T Matching


Fig.1. T-match circuit

A T-matching circuit can be used either for upward or downward matching. It is a dual of \pi-match circuit. A basic T-match circuit is shown in Fig.1. Like in \pi-match, T-match also allows to set impedance transformation ratio and Q-factor/matching BW of circuit independently.

 

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

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

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

 

 

Fig.2. T-match circuit decomposed to two L-match circuits

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

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

 

 

 

 

Fig.3. T-match circuit after series to parallel transformation

Rearranging Eq.(3)

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

The total Q of the circuit is given by

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

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

(7)   \begin{equation*} X_A = X_1(1+{1\over Q_1^2}) \mbox{ \quad\quad\quad\quad\quad\quad } X_B = X(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.
  2. Calculate Q_1 and Q_2 using Eq.(5)
  3. Find X_1 and X_2 using Eq.(1)
  4. Find X_A and X_B using Eq.(2)

 

 

 

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