Tapped Inductor Matching 1 comment

Tapped Inductor Matching

The impedance transformation ratio(R_{in}/R_L) and Q-factor/matching bandwidth of the circuit can be set independently. A tapped inductor matching circuit is shown in Fig.1

Fig.1 Tapped inductor matching circuit

From Fig.1, Q-factor of L_2-R_L branch is given by,

(1)   \begin{equation*} Q_2 = \frac{R_L}{\omega_o L_2} \end{equation*}

At resonant frequency the circuit is redrawn, by parallel-to-series transformation, as shown in Fig.1(b).

(2)   \begin{equation*} R_{L,s} = {R_L \over 1+ Q_2^2} \end{equation*}

(3)   \begin{equation*} L_{2,s}={L_2 \over 1+\frac{1}{Q_2^2}} \end{equation*}

(4)   \begin{equation*} L_{eq}=L_1 + L_{2,s} = {{L_1(1+Q_2^2)+L_2 Q_2^2}\over{1+Q_2^2}} \end{equation*}

(5)   \begin{equation*} Q={\omega_o L_{eq}\over R_{L,s}} = {{\omega_o\left(L_1(1+Q_2^2)+L_2 Q_2^2\right)}\over{R_L}} \end{equation*}

From Eq.(5) and Eq(1)

(6)   \begin{equation*}  Q={{L_1(1+Q_2^2)+L_2 Q_2^2}\over L_2 Q_2} \end{equation*}

If Q_2^2 >> 1,

(7)   \begin{equation*} Q = Q_2\left( {L_1 + L_2 \over L_2 }\right) \end{equation*}

Fig.1(c) illustrates the parallel R-L-C resonant circuit form of tapped inductor matching circuit.

(8)   \begin{equation*} Q = \omega_o C R_{in} \end{equation*}

(9)   \begin{equation*} R_{in } = R_{L,p} = R_{L,s}(1+Q^2) =R_L{(1+Q^2) \over (1+Q_2^2)} \end{equation*}

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

(10)   \begin{equation*} {R_{in} \over R_L} \approx {Q^2 \over Q_2^2} = \left(\frac{L_1+L_2}{L_2}\right)^2 \end{equation*}

When Q-factors are large, from Eq.(7), we can conclude that the impedance transformation ratio R_{in}/R_L is a function of inductors tap ratio. Hence the name tapped inductor matching for this matching circuit.

Design Procedure

  1. Calculate the value of Q_2 using Eq.(9).

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

  2. Calculate L_2 using Eq.(1)

        \begin{equation*} L_2 = {R_L \over \omega_o Q_2} \end{equation*}

  3. Find the value of C using Eq.(8)

        \begin{equation*} C={Q \over \omega_o R_{in}} \end{equation*}

  4. Calculate L_1 using Eq.(6)

        \begin{equation*} L_1 = {QQ_2-Q_2^2 \over 1+Q_2^2} L_2 \end{equation*}

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