# Different 50Ohm-matched condition for different length of metamaterial waveguide

Based on coplanar waveguide, I am designing a meta-material waveguide by inserting lumped-element inductors along the center line.

Here are some parameters:

• Target frequency: 10-20 GHz
• Size of lump capacitor: 10 um, sit along the CPW (insert into a cut along the centerline)
• CPW dimension: CL width = 20 um, CL gap = 11.8 um (nominally 50 Ohm TL)

The lumped-element inductors are separated by 20 um. To achieve 50 Ohm matching, between the inductors I inserted a pair of open stub, extending from the center line into the GND without touching. Together with the inductor, I define a unit cell using T-model as "LCL", where L is half the inductance of each inductor from the front and rear of a cell.

A similar figure can be found in Fig. 1 in this work: https://arxiv.org/pdf/2007.00638.pdf

In COMSOL, I simulated a short array of 20 such cells terminated by 50 Ohm lumped ports, where I swept the stub length as a parameters: 50Ohm_LumpedPort_1 --- (LCL) --- (LCL) --- ... --- (LCL) --- (LCL) --- 50Ohm_LumpedPort_2

With the above 20-cell simulation, I found that I needed a stub length of 100 um (each side) to obtain sufficient shunt capacitance to get good 50 Ohm matching, characterized by a minimum peak reflection over a wide band of freq (8-12 GHz)

Then I expanded the simulation from 20 cells to 200 cells. The models are similar except that now we have a longer chip as the array is straight. I also kept the lumped-element inductor the same. To my surprise, now the stub length that gives 50 Ohm matched condition is significantly shorter at 80 um.

To analyse the issue, I fitted the 20-cell model in Python to an expected transfer matrix to extract the L and C values of the cell at a given stub length. Using the obtained L and C, I formed a 200-cell array and analyse the Bloch impedance, I do not see a big difference from that of a 20-cell array, and they are both very close to Z_0 = sqrt(L/C) expected from a single cell.

Much appreciated if you can advise on it.