Branch-Line Coupler
Four-port 90° hybrid using quarter-wave transmission line branches
Definition & Operating Principle
A branch-line coupler (also called a quadrature hybrid or 90° hybrid) is a four-port passive microwave network that splits an input signal into two output signals of equal amplitude with a 90-degree phase difference. The structure consists of four quarter-wavelength transmission line sections arranged in a rectangular ring. Signal enters port 1, divides through two parallel paths around the ring, and recombines at ports 2 (through) and 3 (coupled) with constructive interference while destructively canceling at port 4 (isolated).
The coupler operates on the principle of superposition: the signal traveling clockwise and counterclockwise around the ring arrives in phase at ports 2 and 3 (with a 90° offset between them) and out of phase at port 4. At the center design frequency, where each arm is exactly λ/4 long, the isolation at port 4 is theoretically infinite and the power split is exactly 3 dB per output. In practice, fabrication tolerances and dielectric variations limit isolation to 20-30 dB and amplitude balance to 0.3-0.5 dB at the design frequency.
Key Formulas
Series Arm Impedance (3 dB coupler):
Zseries = Z0 / √2 = 50 / √2 = 35.36 Ω
Shunt Arm Impedance:
Zshunt = Z0 = 50 Ω
Arm Length (all four):
L = λg / 4
where λg is the guided wavelength in the transmission line medium at f0
General Coupling (unequal split):
Zseries = Z0 / √(1 + k²), Zshunt = Z0 / k
where k = voltage coupling coefficient (k = 1 for 3 dB equal split)
Hybrid Coupler Comparison
| Parameter | Branch-Line | Lange | Rat-Race | Wilkinson |
|---|---|---|---|---|
| Output Phase Difference | 90° | 90° | 180° | 0° |
| Typical Bandwidth | 10-15% | 40-60% | 20-30% | 20-30% |
| Physical Size | λ/4 ring | Compact | 3λ/4 ring | λ/4 arms |
| Coupling Range | 3-10 dB | 3-6 dB | 3 dB only | 3 dB (equal) |
| Fabrication Complexity | Low | High (bonds) | Low | Low |
| Isolation (typical) | 20-30 dB | 20-25 dB | 20-30 dB | 20-30 dB |
| Common Application | Mixer LO feed | Wideband I/Q | Balanced amp | Power divider |
Practical Application
In a Ka-band satellite receiver front end operating at 20 GHz, a branch-line coupler feeds the local oscillator signal to a balanced mixer pair. The coupler splits the 18.5 GHz LO into two equal-amplitude signals with 90° phase offset, driving the two mixer diodes in quadrature. This arrangement cancels LO-to-RF leakage and suppresses even-order intermodulation products, improving spurious-free dynamic range by 15-20 dB compared to a single-ended mixer. The coupler is implemented in microstrip on a 5-mil Rogers RT/duroid 5880 substrate, where the quarter-wave arms measure approximately 2.1 mm at 20 GHz. The 35.36-ohm series arms require 0.38 mm trace width while the 50-ohm shunt arms use 0.15 mm width, both within standard PCB fabrication capability.
Frequently Asked Questions
Why is the branch-line coupler bandwidth limited?
The branch-line coupler relies on quarter-wavelength transmission line sections that are resonant at a single center frequency. Away from that frequency, the electrical length deviates from 90°, degrading amplitude balance and phase quadrature. A standard single-section coupler maintains less than 1 dB amplitude imbalance over roughly 10-15% fractional bandwidth. Multi-section designs extend bandwidth to 30-40% at the cost of increased physical size.
What are the port impedances in a standard 3 dB branch-line coupler?
For a 3 dB equal-split coupler in a 50-ohm system, the series arms have Z0/√2 = 35.36 ohms and the shunt arms have Z0 = 50 ohms. All four arms are one quarter wavelength long at the design frequency. These values ensure equal power division with exactly 90° phase difference between the two output ports.
How does a branch-line coupler compare to a Lange coupler?
The branch-line coupler uses quarter-wave transmission lines in a rectangular ring, while the Lange uses interdigitated edge-coupled lines. Branch-line is easier to fabricate but has narrower bandwidth (10-15% vs 40-60%) and larger footprint. Lange couplers require wire bonds between alternating fingers, increasing manufacturing complexity. For narrowband single-frequency applications like mixer LO feeds, branch-line is preferred for its simplicity.