Double-Box Branch-Line Coupler | Qucs-S S-parameter Viewer Documentation

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# Double-Box Branch-Line Coupler

## Overview

Enhanced quadrature hybrid using two cascaded branch-line sections providing wider bandwidth and better impedance transformation than single-box design. Maintains 90° phase difference while achieving 50-75% wider bandwidth through dual-box structure.

## Topology

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

## Design Equations

For equal split (K = 1):

Let $t = \sqrt{2}$, $r = 1$. The shunt and series arm impedances are:

\[Z_A = Z_D = \frac{Z_0}{\sqrt{2} - 1} = Z_0\,(\sqrt{2} + 1) \approx 2.414 \times Z_0\]
\[Z_B = \frac{Z_0}{\sqrt{2}} \approx 0.707 \times Z_0\]
\[R = Z_0 \quad \text{(isolation resistor)}\]

For Z₀ = 50 Ω:

Parameter | Value
--- | ---
Z_A = Z_D | 120.7 Ω
Z_B | 35.4 Ω
R | 50 Ω

## Advantages Over Single-Box

1. Wider bandwidth: 25–35% vs. 15–20% (50–75% improvement)

## Limitations

1. Larger size: 7 lines vs. 4 — approximately 2× the area of a single-box design

2. More complex layout

3. Higher impedance (Z_A, Z_D): 120.7 Ω results in narrow lines, challenging on thin substrates where fabrication tolerance is critical

4. Still narrowband relative to other broadband coupler topologies

## See Also

- RF Circuit Synthesis Tools