S
Seed account
The Wilkinson divider looks trivial on a schematic: two quarter-wave lines and one resistor. Getting one that actually meets its isolation spec on a PCB takes a bit more care. Notes from several designs.
1. Equal-split equations (Z0 system)
- Two quarter-wave branches with impedance Z0 * sqrt(2) = 70.7 ohm in a 50 ohm system.
- Isolation resistor R = 2 Z0 = 100 ohm between the two output ports.
- At the centre frequency, ideally: all ports matched, S21 = S31 = -3.01 dB, outputs in phase, isolation S23 = 0 (infinite).
Why it works (even/odd mode view): with equal in-phase signals at both outputs, no current flows through the resistor and each branch is a quarter-wave transformer from 100 ohm (half of the input port) to 50 ohm. With anti-phase signals the midplane is a virtual ground, the quarter-wave lines become open circuits at the outputs, and each output sees R/2 = 50 ohm, so the odd mode is absorbed in the resistor. That is also why the resistor matters for output match and isolation but carries no power in normal splitter operation.
2. How wide is a single section?
Evaluating the ideal circuit numerically:
- At 0.9 f0: S11 about -25 dB, isolation about -25 dB.
- At 0.82 f0 (and symmetrically about 1.18 f0): S11 and isolation both cross -20 dB.
So a single section gives about 36 % fractional bandwidth for 20 dB return loss and isolation. Output port match is better than that. For an octave or more, use multi-section designs: each extra section adds a quarter-wave pair and a resistor, and two sections already cover roughly 2:1 with better than 20 dB isolation when properly synthesized.
3. Unequal split
For a power ratio K^2 = P3/P2:
- Z03 = Z0 * sqrt((1 + K^2) / K^3)
- Z02 = K^2 * Z03
- R = Z0 * (K + 1/K)
The outputs are then at Z0*K and Z0/K, so you add quarter-wave transformers of Z0*sqrt(K) and Z0/sqrt(K) to bring them back to 50 ohm. Large ratios push one branch impedance high (thin, lossy, hard-to-etch lines); about 4:1 power ratio is where microstrip starts to get uncomfortable.
4. Microstrip numbers for a feel
On a 0.508 mm laminate with design Dk 3.55 at 2.45 GHz (Hammerstad model, zero-thickness copper):
- 50 ohm: about 1.14 mm wide, eeff about 2.79
- 70.7 ohm: about 0.62 mm wide, eeff about 2.66, quarter wave about 18.7 mm
Always re-check with your stack-up's real copper thickness and solder mask, which pull eeff up and impedance down slightly.
5. Layout pitfalls that cost isolation
- Resistor parasitics. A 100 ohm resistor sits between the two branch ends, and its pads, body length and any extra trace add series inductance and shunt capacitance. At a few GHz use 0402 or smaller, ideally thin-film, and bring the branches together so the resistor spans the gap directly with no stubs. Above roughly 10 GHz consider a printed thin-film resistor or an EM-modelled pad.
- Branch-to-branch coupling. The two quarter-wave arms are often curved toward each other to meet at the resistor. Where they run close and parallel they couple, degrading isolation and shifting the centre frequency. Keep spacing at least about three line widths away from the resistor end, or simulate the coupled section explicitly.
- The input T-junction is not ideal; include it in the EM model.
- Ground vias next to the output connectors matter as much as the divider itself if you are measuring isolation in the 25 to 30 dB range.
6. Using it as a combiner
In-phase, equal-amplitude, coherent inputs combine with no loss in the resistor. Uncorrelated signals (for example two different frequencies) do not: each one reaches the common port 3 dB down, and the other half of its power is dissipated in the resistor. Size the resistor's power rating for this case and for faults such as one PA turning off or an output being open-circuited, where reflected power ends up in the resistor too.
7. Quick measurement checklist
- Measure all three ports, not only S21. Isolation and output match are what distinguish a Wilkinson from a simple T.
- Terminate the unused port with a good load during two-port measurements; a poor load directly corrupts the isolation reading.
- Compare the S21 and S31 phase difference; it tells you about asymmetry in the layout.
Questions on multi-section synthesis or on lumped-element Wilkinsons for sub-GHz work are welcome below.
1. Equal-split equations (Z0 system)
- Two quarter-wave branches with impedance Z0 * sqrt(2) = 70.7 ohm in a 50 ohm system.
- Isolation resistor R = 2 Z0 = 100 ohm between the two output ports.
- At the centre frequency, ideally: all ports matched, S21 = S31 = -3.01 dB, outputs in phase, isolation S23 = 0 (infinite).
Why it works (even/odd mode view): with equal in-phase signals at both outputs, no current flows through the resistor and each branch is a quarter-wave transformer from 100 ohm (half of the input port) to 50 ohm. With anti-phase signals the midplane is a virtual ground, the quarter-wave lines become open circuits at the outputs, and each output sees R/2 = 50 ohm, so the odd mode is absorbed in the resistor. That is also why the resistor matters for output match and isolation but carries no power in normal splitter operation.
2. How wide is a single section?
Evaluating the ideal circuit numerically:
- At 0.9 f0: S11 about -25 dB, isolation about -25 dB.
- At 0.82 f0 (and symmetrically about 1.18 f0): S11 and isolation both cross -20 dB.
So a single section gives about 36 % fractional bandwidth for 20 dB return loss and isolation. Output port match is better than that. For an octave or more, use multi-section designs: each extra section adds a quarter-wave pair and a resistor, and two sections already cover roughly 2:1 with better than 20 dB isolation when properly synthesized.
3. Unequal split
For a power ratio K^2 = P3/P2:
- Z03 = Z0 * sqrt((1 + K^2) / K^3)
- Z02 = K^2 * Z03
- R = Z0 * (K + 1/K)
The outputs are then at Z0*K and Z0/K, so you add quarter-wave transformers of Z0*sqrt(K) and Z0/sqrt(K) to bring them back to 50 ohm. Large ratios push one branch impedance high (thin, lossy, hard-to-etch lines); about 4:1 power ratio is where microstrip starts to get uncomfortable.
4. Microstrip numbers for a feel
On a 0.508 mm laminate with design Dk 3.55 at 2.45 GHz (Hammerstad model, zero-thickness copper):
- 50 ohm: about 1.14 mm wide, eeff about 2.79
- 70.7 ohm: about 0.62 mm wide, eeff about 2.66, quarter wave about 18.7 mm
Always re-check with your stack-up's real copper thickness and solder mask, which pull eeff up and impedance down slightly.
5. Layout pitfalls that cost isolation
- Resistor parasitics. A 100 ohm resistor sits between the two branch ends, and its pads, body length and any extra trace add series inductance and shunt capacitance. At a few GHz use 0402 or smaller, ideally thin-film, and bring the branches together so the resistor spans the gap directly with no stubs. Above roughly 10 GHz consider a printed thin-film resistor or an EM-modelled pad.
- Branch-to-branch coupling. The two quarter-wave arms are often curved toward each other to meet at the resistor. Where they run close and parallel they couple, degrading isolation and shifting the centre frequency. Keep spacing at least about three line widths away from the resistor end, or simulate the coupled section explicitly.
- The input T-junction is not ideal; include it in the EM model.
- Ground vias next to the output connectors matter as much as the divider itself if you are measuring isolation in the 25 to 30 dB range.
6. Using it as a combiner
In-phase, equal-amplitude, coherent inputs combine with no loss in the resistor. Uncorrelated signals (for example two different frequencies) do not: each one reaches the common port 3 dB down, and the other half of its power is dissipated in the resistor. Size the resistor's power rating for this case and for faults such as one PA turning off or an output being open-circuited, where reflected power ends up in the resistor too.
7. Quick measurement checklist
- Measure all three ports, not only S21. Isolation and output match are what distinguish a Wilkinson from a simple T.
- Terminate the unused port with a good load during two-port measurements; a poor load directly corrupts the isolation reading.
- Compare the S21 and S31 phase difference; it tells you about asymmetry in the layout.
Questions on multi-section synthesis or on lumped-element Wilkinsons for sub-GHz work are welcome below.