Coplanar Waveguide - Revision history

Schuetz2 at 05:27, 12 June 2025

2025-06-12T05:27:25Z

← Older revision		Revision as of 05:27, 12 June 2025
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	title = {Coplanar Waveguide},		title = {Coplanar Waveguide},
	howpublished = {{MORwiki} -- Model Order Reduction Wiki},		howpublished = {{MORwiki} -- Model Order Reduction Wiki},
	url = <nowiki>{~~http~~://modelreduction.org/~~index.php~~/Coplanar_Waveguide}</nowiki>,		url = <nowiki>{https://modelreduction.org/morwiki/Coplanar_Waveguide}</nowiki>,
	year = {2018}		year = {2018}
	}		}

Mlinaric: Text fixes, math edits

2023-09-01T18:44:23Z

Text fixes, math edits

← Older revision		Revision as of 18:44, 1 September 2023
Line 5:		Line 5:
	[[Category:affine parameter representation]]		[[Category:affine parameter representation]]
	[[Category:Stationary]]		[[Category:Stationary]]



	==Description==		==Description==

	A '''coplanar waveguide''' (see Fig. 1) is a microwave semiconductor device, which is governed by [[wikipedia:Maxwell's_equations\|Maxwell's equations]].		A '''coplanar waveguide''' (see Fig. 1) is a microwave semiconductor device, which is governed by [[wikipedia:Maxwell's_equations\|Maxwell's equations]].
	The [[wikipedia:Coplanar_waveguide\|coplanar waveguide]] considered with dielectric overlay, i.e. a transmission line shielded within two layers of multilayer board with <math>0.~~5mm~~</math> thickness are buried in a substrate with <math>~~10mm~~</math> thickness and relative permittivity		The [[wikipedia:Coplanar_waveguide\|coplanar waveguide]] considered with dielectric overlay, i.e. a transmission line shielded within two layers of multilayer board with <math>0.5 \, \text{mm}</math> thickness are buried in a substrate with <math>10 \, \text{mm}</math> thickness and relative permittivity
	<math>\epsilon_r = 4.4 </math> and relative permeability <math>\mu_r = 1 </math>, and low conductivity <math>\sigma = 0.02 S/m </math>.		<math>\epsilon_r = 4.4</math> and relative permeability <math>\mu_r = 1</math>, and low conductivity <math>\sigma = 0.02 \, \text{S/m}</math>.
	The low-loss upper layer has low permittivity <math>\epsilon_r = 1.07 </math> and <math>\sigma = 0.01 S/m</math>.		The low-loss upper layer has low permittivity <math>\epsilon_r = 1.07</math> and <math>\sigma = 0.01 \, \text{S/m}</math>.
	The whole structure is ~~enlosed~~ in a metallic box of dimension <math>~~140mm~~</math> by <math>~~100mm~~</math> by <math>~~50mm~~</math>.		The whole structure is enclosed in a metallic box of dimension <math>140 \, \text{mm}</math> by <math>100 \, \text{mm}</math> by <math>50 \, \text{mm}</math>.
	The discrete port with <math>~~50Ohm~~</math> lumped load imposes <math>1 A</math> current as the input to the one side of the strip.		The discrete port with <math>50 \, \Omega</math> lumped load imposes <math>1 \, \text{A}</math> 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.		The voltage along the discrete port 2 at the end of the other side of coupled lines is integrated as the output.

Line 24:		Line 22:
	==Data==		==Data==

	Considered parameters are the frequency <math> \omega </math> and the width <math> \nu </math> of the middle stripline.		Considered parameters are the frequency <math>\omega</math> and the width <math>\nu</math> of the middle stripline.

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

	The discretized bilinear form is <math> a(u, v; \omega, \nu) = \sum_{q=1}^Q \Theta^q(\omega, \nu) A^q </math>, with matrices <math> A^q </math>.		The discretized bilinear form is <math>a(u, v; \omega, \nu) = \sum_{q=1}^Q \Theta^q(\omega, \nu) A^q</math>, with matrices <math>A^q</math>.

	The matrices corresponding to the bilinear forms <math> a^q( \cdot , \cdot ) </math> as well as the input and output forms and H(curl) inner product matrix have been assembled		The matrices corresponding to the bilinear forms <math>a^q(\cdot, \cdot)</math> 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		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.		appearance in the summation.
Line 36:		Line 34:
	The coefficient functions are given by:		The coefficient functions are given by:

	:<math> \Theta^1(\omega, \nu) = 1 ~~</math>~~		:<math>
			\begin{align}
	~~:<math>~~ \Theta^2(\omega, \nu) = \omega ~~</math>~~		\Theta^1(\omega, \nu) &= 1, \\
			\Theta^2(\omega, \nu) &= \omega, \\
	~~:<math>~~ \Theta^3(\omega, \nu) = -\omega^2 ~~</math>~~		\Theta^3(\omega, \nu) &= -\omega^2, \\
			\Theta^4(\omega, \nu) &= \frac{\nu}{6}, \\
	~~:<math>~~ \Theta^4(\omega, \nu) = \frac{\nu}{6} ~~</math>~~		\Theta^5(\omega, \nu) &= \frac{6}{\nu}, \\
			\Theta^6(\omega, \nu) &= \frac{6 \omega}{\nu}, \\
	~~:<math>~~ \Theta^5(\omega, \nu) = \frac{6}{\nu} ~~</math>~~		\Theta^7(\omega, \nu) &= -\frac{6 \omega^2}{\nu}, \\
			\Theta^8(\omega, \nu) &= \frac{\nu \omega}{6}, \\
	~~:<math>~~ \Theta^6(\omega, \nu) = \frac{6 \omega}{\nu} ~~</math>~~		\Theta^9(\omega, \nu) &= -\frac{\nu \omega^2}{6}, \\
			\Theta^{10}(\omega, \nu) &= \frac{16 - \nu}{10}, \\
	~~:<math>~~ \Theta^7(\omega, \nu) = -\frac{6 \omega^2}{\nu} ~~</math>~~		\Theta^{11}(\omega, \nu) &= \frac{10}{16 - \nu}, \\
			\Theta^{12}(\omega, \nu) &= \frac{10 \omega}{16 - \nu}, \\
	~~:<math>~~ \Theta^8(\omega, \nu) = \frac{\nu \omega}{6} ~~</math>~~		\Theta^{13}(\omega, \nu) &= -\frac{10 \omega^2}{16 - \nu}, \\
			\Theta^{14}(\omega, \nu) &= \frac{16 - \nu}{10} \omega, \\
	~~:<math>~~ \Theta^9(\omega, \nu) = -\frac{\nu \omega^2}{6} ~~</math>~~		\Theta^{15}(\omega, \nu) &= -\frac{16 - \nu}{10} \omega^2.
			\end{align}
	~~:<math>~~ \Theta^{10}(\omega, \nu) = \frac{16 - \nu}{10} ~~</math>~~		</math>

	~~:<math>~~ \Theta^{11}(\omega, \nu) = \frac{10}{16 - \nu} ~~</math>~~

	~~:<math>~~ \Theta^{12}(\omega, \nu) = \frac{10 \omega}{16 - \nu} ~~</math>~~

	~~:<math>~~ \Theta^{13}(\omega, \nu) = -\frac{10 \omega^2}{16 - \nu} </math>

	~~:<math> \Theta^{14}(\omega, \nu) = \frac{16 - \nu}{10} \omega </math>~~		The parameter domain of interest is <math>\omega \in [0.6, 3.0] \cdot 10^9 \, \text{Hz}</math>, where the factor of <math>10^9</math> has already been taken into account
			while assembling the matrices, while the geometric variation occurs between <math>\nu \in [2.0, 14.0]</math>.
	~~:<math> \Theta^{15}(\omega, \nu) = -\frac{16 - \nu}{10} \omega^2 </math>~~		The input functional also has a factor of <math>\omega</math>.

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

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

	==Origin==		==Origin==
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	:<math>		:<math>
	\begin{~~array}{rcl~~}		\begin{align}
	\sum_{q=1}^{15} \Theta^q(\omega, \nu) A^q u &=& b \\		\sum_{q = 1}^{15} \Theta^q(\omega, \nu) A^q u(\omega, \nu) &= b \\
	s &=& \sqrt{ (l_1^T * u)^2 + (l_2^T * u)^2 }		s(\omega, \nu) &= \sqrt{(l_1^T u(\omega, \nu))^2 + (l_2^T u(\omega, \nu))^2}
	\end{~~array~~}		\end{align}
	</math>		</math>

Line 97:		Line 84:

	<math>A^q \in \mathbb{R}^{15504 \times 15504}</math>,		<math>A^q \in \mathbb{R}^{15504 \times 15504}</math>,
	<math>b ~~\in \mathbb{R}^{15504}</math>~~,		<math>b, l_1, l_2 \in \mathbb{R}^{15504}</math>.
	~~<math>~~l_1 ~~\in \mathbb{R}^{15504}</math>~~,
	~~<math>~~l_2 \in \mathbb{R}^{15504}</math>.

	==Citation==		==Citation==

	To cite this benchmark, use the following references:		To cite this benchmark, use the following references:

Himpe at 13:11, 25 August 2022

2022-08-25T13:11:17Z

← Older revision		Revision as of 13:11, 25 August 2022
Line 3:		Line 3:
	[[Category:Parametric]]		[[Category:Parametric]]
	[[Category:linear]]		[[Category:linear]]
	~~[[Category:time invariant]]~~
	[[Category:affine parameter representation]]		[[Category:affine parameter representation]]
			[[Category:Stationary]]

Himpe at 13:00, 25 August 2022

2022-08-25T13:00:38Z

← Older revision		Revision as of 13:00, 25 August 2022
Line 1:		Line 1:
	[[Category:benchmark]]		[[Category:benchmark]]
	[[Category:PDE]]		[[Category:PDE]]
	[[Category:~~parametric 2-5 parameters~~]]		[[Category:Parametric]]
	[[Category:linear]]		[[Category:linear]]
	[[Category:time invariant]]		[[Category:time invariant]]

Mitchell: errant comma

2022-03-14T15:14:59Z

errant comma

← Older revision		Revision as of 15:14, 14 March 2022
Line 70:		Line 70:
	The input functional also has a factor of <math> \omega </math>.		The input functional also has a factor of <math> \omega </math>.

	There are two output functionals, which is due to the fact, that the complex system has been rewritten as a real symmetric one.		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		In particular the computation of the output

Himpe: Fix CrossRefs

2021-08-27T07:28:46Z

Fix CrossRefs

← Older revision		Revision as of 07:28, 27 August 2021
Line 10:		Line 10:
	==Description==		==Description==

	A '''coplanar waveguide''' (see ~~<xr id="fig:coplanar"/>~~) is a microwave semiconductor device, which is governed by [[wikipedia:Maxwell's_equations\|Maxwell's equations]].		A '''coplanar waveguide''' (see Fig. 1) is a microwave semiconductor device, which is governed by [[wikipedia:Maxwell's_equations\|Maxwell's equations]].
	The [[wikipedia:Coplanar_waveguide\|coplanar waveguide]] considered with dielectric overlay, i.e. a transmission line shielded within two layers of multilayer board with <math>0.5mm</math> thickness are buried in a substrate with <math>10mm</math> thickness and relative permittivity		The [[wikipedia:Coplanar_waveguide\|coplanar waveguide]] considered with dielectric overlay, i.e. a transmission line shielded within two layers of multilayer board with <math>0.5mm</math> thickness are buried in a substrate with <math>10mm</math> thickness and relative permittivity
	<math>\epsilon_r = 4.4 </math> and relative permeability <math>\mu_r = 1 </math>, and low conductivity <math>\sigma = 0.02 S/m </math>.		<math>\epsilon_r = 4.4 </math> and relative permeability <math>\mu_r = 1 </math>, and low conductivity <math>\sigma = 0.02 S/m </math>.

Hessm at 17:44, 9 August 2019

2019-08-09T17:44:26Z

← Older revision		Revision as of 17:44, 9 August 2019
Line 105:		Line 105:

	* For the benchmark itself and its data:		* For the benchmark itself and its data:
	::The MORwiki Community, '''~~Branchline Coupler~~'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Coplanar_Waveguide		::The MORwiki Community, '''Coplanar Waveguide'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Coplanar_Waveguide

	@MISC{morwiki_waveguide,		@MISC{morwiki_waveguide,

Himpe: Add dimensions section and various other fixes.

2018-11-13T09:57:35Z

Add dimensions section and various other fixes.

@@ Line 10: / Line 10: @@
 ==Description==
-A '''coplanar waveguide''' (see <xr id="fig:coplanar"/>) is a microwave semiconductor device, which is governed by maxwell's equations.
+A '''coplanar waveguide''' (see <xr id="fig:coplanar"/>) is a microwave semiconductor device, which is governed by [[wikipedia:Maxwell's_equations|Maxwell's equations]].
-The '''coplanar waveguide''' considered with dielectric overlay, i.e. a transmission line shielded within two layers of multilayer board with <math>0.5mm</math> thickness are buried in a substrate with <math>10mm</math> thickness and relative permittivity
+The [[wikipedia:Coplanar_waveguide|coplanar waveguide]] considered with dielectric overlay, i.e. a transmission line shielded within two layers of multilayer board with <math>0.5mm</math> thickness are buried in a substrate with <math>10mm</math> thickness and relative permittivity
 <math>\epsilon_r = 4.4 </math> and relative permeability <math>\mu_r = 1 </math>, and low conductivity <math>\sigma = 0.02 S/m </math>.
 The low-loss upper layer has low permittivity <math>\epsilon_r = 1.07 </math> and <math>\sigma = 0.01 S/m</math>.
@@ Line 66: / Line 66: @@
 :<math> \Theta^{15}(\omega, \nu) = -\frac{16 - \nu}{10} \omega^2 </math>
-The parameter domain of interest is <math>\omega \in [0.6, 3.0] * 10^9 </math> Hz, where the factor of <math>10^9 </math> has already been taken into account
+The parameter domain of interest is <math>\omega \in [0.6, 3.0] \cdot 10^9 </math> Hz, where the factor of <math>10^9 </math> has already been taken into account
 while assembling the matrices, while the geometric variation occurs between <math> \nu \in [2.0, 14.0]</math>.
 The input functional also has a factor of <math> \omega </math>.
@@ Line 82: / Line 82: @@
 The files are numbered according to their appearance in the summation and can be found here: [[Media:Matrices_cp.tar.gz|Matrices_cp.tar.gz]].
 ==References==

Himpe: Updated Coplanar Waveguide

2018-02-28T09:21:16Z

Updated Coplanar Waveguide

← Older revision		Revision as of 09:21, 28 February 2018
Line 11:		Line 11:

	A '''coplanar waveguide''' (see <xr id="fig:coplanar"/>) is a microwave semiconductor device, which is governed by maxwell's equations.		A '''coplanar waveguide''' (see <xr id="fig:coplanar"/>) 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		The '''coplanar waveguide''' considered with dielectric overlay, i.e. a transmission line shielded within two layers of multilayer board with <math>0.5mm</math> thickness are buried in a substrate with <math>10mm</math> thickness and relative permittivity
	<math> \epsilon_r = 4.4 </math> and relative permeability <math> \mu_r = 1 </math>, and low conductivity <math> \sigma = 0.02 S/m </math>. The low-loss upper layer has low permittivity <math> \epsilon_r = 1.07 </math> and		<math>\epsilon_r = 4.4 </math> and relative permeability <math>\mu_r = 1 </math>, and low conductivity <math>\sigma = 0.02 S/m </math>.
	<math> \sigma = 0.01 S/m </math>. 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.		The low-loss upper layer has low permittivity <math>\epsilon_r = 1.07 </math> and <math>\sigma = 0.01 S/m</math>.
			The whole structure is enlosed in a metallic box of dimension <math>140mm</math> by <math>100mm</math> by <math>50mm</math>.
			The discrete port with <math>50Ohm</math> lumped load imposes <math>1 A</math> 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 id="fig:coplanar">		<figure id="fig:coplanar">
	[[File:CoplanarWaveguideScaled.jpg\|frame\|<caption>Coplanar Waveguide Model<ref~~>M. W. Hess, P. Benner, "<span class~~="~~plainlinks~~">[http://www.mpi-magdeburg.mpg.de/preprints/2012/MPIMD12-17.pdf Fast Evaluation of Time-Harmonic Maxwell's Equations Using the Reduced Basis Method]</span>", IEEE Transactions on Microwave Theory and Techniques, DOI 10.1109/TMTT.2013.2258167 , 2013.</~~ref~~></caption>]]		[[File:CoplanarWaveguideScaled.jpg\|frame\|<caption>Coplanar Waveguide Model<ref name="hess2003"/></caption>]]
	</figure>		</figure>

Line 31:		Line 34:
	appearance in the summation.		appearance in the summation.

	~~The files are numbered according to their appearance in the summation and can be found here: [[Media:Matrices_cp.tar.gz\|Matrices_cp.tar.gz]]~~		The coefficient functions are given by:


	The coefficient functions are given by

	:<math> \Theta^1(\omega, \nu) = 1 </math>		:<math> \Theta^1(\omega, \nu) = 1 </math>
Line 66:		Line 66:
	:<math> \Theta^{15}(\omega, \nu) = -\frac{16 - \nu}{10} \omega^2 </math>		:<math> \Theta^{15}(\omega, \nu) = -\frac{16 - \nu}{10} \omega^2 </math>

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

	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		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

	<math> s(u) = \| l^T * u \| </math> with complex vector <math> u </math> turns into <math> s(u) = \sqrt{ (l_1^T * u)^2 + (l_2^T * u)^2 } </math> with real vector <math> u </math>.		<math>s(u) = \| l^T * u \|</math> with complex vector <math>u</math> turns into <math>s(u) = \sqrt{ (l_1^T * u)^2 + (l_2^T * u)^2 }</math> with real vector <math> u </math>.

	==Origin==		==Origin==

	The models have been developed within the [http://www.moresim4nano.org MoreSim4Nano project].		The models have been developed within the [http://www.moresim4nano.org MoreSim4Nano] project.

			==Data==

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

	==References==		==References==

	<references/>		<references>

			<ref name="hess2003">M.W. Hess, P. Benner, "<span class="plainlinks">[https://doi.org/10.1109/TMTT.2013.2258167 Fast Evaluation of Time-Harmonic Maxwell's Equations Using the Reduced Basis Method]</span>", IEEE Transactions on Microwave Theory and Techniques, 61(6): 2265--2274, 2013.</ref>

			</references>

	==Contact==		==Contact==

	'' [[User:hessm\|Martin Hess]]''		'' [[User:hessm\|Martin Hess]]''

Hessm: link to matrices corrected

2014-07-14T13:02:47Z

link to matrices corrected

← Older revision		Revision as of 13:02, 14 July 2014
Line 31:		Line 31:
	appearance in the summation.		appearance in the summation.

	The files are numbered according to their appearance in the summation and can be found here: [[Media:~~Matrices~~.tar.gz\|~~Matrices~~.tar.gz]]		The files are numbered according to their appearance in the summation and can be found here: [[Media:Matrices_cp.tar.gz\|Matrices_cp.tar.gz]]