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

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Concept #1: Impedance in AC Circuits

**Transcript**

Hey guys, in this video we want to talk about this quantity of a circuit called impedance it's going to be very similar to reactance, let's get to it. We know how to find the current in any AC circuit with a single element, so we saw an AC source connected to a resistor an AC source connected to a capacitor an AC source connected to an inductor the maximum current in that circuit was just the maximum voltage divided by the reactance and in this case I'm considering the reactance of a resistor to just be its resistance because reactance and resistance are the same for resistors now. There are two types of circuits for combining multiple elements we know there are series circuits and parallel circuits there are also types of circuits that have neither series or parallel connections but those we are not going to encounter when discussing AC circuits they will either be purely series or purely parallel. Whenever an AC circuit has multiple elements in series the current phasors all line up they are all in phase and that just comes from the fact that the current is the same for all elements in series that's the definition of a series connection whenever an AC circuit has multiple elements in parallel the voltage phasors line up the voltage phasors 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. Now let's consider one particular circuit which happens to be an AC source connected in series to a resistor in a capacitor in this case so let's draw that circuit here we have our AC source here we have our resistor and our capacitor connected to our AC source I have defined in this case the voltage across the resistor and a 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 so those have to be the same the same maximum at least.

In this case the maximum voltage across the resistor and capacitor V R C will not simply be the sum of V R and V C remember that we have equations for both of these this is simply I R and this is I X C it's not just going to be the sum of those two because the maximum voltage's don't appear at the same time, instead the maximum voltage's actually going to be the vector sum of voltage phasors this is one of the particular reasons why we use phasors is because you can add them like vectors. So this is a series circuit we are going to have both elements current phasors to be in parallel. Now a resistor always has its voltage phasor in series with its current phasor so right here we have the voltage phasor of the resistor now a capacitor always has its voltage phasor lagging by 90 degrees to its current phasor so right here we have the voltage phasor of the capacitor. So what is the total voltage going to be well V. R. C. Is just going to be Pythagorean theorem V R squared plus V C squared right this is the vector sum of those two phasors so this is going to be IMAX squared R squared which is just the maximum voltage across the resistor plus IMAX squared, X C 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 we want to rewrite this like Ohms law like having a reactance like having a resistance and we rewrite it with this variable Z and Z we call the impedance of this AC circuit which acts like the effective reactance of the entire circuit with all the elements taken into account and the maximum current output by a source is always going to be defined in terms of the reactance it's always going to be defined as V Max divided by Z in this particular case the case of a series R C circuit we saw that the reactance is just R squared plus now I'm going to substitute in the capacitive reactance this is 1 over omega squared C squared this is the impedance of a series R C circuit but that's only for a series R C circuit the impedance can be found for multiple different types of circuits and it's all found the same way you draw the phasor diagram and you do the vector sum of something that you're looking for that will lead you to the impedance.

Let's do an example to illustrate that what's the impedance of an AC circuit with a resistor and an inductor in series ? In this case once again since they are in series the current is going to be the same for both of them. So this is the current now the voltage across the resistor is always in phase with its current so this is a voltage across a resistor and the voltage across an inductor always leads its current by 90. So this is the voltage across the inductor it's leading by 90 so what is the maximum voltage in this circuit, it's just going to be the vector sum of those two voltage phasors. So I'm going to use Pythagorean theorem, now I'm going to substitute N IMAX squared R squared for the max ohm voltage across the resistor and IMAX squared X L squared for the maximum voltage across a inductor this factor of IMAX they both share so I can factor that out and the inductive reactance I can substitute N in terms of the angular frequency and don't forget this term right here is the impedance so the impedance of this circuit is the square root of R squared plus omega squared L squared and this is absolutely different than the impedance of a series R C circuit that we saw before this. Alright guys that wraps up our discussion on impedance Thanks for watching.

Example #1: Impedance of a Parallel LR AC Circuit

**Transcript**

Hey guys, let's do an example using impedance. What's the impudence of a parallel LR circuit? LR AC circuit. So it has an inductor and a resistor in it and let's draw the phasor diagram for this when they are parallel. When they are parallel, don't forget, their voltage phasors are in phase because all elements in parallel have the same voltage. So I'm going to draw some voltage phasor right here. Now for the resistor, the current and voltage phasors are always going to be in phase. So here is the current phasor for the resistor. Now the voltage phasor for the inductor is always going to lead its current phasor so I need to draw the current phasor for the inductor as lagging by 90 degrees. So here is the current phasor for the inductor. Now the maximum current in this circuit is going to be given by the vector sum of those two current phasors so using Pythagorean theorem, I get this. 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 going to be the square root of V max squared over R squared plus V max squared over XL 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 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 reactance and this becomes one over omega squared L squared and this is our impedance for a parallel LR 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. Alright guys, thanks for watching.

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

Practice: 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?

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