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# Q Factor in Electrical and Electronics Engineering

In a Tuned Circuit, The ratio between Reactance and Resistance is called Q Factor or Quality Factor … Or …
Opposite of the Power factor is called the Q-Factor or Quality Factor of a Coil or its figure of merit.
Q Factor = 1/ Power Factor=1/Cosθ= Z/R        (Where Power Factor Cosθ = R/Z)
If R is too small with respect to Reactance
Then Q factor = Z/R = ωL/R = 2πfL / R       (ωL/R = 2πf)
Also, Q Factor may be defined as the ratio between stored energy and Energy dissipated per cycle in a Circuit
Q = 2π x (Stored Energy/ Power loss)
In a Resonator, Q is the ratio between stored energy in resonator and energy supplied by generator to keep signal amplitude constant
Q = 2π (Maximum Energy Stored/Energy dissipate per Cycle) in the coil.
Good to Know *1:
In Electrical System & Circuits, The stored Energy is the sum of stored energies in lossless Inductors and Capacitors. And the lost energy is the sum of the energies dissipated in resistors (Heat, light etc) per cycle
Whereas;
Capacitor absorbs Reactive Power and Stores Energy in the form of Electric field
Inductor absorbs Reactive Power and Stores Energy in the form of Magnetic Field
And
Resistor absorbs the real power and dissipates in the form of heat and light
Q Factor in Pure Capacitive (C) and Pure Inductive (L) Circuits
As we know that the Power in Pure Capacitive and Inductive Circuits are Zero. Thus the Circuit Power factor is also Zero. But the circuit “Q” factor is the inverse of Power factor, thus “Q” factor in both Pure Capacitive and Inductive Circuits are infinite ().
Q Factor in a Series RL Circuit
In Series RL Circuit, Impedance (Z) = the inductive Reactance = XL = 2πfL, Therefore the Quality factor “Q”
= Z/ R → = XL /R = 2πfrL /R
Q Factor in a Series RC Circuit
In Series RC Circuit, Impedance (Z) = Capacitive Reactance = XC = 1/2πfC, Therefore the Quality factor “Q”
= Z/ R → = XC /R = (1/2πfrC) /R → = 1 / 2πfrCR.
Where
Z = Impudence = Resistance in AC Circuits (Z = XL2-XC2 Ω)
R = Resistance in Ω
C = Capacitance in Farads
L = Inductance in Hennery
XL = Inductive reactance in Ω
XC = Capacitive Reactance in Ω
fr= Resonance Frequency in Hz

Q Factor of a tuned circuit = resonance frequency / bandwidth

Q = fr / B
Q = fr / (f2 – f1)
Where
fr = Resonance Frequency in Hertz
B = Bandwidth = the difference between the upper and lower frequencies in a continuous set of frequencies = B = (f2– f1)
Q Factor in a Series RLC Circuit (Voltage input resonance Circuit)
In an ideal series RLC circuit (Also in a (TRF) tuned radio frequency receiver) the Quality “Q” factor is
Q = (1/R) x (√ (L/C) = ω0L/R
It is clear from the above equation that the larger the Series Resistance, the smaller the “Q” factor of the Circuit i.e., the more energy lost and the wider bandwidth.
Good to know*2: A high Q factor of resonant circuit has a narrow bandwidth as compared to a low “Q” factor
Q Factor in a Parallel RLC Circuit (Current input resonance Circuit)
“Q” factor in a Parallel RLC circuit is just the inverse of the “Q” Factor in Series RLC circuit
Q = R x (√ (C /L) = R /ω0L
Where
R = Resistance in Ω
C = Capacitance in Farads
L = Inductance in Henry
It is clear from the above equation that the lower the Resistance, the larger the “Q” factor of the Circuit i.e. the less energy lost and the narrower bandwidth and it would be useful in filter design circuits to determine the bandwidth

Q Factor in a Circuit having Complicated Impedances
As we discussed above that “In a Tuned Circuit, The ratio between Reactance and Resistance is called Q Factor or Quality Factor … Or
Opposite of the Power factor is called the Q-Factor or Quality Factor of a Coil.
Q Factor = 1/ Power Factor=1/Cosθ= Z/R        (Where Power Factor Cosθ = R/Z)”
These for; we can also determine the “Q” factor of a Circuit having Complicated Impedances if we know the Circuit Power factorwhere
Power factor= Cosθ = R/Z   … or…
The Tangent of the phase angle (θ) between current and voltage.
Good to know *3:
A high Q factor of resonant circuit has a narrow bandwidth as compared to a low “Q” factor

A low Q factor gives a broad band (wide bandwidth)
A high Q factor gives a narrow band (small bandwidth
)

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