Difference between revisions of "Passive Matching Networks"

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{{NumBlk|::|<math>\omega^2 R_S C_S R_P C_P = 1 = \frac{Q_P}{Q_S}</math>|{{EquationRef|17}}}}
 
{{NumBlk|::|<math>\omega^2 R_S C_S R_P C_P = 1 = \frac{Q_P}{Q_S}</math>|{{EquationRef|17}}}}
  
Leading to the result that <math>Q_P=Q_S</math>. Using eq. 17 to solve for <math>R_P C_P</math>, and plugging it into eq. 16, we get:
+
Leading to the result that <math>Q_P=Q_S=Q</math>. Using eq. 17 to solve for <math>R_P C_P</math>, and plugging it into eq. 16, we get:
  
{{NumBlk|::|<math>R_S C_S + \frac{1}{\omega^2 R_S C_S} - R_P C_S = R_S + \frac{1}{\omega^2 R_S C_S^2} - R_P =  R_S \left(1+ \frac{1}{\omega^2 R_S^2 C_S^2}\right) - R_P = R_S \left(1+ Q_S^2\right) - R_P = 0</math>|{{EquationRef|18}}}}
+
{{NumBlk|::|<math>R_S C_S + \frac{1}{\omega^2 R_S C_S} - R_P C_S = R_S + \frac{1}{\omega^2 R_S C_S^2} - R_P =  R_S \left(1+ \frac{1}{\omega^2 R_S^2 C_S^2}\right) - R_P = R_S \left(1+ Q^2\right) - R_P = 0</math>|{{EquationRef|18}}}}
  
 
We can then express <math>R_P</math> as a function of <math>R_S</math> and <math>Q_S</math> as:
 
We can then express <math>R_P</math> as a function of <math>R_S</math> and <math>Q_S</math> as:
  
{{NumBlk|::|<math>R_P = R_S \left(1+ Q_S^2\right)</math>|{{EquationRef|19}}}}
+
{{NumBlk|::|<math>R_P = R_S \left(1+ Q^2\right)</math>|{{EquationRef|19}}}}
  
 
Substituting eq. 19 into eq. 17, we get:
 
Substituting eq. 19 into eq. 17, we get:
  
{{NumBlk|::|<math>\omega^2 R_S C_S R_P C_P = \omega^2 R_S^2 \frac{C_S^2}{C_S} \left(1+ Q_S^2\right) C_P =  \frac{1}{Q_S^2 C_S} \left(1+ Q_S^2\right) C_P =1</math>|{{EquationRef|20}}}}
+
{{NumBlk|::|<math>\omega^2 R_S C_S R_P C_P = \omega^2 R_S^2 \frac{C_S^2}{C_S} \left(1+ Q^2\right) C_P =  \frac{1}{Q^2 C_S} \left(1+ Q^2\right) C_P =1</math>|{{EquationRef|20}}}}
  
 
Thus, we can express <math>C_P</math> as:
 
Thus, we can express <math>C_P</math> as:
  
{{NumBlk|::|<math>C_P = \frac{Q_S^2}{1+ Q_S^2} C_S = \frac{1}{1 + \frac{1}{Q_S^2}} C_S</math>|{{EquationRef|21}}}}
+
{{NumBlk|::|<math>C_P = \frac{Q^2}{1+ Q^2} C_S = \frac{1}{1 + \frac{1}{Q^2}} C_S</math>|{{EquationRef|21}}}}
  
For <math>Q_S^2 \gg 1</math>, we get <math>R_P\approx Q_S^2 R_S</math> and <math>C_P\approx C_S</math>.
+
For <math>Q^2 \gg 1</math>, we get <math>R_P\approx Q^2 R_S</math> and <math>C_P\approx C_S</math>.
  
 
=== RL Circuits ===
 
=== RL Circuits ===
Line 104: Line 104:
 
{{NumBlk|::|<math>R_P R_S = \omega^2 L_P L_S </math>|{{EquationRef|24}}}}
 
{{NumBlk|::|<math>R_P R_S = \omega^2 L_P L_S </math>|{{EquationRef|24}}}}
  
{{NumBlk|::|<math>\frac{R_P}{\omega L_P} = \frac{ \omega L_S}{R_S } = Q_P = Q_S</math>|{{EquationRef|25}}}}
+
{{NumBlk|::|<math>\frac{R_P}{\omega L_P} = \frac{ \omega L_S}{R_S } = Q_P = Q_S = Q</math>|{{EquationRef|25}}}}
  
 
And for the imaginary components:
 
And for the imaginary components:
Line 118: Line 118:
 
{{NumBlk|::|<math>R_P L_P =R_S L_P + \frac{\omega^2 L_P L_S}{R_S} L_S</math>|{{EquationRef|28}}}}
 
{{NumBlk|::|<math>R_P L_P =R_S L_P + \frac{\omega^2 L_P L_S}{R_S} L_S</math>|{{EquationRef|28}}}}
  
{{NumBlk|::|<math>R_P =R_S + \frac{\omega^2 L_S^2}{R_S^2} R_S = R_S\left(1 + Q_S^2\right)\approx R_S Q_S^2</math>|{{EquationRef|29}}}}
+
{{NumBlk|::|<math>R_P =R_S + \frac{\omega^2 L_S^2}{R_S^2} R_S = R_S\left(1 + Q^2\right)\approx R_S Q^2</math>|{{EquationRef|29}}}}
  
 
And substituting eq. 29 into eq. 27:
 
And substituting eq. 29 into eq. 27:
  
{{NumBlk|::|<math>R_P = \frac{\omega^2 L_P L_S}{R_S} = R_S\left(1 + Q_S^2\right)</math>|{{EquationRef|30}}}}
+
{{NumBlk|::|<math>R_P = \frac{\omega^2 L_P L_S}{R_S} = R_S\left(1 + Q^2\right)</math>|{{EquationRef|30}}}}
  
{{NumBlk|::|<math>\frac{\omega^2 L_S^2}{R_S^2} \frac{L_P}{L_S} = Q_S^2 \frac{L_P}{L_S}= \left(1 + Q_S^2\right)</math>|{{EquationRef|31}}}}
+
{{NumBlk|::|<math>\frac{\omega^2 L_S^2}{R_S^2} \frac{L_P}{L_S} = Q^2 \frac{L_P}{L_S}= \left(1 + Q^2\right)</math>|{{EquationRef|31}}}}
  
{{NumBlk|::|<math>L_P = L_S \frac{1 + Q_S^2}{Q_S^2} = L_S \left(1 + \frac{1}{Q_S^2}\right)\approx L_S </math>|{{EquationRef|32}}}}
+
{{NumBlk|::|<math>L_P = L_S \frac{1 + Q^2}{Q^2} = L_S \left(1 + \frac{1}{Q^2}\right)\approx L_S </math>|{{EquationRef|32}}}}
  
 
== Basic Matching Networks ==
 
== Basic Matching Networks ==

Revision as of 11:04, 2 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 respectively.

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)

Note that the quality factor is frequency dependent, and in the ideal lossless case, either or , leading to or respectively.

Series-Parallel Conversions

It is very convenient to be able to convert the series RC or RL circuit to its parallel equivalent or vice-versa, especially in the context of matching circuits.

RC Circuits

For RC circuits, the series and parallel impedances are:

 

 

 

 

(11)

 

 

 

 

(12)

Setting and rewriting the espressions, we get:

 

 

 

 

(13)

 

 

 

 

(14)

 

 

 

 

(15)

Equating the real part and the imaginary part to zero, we get:

 

 

 

 

(16)

 

 

 

 

(17)

Leading to the result that . Using eq. 17 to solve for , and plugging it into eq. 16, we get:

 

 

 

 

(18)

We can then express as a function of and as:

 

 

 

 

(19)

Substituting eq. 19 into eq. 17, we get:

 

 

 

 

(20)

Thus, we can express as:

 

 

 

 

(21)

For , we get and .

RL Circuits

In a similar way, we can convert the series RL circuit to its parallel equivalent, and vice versa. Thus, for we get:

 

 

 

 

(22)

 

 

 

 

(23)

And once again, we can separate the real and imaginary components of eq. 23. Looking at the real components:

 

 

 

 

(24)

 

 

 

 

(25)

And for the imaginary components:

 

 

 

 

(26)

From eq. 24, we get:

 

 

 

 

(27)

And plugging this into eq. 26 gives us:

 

 

 

 

(28)

 

 

 

 

(29)

And substituting eq. 29 into eq. 27:

 

 

 

 

(30)

 

 

 

 

(31)

 

 

 

 

(32)

Basic Matching Networks

Loss in Matching Networks

T and Pi Matching Networks