Double-Tapped Resonator Matching

Description

The double-tapped resonator uses a shunt-series-shunt arrangement of two inductors and two capacitors to achieve impedance matching at a target frequency. A user-specified series inductor L2 (Ltap) forms part of a resonant tank with C1, giving an extra degree of freedom over the Tapped-C and Tapped-L networks. The loaded Q controls the bandwidth.

When to Use

RF frequencies.
Narrow bandwidth.
An fixed inductive feature that must be embedded into the matching network.

Design Equations

Auxiliary Q Factor

\[Q_2 = \sqrt{\frac{R_L}{R_S}(Q^2 + 1) - 1}\]

Component Values

\[L_1 = \frac{R_S}{\omega_0\, Q}\]

\[L_{eq} = \frac{L_1\, Q^2}{1 + Q^2} + L_2\]

\[C_{eq} = \frac{1}{\omega_0^2\, L_{eq}}\]

\[C_2 = \frac{Q_2}{\omega_0\, R_L}\]

\[C_2' = \frac{C_2\,(1 + Q_2^2)}{Q_2^2}\]

\[C_1 = \frac{C_{eq}\, C_2'}{C_2' - C_{eq}}\]

Minimum Q Constraint

\[Q > Q_{\min} = \sqrt{\frac{\max(R_S, R_L)}{\min(R_S, R_L)} - 1}\]

Values of Q below Qmin make Q2 imaginary and the design is invalid.

Parameters

Parameter	Description
Z0	Source impedance RS (Ω)
ZL	Load impedance (Ω)
Frequency	Matching frequency (Hz)
Q	Loaded Q factor (Q > Qmin)
Ltap	User-specified series inductor L2 (nH)

Limitations

Narrowband.
Only real-to-real matching.
The value of Ltap directly affects C1 through the resonator equations; very large Ltap values may push C1 to impractical or negative values.