Tee Matching

Description

Tee matching uses a series-shunt-series lumped-element arrangement to achieve impedance matching at a target frequency. The loaded Q can be freely chosen, giving the designer 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

\[R_v = (Q^2 + 1)\,\min(R_S,\, R_L)\]

Rv is always greater than both RS and RL.

Source-side L-section (RS → Rv)

Low-pass solution:

\[B_{1a} = \frac{\sqrt{R_v/R_S}\,\sqrt{R_v^2 - R_S R_v}}{R_v^2}\]

\[X_1 = \frac{1}{B_{1a}} + \frac{R_S}{R_v} - \frac{R_S}{B_{1a} R_v}\]

High-pass solution: negate both B₁ₐ and X₁.

Load-side L-section (Rv → ZL)

Low-pass solution:

\[X_2 = \sqrt{R_L(R_v - R_L)} - X_L\]

\[B_{1b} = \frac{\sqrt{(R_v - R_L)/R_L}}{R_v}\]

High-pass solution: negate both B₁ᵦ and X₂, and subtract XL with the opposite sign.

Combined Centre Susceptance

\[B_1 = B_{1a} + B_{1b}\]

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 imaginary 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	All shunt caps, series inductors
LP-HP	Low-pass	High-pass	Mixed
HP-LP	High-pass	Low-pass	Mixed
HP-HP	High-pass	High-pass	All shunt inductors, series caps

Limitations

Q constraint must be given -> Narrowband.

Reference

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