Single-Stub Matching

Description

Single-stub matching uses one transmission line section followed by a shunt stub (open or short-circuited) to achieve impedance matching. This distributed-element approach is suitable for microwave frequencies.

When to Use

Microwave frequencies (> 1 GHz)
Distributed element implementation required
Moderate bandwidth needed (10-20%)

Design Equations

Transmission Line Distance

\[t = \frac{X_L + \sqrt{(R_L/Z_0)|Z_0^2 - R_L^2 + X_L^2|}}{R_L - Z_0}\]

\[d = \frac{\lambda}{2\pi}\arctan(t)\]

Stub Susceptance

\[B = \frac{R_L^2 t - (Z_0 - X_L t)(Z_0 t + X_L)}{Z_0(R_L^2 + (Z_0 t + X_L)^2)}\]

Stub Length

For open-circuit stub:

\[l_{stub} = -\frac{\lambda}{2\pi}\arctan(BZ_0)\]

For short-circuit stub:

\[l_{stub} = \frac{\lambda}{2\pi}\arctan\left(\frac{1}{BZ_0}\right)\]

where λ is the wavelength at the matching frequency.

Parameters

Parameter	Description
Z0	Characteristic impedance (Ω)
ZL	Load impedance (Ω)
Frequency	Matching frequency (Hz)
Stub Type	Open or short-circuit termination
Implementation	Ideal TL or microstrip

Stub Termination

Open-Circuit Stub

Easier to fabricate in microstrip (no vias nor shunt caps (1 nF or so) required)
End effect requires compensation
Avoid DC paths to ground

Short-Circuit Stub

Requires via to ground plane
No end-effect compensation needed
Better for narrowband matching

Limitations

Frequency-dependent (narrow to moderate bandwidth)
Requires adequate space for transmission line
Stub length can become impractical at low frequencies
Sensitive to fabrication tolerances

Example

Match 75Ω to 50Ω at 1 GHz

Input data

Parameter	Value
Z0	50Ω
ZL	75Ω
frequency	1 GHz
Configuration	Open-circuit stub

Results

Parameter	Value
Line distance (d)	42.3 mm
Stub length (l)	131 mm

Circuit topology:

Port ── TLIN(d) ──┬── Load (75Ω)
                  │
                STUB(l)
                  │
                 O/C

Reference

Pozar, D. M. “Microwave Engineering”, 4th Edition, Wiley, 2011, pp. 234-241