## Quality Factor of Inductor

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 inductor, **quality factor** is expressed as,

Where, L is the effective inductance of the coil in Henrys and R is the effective resistance of the coil in Ohms. As the unit of both resistance and reactance is Ohm, Q is a dimensionless ratio.

The Q factor may also be defined as

Let us prove the above expression. For that let us consider a sinusoidal voltage V of frequency ω radians/seconds applied to an inductor L of effective internal resistance R as shown in Figure 1(a). Let the resulting peak current through the inductor be I_{m}.

Then the maximum energy stored in the inductor

Figure 1. 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,

## Quality Factor of a Capacitor

Figure 1(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 ω 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 since the unit of both reactance and resistance is the same and it is Ohm. Equation (2) giving the alternative definition of Q also holds good in this case. Thus, for the circuit of Figure 1(b), on application of a sinusoidal voltage of value V volts and frequency ω, the maximum energy stored in the capacitor.

Where, V_{m} is the maximum value of voltage across the capacitance C.

But if

then

Where, I_{m} is the maximum value of current through C and R.

Hence, the maximum energy stored in capacitor C is

Energy dissipated per cycle

So, the quality factor of capacitor is

Often a lossy capacitor is represented by a capacitance C with a high resistance R_{p} in shunt as shown in Figure 2.

Then for the capacitor of Figure 2, the maximum energy stored in the capacitor

Where, V_{m} is the maximum value of the applied voltage. The average power dissipated in resistance R_{p}.

Figure 2. Alternative method of representing a lossy capacitor

Energy dissipated per cycle

Hence,