RF Power Dividers & Combiners

Power dividers split one signal into two or more outputs. Power combiners do the reverse: sum multiple signals into one output. The Wilkinson divider is the classic design because it achieves simultaneous matching, isolation, and lossless operation using only transmission line sections and a single resistor.

Wilkinson Power Divider

The equal-split Wilkinson divider uses two λ/4 transmission line arms of impedance \(Z_0\sqrt{2}\) (≈ 70.7 Ω in a 50 Ω system) and an isolation resistor \(R = 2Z_0\) between the two output ports:

Equal-split Wilkinson divider: two λ/4 arms at Z₀√2 ≈ 70.7 Ω, isolation resistor 2Z₀ = 100 Ω (for 50 Ω system).

\(Z_{arm} = Z_0\sqrt{2}, \quad R_{iso} = 2Z_0\)

Key properties: all three ports are simultaneously matched; the output ports are isolated from each other (signals applied to port 2 are absorbed by R and do not appear at port 3); the divider is lossless when port 2 and port 3 signals are in phase (the isolation resistor carries no current).

Unequal-Split Wilkinson

For a power split ratio \(K^2:1\) (port 2 gets \(K^2\) times more power than port 3):

\(Z_1 = Z_0\sqrt{\dfrac{1+K^2}{K^3}}, \quad Z_2 = Z_0\sqrt{K(1+K^2)}, \quad R_{iso} = Z_0\!\left(K + \dfrac{1}{K}\right)\)

For K=1 (equal split): \(Z_1 = Z_2 = Z_0\sqrt{2}\), \(R = 2Z_0\). For K=2 (6 dB more to port 2): \(Z_1 ≈ 35.4\) Ω, \(Z_2 ≈ 112\) Ω, \(R = 125\) Ω (in 50 Ω system).

N-Way Dividers

An N-way Wilkinson divider can be built by cascading 2-way sections or using a star junction with N arms. The N-way lumped design uses N λ/4 arms of impedance \(Z_0\sqrt{N}\) and isolation resistors \(NZ_0\) between adjacent output ports.

Hybrid Couplers

A 90° hybrid (branch-line coupler) is a four-port device where the two outputs are 90° apart in phase. It is used in balanced amplifiers and I/Q mixers. A 180° hybrid (rat-race or magic T) gives outputs that are either in-phase or anti-phase, used for sum-difference networks and balanced mixers.

Comparison of common RF power divider and hybrid coupler types.

Device	Phase split	Bandwidth	Use
Wilkinson divider	0° (in-phase)	Moderate (~20%)	Power splitting, combining
90° hybrid coupler	0°/90°	~20%	Balanced amps, I/Q mixers
180° rat-race	0°/180°	~20%	Balanced mixers, sum/difference
Lange coupler	0°/90° (3 dB)	Wide (octave)	Wideband balanced amps

Coherent Power Combining

When N amplifiers feed a combining network, the combined output power is:

\(P_{out} = \eta N P_{each}\)

where η is the combining efficiency (0.93–0.98 for corporate networks). The gain over a single amplifier is \(10\log_{10}(N) + 10\log_{10}(\eta)\) dB. If k amplifiers fail, coherent combining means power drops as \((N-k)^2/N^2\) of the full output — graceful degradation rather than total failure.

Physical Implementation

PCB microstrip: The most common implementation. λ/4 at 2.45 GHz on FR4 (εᵣ ≈ 4.4) is about 18–20 mm long.
Lumped element: Below ~1 GHz, LC equivalents replace transmission line sections, saving board space.
SMD Wilkinson: Available as pre-packaged components from manufacturers at fixed frequencies.

🔧 Related Calculators

Wilkinson Power Divider → RF Power Combining → Quarter-Wave Transformer →