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Concept #1: Resonance in Series LRC Circuits


Hey guys, in this video we're going to talk about resonance in LRC circuits. Alright, let's get to it. I've graphed on the upper right corner the impedance, the resistance, the capacitive reactance and the inductive reactance all as functions of omega. The impedance depends upon these three things. Now the resistance doesn't change with omega but the capacitive and the inductive reactance do change with omega. The inductive reactance gets larger and larger and larger the larger omega is and the capacitive reactance gets larger and larger and larger the smaller omega is. So the impedance blows up at large or small frequency but there is a minimum in between there. Recall that the impedance is a square root of R squared plus XL squared minus, sorry that's not squared, XL minus XC squared. All of that square rooted. When the two impedances, the capacitive and inductive impedances, equal each other that's when we are at a minimum for the impedance. When the inductive and capacitive reactances equal one another then you lose this term right here and the impedance equals its smallest value which is the resistance. When this occurs we say that the circuit is in resonance. The resonant frequency of an LRC circuit is given by this equation and this is just found by solving XC equals XL for the frequency. Since resonance occurs when the impedance is smallest the current is going to be largest in the circuit when the circuit is in resonance.

Let's do an example. An AC circuit is composed of a 10 ohm resistor, a 2 Henry inductor and a 1.2 millifarad capacitor. If it is connected to a power source that operates at a maximum voltage of 120 volts, what frequency should it operate at to produce the largest possible current in the circuit? What would the value of this current be? What frequency should it operate at is just asking what is the resonant frequency such that it produces the largest possible current. We know that in resonance you have the largest possible current. So the resonant angular frequency is one over the square root of LC so this is one over the square root of 2, it's a 2 Henry inductor, 1.2 times 10 to the -3, it's a 1.2 millifarad capacitor and this whole thing equals 20.4 inverse seconds but if they're asking for a frequency it's better to give this in terms of the linear frequency in case that's what they're looking for. The linear frequency is just omega over 2 Pi and I can call that F not to imply that it's the resonant frequency and this is going to be 20.4 over 2 Pi which equals 3.25 Hertz. That's one answer done. What is the current in this circuit, the maximum current in resonance? Don't forget that the maximum current produced by a source is always going to be the maximum voltage divided by the impedance. In resonance, the impedance just becomes the resistance. So in resonance we have that this is just Z equals R. So it's a maximum voltage of 120 volts divided by a 10 ohm resistor is 12 amps. Very easy, very straightforward. You don't have to use that very complicated resonance equation and the last couple points I want to make is that in a series LRC circuit, the current is the same throughout the inductor and the capacitor. The current's the same through everything, the resistor, the inductor and the capacitor. That's what it means to be in series. In resonance, since their reactances are the same, this must mean that the maximum voltage across the inductor and the capacitor is also the same. Alright guys, that wraps up our discussion on the resonance, sorry, the resonant frequency and resonance in and LRC circuit. Thanks for watching.

Practice: A series LRC circuit is formed with a power source operating at VRMS = 100 V, and is formed with a 15 Ω resistor, a 0.05 H inductor, and a 200 µF capacitor. What is the voltage across the inductor in resonance? The voltage across the capacitor?