Design the 3-dB branch-line coupler (quadrature hybrid) that splits one RF signal into two outputs of equal amplitude and a 90° phase difference. Adjust the system impedance, operating frequency and substrate permittivity to see the series and shunt branch impedances and the quarter-wave line length update in real time.
Parameters
System impedance Z0
Ω
Reference impedance of the input/output ports. 50Ω is standard
Operating frequency f
GHz
Centre frequency for which the coupler is designed
Substrate permittivity εr
Relative permittivity of the PCB dielectric (FR-4 ≈ 4.4)
Results
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Series branch Z Za (Ω)
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Shunt branch Z Zb (Ω)
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λ/4 line length (mm)
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Coupled port (dB)
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Output phase difference (°)
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Effective permittivity εeff
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Branch-line coupler schematic — signal propagation animation
A signal entering the input port travels through the four quarter-wave lines arranged in a square and splits equally to the through and coupled ports. No signal appears at the fourth (isolated) port.
The series branches (top and bottom horizontal arms) are Z0/√2, the shunt branches (left and right vertical arms) are Z0. All four arms are one quarter of the guided wavelength (λ/4) long.
The guided wavelength λg is found from the speed of light c, the frequency f and the effective permittivity εeff. For microstrip the approximation εeff ≈ (εr+1)/2 is used.
The input divides equally into −3 dB at the through and coupled ports, with a 90° phase difference between the two outputs. The fourth port is isolated and carries no signal.
What is the Branch-Line Coupler Simulator?
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A "branch-line coupler" is a part that splits a signal into two, right? How is it different from the Wilkinson divider I learned earlier?
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Both do "one input, two equal outputs", but the output phase is the decisive difference. The Wilkinson gives two outputs that are in phase (0° difference). The branch-line coupler instead puts a precise 90° phase difference between the two outputs — that is why it is called a "90° hybrid" or a "quadrature hybrid". Another characteristic: the branch-line uses no isolation resistor at all. It is just four quarter-wave lines arranged in a square.
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Four quarter-wave lines arranged in a square... so the four corners become the ports?
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Exactly. The four corners of the square become the input port, the through port, the coupled port and the isolated port. The top and bottom horizontal arms (series branches) have a characteristic impedance of Z0/√2, and the left and right vertical arms (shunt branches) have Z0. For Z0 = 50Ω, the series branches are about 35.4Ω and the shunt branches 50Ω. All four are λ/4 long. Move the frequency slider f on the left and you will see that λ/4 line length change in the chart below.
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I see. So why does a 90° difference appear between the two outputs? It seems strange for such a symmetric square.
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The key is that "the number of lines from the input to each output differs". From the input port, the through port (the adjacent corner) is reached via one series branch, while the coupled port is reached via a series plus a shunt branch. Each λ/4 line adds 90° of phase as you pass through it, so the total phase at the two outputs differs by 90°. On top of that, the two paths heading toward the isolated port arrive exactly out of phase and cancel — so no signal appears at the isolated port. That is the isolation.
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If no signal appears at the isolated port, is the fourth port just unused? That feels like a waste.
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No, it has a real job. The basic move is to connect a matched Z0 termination to the fourth port. Then, even if a reflection occurs at an antenna or amplifier on the output side, that reflected wave is cleanly absorbed by the termination on the isolated port. That is exactly why a balanced amplifier is so robust. Alternatively, you can use the fourth port as a second "input", and then it becomes a circuit that takes the sum and difference of two inputs. One square works as a divider or a combiner depending on how you use it.
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One last thing. The coupled port shows −3dB. Is that a loss?
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That is not a "loss" but an "equal split". In a 3-dB hybrid, the input power divides exactly in half between the through port and the coupled port. Half is a power ratio of 0.5, which in decibels is 10·log10(0.5) ≈ −3.01dB. So if both the through and coupled ports read −3dB, that is proof the equal split is working as designed. A real circuit adds a little conductor and dielectric loss, but a good design gives only 0.1–0.5dB at microwave frequencies. So a measurement around −3.5dB is perfectly healthy.
Frequently Asked Questions
For a standard 3-dB (equal-split) 90° hybrid, set the characteristic impedance of the top and bottom horizontal arms (series branches) to Za = Z0/√2 and the left and right vertical arms (shunt branches) to Zb = Z0. For Z0 = 50Ω, the series branches are about 35.36Ω and the shunt branches 50Ω. All four arms are made one quarter-wavelength (λ/4) long at the operating frequency. This splits the input equally between the through and coupled ports with a 90° phase difference.
A branch-line coupler is built from four quarter-wave lines arranged in a square. The path from the input port to the through port and the path to the coupled port traverse a different number of lines, and therefore accumulate a different phase, designed so that the difference is exactly 90°. The through and coupled outputs have equal amplitude and are 90° apart in phase, which is why it is called a quadrature (90°) hybrid. At the fourth port the two paths arrive in anti-phase and cancel, so no signal appears.
Find the guided wavelength λg from the effective permittivity εeff, and one quarter of it is the line length: λg = c/(f·√εeff) (c is the speed of light, f the operating frequency), ℓ = λg/4. For microstrip, εeff ≈ (εr+1)/2 is a useful approximation. For example, on an εr=4.4 substrate (FR-4 equivalent) at 2.4GHz, εeff ≈ 2.7, λg ≈ 76mm, and the λ/4 line length is about 19mm. The higher the frequency and the larger the εr, the shorter the line.
In an ideal branch-line coupler, a signal entering the input port produces no output at all at the fourth port (the isolated port). In practice the basic approach is to connect a matched Z0 termination there. In a balanced amplifier, amplifiers are connected to the two outputs and any reflected waves are absorbed by the termination on the isolated port. Conversely, if the fourth port is used as a second input, the coupler becomes a sum-and-difference (combiner) circuit.
Real-World Applications
Balanced amplifiers: This is the most representative application of the branch-line coupler. The input is split into two by a hybrid, each half passes through an identical amplifier, and the outputs are recombined by another hybrid. Even if a reflection occurs at an amplifier, that reflected wave collects at the termination on the hybrid's isolated port and is absorbed, so it does not return to the input port. As a result, even when the individual amplifiers are poorly matched, the system as a whole achieves a good input/output VSWR.
I/Q (quadrature) modulation and demodulation: The 90° phase difference between the two outputs is at the heart of radios that handle I (in-phase) and Q (quadrature) components. Quadrature modulators and demodulators, image-reject mixers and single-sideband (SSB) circuits all use a branch-line coupler to split a signal into two paths shifted exactly 90° apart. The phase accuracy translates directly into image rejection ratio and EVM.
Antenna polarization control: A circularly polarized antenna must feed two orthogonal radiating elements with equal amplitude and a 90° phase difference. The branch-line coupler is ideal for this feed: by sending 90°-shifted signals into two adjacent edges of a patch antenna, it generates right-hand or left-hand circular polarization. It is widely used in GPS and satellite-communication antennas.
PCB implementation of microwave circuits: A branch-line coupler can be built entirely from microstrip or stripline and needs no extra components such as an isolation resistor. Since it is made simply by drawing four quarter-wave lines as a square on a printed circuit board, it is low cost and well suited to mass production. Estimating the series and shunt branch impedances and the λ/4 line length with this tool gives you a first approximation of the board layout right away.
Common Misconceptions and Pitfalls
The biggest misconception is assuming the branch-line coupler works over a wide frequency range. The basic branch-line coupler uses four λ/4 lines, so it only works correctly at the design frequency. When the line length deviates from λ/4, the split balance, the 90° phase difference and the isolation all degrade. In general, good performance is limited to a band of about ±10–20% around the centre frequency. If you need more than one octave of bandwidth, use a multi-section branch-line coupler with several cascaded stages, or a coupled-line coupler such as the Lange coupler. This tool targets the single-section basic form, so remember its values are design values at the centre frequency.
Next, assuming the effective permittivity εeff = (εr+1)/2 is an accurate value. The formula used in this tool is only a simple microstrip approximation. The real εeff varies with the ratio of line width to substrate thickness, the conductor thickness and the surface roughness, and is accurately found with the Hammerstad-Jensen equations or an electromagnetic simulator (method of moments / FEM). The error of the simple formula is a few to over ten percent, and that error feeds straight into λg and the line length. Moreover, in a branch-line coupler the series branch (a narrow line) and the shunt branch (a wider line) have different widths and therefore slightly different εeff. Always finalize each line dimension with a detailed calculation or simulation before prototyping.
Finally, the misconception that the junctions where the branches meet can be ignored as ideal points. In a real branch-line coupler, parasitic reactances arise at the T- and L-shaped junctions where the four lines cross. If you ignore this discontinuity and design with ideal line lengths, the centre frequency can shift and errors appear in the split balance and phase difference. At microwave frequencies, junction compensation (fine-tuning the line lengths or chamfering the corners) is essential, and the dimensions are ultimately refined with a full-wave electromagnetic simulation. Use the values from this tool as a starting point before junction compensation.
How to Use
Enter system characteristic impedance (Z0) between 25–75 Ω, typically 50 Ω for standard RF systems
Set center frequency (f) in GHz; coupler operates within ±10% of this frequency with minimum loss
Input substrate relative permittivity (εr); common values are 2.2 (FR-4), 3.38 (Rogers RO4003), or 10.2 (alumina)
Simulator calculates series branch impedance Za, shunt branch impedance Zb, and physical λ/4 transmission line length required for 3-dB coupling
Review coupled port isolation (dB), 90° quadrature phase shift, and effective permittivity εeff for layout
Worked Example
Design a 3-dB branch-line coupler at 2.4 GHz on FR-4 (εr=4.4) with Z0=50 Ω. Simulator outputs: Za=70.7 Ω (series arm), Zb=35.4 Ω (shunt arm), λ/4 line length=24.3 mm, coupled port magnitude=−3.0 dB at center frequency, phase difference=90.0°, εeff=3.06. Microstrip trace widths follow Za requiring 0.25 mm width; Zb requires 1.8 mm. Measured insertion loss typically 0.5 dB including connector losses on actual PCB.
Coupled port isolation improves with symmetry; asymmetric feeding reduces directivity by 6–10 dB in practice
Physical line length must account for εeff, not just εr; simulator-predicted 24.3 mm requires confirmation via ADS or HFSS for trace geometry effects
Operating bandwidth narrows at higher frequencies; 10 GHz designs need tight process control (±0.05 mm trace width tolerance) to maintain 90° phase over ±500 MHz