Test: Resonance in LCR Circuit - JEE MCQ

If series LCR circuit is present then z={R²+(xl-xc)²}^1/2
if resonance is present then X_C=C_L or V_C=C_L then Z= Z_o
if LCR circuit is in parallel form then in which circuit resonance is also present firstly 1/Z ={(1/R)₂+ ((1/xc) –(1/xl))²}^1/2 and in resonance conditions X_C=C_L according to this Z= Z_o then option C is the correct answer

Resonance is a condition in LCR circuit where the inductive reactance ( XL) and capacitive reactance (Xc) is equal.
Impedance at resonance is equal to the Resistance of the resistor because the other two cancel themselves off according to the below formula -
Z2=R2+(XL²-Xc²)
As XL = Xc ,
Z = R
 

frequency-to-bandwidth ratio of the resonator:
Q=fr/△f= ωr/△ω
where fr is the resonant frequency, Δf is the resonance width or full width at half maximum (FWHM) i.e. the bandwidth over which the power of vibration is greater than half the power at the resonant frequency, ωr = 2πfr is the angular resonant frequency, and Δω is the angular half-power bandwidth.

If the losses incurred in the electrical system could be compensated, the amplitude of oscillation would remain constant and as such the oscillation would continue indefinitely against both external disturbances and changes in the initial conditions. This type of oscillation is called undamped oscillation.

ω=1/√LC
ω=1√0.09x25
ω=0.666
ω=2πf
f=0.666/(2 x 3.14)
f=0.106 s^-1

Explanation:

• Quality Factor: The measure of sharpness in an AC circuit is given by the Quality Factor. It is a dimensionless number that characterizes the lossiness of a resonator or filter circuit.



 

At resonance, the potential drop across the capacitance is equal and opposite to that of inductance i.e. 110V.

The series resonance circuit is known as an acceptor circuit. The impedance of the acceptor circuit is minimum but the voltage can be magnified.
The Acceptor circuit provides the maximum response to currents at its resonant frequency.
Series resonance circuit is known as acceptor circuit because the impedance at the resonance is at its minimum so as to accept the current easily such that the frequency of the accepted current is equal to the resonant frequency.
 

Given, emf = 0sin(t) At resonance frequency, we have: f = 1/(2π√(LC)) 0sin(t) = Vm sin(2πft) Where Vm is the maximum voltage of the source and f is the frequency of the source. Comparing the two equations, we get: f = 1/(2π√(LC)) Therefore, the maximum current in the circuit is obtained when: L = 1/ C Hence, option 'D' is the correct answer.
 

The most common application of tank circuits is tuning radio transmitters and receivers. For example, when we tune a radio to a particular station, the LC circuits are set at resonance for that particular carrier frequency. A parallel resonant circuit can be used as load impedance in output circuits of RF amplifiers.