Question

What is Q-factor? Write its expression and write the conditions for its maximum value.

Answer 1

Hint: The quality factor or ‘Q’ is a dimensionless quantity that describes the nature of damping in a resonating circuit. It is indeed the ratio of maximum energy stored in the circuit to the energy dissipated during each cycle of oscillation.

Complete step by step solution:
Part 1:
This concept of quality factor Q is applicable to many areas of physics and engineering like for studying the behavior of mechanical pendulum, the similar resonating element of the mechanical structure, or electrical resonating circuits. For electrical circuits, Q is defined as the ratio of maximum energy stored in a resonating circuit to the energy supplied per cycle of oscillation by an external source to keep the net signal amplitude constant. For LCR resonating circuit Q are the ratio of resonating frequency and the difference in both frequencies for which power is \[\dfrac{1}{2}\] of its peak value i.e. current is it’s $\dfrac{1}{{\sqrt 2 }}$times of its peak.
Part 2
Mathematically, the expression of quality factor(Q) is given as:
$Q = \dfrac{{{\omega _0}L}}{R} = \dfrac{1}{{R{\omega _0}C}} = \dfrac{1}{R}\sqrt {\dfrac{L}{C}} $
Where,
${\omega _0}$is the resonating frequency,
R is the total resistance of the circuit,
L is the value of inductance of the circuit,
C is the value of the capacitance of the circuit.
From eq.(1) the condition for maximizing Q is given as:
To minimize the value of R in the circuit.
To maximize the value of $\dfrac{L}{C}$, either by increasing L or by decreasing C or doing both.

Additional information:
The damping nature of a system is characterized by the value of Q. For \[Q > \dfrac{1}{2}\]the system is called under-damped and the system oscillates with slowly decreasing amplitude. For $Q < \dfrac{1}{2}$ the system is called over-damped and the oscillation amplitude exponentially decreases to reach a steady value. For \[Q = \dfrac{1}{2}\] it's called critically damped and the system just oscillates one time with decreasing amplitude and then reaches a steady state. As the damping of the system decreases Q value increases. For an undamped system, Q is infinite.

Note:
From eq.(1) notice that the quality factor is completely the intrinsic property of the circuit. It is not affected by any external factors like external forcing or the frequency of the AC source connected to the given circuit.