Difference between revisions of "Quality Factor"

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(Created page with "Let us review the many definitions of the '''quality factor''', <math>Q</math>. This context-dependent metric can allow us to gain important insights on the behavior and imple...")
 
 
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== Component Quality Factor ==
 
== Component Quality Factor ==
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For a transfer function that we can write as:
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{{NumBlk|::|<math>
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H\left(j\omega\right) = \frac{1}{R\left(\omega\right) + jX\left(\omega\right)}
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</math>|{{EquationRef|1}}}}
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We can define the '''component quality factor''', <math>Q</math>, as:
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{{NumBlk|::|<math>
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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)}
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</math>|{{EquationRef|2}}}}
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=== Example: A Lossy Inductor ===
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[[File:Lossy L.png|thumb|300px|Figure 1: A lossy inductor.]]
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[[File:Lossy C.png|thumb|300px|Figure 2: A lossy capacitor.]]
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For a lossy inductor, modeled as an ideal inductor with a series resistance, <math>R_s</math>, as shown in Fig. 1, we can write the admittance as:
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{{NumBlk|::|<math>
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Y_L = \frac{I\left(\omega\right)}{V\left(\omega\right)} = \frac{1}{R_s + j\omega L}
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</math>|{{EquationRef|3}}}}
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The quality factor of the lossy inductor is then equal to:
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{{NumBlk|::|<math>
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Q_L = \frac{\omega L}{R_s}
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</math>|{{EquationRef|4}}}}
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=== Example: A Lossy Capacitor ===
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For a lossy capacitor, modeled as an ideal capacitor in parallel with a resistance, <math>R_p</math>, as shown in Fig. 2, we can write the impedance as:
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{{NumBlk|::|<math>
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Z_C = \frac{V\left(\omega\right)}{I\left(\omega\right)} = \frac{1}{\frac{1}{R_p} + j\omega C}
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</math>|{{EquationRef|5}}}}
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The quality factor of the lossy capacitor is then equal to:
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{{NumBlk|::|<math>
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Q_C = \omega C\cdot R_p
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</math>|{{EquationRef|6}}}}
  
 
== Pole Quality Factor ==
 
== Pole Quality Factor ==
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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.
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Consider the pole shown in Fig. 3. We define the pole quality factor as:
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{{NumBlk|::|<math>
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Q_\text{pole} = \frac{\omega_p}{2\sigma_x}
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</math>|{{EquationRef|7}}}}
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Where <math>\omega_p</math> is the distance of the pole from the origin, and <math>\sigma_x</math> is the distance of the pole from the <math>j\omega</math>-axis. This means that higher Q poles are closer to the <math>j\omega</math>-axis.
  
 
== Band-Pass Filter Quality Factor ==
 
== Band-Pass Filter Quality Factor ==
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The band-pass filter quality factor, <math>Q_{BP}</math> 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:
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{{NumBlk|::|<math>
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Q_{BP} = \frac{\omega_c}{\omega_2 - \omega_1} =  \frac{\omega_c}{\Delta \omega}
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</math>|{{EquationRef|8}}}}
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Where <math>\omega_c</math> is the center filter frequency, and <math>\Delta \omega = \omega_2 - \omega_1</math> is the filter bandwidth, as shown in Fig. 4.
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{|
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|[[File:Pole Q.svg|thumb|300px|Figure 3: The pole quality factor.]]
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|[[File:Bp filter Q.svg|thumb|400px|Figure 4: The band-pass filter quality factor.]]
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|-
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|}

Latest revision as of 20:51, 16 March 2021

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.