Question

1. An L-R-C circuit consists of a 5 µF capacitor, a 2 mH inductor and a 50 Ω resistor,
connected in series to an ac source of variable angular frequency ω, but of fixed voltage
amplitude V_{​​​​​​s}. At the resonance frequency, the amplitude of the current oscillations in
this circuit is measured to be 100 mA.

(a) In your own words explain what is meant by the resonance behaviour of an L-R-C
series ac circuit. At what frequency of the source will the circuit described above
resonate?

(b) Calculate the amplitude of the voltage that the source supplies (Vs).

If the frequency of the ac source is instead set to ω = 103
rad/s,

(c) What is the sum of voltage amplitudes across the inductor, the capacitor and the
resistor?

(d) What is the sum of the instantaneous voltages across the inductor, the capacitor
and the resistor, 2 ms after the instantaneous circuit current was maximum?
Compare your answer to your answer in part (c) and explain.

(e) What are the maximum values for the electric and magnetic energies stored in
the capacitor and the inductor?

Please answer ALL parts of the question.

Answer 1

(a) Inductor and Capcitor have different characterstics for alternating currents. In inductor current lags behind voltage by phase angle /2 , whereas in capacitor current leads the voltage by phase angle /2 . For a short time, if a transient current is passed through LC circuit when inductor and capacitor are connected parallel, Capacitor is quickly charged and maximum energy is stored in capacitor . In absence of external current , the stored charges are getting discharged through inductor and now energy is transferred slowly to inductor and potential difference is developed across inductor. Now this potential differnce charges back capacitor. Like this, the cycle repeats and this natural frequency of oscillation depends on the value of inductance L and capacitance C.

Just like resonance occurs in a forced oscilation of mechanical system , when the driving force frequency matches the natural frequency of the mechanical system and amplitude of vibration becomes maximum at resosnance, LCR electrical circuit also exhibits similar resoance when the frequency of applied alternating voltage equals the natural frequency of LC circuit oscillation.

Resonance angular frequency is given by,

resonance frequency = / 2 = ( 104 / 2 ) = 1.667 kHz

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if maximum current at resonance is 100 mA , then amplitude of alternating voltage vm is given by

vm = imax R = 100 10-3 50 = 5 V

XL = inductive reactance at angular frequency 103 rad/s = L = 2 10-3 103 = 2

XC = capacitive reactance at angular frequency 103 rad/s = 1/C = 1/ [ 5 10-6 103 ] = 200

Impedence of circuit Z at angular frequency 103 rad/s is given by

maximum current at angular frequency 103 rad/s = 5 / 204 = 2.451 10-2 A

Voltage across resistor, when angular frequency is 103 rad/s

vRm = 50 2.451 10-2 A = 1.225 V

Voltage across inductor, when angular frequency is 103 rad/s

vLm = 22.451 10-2 A = 4.9 10-2 V

Voltage across capacitor, when angular frequency is 103 rad/s

vCm = 200 2.451 10-2 A = 4.9 V

vector sum vm of all maximum votlage is given by

Phase angle is obtained from , tan = [ XC - XL ] / R = (200-2)/50 = 3.96

hence = 76o = 1.323 rad

voltages after 2 ms from maximum

voltage across inductor vLm = 2.451 10-2 sin( 103 2 10-3 + 1.323 ) 2 = -8.844 10-3 V

voltage across capacitor vCm = 2.451 10-2 sin( 103 2 10-3 + 1.323 ) 200 = - 0.884 V

voltage across resistance vRm = 2.451 10-2 sin( 103 2 10-3 + 1.323 ) 50 = - 0.221 V

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maximum stored magnetic energy in inductor at resonance = (1/2) L i2

maximum stored magnetic energy in inductor at resonance = (1/2) 2 10-3 0.1 0.1 = 10-5 J

maximum stored electric energy in capacitor at resonance = (1/2) C v2

maximum stored electric energy in capacitor at resonance = (1/2) 5 10-6 5 5 = 62.5 10-6 J