# Quarter Wave Transformer 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.

### The Physics

A quarter-wave transformer is a simple impedance transformer which is commonly used in impedance matching in order to minimize the energy which is reflected when a transmission line is connected to a load. The quarter-wave transformer uses a transmission line with different characteristic impedance and with a length of one-quarter of the guided-wavelength to match a line to a load. A basic schematic is shown in Fig. 1.

Figure 1: Circuit schematic of a quarter-wave transformer

### The Model

The quarter-wave transformer in this example connects a 50 Ω line to a 100 Ω load at 1 GHz, and is realized using a microstrip technology on an FR-4 substrate and modeled in CST Studio Suite® as shown in Fig 2.

Figure 2: Quarter-wave transformer using microstrip technology modeled in CST Studio Suite.

This model has a microstrip of  50 Ω that needs to be matched to a load of 100 Ω. By using Eq. 1 we find that the transforming line has 70.71 Ω. The microstrips must be realized on a substrate with εr=4.3, a thickness of 1.2 mm and a metallization of 0.035 mm. In Table 1 we summarize all the parameters of this project.

 Parameter Value Description h 1.2 mm Substrate thickness t 0.035 mm Metallization thickness eps_r 4.3 Substrate permittivity Z0 50 Ω System characteristic impedance RL 100 Ω Load impedance W50 2.35 mm 50 Ω line width W70 1.23 mm 70.71 Ω line width l70 41.84 mm Transformer length

### Discussion of results

Figure 3: E-field phase animation of a quarter-wave transformer showing no reflection in the load.

Fig. 3 shows the E-field phase animation. On that animation we can see that at the resonant operating frequency (1 GHz) there is no reflection from the load, and therefore the load of 100 Ω is matched to the input line of 50 Ω. In Fig. 8 we plot the broadband frequency response of the transformer. Because there is a 10 mm microstrip leading up to the transformer, the S-Parameters need to be de-embedded. This is done using “De-embed S-Parameter” under the “S-Parameter Calculations” menu on the “Post Processing” ribbon, with a distance to reference plane of -10 mm.

Figure 4: Broadband S-parameter result of the quarter-wave transformer.

Note that the return is at a minimum at frequencies where the Z1 line has a length of (2n+1)λ/4, n=0,1,2,.... This can be better understood if we see the Smith Chart of the input impedance of the transformer (Fig 5). The normalized load (compared to the 50 ohm line) has a value of 2, and therefore it is the starting point of our analysis. From the load to the source we walk clockwise on a circle of constant VSWR centered at the normalized Z1(=1.41). The longer the electrical length of transforming line, the further we will walk over the constant VSWR circle. The circuit is ideally matched when the normalized impedance passes through the center of the Smith chart (the point Z=1+0i). As a complete rotation in the chart represents a half-wavelength, any integer number of 1/2λ+1/4λ will result in a matched circuit.

Figure 5: Smith chart of the load impedance seen at the beginning of the transforming line. Note the circular pattern centered at 1.41 and passing through 1 as expected.

### Questions

• Model the same quarter-wave transformer, but make a parameter sweep in the permittivity of the substrate εr=(3.3,4.3,5.3) and observe the S-parameter results. What effect does a variation in permittivity have on the performance of the transformer? What implications does this have for the manufacturer?
• Suppose that you have a 50 Ω microstrip line that needs to be connected, with minimum reflection at 1 GHz, to another line with 25 Ω. What impedance would be required in the quarter-wavelength transformer to match these lines? How would you expect the geometry of the new quarter-wave transformer to differ from the old one?

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### References

• [1] D.M. Pozar, Microwave Engineering, 4th Edition, John Wiley & Sons, pp. 72-75