Double-Stub Matching

Description

Double-stub matching uses two shunt stubs separated by a fixed transmission line length (typically λ/8). This configuration allows matching without requiring a variable-length main line, making it practical for tunable matching networks.

Design Equations

Load Admittance Transformation

\[G_L = \frac{R_L}{R_L^2 + X_L^2}, \quad B_L = -\frac{X_L}{R_L^2 + X_L^2}\]

Matchability Condition

The load can be matched if:

\[G_L \leq \frac{Y_0(1 + t^2)}{2t^2}\]

where $t = \tan(\beta d)$ and d = λ/8 (giving t = 1).

Stub Susceptances

\[B_1 = -B_L + \frac{Y_0 + \sqrt{(1+t^2)G_L Y_0 - G_L^2 t^2}}{t}\]

\[B_2 = \frac{Y_0\sqrt{(1+t^2)G_L Y_0 - G_L^2 t^2} + G_L Y_0}{G_L t}\]

Stub Lengths

For open-circuit stubs:

\[l_1 = \frac{\lambda}{2\pi}\arctan(B_1 Z_0), \quad l_2 = \frac{\lambda}{2\pi}\arctan(B_2 Z_0)\]

For short-circuit stubs:

\[l_1 = -\frac{\lambda}{2\pi}\arctan\left(\frac{1}{B_1 Z_0}\right), \quad l_2 = -\frac{\lambda}{2\pi}\arctan\left(\frac{1}{B_2 Z_0}\right)\]

Parameters

Parameter	Description
Z0	Characteristic impedance (Ω)
ZL	Load impedance (Ω)
Frequency	Matching frequency (Hz)
Spacing	Fixed at λ/8
Stub Type	Open or short-circuit
Implementation	Ideal TL or microstrip

Advantages

Adjustable matching: Both stubs can be tuned
Fixed spacing: No need to adjust main line length
Laboratory-friendly: Easy to implement with adjustable stubs
Broadband capability: Can be tuned across frequencies

Limitations

Forbidden Region

Loads with high conductance (low resistance) relative to Y0 cannot be matched:

\[G_L > \frac{Y_0(1 + t^2)}{2t^2}\]

For λ/8 spacing (t = 1), this becomes $G_L > Y_0$.

Stub Terminations

Open-Circuit Stubs

Common in microstrip
End-effect compensation needed
Avoid DC paths to ground

Short-Circuit Stubs

Require vias in microstrip
More predictable at lower frequencies
Better for high-power applications

Example

Match 100 + j50Ω to 50Ω at 3 GHz

Input data

Parameter	Value
Z0	50Ω
ZL	100 + j50Ω
frequency	1 GHz
Configuration	Open-circuit stubs

Results

Parameter	Value
Stub 1 length	59.6 mm
Line length	37.5 mm
Stub 2 length	52.8 mm

Circuit topology:

Port ── STUB1 ── TLIN(λ/8) ── STUB2 ── Load
        │                     │
       O/C                   O/C

Reference

Pozar, D. M. “Microwave Engineering”, 4th Edition, Wiley, 2011, pp. 241-245