Quality Factor

Let us review the many definitions of the quality factor, $Q$. This context-dependent metric can allow us to gain important insights on the behavior and implementation of energy storage and loss in circuits.

Component Quality Factor

For a transfer function that we can write as:

$H\left(j\omega \right)={\frac {1}{R\left(\omega \right)+jX\left(\omega \right)}}$

(1)

We can define the component quality factor, $Q$, as:

$Q=\left.{\frac {\text{energy stored}}{\text{average power dissipated}}}\right|_{\text{over a period of time}}={\frac {X\left(\omega \right)}{R\left(\omega \right)}}$

(2)

Example: A Lossy Inductor

For a lossy inductor, modeled as an ideal inductor with a series resistance, $R_{s}$, as shown in Fig. 1, we can write the admittance as:

$Y_{L}={\frac {I\left(\omega \right)}{V\left(\omega \right)}}={\frac {1}{R_{s}+j\omega L}}$

(3)

The quality factor of the lossy inductor is then equal to:

$Q_{L}={\frac {\omega L}{R_{s}}}$

(4)

Example: A Lossy Capacitor

For a lossy capacitor, modeled as an ideal capacitor in parallel with a resistance, $R_{p}$, as shown in Fig. 2, we can write the impedance as:

$Z_{C}={\frac {V\left(\omega \right)}{I\left(\omega \right)}}={\frac {1}{{\frac {1}{R_{p}}}+j\omega C}}$

(5)

The quality factor of the lossy capacitor is then equal to:

$Q_{C}=\omega C\cdot R_{p}$

(6)

Pole Quality Factor

The quality factor of a pole is a good indicator of the "cost" of implementing a pole. Higher Q poles have more stringent requirements in terms of loss, thus the pole Q allows us to determine which poles require more resources to implement.

Consider the pole shown in Fig. 3. We define the pole quality factor as:

$Q_{\text{pole}}={\frac {\omega _{p}}{2\sigma _{x}}}$

(7)

Where $\omega _{p}$ is the distance of the pole from the origin, and $\sigma _{x}$ is the distance of the pole from the $j\omega$-axis. This means that higher Q poles are closer to the $j\omega$-axis.

Band-Pass Filter Quality Factor

The band-pass filter quality factor, $Q_{BP}$ is a measure of the filter's bandwidth. The higher the quality factor, the more selectivity we have, and hence, lower loss is required to implement the filter. Thus, the band-pass filter quality factor is defined as:

$Q_{BP}={\frac {\omega _{c}}{\omega _{2}-\omega _{1}}}={\frac {\omega _{c}}{\Delta \omega }}$

(8)

Where $\omega _{c}$ is the center filter frequency, and $\Delta \omega =\omega _{2}-\omega _{1}$ is the filter bandwidth, as shown in Fig. 4.

Figure 3: The pole quality factor.

Figure 4: The band-pass filter quality factor.