Pi Matching | Qucs-S S-parameter Viewer Documentation

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# Pi Matching

## Description

Pi matching uses a shunt-series-shunt lumped-element arrangement to achieve impedance matching at a target frequency. Like the Tee network, the loaded Q can be freely chosen, giving direct control over bandwidth. Four LP/HP topology combinations are available for each Q value.

## When to Use

- RF frequencies.

- Narrow bandwidth.

- Small area to be implemented.

- A specific filtering characteristic (low-pass or high-pass) is desired.

## Design Equations

The network is decomposed into two back-to-back L-sections meeting at a real virtual resistance Rv.

### Virtual Intermediate Resistance

\[R_v = \frac{\max(R_S,\, R_L)}{Q^2 + 1}\]

Rv is always smaller than both RS and RL.

### Source-side L-section (RS → Rv)

Low-pass solution:

\[X_{1a} = \sqrt{R_v (R_S - R_v)}\]
\[B_1 = \frac{\sqrt{(R_S - R_v)/R_v}}{R_S}\]

High-pass solution: negate both X₁ₐ and B₁.

### Load-side L-section (Rv → ZL)

Low-pass solution:

\[B_2 = \frac{X_L + \sqrt{R_L/R_v}\,\sqrt{R_L^2 + X_L^2 - R_v R_L}}{R_L^2 + X_L^2}\]
\[X_{1b} = \frac{1}{B_2} + \frac{X_L R_v}{R_L} - \frac{R_v}{B_2 R_L}\]

High-pass solution: negate the square-root term in B₂, then recompute X₁ᵦ.

### Combined Series Reactance

\[X_1 = X_{1a} + X_{1b}\]

### Component Values

Condition | Element | Formula
--- | --- | ---
X > 0 | Inductor | L = X / ω₀
X < 0 | Capacitor | C = −1 / (ω₀ X)
B > 0 | Capacitor | C = B / ω₀
B < 0 | Inductor | L = −1 / (ω₀ B)

### 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 Rv larger than the smaller termination and the design is invalid.

## Parameters

Parameter | Description
--- | ---
Z0 | Source impedance RS (Ω)
ZL | Load impedance (Ω), may be complex
Frequency | Matching frequency (Hz)
Q | Loaded Q factor (Q > Qmin)
Network type | LP-LP, LP-HP, HP-LP, or HP-HP

## Network Type (LP/HP Mask)

Each of the two L-section halves can independently be low-pass or high-pass, giving four topologies for any given Q:

Mask | Source side | Load side | Characteristic
--- | --- | --- | ---
LP-LP | Low-pass | Low-pass | Shunt caps, series inductor
LP-HP | Low-pass | High-pass | Mixed
HP-LP | High-pass | Low-pass | Mixed
HP-HP | High-pass | High-pass | Shunt inductors, series cap

## Tee and Pi matching

The Tee and Pi networks are duals of each other and share the same Q-based design procedure, but with opposite roles for Rv:

| Tee | Pi
--- | --- | ---
Rv relative to terminations | Rv > RS and RL | Rv < RS and RL
Topology | Series-shunt-series | Shunt-series-shunt
Favours | Low-impedance terminations | High-impedance terminations

## Limitations

- Q constraint must be given -> Narrowband.

## Reference

Vizmuller, P. “RF Design Guide: Systems, Circuits and Equations.” Artech House, 1995.

## See Also

- RF Circuit Synthesis Tools
- Tee Matching