The Wilkinson power divider is a power divider circuit that achieves isolation between its two output ports when all ports are matched. Although the Wilkinson power divider can be designed to achieve arbitrary power division (e.g., see Pozar [1]), this example will investigate the equal-split (3dB) case. FDTD will be used to obtain the scattering parameters of the device.
Background
The above figure presents the layout of the Wilkinson power divider which consists of an input transmission line with a characteristic impedance Z0 that branches into two quarter-wavelength transmission-line sections of characteristic impedance √2Z0 that carry power into the output transmission lines of characteristic impedance Z0. A shunt resistor (R) of value 2Z0 is symmetrically placed between the output branches to achieve high isolation between the output ports. Using even- and odd-mode transmission-line analysis, the Wilkinson power divider can be shown to possess the following scattering parameters at the design frequency: S11= 0, S12=S13=-j/√2, S23=S32=0. Using FDTD, we will determine the frequency response of the Wilkinson power divider for Example 7.2 in Pozar [1] where f0=1 GHz, Z0=50 Ω, √2Z0=70.7 Ω, and R=2Z0=100 Ω.
Simulation Setup
The structure group “traces and load” is used to construct and set the physical and electrical parameters of the Wilkinson power divider in the FDTD simulation file wilkinson_power_divider.fsp. The microstrip transmission lines are modeled using 2D perfect electric conductor (PEC) rectangular sheets placed on top of a 1.59mm thick substrate that possesses a relative permittivity of 2.2. The required width of each transmission line section is calculated using Eqs. 3.195 and 3.197 in Pozar [1] (see the microstrip.lms script file in the microstrip example) to be 4.9mm (Z0=50 Ohms) and 2.804 mm (√2Z0=70.7 Ω). The quarter-wave transmission lines are constructed using 2D polygons formed into an annular shape whose circumference is calculated using Eqs. 3.194 in Pozar [1] to be λg/4=55.5 mm. The resistor is modeled using a 2D rectangular sheet that is assigned an material with R=100 Ohms.
Ports are placed over the input and output traces to inject the transmission-line mode over a frequency range of 0.5 – 1.5 GHz and calculate the devices scattering parameters. See the Ports page for more details on their set up. As will be discussed below, the source port will be changed manually to excite one port at a time.
A mesh override region is placed over each trace in order to resolve their length and width. The curved and angled nature of the branching traces requires the mesh size in the x and y directions to be comparable (dx=dy). This is not a constraint for the feed and output traces which align with the coordinate axis. A copy of the mesh override region used for the branching traced is placed to the right of the output traces to maintain a symmetric mesh.
PML absorbing boundary conditions surround the entire simulation region except on the z min boundary, which is assigned as a Metal boundary condition to model the microstrip transmission-line’s ground plane.
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| Isolation Simulation | Transmission Simulation |
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Note : Multiport Simulation To find the generalized scattering parameters Sij, port j is driven with an incident wave and the reflected wave from port i is measured when for all other ports are matched. To obtain the all the S-parameter entries, multiple simulations must be run in. In general, a N-port device will require N separate simulations. However, symmetry and reciprocity of a device allows us to determine all the scattering parameters using a smaller number of simulations. |
To obtain all scattering parameters of the device requires us to run two simulations (here we are exploiting symmetry and reciprocity). This is automated using the script file wilkinson_power_divider.lsf. The script begins by obtaining the isolation (S32 and S23) of the device by setting Port 2 as the source port as indicated in the above image by the purple arrow. In this arrangement, Port 2 injects a transmission-line mode in which Port 1 and 3 measure the transmitted power. When the simulation ends, the script slightly modifies the layout to obtain the transmission (S21 and S31) and reflection (S11) seen from the input (Port 1). This is achieved by setting Port 1 as the source as shown in the above figure on the right. Due to the symmetric nature of the problem, we apply a symmetric boundary condition on y min to speed up the simulation.
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Note : S-parameter sweep vs scripting Although the S-parameter matrix sweep could be used to automate the process of obtaining all the entries of the s-parameter matrix, it is not used here. That is because we are also interested in each simulation's electric-field profiles (see below). Furthermore, we wanted to apply symmetry boundary condition in one of the simulations and not both. |
Results and Analysis
The above figures present the scattering parameter's frequency response and the electric field profile at 1 GHz for the isolation and transmission simulations. These figures are generated by the script after the simulation finishes. It should be noted that these results can be obtained with a finer mesh over the traces than the one specified in the simulation file.
The simulated Wilkinson power divider is well matched at its input (S11=-40 dB, f=1.0 GHz) and output (S22=-32 dB, f=1 GHz) ports, has excellent isolation (S32=-43 dB, f=1 GHz), and has a center frequency of 1.01 GHz which is within 1% of the design operating frequency of 1 GHz. Furthermore, we observe a 3 dB equal split of power (S31=-3 dB at f=1 GHz) which varies by less than 10% across the simulated band.
Related references
- D. M. Pozar, Microwave Engineering, Fourth Edition. John Wiley & Sons (2012).