How are the impedances of a branchline coupler calculated?

The branch-line coupler impedances are determined by the desired coupling ratio and port impedance. For a general coupling ratio C (in linear voltage ratio), with port impedance Z_0: Series arms (through): Z_series = Z_0 / sqrt(1 + C^2). Shunt arms (branch): Z_shunt = Z_0 * C / sqrt(1 + C^2). For the standard 3 dB coupler (equal split, C = 1): Z_series = Z_0 / sqrt(2) = 35.36 ohms for Z_0 = 50 ohms. Z_shunt = Z_0 = 50 ohms. For unequal split designs, different coupling values produce different impedance pairs. Examples for Z_0 = 50 ohms: 3 dB split: Z_series = 35.4 ohms, Z_shunt = 50.0 ohms. 6 dB split: Z_series = 22.4 ohms, Z_shunt = 44.7 ohms. 10 dB split: Z_series = 15.4 ohms, Z_shunt = 49.5 ohms. Each arm is exactly lambda/4 long at the center frequency. The physical length in microstrip is: L = lambda_eff/4 = c/(4*f*sqrt(epsilon_eff)). The line widths are determined by the substrate permittivity and thickness to achieve the required impedances. Practical consideration: the 35.36-ohm line in the series arm is wider than the 50-ohm line, and the T-junctions at the corners introduce parasitic effects (junction capacitance) that must be compensated by shortening the arms slightly or adding chamfers (mitering) at the corners.

How can branchline coupler bandwidth be extended?

The single-section branchline coupler has inherently narrow bandwidth (10-20%) because the quarter-wave arms are frequency-dependent. Several techniques extend the bandwidth. Multi-section branchline: cascading two or more branchline sections with optimized impedances creates a broadband coupler. A two-section design achieves approximately 30-40% bandwidth. Three sections can reach 50%+. The impedances of each section are synthesized using Chebyshev or maximally flat (binomial) matching theory. The trade-off is increased size (each section adds lambda/4 length). Coupled-line hybrid: replacing the series arms with coupled-line sections introduces additional degrees of freedom. Tighter coupling and broader bandwidth are possible. Used in stripline implementations where broadside coupling is achievable. Lumped-element branchline: replacing the quarter-wave lines with equivalent lumped LC pi-networks or T-networks. Dramatically reduces size (especially below 2 GHz). Bandwidth similar to single-section distributed (10-20%) but can be extended with multiple sections. Component Q limits performance. Defected ground structures (DGS): etching patterns in the ground plane beneath the branchline modifies the effective impedance and electrical length, enabling size reduction (30-50%) with maintained or slightly improved bandwidth. Wideband branchline with stubs: adding open or short-circuited stubs at the corners of the ring creates additional resonances that extend the bandwidth to 40-60% in a single layer.

What are the main applications of branchline couplers?

The branchline coupler is one of the most versatile passive components in RF design. Balanced amplifiers: the primary application. Input branchline splits the signal into two paths with 90-degree phase offset feeding identical amplifiers. Output branchline recombines. Provides excellent input/output VSWR and graceful degradation. The branchline bandwidth typically limits the balanced amplifier bandwidth. Single-sideband (SSB) and image-reject mixers: two mixers fed in quadrature (via branchline at the LO or RF port) with IF outputs combined through a 90-degree hybrid. The quadrature relationship causes the image frequency to cancel (15-25 dB image rejection typical, limited by amplitude/phase balance of the hybrids). Butler matrix beam-forming networks: branchline couplers are the core building blocks of Butler matrices that create orthogonal beams in phased array antennas. A 4x4 Butler matrix uses four branchline couplers and two phase shifters. Circularly polarized antenna feeds: feeding two orthogonal antenna elements (or two ports of a dual-polarized patch) through a branchline creates circular polarization. Port 2 feeds one element, Port 3 feeds the orthogonal element with 90-degree phase difference. RHCP or LHCP is selected by choosing the input port. Power monitoring: asymmetric coupling ratios (6, 10, 20 dB) create directional couplers for power monitoring while maintaining the quadrature phase property. Duplexers and switches: combined with reflective elements (PIN diodes, varactors) at the coupled and direct ports, branchline couplers create reflective-type phase shifters and switches.

Passive / Coupler

Branchline Coupler

/BRANCH-lyn KUP-ler/

Four-port 90° quadrature hybrid using λ/4 transmission line sections in a rectangular ring. Series arms: Z₀/√2 (35.4 Ω for 50 Ω). Shunt arms: Z₀. Port 2: −3 dB, 0°. Port 3: −3 dB, −90°. Port 4: isolated. Bandwidth 10–20% (single-section), 40%+ (multi-section). Foundation for balanced amplifiers, SSB mixers, and Butler matrices.

Type: 90° hybrid

Z_series: Z₀/√2

BW: 10–20% (1-sect.)

Understanding the Branchline Coupler

The branchline coupler is arguably the most important planar passive component in microwave engineering. Its ability to split a signal into two equal-amplitude outputs with precisely 90° phase difference underpins balanced amplifier design, image-reject mixer topologies, circular polarization feed networks, and Butler matrix beam-forming.

While conceptually simple (four quarter-wave lines in a ring), practical branchline design demands careful attention to junction parasitics, line dispersion, substrate effects, and bandwidth limitations. Multi-section and lumped-element variants address these constraints for different frequency ranges and integration requirements.

Design Equations

3 dB Coupler (equal split):
Z_series = Z₀/√2 = 35.36 Ω (50 Ω system)
Z_shunt = Z₀ = 50 Ω
Arm length = λ_eff/4 = c/(4f√ε_eff)

General Coupling (voltage ratio C):
Z_series = Z₀/√(1 + C²)
Z_shunt = Z₀·C/√(1 + C²)

Port Outputs (ideal, at center freq.):
S₂₁ = −j/√2 (−3 dB, −90°)
S₃₁ = −1/√2 (−3 dB, −180°)

Coupling Ratio Design Table

Coupling (dB)	C (linear)	Z_series (Ω)	Z_shunt (Ω)	Application
3	1.000	35.4	50.0	Balanced amp, SSB mixer
6	0.500	44.7	22.4	Unequal power split
10	0.316	47.6	15.1	Power monitoring
20	0.100	49.8	5.0	Signal sampling

Bandwidth Extension Methods

Method	Bandwidth	Size Impact	Complexity	Best For
Single-section	10–20%	Baseline	Low	Narrowband, simple
2-section cascade	30–40%	2×	Moderate	Broadband balanced PA
3-section cascade	50%+	3×	High	Wideband systems
Lumped-element	10–20%	80% smaller	Moderate	<2 GHz, RFIC
Stub-loaded	40–60%	1.3×	Moderate	Single-layer broadband

Common Questions

Frequently Asked Questions

Impedance calculation?

For 3 dB: Z_series = Z₀/√2 (35.4 Ω), Z_shunt = Z₀. General coupling C: Z_series = Z₀/√(1+C²). Each arm is λ/4 at center frequency. Corner mitering compensates junction capacitance.

Bandwidth extension?

Multi-section cascading (2–3 sections for 30–50%+ BW) with Chebyshev impedance synthesis. Lumped-element for compact <2 GHz designs. Stub-loaded rings for single-layer broadband (40–60%). DGS for 30–50% size reduction.

Applications?

Balanced amplifiers (VSWR improvement), SSB/image-reject mixers (quadrature LO feed), Butler matrices (phased array beam-forming), CP antenna feeds (orthogonal 90° excitation), reflective phase shifters, and power monitoring.

Related Terms

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