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31. Alternating Current

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Hey, guys, in this video, we're gonna talk about Resonance and Larson circuits. All right, let's get to it. I've graphed on the upper right corner, the impedance, the resistance, the capacitive reactant since and the inductive reactant all as functions of omega. Okay, the impedance depends upon these three things. Now, the resistance doesn't change with omega, but the capacitive and inductive reactant do change with omega. The inductive reactions gets larger and larger and larger, the larger omega is, and the capacity of reactions gets larger and larger and larger. The smaller omega is okay, so the impedance blows up at large or small frequency. But there is a minimum in between their recall that the impedance is the square root of R squared plus x l squared mine. Sorry, that's not squared. X l minus X c squared all of that square rooted. Okay, when the two impedance is the capacitive and inductive impedance is equal each other. That's when we were at a minimum four. The impedance. When the inductive and the capacity capacity of react Ince's equal one another. Then you lose this term right here. And the impedance equals its smallest value, which is the resistance. Okay, When this occurs, we say that the circuit is in resonance. Okay, The resonant frequency often circuit is given by this equation, and this is just found by solving XY equals X l for the frequency. Okay. Since resonance occurs when the impedance is smallest, the current is going to be largest in the circuit when the circuit is in resonance. Okay, let's do an example. And a C circuit is composed of a 10 ohm resistor A to Henry in Doctor and a 1.2 million Farid capacitor. If it is connected to a power source that operates at a maximum voltage of volts, what frequency should operate at to produce the largest possible current in that's in the circuit. What would the value of this current B okay, What frequency should operate at is just asking what is the resonant frequency such that it produces the largest possible current right? We know that in resonance you have the largest possible current. So the resident angular frequency is one over spirit of Elsie. So this is one over the square root of two. It's A to Henry and Dr 1.2 times 10 to the negative three. It's a 1.2 million fair at capacitor, and this holding equals 24 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 two pi and I can call, though that s not Thio imply that it's the resonant frequency. And this is gonna be 24/2 pi, which equals 3.25 hertz. Okay, 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 residence we have that This is just Z equals R, right? So it's a maximum voltage of 120 volts divided by a 10 home resistor is 12 amps. Very easy. Very straightforward. You don't have to use that very complicated resonance equation. Okay, guys. And the last couple points I want to make is that in a serious LLC circuit, the current is the same throughout the induct er and the capacitor, the current the same through everything to resist or the conductor and the capacitor. That's what it means to be in Siris in resonance since their react, Ince's are the same. This must mean that the maximum voltage across the induct er and the capacitor is also the same. Alright, guys, that wraps up our discussion on the resonance Sorry, the resonant frequency and residents in an l. R C circuit. Thanks for watching.

2

Problem

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?

A

V_{L} = 141 V; V_{C} = 107 V

B

V_{L} = 107 V; V_{C} = 107 V

C

V_{L} = 141 V; V_{C} = 141 V

D

V_{L} = 149 V; V_{C} = 149 V

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