 ## Physics

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31. Alternating Current

# Series LRC Circuits

1
concept

## LRC Circuits in Series 5m
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Hey, guys, in this video, we're going to start our discussion on Siri's L. R C. Circuits. Okay, these are circuits that are composed of induct er's resistors and capacitors, all connected in Siris to an A C source. All right, let's get to it. In a serious circuit, the current through each element is the same. This is true for any Siri's circuit in the D. C circuit. We would simply say that the voltage across all three of these elements V. L R C the maximum voltage is just equal to the sum of the maximum voltage is across each of the individual elements. In this case, we would call this I excel. I are, and I xsi Okay, now this is not true in a circus because each of the peak voltages each of the maximum voltage is peaks at a different time, so you cannot simply add them all up. Okay, In an l r C 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 induct er and the capacitor. This is actually the relationship between the maximum voltage is across all three elements, which, by the way, is the same as the maximum voltage produced by the A C source. That's just Kirchoff Luke rule. This relates the maximum voltage across all three elements with the maximum voltage across each element. It's not the some of them. It's this weird square root equation because each of the maximum peaks at a different time okay, we want to define something called the impedance off this circuit, which acts as the effective reactant of this circuit in a Siri's circuit. The impedance is defined as this. This is a very, 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 will tell you the maximum current produced by the source. Let's do a quick example. Ah circuit is formed by attaching an A C source in Siris to a 0. Henry and Dr a 10 ohm resistor and a 500 micro fair. Add capacitor. If the source operates at a 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 gonna be dealing with angular frequency. So we should just convert these right off the bat to get them out of the way. Okay, The maximum voltage is just gonna be the square root of two times the RMS voltage, which is the square root of two times 120 volts, which is 170 volts. Okay. And the angular frequency is two pi times the linear frequency, which is two 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 squared plus omega L minus one over Omega C squared, which is the square root of 10 OEMs squared. Plus, remember 3 77 times the induct in Swiss, half of Henry minus one over 3 77 times 500. Micro fair adds microbes tend to the negative six and that whole thing squared and the square root of this whole thing. So the impedance is 183 owns. 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 gonna be the 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 1 83 owns, 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 serious L. R. C. Circuits. Thanks for watching.
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Problem

An AC source operates at an RMS voltage of 70 V and a frequency of 85 Hz. If the source is connected in series to a 20 Ω resistor, a 0.15 H inductor and a 500 µF capacitor, answer the following questions:

a) What is the maximum current produced by the source?

b) What is the maximum voltage across the resistor?

c) What is the maximum voltage across the inductor?

d) What is the maximum voltage across the capacitor? 