A.C. Fundamentals - Q-factor of a Parallel Resonant Circuit
1 Answer
The Q-factor (Quality Factor) of a parallel resonant circuit, also known as a parallel RLC circuit, is a measure of its efficiency and selectivity. It indicates how well the circuit can resonate at a specific frequency and how sharply it responds to frequencies close to the resonant frequency. The Q-factor is defined as the ratio of the reactance (X) of the inductor or capacitor to the resistance (R) of the circuit:
Q = X / R
Where:
X is the reactance of the inductor or capacitor at the resonant frequency.
R is the resistance of the circuit.
For a parallel resonant circuit, the total admittance (Y) of the circuit is given by the sum of the admittances of the resistor (G), capacitor (C), and inductor (L):
Y = G + jωC + 1 / (jωL)
Where:
G is the conductance (reciprocal of resistance) of the resistor.
j is the imaginary unit (√(-1)).
ω is the angular frequency (2π times the frequency).
C is the capacitance of the capacitor.
L is the inductance of the inductor.
The resonance frequency (ω₀) of the parallel resonant circuit occurs when the imaginary part of the total admittance is zero, which means that the reactance of the capacitor and the reactance of the inductor cancel each other out:
jω₀C + 1 / (jω₀L) = 0
Solving for ω₀:
ω₀² = 1 / (LC)
Now, let's calculate the Q-factor using the formula Q = X / R. At resonance (ω = ω₀), the reactance of the capacitor (Xc) and the reactance of the inductor (Xl) are equal in magnitude but opposite in sign:
Xc = -Xl
1 / (jω₀C) = jω₀L
Solving for ω₀:
ω₀ = 1 / √(LC)
The reactance of the inductor and capacitor at resonance is given by:
X = Xc = -Xl = 1 / (jω₀C) = -j / ω₀C
Now, substituting the values of X and R into the Q-factor formula:
Q = X / R = (-j / ω₀C) / R = -j / (Rω₀C)
Therefore, the Q-factor of a parallel resonant circuit can be expressed as:
Q = -1 / (Rω₀C)
A higher Q-factor indicates a sharper resonance and better selectivity in the circuit. It implies that the circuit can efficiently store energy at the resonant frequency and attenuate frequencies that are not close to the resonant frequency.
Q = X / R
Where:
X is the reactance of the inductor or capacitor at the resonant frequency.
R is the resistance of the circuit.
For a parallel resonant circuit, the total admittance (Y) of the circuit is given by the sum of the admittances of the resistor (G), capacitor (C), and inductor (L):
Y = G + jωC + 1 / (jωL)
Where:
G is the conductance (reciprocal of resistance) of the resistor.
j is the imaginary unit (√(-1)).
ω is the angular frequency (2π times the frequency).
C is the capacitance of the capacitor.
L is the inductance of the inductor.
The resonance frequency (ω₀) of the parallel resonant circuit occurs when the imaginary part of the total admittance is zero, which means that the reactance of the capacitor and the reactance of the inductor cancel each other out:
jω₀C + 1 / (jω₀L) = 0
Solving for ω₀:
ω₀² = 1 / (LC)
Now, let's calculate the Q-factor using the formula Q = X / R. At resonance (ω = ω₀), the reactance of the capacitor (Xc) and the reactance of the inductor (Xl) are equal in magnitude but opposite in sign:
Xc = -Xl
1 / (jω₀C) = jω₀L
Solving for ω₀:
ω₀ = 1 / √(LC)
The reactance of the inductor and capacitor at resonance is given by:
X = Xc = -Xl = 1 / (jω₀C) = -j / ω₀C
Now, substituting the values of X and R into the Q-factor formula:
Q = X / R = (-j / ω₀C) / R = -j / (Rω₀C)
Therefore, the Q-factor of a parallel resonant circuit can be expressed as:
Q = -1 / (Rω₀C)
A higher Q-factor indicates a sharper resonance and better selectivity in the circuit. It implies that the circuit can efficiently store energy at the resonant frequency and attenuate frequencies that are not close to the resonant frequency.