Single-Stub Matching | Qucs-S S-parameter Viewer Documentation

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# 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

## See Also

- RF Circuit Synthesis Tools