## Thursday, May 06, 2010

### Alternating Current Circuits – Two Questions (MCQ) on Resonance

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"We have to do the best we can. This is our sacred human responsibility."
Albert Einstein
The phenomenon of resonance in alternating current circuits is an interesting one with many practical applications and questions involving resonance are often seen in entrance question papers. You may try searching for ‘resonance’ on this blog (making use of the search box provided on this page) to access all previous posts related to resonance.
Today we will discuss two multiple choice questions meant for high lighting the behaviour of AC circuits at resonance:
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(1) An alternating current source of negligible impedance has an emf 6 volt. It is connected in series with a series LCR circuit consisting of a 72 Ω resistor and an inductor-capacitor combination. The inductor offers a reactance of 120 Ω at the operating frequency and the power dissipated in the LCR circuit is 0.5 W. What is the reactance offered by the capacitor?
(a) 60 Ω
(b) 72 Ω
(c) 120 Ω
(d) 192 Ω
(e) Zero
It is always a good idea to check whether the circuit is at resonance (Question setters have a craze to set questions involving resonance).
Since the power is dissipated in the resistor alone, we have
V2/72 = 0.5 where V is the voltage across the resistor.
[Remember the power equation, P = V2/R]
We obtain V2 = 36 from which V = 6 volt.
The circuit is indeed at resonance since the entire supply voltage appears across the resistance.
At resonance the inductive reactance is equal to the capacitive reactance and hence the correct option is 120 Ω.
(2) In a series LCR circuit R = 200 Ω and the voltage and the frequency of the main supply are 220 V and 50 Hz respectively. On taking out the capacitance from the circuit the current lags behind the voltage by 30º. On taking out the inductor from the circuit the current leads the voltage by 30º. The power dissipated in the LCR circuit is
(1) 242 W
(2) 305 W
(3) 210 W
(4) 0 W
This question appeared in AIEEE 2010 question paper.
The phase lag of current produced by the inductance is equal in value to the phase lead of current produced by the capacitor. This means that in the LCR circuit connected to the 220 V, 50 Hz supply, there is no phase difference between the supply voltage and the current and the circuit behaves as a pure resistor. Indeed the circuit is at resonance and the entire supply voltage appears across the resistor.
Therefore, the power P dissipated in the LCR circuit is given by
P = V2/R = 2202/200 = 242 W.

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