Hey guys. In this video, we want to talk about this quantity of a circuit called impedance. Okay? It's going to be very similar to reactants. So, let's get to it. We know how to find the current in any AC circuit with a single element. Okay? 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 of a resistor to just be its resistance. Okay, because reactance and resistance are the same for resistors. Now there are 2 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 nor parallel connections but those you are not going to encounter when discussing AC circuits. They will either be purely series or purely parallel. Okay? Whenever an AC circuit has multiple elements in series, the current phasors 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 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. Okay?

Now let's consider one particular circuit, which happens to be an AC source connected in series to a resistor and a capacitor. Okay. 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 the capacitor to be V_{RC}. 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_{RC} will not simply be the sum of V_{R} and V_{C}. Okay? Remember that we have equations for both of these. This is simply I_{R} and this is I_{XC}. It's not just going to be the sum of those 2 because the maximum voltages don't appear at the same time. Okay? Instead, of voltage phasors. This is one of the particular reasons why we use phasors because you can add them like vectors. Okay? 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. Okay? So what is the total voltage going to be? Well, Ǳ_{RC} is going to be the Pythagorean theorem, Ǳ_{R} squared plus Ǳ_{C} squared. Right? This is the vector sum of those 2 phasors. So this is going to be I_{max}^{2}R^{2}, which is just the maximum voltage across the resistor plus I_{max}^{2}XC^{2}, which is just the maximum voltage across the capacitor. Okay? We want to rewrite this like Ohm's 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. Okay? And the maximum current output by a source is always going to be defined in terms of the reactants. It's always going to be defined as V _{max/z}. Okay? In this particular case, the case of a series RC circuit, we see that the reactance is just R^{2} plus and I'm going to substitute in the capacitive reactance. This is 1 over omega^{2} c^{2}. Okay. This is the impedance of a series RC circuit. But that's only for a series RC 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. Okay? Let's do an example to illustrate that. What's the impedance of an AC circuit with a resistor and an inductor in series? Okay? In this case, once again, since they are in series, the current is going to 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 the 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 2 voltage phasors. So I'm going to use the Pythagorean theorem. Right? Now I'm going to substitute in I_{max}^{2}R^{2} for the maximum voltage across the resistor, and I_{max}^{2}XL^{2} for the maximum voltage across an inductor. This factor of I_{max} they both share so I can factor that out. And the inductive reactance, I can substitute in terms of the angular frequency. And don't forget, this term right here is the reactance, sorry, is the impedance. So the impedance of this circuit is the square root of R^{2} plus omega^{2} L^{2}. Okay and this is absolutely different from the impedance of a series RC circuit that we saw before this. Alright guys, that wraps up our discussion on impedance. Thanks for watching.