit? Okay. L r a c circuit. Okay, so it has an induct er and a resistor in it. And let's draw the phaser diagram for this. When they're parallel when they're parallel, don't forget their voltage. Phasers are in phase because all elements in parallel have the same voltage. So I'm gonna draw some voltage phaser right here Now for the resistor, the current and voltage phasers air always going to be in phase. So here is the current fazer for the resistor. Now, the voltage phaser for the induct er is always going to lead its current fazer. So I need to draw the current phaser for the induct er as lagging by degrees. So here's the current fazer for the induct er. Now the maximum current in this circuit is going to be given by the vector sum of those two current phasers using Pythagorean theory. Um, I get this. Okay? Okay. Now I want to rewrite everything in terms of the maximum voltage. So this is V Max over the impedance. Remember, this is the definition of the impedance that the maximum current produced by the source is just the maximum voltage of the source divided by the impedance. This is gonna be the square root of V. Max squared over R squared, plus the max squared over X l squared. Why do both of these terms also have V Max? Because this is a parallel circuit. So everything has the same voltage as the source, right? So I can cancel all of these terms of V max. So this tells me that one over Z, the impedance is the square root of one over r squared. And I can substitute in the equation for the inductive reactant since, and this becomes one over omega squared elsewhere. And this is our impedance for a parallel L R circuit. For parallel circuits, you're never quite going to define the impedance. You're always going to find one over the impedance like this, but this is a perfectly fine way of representing the answer. All right, guys, Thanks for watching

3

Problem

What’s the impedance of a parallel RC AC circuit?

A

Z = sqrt(R

^{2}- 1/ω^{2}C^{2})B

Z = sqrt(1/R

^{2}+ ω^{2}C^{2})C

1/Z = sqrt(1/R

^{2}+ ω^{2}C^{2})D

1/Z = sqrt(1/R

^{2}+ 1/ω^{2}C^{2})4

Problem

An AC source operates at a maximum voltage of 120 V and an angular frequency of 377 s^{-1} . If this source is connected in parallel to a 15 Ω resistor and in parallel to a 0.20 mF capacitor, answer the following questions:

a) What is the maximum current produced by the source?

b) What is the maximum current through the resistor?

c) What is the maximum current through the capacitor?

A

a) 12 V b) 8A c) 4A

B

a) 12 V b) 8A c) 9A

C

a) 7.8 V b) 7.6A c) 0.2A

D

a) 8.9 V b) 8A c) 4A

Additional resources for Impedance in AC Circuits

PRACTICE PROBLEMS AND ACTIVITIES (3)

- You have a 200-Ω resistor, a 0.400-H inductor, and a 6.00-μF capacitor. They are connected to form an L-R-C se...
- You have a 200-Ω resistor, a 0.400-H inductor, and a 6.00-μF capacitor. They are connected to form an L-R-C se...
- You have a 200-Ω resistor, a 0.400-H inductor, and a 6.00-μF capacitor. Suppose you take the resistor and indu...