Tapped Capacitor Matching


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

Tapped capacitor matching

Figure 1. Tapped capacitor impedance matching network

STEP-1
The Q-factor(Q_2) of parallel R_LC_2 branch is,

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

At resonant frequency(\omega_o), the parallel R_LC_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_1C_{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_1C_{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*}

When Q_2^2>>1,

(7)   \begin{equation*}Q \approx Q_2 \left({C_1 + C_2 \over C_1} \right) \end{equation*}

 

STEP-3

To present the circuit in parallel resonant form, series to parallel transformation is performed. The equivalent circuit is shown in Figure 1(c).

(8)   \begin{equation*}Q=\omega_o R_{in}C_t={R_{in}\over \omega_o L}\end{equation*}

(9)   \begin{equation*}R_{in}=R_{Lp}^{'}=R_L^{'}(1+Q^2)=R_L\left( 1+Q^2 \over {1+Q_2^2}\right)\end{equation*}

(10)   \begin{equation*}Q_2 = \sqrt{{R_L\over R_{in}}(1+Q^2)-1}\end{equation*}

(11)   \begin{equation*}C_t = \frac{C_{eq}}{1+{1\over Q^2}}\end{equation*}

When Q^2>>1 and Q_2^2>>1, from Eq.(7) and Eq.(9),

(12)   \begin{equation*} \frac{R_{in}}{R_L} \approx {\left(1 + \frac{C_2}{C_1}\right)}^2 \end{equation*}

Therefore the impedance transformation ratio is mainly a function of capacitance tap ratio. Hence tapped-capacitor matching network.

DESIGN PROCEDURE

  • Find Q of the circuit, from the desired bandwidth, Q=\frac{\omega_o}{BW}
  • Determine L and C_t from Eq.(8)
  • Find Q_2 from Eq.(10)
  • Knowing Q_2, now find the value of C_2 from Eq.(1)
  • Finally find the value of C_1 using Eq.(4)

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