Wilkinson Power Divider | Qucs-S S-parameter Viewer Documentation

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# Wilkinson Power Divider

## Overview

The Wilkinson power divider is the most widely used RF power splitter topology. It provides matched ports, high isolation between outputs, and can be designed for equal or unequal power split ratios.

## Topology

               ┌───[Zbr, λ/4]──── Output #1
               │               |
    Input  ────┤              [R]
               │               |
               └───[Zbr, λ/4]──── Output #2

Key components:

- Two λ/4 transmission line branches

- One isolation resistor (R) between outputs. If the balance is perfect, no current flows through it.

## Design Equations

### Equal Power Split (K = 1, 0 dB ratio)

Branch impedance and isolation resistor:

\[Z_2 = Z_3 = \sqrt{2} \times Z_0 \approx 70.7 \ \Omega \quad (Z_0 = 50 \ \Omega)\]
\[R = 2 \times Z_0 = 100 \ \Omega\]

Power division (3 dB to each output):

\[P_2 = P_3 = \frac{P_{in}}{2}\]

### Unequal Power Split (K ≠ 1)

Power ratio K = P₂/P₃ (linear), or K_dB = 10 × log₁₀(K).

Branch impedances:

\[Z_2 = Z_0 \sqrt{\frac{2\,(1 + K)}{K}}\]
\[Z_3 = Z_0 \sqrt{2\,(1 + K)}\]

Isolation resistor (sum of two series sections):

\[R_2 = Z_0 \times K \qquad R_3 = \frac{Z_0}{K} \qquad R = R_2 + R_3 = Z_0 \left(K + \frac{1}{K}\right)\]

Power division:

\[P_2 = P_{in} \times \frac{K}{K + 1} \qquad P_3 = P_{in} \times \frac{1}{K + 1}\]

## Example: Equal Split, Z₀ = 50 Ω, f₀ = 2 GHz

\[Z_2 = Z_3 = 50\sqrt{2} \approx 70.7 \ \Omega\]
\[R = 2 \times 50 = 100 \ \Omega\]
\[\frac{\lambda}{4} = \frac{3 \times 10^8}{4 \times 2 \times 10^9} = 37.5 \ \text{mm}\]

## Advantages

1. Matched ports: All ports matched to Z₀

2. Good isolation: Typically 20–30 dB between outputs

3. Good bandwidth: 20–40% fractional

4. Scalable: Can cascade for N-way division (N = 2ⁿ)

5. Well-established: Mature design, extensive literature

## Limitations

1. Resistor loss: Isolation resistor may dissipate power if the circuit is not perfectly balanced

2. Physical size: λ/4 lines large at low frequency

3. Unequal split limitations:

   - Wide Z₂/Z₃ ratios → fabrication challenges

   - Large K → very high/low impedances

Rule of thumb: Keep 30 Ω < Z₂, Z₃ < 150 Ω for practical fabrication.

## References

[1] Wilkinson, E. J. (1960). “An N-Way Hybrid Power Divider.” IRE Trans. Microwave Theory Tech., MTT-8, pp. 116-118.

[2] Pozar, D. M. (2012). Microwave Engineering (4th ed.), Section 7.3, pp. 336-340. Wiley.

## See Also

- RF Circuit Synthesis Tools