Silicon Nitride Membrane - Revision history

Malhotra: infobox string

2023-11-30T09:40:37Z

infobox string

@@ Line 4: / Line 4: @@
 [[Category:first differential order]]
 [[Category:time invariant]]
 ==Description==

Lund: /* Dimensions */ re-index

2023-05-09T14:42:14Z

Dimensions: re-index

← Older revision		Revision as of 14:42, 9 May 2023
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	:<math>		:<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	(~~E_0~~ + \rho c_p ~~E_1~~)\dot{x}(t) &=& -(~~A_0~~ + \kappa ~~A_1~~ + h ~~A_2~~)x(t) + Bu(t) \\		(E_1 + \rho c_p E_2)\dot{x}(t) &=& -(A_1 + \kappa A_2 + h A_3)x(t) + Bu(t) \\
	y(t) &=& Cx(t)		y(t) &=& Cx(t)
	\end{array}		\end{array}
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	System dimensions:		System dimensions:

	<math>~~E_0 \in \mathbb~~{R}~~^{60020 \times 60020}</math>,~~		<math>E_{1,2} \in \mathbb{R}^{60020 \times 60020}</math>,
	~~<math>E_1~~ \in \mathbb{R}^{60020 \times 60020}</math>,		<math>A_{1,2,3} \in \mathbb{R}^{60020 \times 60020}</math>,
	<math>~~A_0 \in \mathbb~~{~~R}^{60020 \times 60020}</math>~~,
	~~<math>A_1 \in \mathbb{R~~}~~^{60020 \times 60020}</math>,~~
	~~<math>A_2~~ \in \mathbb{R}^{60020 \times 60020}</math>,
	<math>B \in \mathbb{R}^{60020 \times 1}</math>,		<math>B \in \mathbb{R}^{60020 \times 1}</math>,
	<math>C \in \mathbb{R}^{2 \times 60020}</math>.		<math>C \in \mathbb{R}^{2 \times 60020}</math>.

Lund: /* Dimensions */ typo

2023-04-24T12:17:13Z

Dimensions: typo

← Older revision		Revision as of 12:17, 24 April 2023
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	:<math>		:<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	(E_0 + \rho c_p E_1)\dot{x}(t) &=& -(A_0 + \kappa A_1 + h ~~A_3~~)x(t) + Bu(t) \\		(E_0 + \rho c_p E_1)\dot{x}(t) &=& -(A_0 + \kappa A_1 + h A_2)x(t) + Bu(t) \\
	y(t) &=& Cx(t)		y(t) &=& Cx(t)
	\end{array}		\end{array}

Lund: /* Dimensions */ Correct number of inputs (data shows one)

2023-04-20T13:32:00Z

Dimensions: Correct number of inputs (data shows one)

← Older revision		Revision as of 13:32, 20 April 2023
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	<math>A_1 \in \mathbb{R}^{60020 \times 60020}</math>,		<math>A_1 \in \mathbb{R}^{60020 \times 60020}</math>,
	<math>A_2 \in \mathbb{R}^{60020 \times 60020}</math>,		<math>A_2 \in \mathbb{R}^{60020 \times 60020}</math>,
	<math>B \in \mathbb{R}^{60020 \times 2}</math>,		<math>B \in \mathbb{R}^{60020 \times 1}</math>,
	<math>C \in \mathbb{R}^{2 \times 60020}</math>.		<math>C \in \mathbb{R}^{2 \times 60020}</math>.

Himpe at 13:01, 25 August 2022

2022-08-25T13:01:20Z

← Older revision		Revision as of 13:01, 25 August 2022
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	[[Category:benchmark]]		[[Category:benchmark]]
	[[Category:~~parametric 2-5 parameters~~]]		[[Category:Parametric]]
	[[Category:linear]]		[[Category:linear]]
	[[Category:first differential order]]		[[Category:first differential order]]

Mitchell: I believe "optimical" should be "optical" but a bit unclear (optimal?)

2022-03-15T11:57:50Z

I believe "optimical" should be "optical" but a bit unclear (optimal?)

← Older revision		Revision as of 11:57, 15 March 2022
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	A '''silicon nitride membrane''' ([[wikipedia:Silicon_nitride\|SiN]] membrane) <ref name="bechthold10"/> can be a part of a gas sensor,		A '''silicon nitride membrane''' ([[wikipedia:Silicon_nitride\|SiN]] membrane) <ref name="bechthold10"/> can be a part of a gas sensor,
	but also a part of an infra-red sensor, a microthruster, an ~~optimical~~ filter etc.		but also a part of an infra-red sensor, a microthruster, an optical filter etc.
	This structure resembles a microhotplate similar to other micro-fabricated devices such as gas sensors <ref name="spannhake05"/> and infrared sources <ref name="graf04"/> (See also [[Gas_Sensor\|Gas Sensor Benchmark]]).		This structure resembles a microhotplate similar to other micro-fabricated devices such as gas sensors <ref name="spannhake05"/> and infrared sources <ref name="graf04"/> (See also [[Gas_Sensor\|Gas Sensor Benchmark]]).
	See Fig. 1, the temperature profile for the SiN membrane.		See Fig. 1, the temperature profile for the SiN membrane.

Mitchell: typo

2022-03-15T11:56:28Z

typo

← Older revision		Revision as of 11:56, 15 March 2022
Line 10:		Line 10:

	A '''silicon nitride membrane''' ([[wikipedia:Silicon_nitride\|SiN]] membrane) <ref name="bechthold10"/> can be a part of a gas sensor,		A '''silicon nitride membrane''' ([[wikipedia:Silicon_nitride\|SiN]] membrane) <ref name="bechthold10"/> can be a part of a gas sensor,
	but also a part of an infra-red sensor, a ~~michrothruster~~, an optimical filter etc.		but also a part of an infra-red sensor, a microthruster, an optimical filter etc.
	This structure resembles a microhotplate similar to other micro-fabricated devices such as gas sensors <ref name="spannhake05"/> and infrared sources <ref name="graf04"/> (See also [[Gas_Sensor\|Gas Sensor Benchmark]]).		This structure resembles a microhotplate similar to other micro-fabricated devices such as gas sensors <ref name="spannhake05"/> and infrared sources <ref name="graf04"/> (See also [[Gas_Sensor\|Gas Sensor Benchmark]]).
	See Fig. 1, the temperature profile for the SiN membrane.		See Fig. 1, the temperature profile for the SiN membrane.

Himpe: Fix CrossRefs

2021-08-27T07:55:35Z

Fix CrossRefs

← Older revision		Revision as of 07:55, 27 August 2021
Line 12:		Line 12:
	but also a part of an infra-red sensor, a michrothruster, an optimical filter etc.		but also a part of an infra-red sensor, a michrothruster, an optimical filter etc.
	This structure resembles a microhotplate similar to other micro-fabricated devices such as gas sensors <ref name="spannhake05"/> and infrared sources <ref name="graf04"/> (See also [[Gas_Sensor\|Gas Sensor Benchmark]]).		This structure resembles a microhotplate similar to other micro-fabricated devices such as gas sensors <ref name="spannhake05"/> and infrared sources <ref name="graf04"/> (See also [[Gas_Sensor\|Gas Sensor Benchmark]]).
	See ~~<xr id="fig:tempprof"/>~~, the temperature profile for the SiN membrane.		See Fig. 1, the temperature profile for the SiN membrane.

	The governing heat transfer equation in the membrane is:		The governing heat transfer equation in the membrane is:

Himpe: Add dimensions section and various other fixes.

2018-11-07T10:08:18Z

Add dimensions section and various other fixes.

← Older revision		Revision as of 10:08, 7 November 2018
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	[[Category:time invariant]]		[[Category:time invariant]]

	==Description==		==Description==

	<figure id="fig:tempprof">[[File:Fig_1.png\|right\|frame\|<caption>silicon nitride membrane temperature profile</caption>]]</figure>		<figure id="fig:tempprof">[[File:Fig_1.png\|right\|frame\|<caption>silicon nitride membrane temperature profile</caption>]]</figure>

	A '''silicon nitride membrane''' (SiN membrane) <ref~~>T. Bechtold, D. Hohfeld, E. B. Rudnyi and M. Guenther, "<span class~~="~~plainlinks~~"~~>[http://dx.doi.org/10.1088/0960-1317/20~~/~~4/045030 Efficient extraction of thin-film thermal parameters from numerical models via parametric model order reduction]</span>", J. Micromech. Microeng. 20(2010) 045030 (13pp).</ref~~> can be a part of a gas sensor, but also a part of an infra-red sensor, a michrothruster, an optimical filter etc. This structure resembles		A '''silicon nitride membrane''' ([[wikipedia:Silicon_nitride\|SiN]] membrane) <ref name="bechthold10"/> can be a part of a gas sensor,
	a microhotplate similar to other micro-fabricated devices		but also a part of an infra-red sensor, a michrothruster, an optimical filter etc.
	such as gas sensors <ref~~>J. Spannhake, O. Schulz, A. Helwig, G. Müller and T. Doll, "<span class~~="~~plainlinks">[http://dx.doi.org/10.1109/ICSENS.2005.1597811 Design, development and operational concept of an advanced MEMS IR source for miniaturized gas sensor systems]</span>~~"~~, Proc. Sensors, 762-765, 2005.<~~/~~ref~~> and infrared sources <ref~~>M. Graf, D. Barrettino, S. Taschini, C. Hagleitner, A. Hierlemann and H. Baltes, "<span class~~="~~plainlinks~~">[~~http://dx.doi.org/10.1021/ac035432h Metal oxide-based monolithic complementary metal oxide semiconductor gas sensor microsystem~~]~~</span>", Anal. Chem., 76:4437-4445, 2004.</ref>~~. See <xr id="fig:tempprof"/>, the temperature profile for the SiN membrane.		This structure resembles a microhotplate similar to other micro-fabricated devices such as gas sensors <ref name="spannhake05"/> and infrared sources <ref name="graf04"/> (See also [[Gas_Sensor\|Gas Sensor Benchmark]]).
			See <xr id="fig:tempprof"/>, the temperature profile for the SiN membrane.

	The governing heat transfer equation in the membrane is:		The governing heat transfer equation in the membrane is:
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	:<math> \nabla \cdot (\kappa \nabla T)+Q - \rho c_p \cdot \frac{\partial T}{\partial t}=0 </math>		:<math> \nabla \cdot (\kappa \nabla T)+Q - \rho c_p \cdot \frac{\partial T}{\partial t}=0 </math>

	where <math>\kappa </math> is the thermal conductivity in <math>W m^{-1} K^{-1}</math>, cp is the specific heat capacity in <math> J kg^{-1} K^{-1}</math>, <math>\rho</math> is the mass density in <math>kg m^{-3}</math> and <math>T</math> is the temperature distribution. We assume a homogeneous heat generation rate over a lumped resistor:		where <math>\kappa </math> is the thermal conductivity in <math>W m^{-1} K^{-1}</math>,
			<math>c_p</math> is the specific heat capacity in <math>J kg^{-1} K^{-1}</math>,
			<math>\rho</math> is the mass density in <math>kg m^{-3}</math> and <math>T</math> is the temperature distribution.
			We assume a homogeneous heat generation rate over a lumped resistor:

	:<math>Q = \frac{u^2(t)}{R(T)}</math>		:<math>Q = \frac{u^2(t)}{R(T)}</math>

	with <math>Q</math> the heat generation rate per unit volume in <math>W m^{-3}</math>.		with <math>Q</math> the heat generation rate per unit volume in <math>W m^{-3}</math>.
	We use the initial condition <math> T_0 = 273K </math>, and the		We use the initial condition <math>T_0 = 273K</math>,
	Dirichlet boundary condition <math> T = 273 K </math> at the bottom of		and the Dirichlet boundary condition <math>T = 273 K</math> at the bottom of the computational domain.
	the computational domain.

	The convection boundary condition at the top of the membrane is		The convection boundary condition at the top of the membrane is
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	==Discretization==		==Discretization==

	Under the above convection boundary condition and assuming <math>T_{air}=0</math>, finite element discretization of the heat transfer model leads to the parametrized system as below,		Under the above convection boundary condition and assuming <math>T_{air}=0</math>,
			a finite element discretization of the heat transfer model leads to the parametrized system as below,

	:<math>		:<math>
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	</math>		</math>

	where the volumetric heat capacity <math>\rho c_p</math>, thermal conductivity		where the volumetric heat capacity <math>\rho c_p</math>,
	<math>\kappa</math> and the heat transfer coefficient <math>h</math> between the membrane		thermal conductivity <math>\kappa</math> and the heat transfer coefficient <math>h</math> between the membrane are kept as parameters.
	are kept as parameters. The volumetric hear capacity <math>\rho c_p</math> is the product of two independent variables, i.e. the specific hear capacity <math>c_p</math> and the density <math>\rho</math>. The range of interest for the four independent variables are ~~respectively~~ <math>\kappa \in [2, 5]</math>, <math>c_p \in [400, 750]</math>, <math>\rho \in [3000,3200]</math>, <math> h \in [10, 12]</math>. The frequency range is <math>f \in [0,25]Hz</math>.		The volumetric hear capacity <math>\rho c_p</math> is the product of two independent variables,
	What is of interest is the output in time domain. The interesting time interval is <math>t \in [0,0.04]s</math>		i.e. the specific hear capacity <math>c_p</math> and the density <math>\rho</math>.
			The range of interest for the four independent variables are <math>\kappa \in [2, 5]</math>, <math>c_p \in [400, 750]</math>, <math>\rho \in [3000,3200]</math> and <math> h \in [10, 12]</math> respectively.
			The frequency range is <math>f \in [0,25]Hz</math>.
			What is of interest is the output in time domain.
			The interesting time interval is <math>t \in [0,0.04]s</math>.

	Here <math>R(T)</math> is either a constant heat resistivity <math>R(T)=R_0</math>, or <math>R(T)=R_0(1+\alpha T)</math>, which depends linearly on the temperature. Here we use <math>R_0=274.94 \Omega</math> and temperature coefficient <math>\alpha=2.293 \pm 0.006 \times 10^{-4}</math>. The model was created and meshed in ANSYS. It contains a constant load vector <math>B</math> corresponding to the constant input power of <math>2.49mW</math>. The number of degrees of freedom is <math>n=60,020</math>.		Here <math>R(T)</math> is either a constant heat resistivity <math>R(T)=R_0</math>, or <math>R(T)=R_0(1+\alpha T)</math>,
			which depends linearly on the temperature.
			Here we use <math>R_0=274.94 \Omega</math> and temperature coefficient <math>\alpha=2.293 \pm 0.006 \times 10^{-4}</math>.
			The model was created and meshed in ANSYS. It contains a constant load vector <math>B</math> corresponding to the constant input power of <math>2.49mW</math>.
			The number of degrees of freedom is <math>n=60,020</math>.

	The input function <math>u(t)</math> is a step function with the value <math>1</math>, which disappears at the time <math>0.02s</math>. This means between <math>0s</math> and <math>0.02s</math> input is one and after that it is zero. However, be aware that <math>u(t)</math> is just a factor with which the load vector B is multiplied and which corresponds to the heating power of <math>2.49mW</math>. This means if one keeps <math>u(t)</math> as suggested above, the device is heated with <math>2.49mW</math> for the time length of 0.02s and after that the heating is turned off. If for whatever reason, one wants the heating power to be <math>5mW</math>, then <math>u(t)</math> has to be set equal to two, etc...		The input function <math>u(t)</math> is a step function with the value <math>1</math>,
	When <math>R(T)=R_0(1+\alpha T)</math>, it is a function of the state vector <math>T</math> and hence, the system has non-linear input. (It is also called a weakly nonlinear system.)		which disappears at the time <math>0.02s</math>.
			This means between <math>0s</math> and <math>0.02s</math> input is one and after that it is zero.
			However, be aware that <math>u(t)</math> is just a factor with which the load vector B is multiplied and which corresponds to the heating power of <math>2.49mW</math>.
			This means if one keeps <math>u(t)</math> as suggested above, the device is heated with <math>2.49mW</math> for the time length of 0.02s and after that the heating is turned off.
			If for whatever reason, one wants the heating power to be <math>5mW</math>, then <math>u(t)</math> has to be set equal to two, etc.
			When <math>R(T)=R_0(1+\alpha T)</math>, it is a function of the state vector <math>T</math> and hence, the system has non-linear input (It is also called a weakly nonlinear system.).

	==Data==		==Data==

	The model is generated in ANSYS. The system matrices are in <span class="plainlinks">[http://math.nist.gov/MatrixMarket/ MatrixMarket]</span> format and can be downloaded here: [[Media: SiN_membrane.tgz\|SiN_membrane.tgz]].		The model is generated in ANSYS. The system matrices are in <span class="plainlinks">[http://math.nist.gov/MatrixMarket/ MatrixMarket]</span> format and can be downloaded here: [[Media: SiN_membrane.tgz\|SiN_membrane.tgz]].

			==Dimensions==

			System structure:

			:<math>
			\begin{array}{rcl}
			(E_0 + \rho c_p E_1)\dot{x}(t) &=& -(A_0 + \kappa A_1 + h A_3)x(t) + Bu(t) \\
			y(t) &=& Cx(t)
			\end{array}
			</math>

			System dimensions:

			<math>E_0 \in \mathbb{R}^{60020 \times 60020}</math>,
			<math>E_1 \in \mathbb{R}^{60020 \times 60020}</math>,
			<math>A_0 \in \mathbb{R}^{60020 \times 60020}</math>,
			<math>A_1 \in \mathbb{R}^{60020 \times 60020}</math>,
			<math>A_2 \in \mathbb{R}^{60020 \times 60020}</math>,
			<math>B \in \mathbb{R}^{60020 \times 2}</math>,
			<math>C \in \mathbb{R}^{2 \times 60020}</math>.

	==References==		==References==

	<references/>		<references>

			<ref name="bechthold10">T. Bechtold, D. Hohfeld, E. B. Rudnyi and M. Guenther, "<span class="plainlinks">[https://doi.org/10.1088/0960-1317/20/4/045030 Efficient extraction of thin-film thermal parameters from numerical models via parametric model order reduction]</span>", J. Micromech. Microeng. 20(4): 045030, 2010.</ref>

			<ref name="spannhake05">J. Spannhake, O. Schulz, A. Helwig, G. Müller and T. Doll, "<span class="plainlinks">[https://doi.org/10.1109/ICSENS.2005.1597811 Design, development and operational concept of an advanced MEMS IR source for miniaturized gas sensor systems]</span>", Proc. Sensors: 762--765, 2005.</ref>

			<ref name="graf04">M. Graf, D. Barrettino, S. Taschini, C. Hagleitner, A. Hierlemann and H. Baltes, "<span class="plainlinks">[https://doi.org/10.1021/ac035432h Metal oxide-based monolithic complementary metal oxide semiconductor gas sensor microsystem]</span>", Anal. Chem., 76(15): 4437--4445, 2004.</ref>

			</references>

	==Contact==		==Contact==

Himpe: Himpe moved page Silicon nitride membrane to Silicon Nitride Membrane: consistent capitalization

2018-08-01T11:38:15Z

Himpe moved page Silicon nitride membrane to Silicon Nitride Membrane: consistent capitalization

← Older revision	Revision as of 11:38, 1 August 2018
(No difference)