Double-Tapped Resonator Matching | Qucs-S S-parameter Viewer Documentation

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# 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.

## See Also

- RF Circuit Synthesis Tools
- Tapped-C Transformer Matching
- Tapped-L Transformer Matching