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

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

Calculate the value of $Q_2$ using Eq.(9).
$\begin{equation*} Q_2 = \sqrt{{R_{in } \over R_L}(1+Q^2)-1} \end{equation*}$
Calculate $L_2$ using Eq.(1)
$\begin{equation*} L_2 = {R_L \over \omega_o Q_2} \end{equation*}$
Find the value of C using Eq.(8)
$\begin{equation*} C={Q \over \omega_o R_{in}} \end{equation*}$
Calculate $L_1$ using Eq.(6)
$\begin{equation*} L_1 = {QQ_2-Q_2^2 \over 1+Q_2^2} L_2 \end{equation*}$