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

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

(1)

The quality factor can then be expressed as

$Q={\frac {\left|B\right|}{A}}$

(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

$Y_{S}={\frac {1}{R_{S}+{\frac {1}{j\omega C}}}}$

(3)

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

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

(4)

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

Parallel RC Circuit

A lossy capacitor can also be modeled as a parallel RC circuit, with the parallel resistance, $R_{P}$ 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:

$Z_{P}={\frac {1}{{\frac {1}{R_{P}}}+j\omega C}}$

(5)

Thus, the quality factor is:

$Q_{P}={\frac {\omega C}{\frac {1}{R_{P}}}}=\omega R_{P}C$

(6)

We can see that in the lossless case, $R_{P}\rightarrow \infty$, resulting in $Q_{P,\mathrm {ideal} }\rightarrow \infty$.

RL Circuits

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

$Y_{S}={\frac {1}{R_{S}+j\omega L}}$

(7)

with quality factor:

$Q_{S}={\frac {\omega L}{R_{S}}}$

(8)

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

$Z_{P}={\frac {1}{{\frac {1}{R_{P}}}+{\frac {1}{j\omega L}}}}$

(9)

$Q_{P}={\frac {\frac {1}{\omega L}}{\frac {1}{R_{P}}}}={\frac {R_{P}}{\omega L}}$

(10)

Note that the quality factor is frequency dependent, and in the ideal lossless case, either $R_{S}=0$ or $R_{P}\rightarrow \infty$, leading to $Q_{S,\mathrm {ideal} }\rightarrow \infty$ or $Q_{P,\mathrm {ideal} }\rightarrow \infty$.