In this example we study a microstrip transmission line with a lumped RLC element in the middle. We calculate the reflected power for different settings of the RLC element in a 3D FDTD simulation and we compare with analytical results. We also show how the results can be visualized in a Smith chart.

## Background

For a transmission line terminated in a load impedance Z_{L}, the reflection coefficient at the position of the load is

$$ \Gamma_{L}=\frac{Z_{L}-Z_{0}}{Z_{L}+Z_{0}} $$

where Z_{0} is the characteristic impedance of the transmission line. The lumped RLC element is formed by a sum in parallel of a resistance (R), an inductance (L) and a capacitance (C); therefore, the load impedance is given by

$$ Z_{L}=Z_{0}+\left(\frac{1}{R}+\frac{1}{j \omega L}+j \omega C\right)^{-1} $$

where ω is the angular frequency of the incident wave. For a lossless transmission line the reflected power at any position along the line is given by |Γ|^{2} = |Γ_{L}|^{2}.

The characteristic impedance Z0 can be found using a FDE simulation as explained in the microstrip example, so it is possible to calculate the power reflected by the load analytically and compare it with the results from a FDTD simulation.

## FDE simulation setup

First we estimate the characteristic impedance of the transmission line in the file microstrip_rlc.lms. The transmission line is formed by a metal strip 2.4mm wide on top of FR4 substrate (permittivity 4.3) with thickness 1.2 mm. For the metal strip we use a 2D rectangle primitive with Perfect Electric Conductor (PEC) material. The propagation direction is assumed to be the z direction and we use metal boundary conditions.

## FDE results

The fundamental mode of the transmission line at 1GHz is shown in Fig.1. This is a quasi-TEM mode (waveguide TE/TM fraction = 100/99.7%) with no loss (since the strip material is PEC). The characteristic impedance can be calculated with the rectangular box shown in Fig.1. The value Z_{0} = 40.4 Ω will be used in the analysis of the FDTD results discussed in the next sections.

Fig.1 Screenshot of Eigenmode solver showing the transmission line mode and its characteristic impedance.

Note : Mesh accuracy Since the microstrip mode is strongly confined at the edges of the metal strip, the impedance results are very sensitive to the mesh step in that region. To obtain a more accurate value of the characteristic impedance use the convergence testing procedure described in the microstrip example. |

## FDTD simulation setup

The full structure of the transmission line in *microstrip_rlc.fsp* is composed of two metal traces and a lumped RLC element 2 mm long placed in between; the metal traces and the substrate have the same properties as in the FDE simulation. The lumped element is modeled using the RLC material available in the Material tab of the 2D rectangle primitive (see Fig. 2 below, for more details visit this page). The length of the lumped element should be electrically small, which means that the length should be small compared to the wavelength.

Fig. 2 Snapshot of the Material tab for 2D rectangle showing the RLC settings. .

We use PML boundary conditions for the propagation direction (z axis) and metal boundaries everywhere else. In the PML settings we increase the number of layers of the standard profile to 28 in order to improve the absorption of the reflected waves at the PML. The mesh override regions "trace mesh" and "load mesh" provide control over the mesh for the cross section of the transmission line and the 2mm gap where the RLC element is located. Compared to the FDE simulation settings, here we use a coarser mesh to reduce the execution time of the FDTD simulation. The simulation region is shown in Fig. 3.

Fig. 3 FDTD simulation setup (XZ view) .

The desired mode is calculated and injected using a port at the feeding metal trace. The port also calculates the forward and backward transmission in that mode to find the complex reflection coefficient (S_{11}); the reflected power is the absolute value squared of this coefficient (|S_{11}|^{2}). In the port settings we use 3 frequency points for the injection and mode expansion calculation. In addition, we have enabled the "calculate characteristic impedance" option to find the impedance of the microstrip mode using the same integration region as in the FDE simulation. For more information about port settings, see Ports.

## Running the FDTD simulation and analysis

The script *microstrip_rlc.lsf* calculates the reflected power from the FDTD simulation and compares it with the analytical results. The results for four different RLC configurations are shown in Fig. 4: a pure resistance R = 1.5 Z_{0}, a pure inductance L=6.5 nH, a pure capacitance C = 2 pF, and a parallel combination of these three components. Besides the numerical and analytical results for the reflected power, we include two curves ("upper bound" and "lower bound") that correspond to the analytical results with a variation of +/- 15% in the values of R, L and C. We also show the normalized impedance and corresponding reflected power for each case at the center frequency (1GHz) in a Smith chart.

Fig.4 Reflected power calculated in FDTD and comparison with analytical results (left column). The impedance values in each case at 1GHz are also plotted (right column).

For the analytical results above we assume that the transmission line has the characteristic impedance Z_{0} = 40.4 Ω calculated from the FDE simulation. Since the mesh used for the FDTD simulation is coarser than in the FDE simulation, the impedance returned by the port results is approximately 31.2 Ω. As the mesh in the FDTD simulation is refined, the impedance and phase velocity of the injected mode become more accurate, but the simulation takes a longer time to run. Nevertheless, for the coarse FDTD mesh we find that the numerical results are within +/-15% of the analytical ones. In addition, the slope of the numerical and analytical results agree well, which shows that the resistive, capacitive and inductive behaviors are correctly reproduced. Increasing the number of PML layers can also be beneficial as it improves the slope of the results.

The Smiths charts shown in Fig.4 can be obtained by opening the Visualizer for the variable Zcent in the Script workspace. It is important to set the normalizing impedance in the visualizer settings to be the characteristic impedance Z_{0}.

### Related references

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