Quality Factor

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Let us review the many definitions of the quality factor, . 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:

 

 

 

 

(1)

We can define the component quality factor, , as:

 

 

 

 

(2)

Example: A Lossy Inductor

Figure 1: A lossy inductor.
Figure 2: A lossy capacitor.

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

 

 

 

 

(3)

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

 

 

 

 

(4)

Example: A Lossy Capacitor

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

 

 

 

 

(5)

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

 

 

 

 

(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:

 

 

 

 

(7)

Where is the distance of the pole from the origin, and is the distance of the pole from the -axis. This means that higher Q poles are closer to the -axis.

Band-Pass Filter Quality Factor

The band-pass filter quality factor, 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:

 

 

 

 

(8)

Where is the center filter frequency, and is the filter bandwidth, as shown in Fig. 4.

Figure 3: The pole quality factor.
Figure 4: The band-pass filter quality factor.