Chapter 12.7 – Quality Factor or Q-Factor | GlobalSpec

12.7 QUALITY FACTOR OR Q-FACTOR

(a) Q of a Coil. Every inductor possesses a small resistance in addition to its
inductance. The lower the value of this resistance R, the better the quality
of the coil. The quality factor or the Q-factor of an inductor at the operating
frequency ω is defined as the ratio of reactance of the coil to its resistance.

Thus for a coil,

where L is the effective inductance of the coil in Henrys and R is the effective
resistance of the coil in Ohms.

Obviously, Q is a dimensionless ratio.

The Q-factor may be defined as

Thus, consider a sinusoidal voltage V of frequency ω radians/seconds
applied to an inductor L of effective internal resistance R as shown in
Figure 12.10(a). Let the resulting peak current through the coil be Im.

Then the maximum energy stored in the inductor

FIGURE 12.10   RL and RC circuits connected to a sinusoidal voltage sources.

The average power dissipated in the inductor per cycle

Hence, the energy dissipated in the inductor per cycle

Hence,

(b) Q of a capacitor. Figure 12.10(b) shows a capacitor C with small
series resistance R associated within. The Q-factor or the quality factor of a
capacitor at the operating frequency is defined as the ratio of the reactance
of the capacitor to its series resistance.

Thus,

In this case also, the Q is a dimensionless quantity. Equation (12.19)
giving the alternative definition of Q also holds good in this case. Thus, for
the circuit of Figure 12.10(b), on application of a sinusoidal voltage of value
V volts and frequency ω, the maximum energy stored in the capacitor

where Vmax is the maximum value of voltage across the capacitance C.

But if

then

where Im is the maximum value of current through C and R.

Hence, the maximum energy stored in capacitor C is

Energy dissipated per cycle

Often a lossy capacitor is represented by a capacitance C with a high
resistance Rp in shunt as shown in Figure 12.11.

Then for the capacitor of Figure 12.11, the maximum energy stored in
the capacitor

where Vmaxis the maximum value of the applied voltage. The average power
dissipated in resistance Rp

FIGURE 12.11   Alternative method of representing a lossy capacitor.

Energy dissipated per cycle

Hence,

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