Tapped Capacitor Matching

The impedance transformation ratio ( $R_{in}/R_L$ ) and bandwidth/Q-factor of the circuit can be set independently.

Figure 1. Tapped capacitor impedance matching network

STEP-1
The Q-factor( $Q_2$ ) of parallel $R_L$ – $C_2$ branch is,

(1) $\begin{equation*} Q_2=\omega_o C_2 R_L \end{equation*}$

At resonant frequency( $\omega_o$ ), the parallel $R_L$ – $C_2$ branch can be represented with series equivalent $R_L^{'}$ – $C_{2s}$ as shown if Figure 1(b)

STEP-2

After parallel to series transformation,

(2) $\begin{equation*}{R_L}^{'} = {R_L\over 1+Q_2^2}\end{equation*}$

(3) $\begin{equation*}C_{2s} = C_2\left(1+{1\over Q_2^2}\right)\end{equation*}$

(4) $\begin{equation*} {1\over C_{eq}}={1\over C_1}+{1\over C_{2}\left(1+{1\over Q_2^2}\right)} \end{equation*}$

The total capacitance in the branch ( $C_1$ – $C_{2s}$ -R_{Lp}^{‘}) is

(5) $\begin{eqnarray*} C_{eq} &=& {C_1C_2(1+Q_2^2) \over {C_1Q_2^2+C_2(1+Q_2^2)}}\\ &\approx& {C_1C_2\over {C_1+C_2}}{(1+Q_2^2)\over Q_2^2} \mbox{, if } Q_2^2>>1 \end{eqnarray*}$

The Q-factor of the branch ( $C_1$ – $C_{2s}$ -R_{Lp}^{‘}) is

(6) $\begin{eqnarray*} Q &=& \frac{1}{\omega_o R_L^{'} C_{eq}} = {1 \over \omega_o {Q_2 /\omega_o C_2 \over 1+Q_2^2} C_{eq} } \\ &=& \frac{C_1Q_2^2+C_2(1+Q_2^2)}{Q_2C_1} \end{eqnarray*}$