How do I calculate the output power at each port of an unequal power divider?
Unequal Power Divider Design
Unequal power dividers are essential in antenna feed networks where different elements require different power levels (for beam shaping or sidelobe control) and in system architectures where different paths have different gain/loss budgets.
Technical Considerations
The unequal Wilkinson divider is derived from the equal Wilkinson by modifying the arm impedances while maintaining the simultaneous matching and isolation conditions: (1) Port matching: each port must present Z0 when the other two ports are terminated in Z0. This requires the quarter-wave arms to transform the output impedance to the correct value at the common junction. (2) For the high-power arm (Port 2): the arm impedance is lower (lower impedance = lower loss = more power). Z_arm2 = Z0 × sqrt((1 + k^2) / k^3). (3) For the low-power arm (Port 3): the arm impedance is higher (higher impedance = more attenuation = less power). Z_arm3 = Z0 × k × sqrt(1 + k^2) = Z_arm2 × k^2. Note: Z_arm2 × Z_arm3 = Z0^2 × (1 + k^2). (4) Isolation resistor: R = Z0 × (k + 1/k). For equal split (k = 1): R = 2×Z0 = 100 ohms (standard Wilkinson). As the split becomes more unequal (k increases): R increases. (5) The output port impedances are Z0 at all ports (the quarter-wave transformers provide the impedance transformation).
- Performance verification: confirm specifications against the application requirements before finalizing the design
- Environmental factors: temperature range, humidity, and vibration affect long-term reliability and parameter drift
- Cost vs. performance: evaluate whether the application demands premium components or standard commercial grades
- Interface compatibility: verify impedance, connector type, and mechanical form factor match the system architecture
- Margin allocation: include sufficient design margin to account for manufacturing tolerances and aging effects
Performance Analysis
(1) Extreme split ratios: for k > 3 (> 10:1 power ratio): the high-power arm impedance becomes very low (< 20 ohms in 50-ohm system) and the low-power arm impedance becomes very high (> 200 ohms). These extreme impedances are difficult to realize in microstrip: low impedance requires very wide traces (possibly wider than the substrate), and high impedance requires very narrow traces (limited by fabrication resolution). Solutions: use a cascaded design (two stages of moderate split: 3:1 × 3:1 = 9:1). Use a coupled-line unequal divider (the coupling controls the split ratio). Use an asymmetric directional coupler (coupling factor = split ratio). (2) Bandwidth: the unequal Wilkinson has similar bandwidth to the equal Wilkinson (20-40% for a single section). Multi-section designs extend the bandwidth, but each section has a different split ratio (more complex design). (3) Amplitude and phase balance: "balance" is not a goal for an unequal divider (the outputs are intentionally unequal). However: the split ratio should be stable over the operating bandwidth. The ratio stability depends on the impedance flatness of each arm. For a single-section design: the split ratio variation is typically ±0.3-0.5 dB across 30% bandwidth.
Frequently Asked Questions
How do I create a 6 dB split?
A 6 dB split means one port gets 4× the power of the other: k^2 = 4, k = 2. P2 = P_in × 4/5 = -0.97 dB. P3 = P_in × 1/5 = -6.99 dB. Arm impedances: Z_arm2 = 50 × sqrt(5/8) = 39.5 ohms. Z_arm3 = 50 × 2 × sqrt(5) = 223.6 ohms. R_iso = 50 × (2 + 0.5) = 125 ohms. The 223.6-ohm arm is very narrow in microstrip (approximately 0.15 mm on 0.5 mm FR-4). This may be at the limit of standard PCB fabrication. Alternative: use a directional coupler with 6 dB coupling instead (the through port gets -1 dB and the coupled port gets -6 dB, approximate equivalent).
What about N-way unequal dividers?
For N > 2 outputs with arbitrary power levels: use a corporate feed network (binary tree of 2-way unequal dividers). Each level of the tree splits the power further, with each divider having a split ratio calculated to deliver the correct power to its subtree. For 4 outputs with power ratio 1:2:3:4: Level 1: 3:7 split (Ports A and B get 3/10 and 7/10 of the input). Level 2: Port A splits 1:2 (producing outputs 1 and 2). Port B splits 3:4 (producing outputs 3 and 4). Each 2-way divider is designed independently as an unequal Wilkinson. Alternative: use a single-stage multi-way divider (star topology) where each arm has a different impedance to achieve the desired power ratio. This is more compact but harder to design for good isolation between all output pairs.
Can I make an unequal split with a coupler instead?
Yes. A directional coupler inherently has an unequal split: Through port: P_in - coupling (e.g., -1 dB for a 10 dB coupler). Coupled port: coupling (e.g., -10 dB). This provides a 10:1 power split with excellent through-port efficiency. Adjusting the coupling value changes the split ratio. Advantages over an unequal Wilkinson: the through port has very low loss (< 0.5 dB), and the coupler provides directional (frequency-selective) coupling. Disadvantage: the coupler is a 4-port device (requires termination of the isolated port), while the Wilkinson is a 3-port device (simpler to integrate).