Coplanar Waveguide

Model Description

A coplanar waveguide is a microwave semiconductor device, which is governed by maxwell's equations. The coplanar waveguide considered with dielectric overlay, i.e. a transmission line shielded within two layers of multilayer board with 0.5mm thickness are buried in a substrate with 10mm thickness and relative permittivity $\epsilon_r = 4.4$ and relative permeability $\mu_r = 1$, and low conductivity $\sigma = 0.02 S/m$. The low-loss upper layer has low permittivity $\epsilon_r = 1.07$ and $\sigma = 0.01 S/m$. The whole structure is enlosed in a metallic box of dimension 140mm by 100mm by 50mm. The discrete port with 50ohm lumped load imposes 1 A current as the input to the one side of the strip. The voltage along the discrete port 2 at the end of the other side of coupled lines is integrated as the output.

Matrices and Data

Considered parameters are the frequency $\omega$ and the width $\nu$ of the middle stripline.

The affine form $a(u, v; \omega, \nu) = \sum_{q=1}^Q \Theta^q(\omega, \nu) a^q(u, v)$ can be established using $Q = 15$ affine terms.

The matrices corresponding to the bilinear forms $a^q( \cdot , \cdot )$ as well as the input and output forms have been assembled using the Finite Element Method, resulting in 7754 degrees of freedom, after removal of boundary conditions.

File:Matrices CoplanarWaveguide.tar.gz

The coefficient functions are given by

$\Theta^1(\omega, \nu) = 1$

$\Theta^2(\omega, \nu) = \omega$

$\Theta^3(\omega, \nu) = -\omega^2$

$\Theta^4(\omega, \nu) = \frac{\nu}{6}$

$\Theta^5(\omega, \nu) = \frac{6}{\nu}$

$\Theta^6(\omega, \nu) = \frac{6 \omega}{\nu}$

$\Theta^7(\omega, \nu) = -\frac{6 \omega^2}{\nu}$

$\Theta^8(\omega, \nu) = \frac{\nu \omega}{6}$

$\Theta^9(\omega, \nu) = -\frac{\nu \omega^2}{6}$

$\Theta^{10}(\omega, \nu) = \frac{16 - \nu}{10}$

$\Theta^{11}(\omega, \nu) = \frac{10}{16 - \nu}$

$\Theta^{12}(\omega, \nu) = \frac{10 \omega}{16 - \nu}$

$\Theta^{13}(\omega, \nu) = -\frac{10 \omega^2}{16 - \nu}$

$\Theta^{14}(\omega, \nu) = \frac{16 - \nu}{10} \omega$

$\Theta^{15}(\omega, \nu) = -\frac{16 - \nu}{10} \omega^2$

The parameter domain of interest is $\omega \in [0.6, 3.0] 10^9 Hz$, where the factor of $10^9$ has already been taken into account while assembling the matrices. The input functional also has a factor of $\omega$

References

The models have been developed within the MoreSim4Nano project.

[1] www.moresim4nano.org

[2] M. W. Hess, P. Benner, Fast Evaluation of Time-Harmonic Maxwell's Equations Using the Reduced Basis Method, MPI preprint http://www.mpi-magdeburg.mpg.de/preprints/2012/MPIMD12-17.pdf