Branch-Line Coupler (Quadrature Hybrid) | Qucs-S S-parameter Viewer Documentation

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# Branch-Line Coupler (Quadrature Hybrid)

## Overview

The branch-line coupler is a four-port network that provides 90° phase difference (quadrature) between output ports. Also called a quadrature hybrid or 3 dB coupler, it’s essential for applications requiring phase quadrature.

## Topology

    Port 1 ─────────[ZA]──────── Port 2
    (Input)    │           |    (Through, 0°)
               │           │
             [ZB]        [ZB]
               │           │
               │           │
    Port 4 ─────────[ZA]──────── Port 3
        (Isolated)            (Coupled, -90°)

## Design Equations

### Power Split Ratio

K = P₂/P₃

### Transmission Line Impedances

\[Z_A = \frac{Z_0}{\sqrt{2}} \approx 0.707 \cdot Z_0 ZB = Z₀ R = Z₀\]

For Z₀ = 50 Ω:

Parameter | Value
--- | ---
ZA | 35.4 Ω
ZB | 50 Ω
R | 50 Ω

## Advantages

1. Quadrature outputs

2. Broader BW: with respect to the single box Branch-Line coupler

## Limitations

1. Narrowband: 10-20% fractional BW (single octave max)

2. Large size: Four λ/4 lines (larger than Wilkinson)

3. Low ZA for large K: ZA → 0 as K → ∞ (hard to fabricate)

## Comparison with Wilkinson

Feature | Branch-Line | Wilkinson
--- | --- | ---
Phase | 90° (quadrature) | 0° (in-phase)
Outputs | 2 | 2
Bandwidth | 10-20% | 20-40%
Size | Larger (4 lines) | Smaller (2 lines)
Isolation | 15-25 dB | 20-30 dB
Application | IQ, mixers | General power split

## References

[1] Pozar, D. M. (2012). Microwave Engineering (4th ed.), Section 7.5, pp. 347-352. Wiley.

## See Also

- RF Circuit Synthesis Tools