**Concept:** 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.

**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?

Calculate the resonance frequency of a series RLC circuit for which the capacitance is 72 μF , the resistance is 44 kΩ, and the inductance is 135 mH.

1. 53.7735

2. 59.9157

3. 133.185

4. 46.7497

5. 99.6276

6. 67.1056

7. 138.843

8. 51.049

9. 48.436

10. 59.2477

Watch Solution

A series RLC circuit with L = 25 mH, C = 0.8 μF and R = 7Ω is driven by a generator with a maximum emf of 12V and a variable angular frequency ω. At resonance, the current

a) leads the generator voltage by 90°.

b) lags the generator voltage by 90°.

c) is in phase with the generator voltage.

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An *R*-*L*-*C *series ac circuit has *R* = 400 Ω, *L* = 0.600 H, and *C* = 5.00 x 10^{-8} F. The voltage amplitude of the source is 80.0 V. When the ac source operates at the resonance frequency of the circuit, what is the average power delivered by the source?

(a) zero

(b) 4.0 W

(c) 8.0 W

(d) 16.0 W

(e) 32.0 W

(f) none of the above answers

Watch Solution

In the figure, which of the phasor diagrams represents a series RLC circuit driven at resonance?

A) 5

B) 2

C) 1

D) 3

E) 4

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An *RLC* circuit is used in a radio to tune into an FM station broadcasting at 99.7 MHz. The resistance in the circuit is 18.0Ω, and the inductance is 1.20 μH. What capacitance should be used?

A) 2.12 pF

B) 25 pF

C) 0.30 pF

D) 8 pF

Watch Solution

The impedance of a series RLC circuit at resonance is described by which of the following?

A) less than *R*

B) impossible to determine

C) equal to *R*

D) larger than *R*

Watch Solution

a) What would have to be the self-inductance of a solenoid for it to store 11.0 J of energy when a 1.50A current runs through it?

b) If such an inductor is connected to a capacitor of 50 *uF*, what will be the resonance frequency in rad/s? and in Hz?

Watch Solution