pi-matching


A pi-match network is shown in Figure-1. Two back-to-back connected l-match circuits form a \pi-match network. The additional element in \pi-match, compared with L-macth, allows independently setting impedance transformation ratio(R_L/R_{in}) and Q-factor of the circuit.

The analysis of \pi-match circuit shown in Figure-1 can be simplified by redrawing the circuit into two L-match sections as shown in Figure-2.

If v_I is the voltage at the input of L-match circuit, and R_I is the impedance seen looking into each L-match circuit, then current flowing through inductor of each L-match circuit is I_L={v_L \over R_I}.

The Q-factor looking into each parallel R-X branch is given by

(1)   \begin{equation*}  Q_1 = {R_{in} \over X_1} \mbox{\quad\quad\quad\quad\quad\quad} Q_2={R_L \over X_2 } \end{equation*}

By series-to-parallel transformation as shown in From Fig.3,

(2)   \begin{equation*} Q_1 = {X_A \over R_I } \mbox{\quad\quad\quad\quad\quad\quad} Q_2 = {X_B \over R_I} \end{equation*}

(3)   \begin{equation*}  R_{in,s} = R_I = {R_{in}\over (1+Q_1^2)} \mbox{\quad\quad\quad\quad\quad\quad} R_{L,s} = R_I = {R_{L}\over(1+Q_2^2)} \end{equation*}

(4)   \begin{equation*} X_{1,s} = {X_1\over 1+{1\over Q_1^2}} \mbox{\quad\quad\quad\quad\quad\quad } X_{2,s} = {X_2\over 1+{1 \over Q_2^2}} \end{equation*}

 

Rearranging Eq.(3)

(5)   \begin{equation*}  Q_1 = \sqrt{\frac{R_{in}}{R_I}-1} \mbox{\quad\quad\quad\quad\quad\quad} Q_2 = \sqrt{\frac{R_{L}}{R_I}-1} \end{equation*}

The total Q of the circuit is given by

(6)   \begin{equation*}  Q = Q_1 + Q_2 = \sqrt{\frac{R_{in}}{R_I}-1} + \sqrt{\frac{R_{L}}{R_I}-1} \end{equation*}

At resonant frequency, |X_A| = |X_{1,s}| and |X_B| = |X_{2,s}|. Therefore,

(7)   \begin{equation*} X_A = {X_1 \over 1+{1\over Q_1^2}} \mbox{ \quad\quad\quad\quad\quad\quad } X_B = {X \over 1+Q_2^2} \end{equation*}

This equation can be used as a sanity check for the calculated T-matching network element values.

DESIGN PROCEDURE

  1. Find R_I using Eq.(6) from given R_L, R_{in} and Q.

        \begin{equation*} Q = Q_1 + Q_2 = \sqrt{\frac{R_{in}}{R_I}-1} + \sqrt{\frac{R_{L}}{R_I}-1} \end{equation*}

  2. Calculate Q_1 and Q_2 using Eq.(5)

        \begin{equation*} Q_1 = \sqrt{\frac{R_{in}}{R_I}-1} \mbox{\quad\quad\quad\quad\quad\quad } Q_2 = \sqrt{\frac{R_{L}}{R_I}-1} \end{equation*}

  3. Lowpass pi-match
    Find X_1 and X_2 using Eq.(1). X_1 = {1\over \omega_o C_1} and X_2 = {1\over \omega_o C_2}
    Find X_A and X_B using Eq.(??). X_A=\omega_o L_A \mbox{ and } X_A=\omega_o L_B
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  4. Highpass pi-match
    Find X_1 and X_2 using Eq.(1). X_1=\omega_o L_1 and X_2 = \omega_o L_2
    Find X_A and X_B using Eq.(??) X_A={1 \over \omega_o C_A} \mbox{ and } X_B={1 \over \omega_o C_B}

 

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