Tapped-L Transformer Matching | Qucs-S S-parameter Viewer Documentation

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# Tapped-L Transformer Matching

## Description

The Tapped-L transformer uses a shunt-series-shunt arrangement of two inductors and one capacitor to achieve impedance matching at a target frequency. It is the dual of the Tapped-C: the series element is always an inductor (L1), and the single capacitor C is split across the two shunt arms together with a second inductor L2. The loaded Q controls the bandwidth.

## When to Use

- RF frequencies.

- Narrow bandwidth.

- Small area to be implemented.

## Design Equations

When RL > RS the roles of the two ports are internally swapped before computing component values, and C and L2 are placed on opposite sides of the network (see Topology section).

### Auxiliary Q Factor

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

### Component Values

\[C = \frac{Q}{\omega_0\, R_S}\]
\[L_2 = \frac{R_L}{Q_2\, \omega_0}\]
\[L_1 = \frac{L_2\,(Q\,Q_2 - Q_2^2)}{Q_2^2 + 1}\]

### 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)

## Tapped-C vs Tapped-L

The Tapped-C and Tapped-L are duals of each other. Both use a shunt-series-shunt Pi arrangement with one reactive element type forming the series arm and the other split across the two shunt arms:

| Tapped-C | Tapped-L
--- | --- | ---
Series element | C1 (capacitor) | L1 (inductor)
Shunt elements | L and C2 | C and L2
Preferred use | Tank circuits, oscillators |

## Limitations

- Q constraint must be given -> Narrowband.

- Only real-to-real matching.

## See Also

- RF Circuit Synthesis Tools
- Tapped-C Transformer Matching
- Double-Tapped Resonator Matching