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.