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A coil of inductance 10 H and resistance 40 ohm is connected in series with a capacitance and supplied by a source of variable frequency. If the maximum current is found at frequency 1000 rad/sec, then Q-factor of the circuit will be _________.
100
250
200
25
250
The Q-factor (Quality Factor) of a series RLC circuit is a measure of the sharpness of resonance and indicates how much energy is dissipated relative to the stored energy. It is defined as the ratio of inductive reactance to resistance at the resonant frequency.
The Q-factor (Quality Factor) of a series RLC circuit is a measure of the sharpness of resonance and indicates how much energy is dissipated relative to the stored energy. It is defined as the ratio of inductive reactance to resistance at the resonant frequency.
Think of a radio tuner; the Q-factor is like the precision of the knob. A high Q-factor means the radio isolates a specific station very sharply, rejecting all other overlapping frequencies, while a low Q-factor would make the radio sound like it is playing multiple stations at once.
╧Й0тАЛ=LCтАЛ1тАЛ тАФ Resonant angular frequency
Q=R╧Й0тАЛLтАЛ тАФ Quality factor formula
In a series RLC circuit, resonance occurs when the inductive reactance XLтАЛ=╧ЙL equals the capacitive reactance XCтАЛ=╧ЙC1тАЛ. At this resonant frequency, the circuit impedance is purely resistive (Z=R), resulting in maximum current. The Q-factor is calculated using the formula Q=R╧Й0тАЛLтАЛ, where ╧Й0тАЛ is the resonant angular frequency, L is the inductance, and R is the resistance.
At resonance, the phase angle between voltage and current is zero.
The Q-factor is dimensionless.
A higher Q-factor results in a narrower bandwidth and sharper resonance peak.
High Q-factor circuits are essential for high-selectivity signal filtering.
High Q circuits are more sensitive to component tolerance variations.
Radio frequency (RF) tuning circuits
Signal processing filters
Given values: L=10┬аH, R=40┬а╬й, ╧Й0тАЛ=1000┬аrad/sec.
Calculation: Q=401000├Ч10тАЛ=4010000тАЛ=250.
Option A is incorrect as it results from a calculation error; Option C and D are incorrect as they do not match the product of ╧Й and L divided by R.
B is correct тАФ Using the formula Q=R╧ЙLтАЛ, substituting the given values yields Q=401000├Ч10тАЛ=250.
Remember that Q-factor is also defined as R1тАЛCLтАЛтАЛ. If capacitance C were provided instead of frequency, you could use this version of the formula.