Lim-Eom Power Divider | Qucs-S S-parameter Viewer Documentation

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# Lim-Eom Power Divider

## Overview

Wideband 4-port power divider with unequal power split capability and excellent bandwidth performance. Developed by Lim and Eom, this topology provides multi-octave operation while maintaining arbitrary power ratios between three output ports.

## Topology

                               Output #1
                                  |
                                  |
    Input     ────────[Z2, λ/4]────────[Z1, λ/4]──────── [R0]
               |                                  |
               |                                  |
          [Z1, λ/4]                          [Z2, λ/4]
               |                                  |
               |                                  |
               |─────────────[Z0, λ/2]────────────|
               |                                  |
               |                                  |
          [Z4, λ/4]                           [Z5, λ/4]
               |                                  |
               |                                  |
    Output #2 ────────[Z5, λ/4]────────[Z4, λ/4]────────  Output #3
                                  |
                                 [R0]

Key features:

- Four ports: 1 input, 3 outputs

- Two grounded isolation resistors (R1, R2)

- Center λ/2 section (Z3) connects upper and lower halves

- Arbitrary power ratios M:N:K at outputs

## Design Equations

### Power Split Ratios

Define output power ratios:

    M = P₂/P_total    (Port 2 power ratio)
    N = P₃/P_total    (Port 3 power ratio)
    K = P₄/P_total    (Port 4 power ratio)

    where M + N + K = 1

### Impedance Calculations

    d₁ = M + N + K = 1
    d₂ = N + K

    Z₁ = Z₀ × √(d₁/d₂) = Z₀ × √(1/(N + K))
    Z₂ = Z₀ × √(d₁/M) = Z₀ × √(1/M)
    Z₃ = Z₀
    Z₄ = Z₀ × √(d₂/N) = Z₀ × √((N + K)/N)
    Z₅ = Z₀ × √(d₂/K) = Z₀ × √((N + K)/K)

    R₁ = R₂ = Z₀

### Line Lengths

    Z₁, Z₂, Z₄, Z₅: λ/4 at f₀
    Z₃: λ/2 at f₀ (impedance repeater)

## Example 1: Equal 3-Way Split (M = N = K = 1/3)

Design:

    M = N = K = 1/3
    d₂ = 2/3

    Z₁ = 50 × √(1/(2/3)) = 50 × √1.5 ≈ 61.2 Ω
    Z₂ = 50 × √(1/(1/3)) = 50 × √3 ≈ 86.6 Ω
    Z₃ = 50 Ω
    Z₄ = 50 × √((2/3)/(1/3)) = 50 × √2 ≈ 70.7 Ω
    Z₅ = 50 × √((2/3)/(1/3)) = 50 × √2 ≈ 70.7 Ω

    R₁ = R₂ = 50 Ω

## Example 2: Unequal Split (M = 1/2, N = 1/3, K = 1/6)

Power distribution:

    Port 2: 50% (3 dB)
    Port 3: 33% (4.8 dB)
    Port 4: 17% (7.8 dB)

Design:

    M = 0.5, N = 0.333, K = 0.167
    d₂ = 0.5

    Z₁ = 50 × √(1/0.5) = 50 × √2 ≈ 70.7 Ω
    Z₂ = 50 × √(1/0.5) = 50 × √2 ≈ 70.7 Ω
    Z₃ = 50 Ω
    Z₄ = 50 × √(0.5/0.333) ≈ 61.2 Ω
    Z₅ = 50 × √(0.5/0.167) ≈ 86.6 Ω

    R₁ = R₂ = 50 Ω

## Advantages

1. Wideband operation:

   - Wider than standard Wilkinson

2. Arbitrary power split: Any M:N:K ratio

   - Full design flexibility

3. High isolation: >20 dB between all outputs

   - Good for amplifier combining

   - Three-way isolation maintained

4. Grounded resistors:

   - Direct heat-sinking possible

   - Similar to Gysel topology -> Better thermal management

   - Higher power capability

5. All ports matched: Good return loss

   - Input and all outputs matched to Z₀

   - Low VSWR

## Limitations

1. Complexity: More complex than standard Wilkinson

   - Five different impedances

   - λ/2 section adds length

2. Size:

   - λ/2 center section

   - More PCB area required

3. Limited to 3 outputs: Fixed 4-port design

   - Cannot extend to N-way easily

   - For N≠3, use tree or other topology

## Comparison with Other 3-Way Dividers

Feature | Lim-Eom | 3-Way Wilkinson | Tree (2× Wilkinson)
--- | --- | --- | ---
Outputs | 3 | 3 | 3
Bandwidth | 100-150% | 20-40% | 20-40%
Unequal split | Arbitrary | Difficult | Limited
Size | Medium | Small | Large (2 stages)
Complexity | High | Medium | Medium
Isolation | >20 dB | >20 dB | >20 dB
Resistors | 2 (grounded) | 3 (floating) | 2 (floating)
Power handling | High | Moderate | Moderate

## References

[1] J.-S. Lim and S.-Y. Eom.”A New 3-Way Power Divider with Various Output Power Ratios,” 1996 IEEE MTT-S Int. Microwave Symp. Dig., pp. 785-788.

[2] Andrei Grebennikov. “Power combiners, impedance transformers and directional couplers: part II”. High Frequency Electronics. 2008

## See Also

- RF Circuit Synthesis Tools