Passive Matching Networks

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

$Y\,\mathrm {or} \,Z={\tfrac {1}{A+jB}}$

(1)

The quality factor can then be expressed as

$Q={\tfrac {B}{A}}$

(2)

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

Series RC Circuit

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

$Y={\tfrac {1}{R_{S}+j{\tfrac {1}{\omega C}}}}$

(3)

We get therefore write the quality factor as:

$Q={\frac {\frac {1}{\omega C}}{R_{S}}}={\frac {1}{\omega R_{S}C}}$

(4)

Note that for the lossless case, $R_{S}=0$, leading to $Q_{\mathrm {ideal} }\rightarrow \infty$.