Quarter-wave transformers are used to match a load impedance with a transmission line in order to minimize reflections. In this example, the load is represented by a 2D RLC lumped element and the load is matched to a microstrip transmission line.

## Background

A quarter-wave impedance transformer is a useful and simple matching network that is used to match the impedance of a terminating load (Z_{L}) to the characteristic impedance of a feeding transmission-line (Z_{0}) . It consists of a quarter-wavelength transmission-line, whose input impedance can be easily calculated using the expression

$$ Z_{i n}=\frac{Z_{0, \lambda / 4}^{2}}{Z_{L}}$$

where Z_{0,}_{λ}_{/}_{4} is line’s characteristic impedance. In order to match the feeding transmission line to the load (i.e. Z_{in}=Z_{0}), the quarter-wavelength transformer’s characteristic impedance is engineered to satisfy the condition \( Z_{0,\lambda / 4} = sqrt{Z_0 Z_l} \). In this example, the characteristic impedance of the microstrip transmission line is controlled through the trace’s width.

## Simulation Setup

This example is based on Problem 5.13 from Pozar [1] in which a single-section quarter-wavelength transformer is designed to match a 350 Ohms load to a 100 Ohms microstrip transmission line at an operating frequency of 4 GHz. We are asked to calculate the percent bandwidth of this transformer corresponding to a standing wave ratio (SWR) of SWR ≤ 2.

To model the microstrip transmission lines, this example makes use of 2D perfect electric conductor (PEC) rectangular sheets placed on top of a 1.59 mm thick substrate that possesses a relative permittivity of 2.2.

The required width of each transmission line section is calculated using Eq. 5.25, 3.195, and 3.197 in Pozar [1] (see the microstrip.lms script file in the microstrip example) to be 1.42mm (Z_{0}=100 Ohms) and 0.22mm (Z_{0,}_{λ}_{/}_{4}=187 Ohms).

The load is modelled using a 2D rectangular sheet that is assigned an

A port is used to calculate and inject the transmission-line mode of the feeding microstrip towards the transformer and load over a frequency range of 2 - 6 GHz. The port has been set up to use 5 frequency points to calculate the broadband mode to inject as well as the modes for the expansion calculation. The "calculate characteristic impedance" option in the port has been enabled so that the characteristic impedance of the injected mode is returned as a result of the port object. More information about port settings, see Ports.

A mesh override region is placed over the microstrips in order to resolve their length and width. It is worth noting that that the simulated characteristic impedance of the microstrip transmission line will depend on the specified mesh size in the mesh override region and a finer mesh will give more accurate results.

PML absorbing boundaries surround the simulation region to absorb any light which may be scattered.

Note : Mesh step size and PML The default number of PML layers used is optimized for cases where the layer thickness is on the order of lambda/10, and the thickness of the PML layers is determined by the mesh step size at the PML boundary. For simulations in the RF wavelength range where the size of the structures is much smaller than the wavelength, mesh override regions are often used to set the mesh step size to values which are hundreds of times smaller than the wavelength. If the mesh step size next to the PML boundary is hundreds of times smaller than the wavelength, the resulting thickness of the PML may be too small to effectively absorb incident light. In this case, you may either want to increase the number of PML layers used, or move the boundaries farther from the mesh override region so the mesh can be graded to a larger size before reaching the PML boundary. |

Under the boundary conditions tab of the FDTD solver region object, symmetry is used across the y-direction to reduce memory requirements.

## Results and Analysis

Running the *quarter_wave_transformer_RLC.lsf* script file after performing the simulation generates a plot of the simulated reflection coefficient (S11) compared to the theoretical reflection coefficient. This theory assumes that the microstrip transmission lines possess the specified characteristic impedance, supports an ideal TEM mode (recall that the microstrip supports a quasi-TEM mode), and neglects the parasitic effects from the step discontinuity in the x span of the transmission lines. In addition, the script displays in the script prompt the SWR ≤ 2 percent bandwidth from theory and simulation.

The simulated and theoretical reflection coefficients’ frequency response shown above generally agree and exhibit a SWR≤2 percent bandwidth of 62% and 68%, respectively. The simulated reflected power (|S11|^{2})is less than 1% at the design frequency (4GHz).

The match can be further improved by using a finer simulation mesh to resolve the microstrip, thereby obtaining a more accurate characteristic impedance and phase velocity in simulations, and by using a more complex theoretical model to account for the parasitic effects of the step discontinuity and fringing fields extending out from the termination of the transformer.

### Additional port results

In addition to the S parameter result, the port1 object also returns the effective index, characteristic impedance (if enabled in the port object) and field profile of the calculated modes at the frequencies used by the internal source and mode expansion monitor. These results are available even before running the simulation and are listed in the Result View window shown below:

Since the port is using the multi-frequency injection option the source modes are calculated at the same frequency points as the mode expansion monitor modes the mode fields are the same.

The supported microstrip mode has power strongly confined at the edges of the strip. Since the mode is strongly confined at the edges, the results can be sensitive to how fine the meshing is around those areas.

As a result, the characteristic impedance returned from the port results using the current mesh is approximately 83 Ohms instead of the 100 Ohms that the microstrip was designed for. Using a finer mesh can result in a closer value of characteristic impedance.

### Related references

[1] D.M. Pozar, Microwave Engineering, Fourth Edition. John Wiley & Sons (2012).