Hey, guys. In this video, we're going to start our discussion on series LRC circuits. Okay. These are circuits that are composed of inductors, resistors, and capacitors all connected in series to an AC source. Alright. Let's get to it.
In a series LRC circuit, the current through each element is the same. This is true for any series circuit. Okay? In a DC circuit, we would simply say that the voltage across all three of these elements, \( V_{LRC} \), the maximum voltage is just equal to the sum of the maximum voltages across each of the individual elements. In this case, we would call this \( IXL \), \( ir \), and \( ixc \). Okay? Now this is not true in AC circuits because each of the peak voltages, each of the maximum voltages peaks at a different time so you cannot simply add them all up. Okay?
In an LRC circuit the maximum voltage is actually going to be given by this weird square root thing. It's the square of the voltage across the resistor plus the square of the difference between the voltages across the inductor and the capacitor. This is actually the relationship between the maximum voltages across all three elements which by the way is the same as the maximum voltage produced by the AC source, that's just Kirchhoff's loop rule. This relates the maximum voltage across all three elements with the maximum voltage across each element. It's not the sum of them. It's this weird square root equation because each of the maxima peaks at a different time. Okay?
We want to define something called the impedance of this circuit, which acts as the effective reactance of this circuit. In a series LRC circuit, the impedance is defined as this: 1. This is a very important equation and the maximum current produced by the source is always going to be given by the maximum voltage of the source divided by the impedance. This is why the impedance is so important because once you calculate it you can simply take the maximum voltage produced by the source divided by the impedance and that'll tell you the maximum current produced by the source. Okay, let's do a quick example.
A circuit is formed by attaching an AC source in series to a 0.5 Henry inductor, a 10 ohm resistor, and a 500 microfarad capacitor. If the source operates at an RMS voltage of 120 volts and at a frequency of 60 Hertz, what is the maximum current in the circuit? Okay. Before continuing with the solution of this problem, we should really address the RMS voltage and the frequency. Remember guys that you're always really going to be dealing with the maximum voltage and you're always going to be dealing with angular frequency. Voltage is just going to be the square root of 2 times the RMS voltage which is the square root of 2 times 120 volts which is 170 volts. Okay? And the angular frequency is \(2\pi\) times the linear frequency which is \(2\pi \times 60\) Hertz which is about 377 inverse seconds. So, that right there tells us the values that we actually need to know.
Now, let's solve this problem. What is the maximum current in this circuit? The maximum current in the circuit is going to be given by this equation. The maximum voltage produced by the battery divided by the impedance so the impedance is going to be given by \( R^2\), which is the square root of \(10 \, \text{ohms}^2\) plus remember \( 377 \times\) the inductance was half a Henry minus \( \frac{1}{377 \times 500 \, \mu\text{F}}\). Micros \(10^{-6}\) and that whole thing squared and the square root of this whole thing so the impedance is 183 ohms. Now that we know the impedance, we can simply use this equation up here to find the maximum current produced by this source or the maximum current in the circuit. That's going to be \( V_{max} \) divided by \( Z \), which is going to be 170 volts. Right? 170 volts not 120 volts because 120 volts is the RMS voltage not the maximum voltage. This is divided by 183 ohms and that is 0.93 amps. One other thing to discuss is I use 377 in here, not 60 because we need the angular frequency, not the linear frequency. Alright, guys. That wraps up our discussion on series LRC circuits. Thanks for watching.
