Difference between revisions of "Passive Matching Networks"

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The quality factor, Q, is a measure of how good a device is at storing energy. Thus, the lower the losses, the higher the Q. In general, if we can express the impedance or admittance of a device as  
 
The quality factor, Q, is a measure of how good a device is at storing energy. Thus, the lower the losses, the higher the Q. In general, if we can express the impedance or admittance of a device as  
  
{{NumBlk|::|<math>Y\,\mathrm{or}\,Z=\tfrac{1}{A + jB}</math>|{{EquationRef|1}}}}
+
{{NumBlk|::|<math>Y\,\mathrm{or}\,Z=\frac{1}{A + jB}</math>|{{EquationRef|1}}}}
  
 
The quality factor can then be expressed as  
 
The quality factor can then be expressed as  
  
{{NumBlk|::|<math>Q=\tfrac{\left|B\right|}{A}</math>|{{EquationRef|2}}}}
+
{{NumBlk|::|<math>Q=\frac{\left|B\right|}{A}</math>|{{EquationRef|2}}}}
  
 
The imaginary component, <math>B</math> represents the energy storage element, and the real component, <math>A</math>, represents the loss (resistive) component.  
 
The imaginary component, <math>B</math> represents the energy storage element, and the real component, <math>A</math>, represents the loss (resistive) component.  
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A lossy capacitor can be modeled as a series RC circuit with the series resistance, <math>R_S</math>, could represent the series loss due to interconnect resistances. Thus, the admittance of the lossy capacitor can be expressed as  
 
A lossy capacitor can be modeled as a series RC circuit with the series resistance, <math>R_S</math>, could represent the series loss due to interconnect resistances. Thus, the admittance of the lossy capacitor can be expressed as  
  
{{NumBlk|::|<math>Y_S =\tfrac{1}{R_S+\tfrac{1}{j\omega C}}</math>|{{EquationRef|3}}}}
+
{{NumBlk|::|<math>Y_S =\frac{1}{R_S+\frac{1}{j\omega C}}</math>|{{EquationRef|3}}}}
 
   
 
   
 
We can therefore express the quality factor of the series circuit as:
 
We can therefore express the quality factor of the series circuit as:
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A lossy capacitor can also be modeled as a parallel RC circuit, with the parallel resistance, <math>R_P</math> could represent the energy loss due to the dielectric leakage of the capacitor. In this case, we can write the impedance of the circuit as:
 
A lossy capacitor can also be modeled as a parallel RC circuit, with the parallel resistance, <math>R_P</math> could represent the energy loss due to the dielectric leakage of the capacitor. In this case, we can write the impedance of the circuit as:
  
{{NumBlk|::|<math>Z_P =\tfrac{1}{\frac{1}{R_P}+j\omega C}</math>|{{EquationRef|5}}}}
+
{{NumBlk|::|<math>Z_P =\frac{1}{\frac{1}{R_P}+j\omega C}</math>|{{EquationRef|5}}}}
  
 
Thus, the quality factor is:
 
Thus, the quality factor is:
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Applying the ideas above to RL circuits, we can get the admittance of a series RL circuit as:
 
Applying the ideas above to RL circuits, we can get the admittance of a series RL circuit as:
  
{{NumBlk|::|<math>Y_S =\tfrac{1}{R_S+j\omega L}</math>|{{EquationRef|7}}}}
+
{{NumBlk|::|<math>Y_S =\frac{1}{R_S+j\omega L}</math>|{{EquationRef|7}}}}
  
 
with quality factor:
 
with quality factor:
  
{{NumBlk|::|<math>Q_S = \frac{\omega L}{R_S}</math>|{{EquationRef|4}}}}
+
{{NumBlk|::|<math>Q_S = \frac{\omega L}{R_S}</math>|{{EquationRef|8}}}}
 +
 
 +
Similarly, for parallel RL circuits, the impedance and quality factor can be expressed as:
 +
 
 +
{{NumBlk|::|<math>Z_P =\frac{1}{\frac{1}{R_P}+\frac{1}{j\omega L}}</math>|{{EquationRef|9}}}}
 +
 
 +
{{NumBlk|::|<math>Q_P = \frac{\frac{1}{\omega L}}{\frac{1}{R_P}}=\frac{R_P}{\omega L}</math>|{{EquationRef|10}}}}
  
 
== Series-Parallel Conversions ==
 
== Series-Parallel Conversions ==

Revision as of 18:18, 1 September 2020

Controlling impedances in RF circuits are essential to maximize power transfer between blocks and to reduce reflections caused by impedance discontinuities along a signal path. In this module, we will define the quality factor, Q, for a device or a circuit, and use this Q to build circuits that modify the impedance seen across a port.

Device Quality Factor

The quality factor, Q, is a measure of how good a device is at storing energy. Thus, the lower the losses, the higher the Q. In general, if we can express the impedance or admittance of a device as

 

 

 

 

(1)

The quality factor can then be expressed as

 

 

 

 

(2)

The imaginary component, represents the energy storage element, and the real component, , represents the loss (resistive) component.

Series RC Circuit

A lossy capacitor can be modeled as a series RC circuit with the series resistance, , could represent the series loss due to interconnect resistances. Thus, the admittance of the lossy capacitor can be expressed as

 

 

 

 

(3)

We can therefore express the quality factor of the series circuit as:

 

 

 

 

(4)

Note that for the lossless case, , leading to .

Parallel RC Circuit

A lossy capacitor can also be modeled as a parallel RC circuit, with the parallel resistance, could represent the energy loss due to the dielectric leakage of the capacitor. In this case, we can write the impedance of the circuit as:

 

 

 

 

(5)

Thus, the quality factor is:

 

 

 

 

(6)

We can see that in the lossless case, , resulting in .

RL Circuits

Applying the ideas above to RL circuits, we can get the admittance of a series RL circuit as:

 

 

 

 

(7)

with quality factor:

 

 

 

 

(8)

Similarly, for parallel RL circuits, the impedance and quality factor can be expressed as:

 

 

 

 

(9)

 

 

 

 

(10)

Series-Parallel Conversions

Basic Matching Networks

Loss in Matching Networks

T and Pi Matching Networks