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 $Q_{P,\mathrm {ideal} }\rightarrow \infty$ respectively.

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 $Q_{P,\mathrm {ideal} }\rightarrow \infty$ 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:

$Z_{S}=R_{S}+{\frac {1}{j\omega C_{S}}}={\frac {j\omega R_{S}C_{S}+1}{j\omega C_{S}}}$

(11)

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

(12)

Setting $Z_{S}=Z_{P}$ and rewriting the espressions, we get:

${\frac {j\omega R_{S}C_{S}+1}{j\omega C_{S}}}={\frac {R_{P}}{1+j\omega R_{P}C_{P}}}=\left(j\omega R_{S}C_{S}+1\right)\left(1+j\omega R_{P}C_{P}\right)=j\omega R_{P}C_{S}$

(13)

$j\omega R_{S}C_{S}+j\omega R_{P}C_{P}-\omega ^{2}R_{S}C_{S}R_{P}C_{P}+1=j\omega R_{P}C_{S}$

(14)

$j\omega \left(R_{S}C_{S}+R_{P}C_{P}-R_{P}C_{S}\right)-\omega ^{2}R_{S}C_{S}R_{P}C_{P}+1=0$

(15)

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

$R_{S}C_{S}+R_{P}C_{P}-R_{P}C_{S}=0$

(16)

$\omega ^{2}R_{S}C_{S}R_{P}C_{P}=1={\frac {Q_{P}}{Q_{S}}}$

(17)

Leading to the result that $Q_{P}=Q_{S}=Q$. Using eq. 17 to solve for $R_{P}C_{P}$, and plugging it into eq. 16, we get:

$R_{S}C_{S}+{\frac {1}{\omega ^{2}R_{S}C_{S}}}-R_{P}C_{S}=R_{S}+{\frac {1}{\omega ^{2}R_{S}C_{S}^{2}}}-R_{P}=R_{S}\left(1+{\frac {1}{\omega ^{2}R_{S}^{2}C_{S}^{2}}}\right)-R_{P}=R_{S}\left(1+Q^{2}\right)-R_{P}=0$

(18)

We can then express $R_{P}$ as a function of $R_{S}$ and $Q_{S}$ as:

$R_{P}=R_{S}\left(1+Q^{2}\right)$

(19)

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

$\omega ^{2}R_{S}C_{S}R_{P}C_{P}=\omega ^{2}R_{S}^{2}{\frac {C_{S}^{2}}{C_{S}}}\left(1+Q^{2}\right)C_{P}={\frac {1}{Q^{2}C_{S}}}\left(1+Q^{2}\right)C_{P}=1$

(20)

Thus, we can express $C_{P}$ as:

$C_{P}={\frac {Q^{2}}{1+Q^{2}}}C_{S}={\frac {1}{1+{\frac {1}{Q^{2}}}}}C_{S}$

(21)

For $Q^{2}\gg 1$, we get $R_{P}\approx Q^{2}R_{S}$ and $C_{P}\approx C_{S}$.

RL Circuits

In a similar way, we can convert the series RL circuit to its parallel equivalent, and vice versa. Thus, for $Z_{P}=Z_{S}$ we get:

${\frac {1}{{\frac {1}{R_{P}}}+{\frac {1}{j\omega L_{P}}}}}={\frac {j\omega R_{P}L_{P}}{R_{P}+j\omega L_{P}}}=R_{S}+j\omega L_{S}$

(22)

$j\omega R_{P}L_{P}=\left(R_{P}+j\omega L_{P}\right)\left(R_{S}+j\omega L_{S}\right)=R_{P}R_{S}-\omega ^{2}L_{P}L_{S}+j\omega R_{S}L_{P}+j\omega R_{P}L_{S}$

(23)

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

$R_{P}R_{S}=\omega ^{2}L_{P}L_{S}$

(24)

${\frac {R_{P}}{\omega L_{P}}}={\frac {\omega L_{S}}{R_{S}}}=Q_{P}=Q_{S}=Q$

(25)

And for the imaginary components:

$R_{P}L_{P}=R_{S}L_{P}+R_{P}L_{S}$

(26)

From eq. 24, we get:

$R_{P}={\frac {\omega ^{2}L_{P}L_{S}}{R_{S}}}$

(27)

And plugging this into eq. 26 gives us:

$R_{P}L_{P}=R_{S}L_{P}+{\frac {\omega ^{2}L_{P}L_{S}}{R_{S}}}L_{S}$

(28)

$R_{P}=R_{S}+{\frac {\omega ^{2}L_{S}^{2}}{R_{S}^{2}}}R_{S}=R_{S}\left(1+Q^{2}\right)\approx R_{S}Q^{2}$

(29)

And substituting eq. 29 into eq. 27:

$R_{P}={\frac {\omega ^{2}L_{P}L_{S}}{R_{S}}}=R_{S}\left(1+Q^{2}\right)$

(30)

${\frac {\omega ^{2}L_{S}^{2}}{R_{S}^{2}}}{\frac {L_{P}}{L_{S}}}=Q^{2}{\frac {L_{P}}{L_{S}}}=\left(1+Q^{2}\right)$

(31)

$L_{P}=L_{S}{\frac {1+Q^{2}}{Q^{2}}}=L_{S}\left(1+{\frac {1}{Q^{2}}}\right)\approx L_{S}$

(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, $R_{S}$ to a load resistance, $R_{L}$ using a lossless matching network:

Case 1: $R_{S}<R_{L}$

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:

$R_{L}^{\prime }={\frac {R_{L}}{1+Q^{2}}}$

(33)

Where $Q=\omega R_{L}C_{1}$. The capacitor $C_{1}$ also gets transformed into:

$C_{1}^{\prime }=C_{1}\left(1+{\frac {1}{Q^{2}}}\right)\approx C_{1}$

(34)

Thus, to keep the impedance seen by the source real, we can cancel out the capacitance $C_{1}^{\prime }$ by the inductance $L_{1}$:

$L_{1}={\frac {1}{\omega ^{2}C_{1}^{\prime }}}$

(35)

If we set $R_{S}=R_{L}^{\prime }$ and define $m={\tfrac {R_{L}}{R_{S}}}$ for $R_{S}<R_{L}$, we can write eq. 33 as

$Q={\sqrt {m-1}}=\omega R_{L}C_{1}$

(36)

We can then solve for the values of $C_{1}$ and $L_{1}$ for a certain frequency $\omega _{0}$ as

$C_{1}={\frac {\sqrt {m-1}}{\omega _{0}R_{L}}}$

(37)

$L_{1}={\frac {1}{\omega _{0}^{2}C_{1}^{\prime }}}$

(38)