# Wilkinson Power Divider Simulation

## Self-guided learning

This example of a Wilkinson power divider contains a short description of the theory, detailed information on how to construct the model, a video showing how to construct the model, and the fully constructed model ready for you to download.

When working through the example, you may notice some small differences in your model compared to ours – this is usually simply due to the fact that you are using a different software version.

Figure 1: E-field phase animation of a Wilkinson power divider.

### The Physics

Wilkinson power divider is an N-port passive device [1], however it is commonly found as a 2-way divider (3 ports). By manipulating the S-parameter matrix, it can be shown that it is unfeasible to achieve a 3-port device that it is at the same time:

• reciprocal (Sij = Sji)
• matched in all the ports (Sii = 0)
• lossless ([S]H[S] = [I])

A Wilkinson power divider is a device that is matched at all ports, lossless when excited in the input port and the output ports are kept isolated. Its S-Parameter matrix for equal power division is given by Eq. 1 and its basic layout is shown in Fig. 2. In order to fully analyze this structure, one can use an "odd-even analysis" and for a complete treatment of it, please refer to [2]

Equation 1
Figure 2: Schematic diagram of Wilkinson power divider

### The Model

Fig. 2 shows a schematic of a Wilkinson divider that can easily be constructed in CST Studio Suite® (Fig. 3). Full dimensions are provided in the model construction notes. The model is simulated with the time domain solver in the frequency range 0 to 2 GHz.

 Parameter Value Description h 1.2 mm Substrate thickness eps_r 4.3 Substrate permittivity t 0.035 mm Metallization thickness W50 2.35 mm 50 Ω (Z0) line width W70 1.23 mm 70.71 Ω (Z0√2) l70 42.54 mm Length of λ/4 of the Z0√2 line width
Figure 3: Wilkinson power divider constructed in CST Studio Suite

### Discussion of Results

In Fig. 1 we see the E-field animation when the port 1 (input) is excited. The input signal flows through the split and is divided in ports 2 and 3. Fig. 4 shows the E-field animation when port 2 is excited. Note that the signal flows through the split, but practically nothing is coupled to port 3.

Figure 4: E-field phase animation when port 2 is excited.

The S-parameters results are shown in Fig. 5. Note that ports 1, 2 and 3 are matched. The equal power distribution can be seen by the overlapping curves of S2,1 and S3,1 and the isolation is shown by the curve S3,2.

Figure 5: S-parameters of the Wilkinson power divider

### Questions

1. The isolation of the output ports is certainly an important trait of this device. Fig. 6 shows the S-parameters when port 2 is excited. As the device is matched for all ports, there is effectively no reflection in S2,2, and the isolation S3,2 tells us that no power was transmitted to the port 3, yet S1,2 shows that only approximately -3 dB was transmitted to port 1. In other words, only half of the power was transmitted. In the Physics section we gave three goals that could not be simultaneously achieved by a 3-port device. Having this in mind, take a look in the power folder of your simulation and figure out where the missing power has gone.
2. What would happen if ports 2 and 3 were excited at the same time?
Figure 6: S-parameters of the Wilkinson power divider for the excitation in port 2

### References

• [1] E.J. Wilkinson, "An N-way Power Divider", IRE Trans. on Microwave Theory and Techniques, vol. 8, p. 116-118, Jan. 1960, doi: 10.1109/TMTT.1960.1124668
• [2] D.M. Pozar, Microwave Engineering, 4th Edition, John Wiley & Sons: New York, 1998, pp. 328-333