Coplanar Waveguide

Description

A coplanar waveguide (see Fig. 1) 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 $0.5\text{mm}$ thickness are buried in a substrate with $10\text{mm}$ thickness and relative permittivity ${ϵ}_r=4.4$ and relative permeability ${\mu }_r=1$, and low conductivity $\sigma =0.02\text{S/m}$. The low-loss upper layer has low permittivity ${ϵ}_r=1.07$ and $\sigma =0.01\text{S/m}$. The whole structure is enclosed in a metallic box of dimension $140\text{mm}$ by $100\text{mm}$ by $50\text{mm}$. The discrete port with $50\mathrm{\Omega }$ lumped load imposes $1\text{A}$ 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.

Figure 1: Coplanar Waveguide Model

Data

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

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

The discretized bilinear form is $a(u,v;\omega ,\nu )=\sum _{q=1}^Q{\mathrm{\Theta }}^q(\omega ,\nu )A^q$, with matrices $A^q$.

The matrices corresponding to the bilinear forms $a^q(\cdot ,\cdot )$ as well as the input and output forms and H(curl) inner product matrix have been assembled using the Finite Element Method, resulting in 7754 degrees of freedom, after removal of boundary conditions. The files are numbered according to their appearance in the summation.

The coefficient functions are given by:

$\begin{array}{rl}{\mathrm{\Theta }}^1(\omega ,\nu )& =1,\\ {\mathrm{\Theta }}^2(\omega ,\nu )& =\omega ,\\ {\mathrm{\Theta }}^3(\omega ,\nu )& =-{\omega }^2,\\ {\mathrm{\Theta }}^4(\omega ,\nu )& =\frac{\nu }{6},\\ {\mathrm{\Theta }}^5(\omega ,\nu )& =\frac{6}{\nu },\\ {\mathrm{\Theta }}^6(\omega ,\nu )& =\frac{6\omega }{\nu },\\ {\mathrm{\Theta }}^7(\omega ,\nu )& =-\frac{6{\omega }^2}{\nu },\\ {\mathrm{\Theta }}^8(\omega ,\nu )& =\frac{\nu \omega }{6},\\ {\mathrm{\Theta }}^9(\omega ,\nu )& =-\frac{\nu {\omega }^2}{6},\\ {\mathrm{\Theta }}^{10}(\omega ,\nu )& =\frac{16-\nu }{10},\\ {\mathrm{\Theta }}^{11}(\omega ,\nu )& =\frac{10}{16-\nu },\\ {\mathrm{\Theta }}^{12}(\omega ,\nu )& =\frac{10\omega }{16-\nu },\\ {\mathrm{\Theta }}^{13}(\omega ,\nu )& =-\frac{10{\omega }^2}{16-\nu },\\ {\mathrm{\Theta }}^{14}(\omega ,\nu )& =\frac{16-\nu }{10}\omega ,\\ {\mathrm{\Theta }}^{15}(\omega ,\nu )& =-\frac{16-\nu }{10}{\omega }^2.\end{array}$

The parameter domain of interest is $\omega \in [0.6,3.0]\cdot 10^9\text{Hz}$, where the factor of $10^9$ has already been taken into account while assembling the matrices, while the geometric variation occurs between $\nu \in [2.0,14.0]$. The input functional also has a factor of $\omega$.

There are two output functionals, which is due to the fact that the complex system has been rewritten as a real symmetric one. In particular the computation of the output $s(u)=|l^{T}u|$ with complex vector $u$ turns into $s(u)=\sqrt{(l_1^{T}u{)}^2+(l_2^{T}u{)}^2}$ with real vector $u$.

Origin

The models have been developed within the MoreSim4Nano project.

Data

The files are numbered according to their appearance in the summation and can be found here: Matrices_cp.tar.gz.

Dimensions

System structure:

$\begin{array}{rl}\sum _{q=1}^{15}{\mathrm{\Theta }}^q(\omega ,\nu )A^{q}u(\omega ,\nu )& =b\\ s(\omega ,\nu )& =\sqrt{(l_1^{T}u(\omega ,\nu ){)}^2+(l_2^{T}u(\omega ,\nu ){)}^2}\end{array}$

System dimensions:

$A^q\in {\mathbb{ℝ}}^{15504\times 15504}$, $b,l_1,l_2\in {\mathbb{ℝ}}^{15504}$.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

The MORwiki Community, Coplanar Waveguide. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Coplanar_Waveguide

@MISC{morwiki_waveguide,
  author =       {{The MORwiki Community}},
  title =        {Coplanar Waveguide},
  howpublished = {{MORwiki} -- Model Order Reduction Wiki},
  url =          {https://modelreduction.org/morwiki/Coplanar_Waveguide},
  year =         {2018}
}

• For the background on the benchmark:

   @ARTICLE{morHesB13,
     author =  {Hess, M.~W. and Benner, P.},
     title =   {Fast Evaluation of Time-Harmonic {M}axwell's Equations Using the Reduced Basis Method},
     journal = {{IEEE} Trans. Microw. Theory Techn.},
     volume =  61,
     number =  6,
     pages =   {2265--2274},
     year =    2013,
     doi =     {10.1109/TMTT.2013.2258167}
   }

Contact

Martin Hess