Resonance in a driven RLC circuit A series RLC circuit has component values R =

Question

Resonance in a driven RLC circuit

A series RLC circuit has component values R = 8.9 , L = 2.12 H and C = 12.0 µF, as well as an oscillating voltage (emf) source of the form V(t)= (12.0 V) sin(wt) with a tunable frequency w.

For the first five parts, suppose the frequency has been set to w = 315 rad/s.

(e) What is the maximum potential difference across the capacitor as the circuit oscillates?

For the remaining parts, the frequency is tuned to the resonant frequency of the circuit.

(f) What is the resonant frequency of the circuit? Express your answer either as an angular frequency or as a cycles-per-second frequency, indicating which you have chosen. (Hint: one is more convenient than the other.)

(g) At this frequency, what is the reactance of the capacitor, and what is the reactance of the inductor?

(h) What is the impedance of the RLC circuit as a whole at this frequency? How does it compare to what you got in part b ?

(i) What is the maximum current that flows in the circuit? (In other words, the amplitude of the oscillating current that flows in the circuit.) How does it compare to what you got in part c ?

(j) What is the maximum potential difference across the resistor as the circuit oscillates?

(k) What is the maximum potential difference across the capacitor as the circuit oscillates?

In part k, you should have found that the amplitude of the voltage across the capacitor is significantly larger than the 12-volt amplitude driving the circuit! Continuous small “pushing” at the resonant frequency can produce large oscillations, kind of like “pumping” a playground swing with your legs.

Explanation / Answer

e)

Max potential difference across capacitor,

Vc = (V/I)*Xc = 12*(1/(315*12*10^-6))/sqrt(8.9^2+(315*2.13 - 1/(315*12*10^-6))^2)

= 7.81 V

f)

For resosnance,

Xl = Xc

So, W*L = 1/WC

So, W = sqrt(1/LC)

So, f = (1/(2*pi))*sqrt(1/LC)

So, f = (1/(2*pi))*sqrt(1/(2.12*12*10^-6))

= 31.6 Hz

g)

Reactance of capacitor = Xc = 1/(2*pi*f*C)

So, Xc = 1/(2*pi*31.6*12*10^-6) = 420.1 ohm

Reactance of inductor,

Xl = 2*pi*f*L = 2*pi*31.6*2.12 = 420.8 ohm

h)

At this frequency impedance of circuit = resistance = 8.9 ohm

i)

Maximum current = V/R = 12/8.9 = 1.34 A

j)

Maximum potential difference across resistor = 12 V

k)

Across capacitor = Xl*I = 420.8*1.34 = 563.9 V

Across inductor = 420.1*1.34 = 563 V

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