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Concept #1: LRC Circuits in Series

Transcript

Hey guys, in this video were going to start our discussion on series LRC circuits. 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. In a DC circuit we would simply say that the voltage across all three of these elements VLRC, 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. 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. 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 Kirchoff'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. 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, 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. 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? 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 so we should just convert these right off the bat to get them out of the way. The maximum 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 170 volts 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 squared plus omega L minus one over omega C squared which is the square root of 10 ohms squared plus remember 377 times the inductance was half a Henry minus one over 377 times 500 microfarads, micro is 10 to the -6, and that whole thing squared and the square root of the 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. 170 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 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.

Practice: 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?

Additional Problems
A 250Ω resister, a 20mH inductor, and a 4.5uF capacitor are in series, and are connected to an AC source: V(t) = V0cos(wt + 45°). The maximum voltage Vo is 45V and the angular frequency w is 350 rad/s. a) Calculate the rms current. b) Calculate the power factor. c) How much power is delivered by the AC source? d) Is this power delivered by the source more or less than that consumed by the resistor?
An RLC circuit has a capacitance reactance, due to its capacitance, of 11 kΩ; an inductive reactance, due to its inductance, of 3 kΩ; and a resistance of 29 kΩ. What is the power factor of the circuit? a) 0.48 b) 0.27 c) 0.96 d) 1.04
In an R-L-C ac series circuit the source voltage amplitude is 360 V and its angular frequency ω is equal to the resonance frequency of the circuit. R = 90.0 Ω, C = 8.00 x 10-6 F, and L = 2.0 H. What is the voltage amplitude VL for the inductor?
In an R-L-C ac series circuit the source voltage amplitude is 360 V and its angular frequency ω is equal to the resonance frequency of the circuit. R = 90.0 Ω, C = 8.00 x 10-6 F, and L = 2.0 H. What is the voltage amplitude VR for the resistor?
The voltage source for a series R-L-C ac circuit is operated at the resonance frequency for the circuit. The amplitude of the voltage across the resistor is 400 V and the amplitude of the voltage across the inductor is 600 V. The voltage amplitude of the source is  A) 1600 V B) 1000 V C) 600 V D) 400 V E) 200 V F) None of the above answers.
In a driven, series RLC circuit, the current leads the voltage of the function generator by 60.0° at ω = 40.0 rad/s. If R = 5.00 Ω, what should be added (in series) to the circuit to maximize the power supplied by the function generator? A. 108 mH inductor B. 2.89 mF capacitor C. 8.66 mF capacitor D. 72.2 mH inductor E. 217 mH inductor
An AC generator of unknown angular frequency a produces a voltage with multiple εmax. The inductance and capacitance values appear on the figure. A student measures the maximum AC voltage across each of the three circuit elements to be VL,max = 6 V, Vc,max = 3 V, and VR,max = 4V, respectively.  By what phase ∅ does the generator voltage lead the current? a. ∅  = 36.9° b. ∅  = 30.0° c. ∅ = 41.4°
In the circuit shown below, the AC generator supplies an EMF of the form ε = 15sin(100t - π/3) volts. A student measures the current to be I = 6 sin(100t) amps. Thus the generator voltage lags the current by π/3 radian (i.e. 60°). The resistance R must be:a. 1.25 Ωb. 2.5 Ωc. 5.0 Ωd. 7.5 Ωe. cannot be determined from the information given.