WILKINSON Power Splitter – Theory

Wilkinson power splitter is a common radio device, especially at microwave and mmWave frequency regions. It allows an even power splitting of RF power while maximizing output isolation and impedance match at all of its ports. The circuit in Figure 1 describes the basic structure of a Wilkinson splitter. It includes couple of transmission lines serving as a quarter wave transformer with √2Z₀ and a matching resistor.

The analysis of the circuit is based on the orthogonal excitation method of a circuit where both odd and even excitation modes are present at the circuit’s ports and their outputs are evaluated. The following is an analysis of the circuit based on those excitation modes.

Odd Mode:

Due to design symmetry, we can divide the complete circuit along an imaginary horizontal line where we will now have two isolated circuits that will reduce analysis complications. As depicted in Figure 2, in an odd mode excitation we set both port 2 and port 3 of the splitter to be identical in voltage but 180⁰ out of phase. Due to this, the electrical potential along the symmetry line is 0V serving as a short circuit.

• Voltages at port 2 and 3 are set to 2V₀.
• With a matched port condition, input voltage will be half (+/- V₀).
• Port 1 has a virtual ground thus excitation voltages do not contribute voltage to port 1.
• Tline’s electrical length is 90⁰ at the central frequency of the circuit, serving as an impedance transformer. A short circuit at its input will be translated to open circuit at its output (port 2 or 3).
• As there is no electrical loading of the Tline at port 2 and 3, the 2Z₀ resistor will assure a matching condition in those ports. Each port is internally matched with a Z₀ resistor.
• Due to virtual ground along the symmetry line, at the center frequency an infinite isolation is achieved between port 2 and port 3.

To conclude, in an odd mode excitation at ports 2 and 3 of the splitter:

• All ports are matched.
• Maximal isolation achieved between ports 2 and 3.
• No voltage contribution to port 1 (input port).

Even Mode:

As was presented in the odd mode excitation analysis, due to design symmetry we now have two isolated circuits. As depicted in Figure 3, in an even mode excitation we set both port 2 and port 3 of the splitter to be identical in voltage and 0⁰ in phase. Due to this, there is no voltage difference between port 2 and port 3 which cause the symmetry line to serve as an open circuit.

• Voltages at port 2 and 3 are set to 2V₀.
• No voltage difference between ports 2 and 3.
• No current is flowing through the resistor; thus, it can be omitted for this analysis.
• As both divided circuits are connected in parallel at Port 1, for each half circuit, the matched resistor is set to 2Z₀.
• For a matched condition for ports 2 and 3, the transformer is required to transform its input load of 2Z₀ to a Z₀. This will happen in the case where its characteristic impedance is set to √2Z₀.
• Due to circuit symmetry and its open circuit condition along it, isolation between ports 2 and 3 is maximized at the central frequency of the circuit.
• Ports 2 and 3 excitation voltages will generate voltage at port 1. As both ports 2 and 3 are connected in parallel at port1, their voltage contribution is identical.
  • Due to transformer length, there is a 90⁰ of phase difference between voltage at port 1 and port 2/3.
  • Port 2/3 voltage level in the matched condition is V₀. Due to power conservation of the transformer and as the load resistor at port 1 is 2Z₀, voltage at port 1 is √2V₀.

To conclude, in an even mode excitation at ports 2 and 3 of the splitter:

• All ports are matched.
• Maximal isolation achieved between ports 2 and 3.
• In case where voltage level at port 2 and 3 is , voltage level at port 1 is √2V₀.
• Following the above, in case where voltage level at port 1 is , voltage levels at port 2 or 3 are V₀/√2 and in phase between them and 90⁰ phase difference to port 1.