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

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# 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

## See Also

- RF Circuit Synthesis Tools