Passive Matching Networks

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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

We can use our series-parallel conversions to increase or decrease resistances based on the quality factor. Consider the circuit below, where we want to match the source resistance, to a load resistance, using a lossless matching network:

Case 1:

Consider the circuit below. Since we know that the parallel resistance of the RC circuit gets reduced when converting it to a series RC circuit:



Case 2:

Loss in Matching Networks

T and Pi Matching Networks