Difference between revisions of "Gas Sensor"

Latest revision as of 07:31, 17 June 2025

Gas Sensor
Background
Benchmark ID	gasSensor_n66917m1q28
Category	oberwolfach
System-Class	LTI-FOS
Parameters
nstates	66917
ninputs	1
noutputs	28
nparameters	0
components	A, B, C, E
Copyright
License	NA
Creator	Lihong Feng
Editor	Lihong Feng Christian Himpe Judith Schneider
Location
NA

Description: Microhotplate Gas Sensor

Figure 1

Figure 2: Masks disposition (left) and the schematical position of the chosen output nodes (right).

The goal of European project Glassgas (IST-99-19003) was to develop a novel metal oxide low power microhotplate gas sensor ^[1]. In order to assure a robust design and good thermal isolation of the membrane from the surrounding wafer, the silicon microhotplate is supported by glass pillars emanating from a glass cap above the silicon wafer, as shown in Fig. 1. In this design, four different sensitive layers can be deposited on the membrane. The thermal management of a microhotplate gas sensor is of crucial importance.

The benchmark contains a thermal model of a single gas sensor device with three main components: a silicon rim, a silicon hotplate and glass structure ^[2]. It allows us to simulate important thermal issues, such as the homogeneous temperature distribution over gas sensitive regions or thermal decoupling between the hotplate and the silicon rim. The original model is the heat transfer partial differential equation.

The device solid model has been made and then meshed and discretized in ANSYS 6.1 by means of the finite element method (SOLID70 elements were used). It contains 68000 elements and 73955 nodes. Material properties were considered as temperature independent. Temperature is assumed to be in degree Celsius with the initial state of $0 C$ . The Dirichlet boundary conditions of $T = 0 C$ is applied at the top and bottom of the chip (at 7038 nodes).

The output nodes are described in Table 1. In Fig. 2 the red marked nodes are positioned on the silicon rim. Their temperature should be close to the initial temperature in the case of good thermal decoupling between the membrane and the silicon rim. The black marked nodes are placed on the sensitive layers above the heater and are numbered from left to right row by row, as schematically shown in Fig 2. They allow us to prove whether the temperature distribution over the gas sensitive layers is homogeneous (maximum difference of $10C$ is allowed by design).

*Table 1: Inputs and outputs for the gas sensor model.*
Number	Code	Comment
1	aHeater	within a heater, to be used for nonlinear input
2-7	SiRim1 to SiRim7	silicon rim
8-28	Memb1 to Memb21	gas sensitive layer

The benchmark contains a constant load vector. The input function equal to $u(t) = 1$ corresponds to the constant input power of $340 mW$ . One can insert a weak input nonlinearity related to the dependence of heater's resistivity on temperature given as:

R(T) = R_{0}(1 + \alpha T)

where $\alpha =1.469 \cdot 10^{-3} K^{-1}$ . To this end, one has to multiply the load vector by a function:

\frac{U^2 274.94 (1 + \alpha T)}{0.34 (274.94 (1 + \alpha T)+148.13)^2}

where $U$ is a desired constant voltage. The temperature in the equation above should be replaced by the temperature at the input 1 (aHeater).

The linear ordinary differential equations of the first order are written as:

\begin{array}{rcl} E \frac{\partial T}{\partial t} &=& A T(t) + B u(t) \\ y(t) &=& C T(t) \end{array}

where $E$ and $A$ are the symmetric sparse system matrices (heat capacity and heat conductivity matrix), $B$ is the load vector, $C$ is the output matrix, and $T$ is the vector of unknown temperatures. The dimension of the system is $66917$ , the number of nonzero elements in matrix $E$ is $66917$ , in matrix $A$ is $885141$ .

The outputs of the transient simulation at output 18 (Memb11) over the rise time of the device of $5 s$ for the original linear (with constant input power of $340 mW$ ) and nonlinear (with constant voltage of $14 V$ ) model are placed in files LinearResults and NonlinearResults respectively. The results can be used to compare the solution of a reduced model with the original one. The time integration has been performed in ANSYS with accuracy of about $0.1 \%$ . The results are given as matrices where the first row is made of times, the second of the temperatures.

More information can also be found in ^[3].

Data

Download matrices in the Matrix Market format: GasSensor-dim1e5-GasSensor.tar.gz, (8 MB).

The matrix name is used as an extension of the matrix file. File *.C names contains a list of output names written consecutively. The system matrices have been extracted from ANSYS models by means of mor4fem.

The discussion of electro-thermal modeling related to the benchmark including the nonlinear input function can be found in ^[4].

Origin

This benchmark is part of the Oberwolfach Benchmark Collection^[5]; No. 38880, see ^[6].

Dimensions

System structure:

\begin{align} E \dot{x}(t) &= Ax(t) + B \\ y(t) &= Cx(t) \end{align}

System dimensions:

$E \in \mathbb{R}^{66917 \times 66917}$ , $A \in \mathbb{R}^{66917 \times 66917}$ , $B \in \mathbb{R}^{66917 \times 1}$ , $C \in \mathbb{R}^{1 \times 66917}$ .

Citation

To cite this benchmark, use the following references:

For the benchmark itself and its data:

The MORwiki Community, Gas Sensor. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Gas_Sensor

@MISC{morwiki_gas,
  author =       {{The MORwiki Community}},
  title =        {Gas Sensor},
  howpublished = {{MORwiki} -- Model Order Reduction Wiki},
  url =          {https://modelreduction.org/morwiki/Gas_Sensor},
  year =         {20XX}
}

For the background on the benchmark:

@INPROCEEDINGS{BecHWetal04,
  author =       {T. Bechtold, J. Hildenbrand, J. Wöllenstein, J. G. Korvink},
  title =        {Model Order Reduction of 3D Electro-Thermal Model for a Novel, Micromachined Hotplate Gas Sensor},
  booktitle =    {Proceedings of 5th International Conference on Thermal and Mechanical Simulation and Experiments in Microelectronics and Microsystems},
  pages =        {263--267},
  year =         {2004},
  doi =          {10.1109/ESIME.2004.1304049}
}

References

↑ J. Wöllenstein, H. Böttner, J.A. Pláza, C. Carné, Y. Min, H.L. Tuller, A novel single chip thin film metal oxide array, Sensors and Actuators B: Chemical 93 (1-3): 350--355, 2003.
↑ J. Hildenbrand, Simulation and Characterisation of a Gas sensor and Preparation for Model Order Reduction, Diploma Thesis, University of Freiburg, Germany, 2003.
↑ T. Bechtold, "Model Order Reduction of Electro-Thermal MEMS", PhD thesis, Department of Microsystems Engineering, University of Freiburg, 2005.
↑ T. Bechtold, J. Hildenbrand, J. Wöllenstein, J. G. Korvink, Model Order Reduction of 3D Electro-Thermal Model for a Novel, Micromachined Hotplate Gas Sensor, Proceedings of 5th International Conference on Thermal and Mechanical Simulation and Experiments in Microelectronics and Microsystems, EUROSIME2004, May 10-12, 2004, Brussels, Belgium: 263--267, 2004.
↑ J.G. Korvink, E.B. Rudnyi, Oberwolfach Benchmark Collection, Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.
↑ Hildenbrand J., Bechtold T., J. Wöllenstein, Microhotplate Gas Sensor. In: Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol 45: 333-336, 2005.

Contact

Lihong Feng

Tamara Bechtold

[wollenstein03-1] J. Wöllenstein, H. Böttner, J.A. Pláza, C. Carné, Y. Min, H.L. Tuller, A novel single chip thin film metal oxide array, Sensors and Actuators B: Chemical 93 (1-3): 350--355, 2003.

[hildenbrand03-2] J. Hildenbrand, Simulation and Characterisation of a Gas sensor and Preparation for Model Order Reduction, Diploma Thesis, University of Freiburg, Germany, 2003.

[bechthold05-3] T. Bechtold, "Model Order Reduction of Electro-Thermal MEMS", PhD thesis, Department of Microsystems Engineering, University of Freiburg, 2005.

[bechthold04-4] T. Bechtold, J. Hildenbrand, J. Wöllenstein, J. G. Korvink, Model Order Reduction of 3D Electro-Thermal Model for a Novel, Micromachined Hotplate Gas Sensor, Proceedings of 5th International Conference on Thermal and Mechanical Simulation and Experiments in Microelectronics and Microsystems, EUROSIME2004, May 10-12, 2004, Brussels, Belgium: 263--267, 2004.

[korvink2005-5] J.G. Korvink, E.B. Rudnyi, Oberwolfach Benchmark Collection, Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.

[hildenbrand2005-6] Hildenbrand J., Bechtold T., J. Wöllenstein, Microhotplate Gas Sensor. In: Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol 45: 333-336, 2005.

[1]

[2]

[3]

[4]

[5]

[6]

@@ Line 1: / Line 1: @@
-[[Category:PMOR benchmark, linear, time invariant, four physical parameters, first order system]]
+[[Category:benchmark]]
+[[Category:Oberwolfach]]
+[[Category:linear]]
+[[Category:ODE]]
+[[Category:first differential order]]
+[[Category:time invariant]]
+{{Infobox
-This is an extension of the non-parametrized model of Gas sensor in the Oberwolfach Model Reduction Benchmark Collection (http://simulation.uni-freiburg.de/downloads/benchmark Gas sensor(38880)) to a parametrized model.
+|Title           = Gas Sensor
+|Benchmark ID    = gasSensor_n66917m1q28
+|Category        = oberwolfach
+|System-Class    = LTI-FOS
+|nstates         = 66917
+|ninputs         = 1
+|noutputs        = 28
+|nparameters     = 0
+|components      = A, B, C, E
+|License         = NA
+|Creator         = [[User:Feng]]
+|Editor          =
+* [[User:Feng]]
+* [[User:Himpe]]
+* [[User:Will]]
+|Zenodo-link     = NA
+}}
-==Description of the device==
+==Description: Microhotplate Gas Sensor==
+<figure id="fig1">[[File:GasSensor1.jpg|490px|thumb|right|Figure 1]]</figure>
+<figure id="fig2">[[File:GasSensor2.jpg|490px|thumb|right|<caption>Masks disposition (left) and the schematical position of the chosen output nodes (right).</caption>]]</figure>
+The goal of European project [https://web.archive.org/web/20050826075758/http://www.cnm.es:80/imb/glassgas/index.htm Glassgas] (IST-99-19003) was to develop a novel metal oxide low power '''microhotplate gas sensor''' <ref name="wollenstein03"/>.
-There is a large demand for gas sensing devices in various domains. They are
+In order to assure a robust design and good thermal isolation of the membrane from the surrounding wafer, the silicon microhotplate is supported by glass pillars emanating from a glass cap above the silicon wafer, as shown in Fig.&nbsp;1.
-desired in e. g. safety applications where combustible or toxic gases are present or
+In this design, four different sensitive layers can be deposited on the membrane.
-in comfort applications, such as climate controls of buildings and vehicles where
+The thermal management of a '''microhotplate gas sensor''' is of crucial importance.
-good air quality is required. Additionally, gas monitoring is needed in process
-control and laboratory analytics. All of these applications demand cheap, small
-and user-friendly gas sensing devices which show high sensitivity, selectivity and
-stability with respect to a given application.
+The benchmark contains a thermal model of a single gas sensor device with three main components:
-A micromachined gas sensor is not only a challenge with respect to thermal
+a silicon rim, a silicon hotplate and glass structure <ref name="hildenbrand03"/>.
-design but also with respect to mechanical design. Only by choosing the right
+It allows us to simulate important thermal issues, such as the homogeneous temperature distribution over gas sensitive regions or thermal decoupling between the hotplate and the silicon rim.
-mechanical design a large intrinsic or thermal-induced membrane stress leading
+The original model is the heat transfer partial differential equation.
-to membrane deformation/ breaking of the membrane can be avoided. It is further
-necessary to build a chemometrics calibration model which correlates the set of
-sensor resistance measurements to the sensed gas concentration. Prior to fabrication,
-a thermal simulation is performed to determine the heating efficiency and
-temperature homogeneity of the gas sensitive regions. As the device is connected to circuitry for heating power
-control and sensing resistor readout, a system-level simulation is also needed.
-Hence, a compact thermal model must be generated. (The text above is taken from [1].)
+The device solid model has been made and then meshed and discretized in [http://www.ansys.com ANSYS] 6.1 by means of the finite element method (<tt>SOLID70</tt> elements were used).
-==Description of the model==
+It contains 68000 elements and 73955 nodes.
+Material properties were considered as temperature independent.
+Temperature is assumed to be in degree Celsius with the initial state of <math>0 C</math>.
+The Dirichlet boundary conditions of <math>T = 0 C</math> is applied at the top and bottom of the chip (at 7038 nodes).
+The output nodes are described in Table&#160;1.
-The heat transfer within a hotplate is described through the governing heat transfer equation [2]
+In Fig.&nbsp;2 the red marked nodes are positioned on the silicon rim.
+Their temperature should be close to the initial temperature in the case of good thermal decoupling between the membrane and the silicon rim.
+The black marked nodes are placed on the sensitive layers above the heater and are numbered from left to right row by row, as schematically shown in Fig 2.
+They allow us to prove whether the temperature distribution over the gas sensitive layers is homogeneous (maximum difference of <math>10C</math> is allowed by design).
+{| class="wikitable" style="margin: auto;"
-<math>
+|+ style="caption-side:bottom;"|''Table 1: Inputs and outputs for the gas sensor model.''
-\nabla \cdot (\kappa \nabla T) + Q- \rho c_p \frac{\partial T}{\partial t}=0, \quad
+|-
-Q=j^2R(T), \quad (1)
+|Number
+|Code
+|Comment
+|-
+|1
+|aHeater
+|within a heater, to be used for nonlinear input
+|-
+|2-7
+|SiRim1 to SiRim7
+|silicon rim
+|-
+|8-28
+|Memb1 to Memb21
+|gas sensitive layer
+|}
+The benchmark contains a constant load vector.
+The input function equal to <math>u(t) = 1</math> corresponds to the constant input power of <math>340 mW</math>.
+One can insert a weak input nonlinearity related to the dependence of heater's resistivity on temperature given as:
+:<math>
+R(T) = R_{0}(1 + \alpha T)
 </math>
+where <math>\alpha =1.469 \cdot 10^{-3} K^{-1}</math>.
-where <math>\kappa(r)</math> is the thermal conductivity in <math>W/(m*K)</math> at the position <math>r, \, c_p</math> is the
+To this end, one has to multiply the load vector by a function:
-specific heat capacity in <math>J /(kg*K), \, \rho(r)</math> is the mass density in
-<math>kg /m^3</math> and <math>T(r,t)</math> is the temperature distribution. We assume a
-homogeneous heat generation rate over a lumped resistor:
-<math>
+:<math>
-Q = \frac{u^2(t)}{R(T)}
+\frac{U^2 274.94 (1 + \alpha T)}{0.34 (274.94 (1 + \alpha T)+148.13)^2}
 </math>
-with unit <math>W/m^3</math>.
+where <math>U</math> is a desired constant voltage.
+The temperature in the equation above should be replaced by the temperature at the input 1 (aHeater).
-We use the initial condition <math> T_0 = 273K </math>, and the
-Dirichlet boundary condition <math> T = 273 K </math> at the bottom of
-the computational domain. The convection boundary condition at the top of the membrane is
+The linear ordinary differential equations of the first order are written as:
-<math>
-q=h(T-T_{air}),
+:<math>
+\begin{array}{rcl}
+E \frac{\partial T}{\partial t} &=& A T(t) + B u(t) \\
+y(t) &=& C T(t)
+\end{array}
 </math>
-where <math>h</math> is the heat transfer coefficient between the membrane and the ambient air in <math>W/m^2/K</math>.
+where <math>E</math> and <math>A</math> are the symmetric sparse system matrices (heat capacity and heat conductivity matrix), <math>B</math> is the load vector, <math>C</math> is the output matrix, and <math>T</math> is the vector of unknown temperatures.
+The dimension of the system is <math>66917</math>, the number of nonzero elements in matrix <math>E</math> is <math>66917</math>, in matrix <math>A</math> is <math>885141</math>.
+The outputs of the transient simulation at output 18 (Memb11) over the rise time of the device of <math>5 s</math> for the original linear (with constant input power of <math>340 mW</math>) and nonlinear (with constant voltage of <math>14 V</math>) model are placed in files <tt>LinearResults</tt> and <tt>NonlinearResults</tt> respectively.
-Assuming <math>T_{air}=0</math>, spatial discretization of the heat transfer model in (1) leads to the parametrized system as below,
+The results can be used to compare the solution of a reduced model with the original one.
+The time integration has been performed in ANSYS with accuracy of about <math>0.1 \%</math>.
+The results are given as matrices where the first row is made of times, the second of the temperatures.
+More information can also be found in <ref name="bechthold05"/>.
-<math>
-(E_0+\rho c_p \cdot E_1) \dot{T} +(A_0 +\kappa \cdot A_1 +h \cdot A_2)T = B \frac{u^2(t)}{R(T)}, \quad
+==Data==
-y=C^T \cdot T.
+Download matrices in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format: [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/GasSensor-dim1e5-GasSensor.tar.gz GasSensor-dim1e5-GasSensor.tar.gz], (8 MB).
+The matrix name is used as an extension of the matrix file.
+File <tt>*.C</tt> names contains a list of output names written consecutively.
+The system matrices have been extracted from ANSYS models by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem].
+The discussion of electro-thermal modeling related to the benchmark including the nonlinear input function can be found in <ref name="bechthold04"/>.
+==Origin==
+This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38880, see <ref name="hildenbrand2005"/>.
+==Dimensions==
+System structure:
+:<math>
+\begin{align}
+E \dot{x}(t) &= Ax(t) + B \\
+y(t) &= Cx(t)
+\end{align}
 </math>
+System dimensions:
-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=1.469  \times 10^{-3}</math>. The model was created and meshed in ANSYS. It contains a constant load vector corresponding to the constant input power of <math>2.49mW</math>. The number of degrees of freedom is <math>n=60,020</math>.
+<math>E \in \mathbb{R}^{66917 \times 66917}</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...
+<math>A \in \mathbb{R}^{66917 \times 66917}</math>,
-Indeed <math>R(T)</math> is a function of the state vector <math>T</math> and hence, the system has non-linear input. (It is also called a weak nonlinear system.)
+<math>B \in \mathbb{R}^{66917 \times 1}</math>,
+<math>C \in \mathbb{R}^{1 \times 66917}</math>.
+==Citation==
-==Data information==
+To cite this benchmark, use the following references:
-The system matrices are in MatrixMarket format(http://math.nist.gov/MatrixMarket/), and can be downloaded here [[File: Matrices_gassensor.tgz]]. The files named by *.<math>A_i, \, i=0,1,2</math> correspond to the system matrices <math>A_i, \, i=0,1,2</math>, respectively. The files named by <math>*.E_i, \, i=0,1,2</math> correspond to <math>E_i, \, i=0,1</math>. The file named by <math>*.B</math> corresponds to the load vector <math>B</math> and the file named by <math>*.C</math> corresponds to the output matrix <math>C</math>.
+* For the benchmark itself and its data:
+::The MORwiki Community, '''Gas Sensor'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Gas_Sensor
+ @MISC{morwiki_gas,
+   author =       <nowiki>{{The MORwiki Community}}</nowiki>,
+   title =        {Gas Sensor},
+   howpublished = {{MORwiki} -- Model Order Reduction Wiki},
+   url =          <nowiki>{https://modelreduction.org/morwiki/Gas_Sensor}</nowiki>,
+   year =         {20XX}
+ }
+* For the background on the benchmark:
+ @INPROCEEDINGS{BecHWetal04,
+   author =       <nowiki>{T. Bechtold, J. Hildenbrand, J. Wöllenstein, J. G. Korvink}</nowiki>,
+   title =        {Model Order Reduction of 3D Electro-Thermal Model for a Novel, Micromachined Hotplate Gas Sensor},
+   booktitle =    {Proceedings of 5th International Conference on Thermal and Mechanical Simulation and Experiments in Microelectronics and Microsystems},
+   pages =        {263--267},
+   year =         {2004},
+   doi =          {10.1109/ESIME.2004.1304049}
+ }
 ==References==
-[1] T. Bechtold, "Model Order Reduction of Electro-Thermal MEMS", PhD thesis, Department of Microsystems Engineering,
-University of Freiburg, 2005.
+<references>
-[2] T. Bechtold, D. Hohfel, E. B. Rudnyi and M. Guenther, "Efficient extraction of thin-film thermal parameters from    numerical models via parametric model order reduction," J. Micromech. Microeng. 20(2010) 045030 (13pp).
+<ref name="korvink2005"> J.G. Korvink, E.B. Rudnyi, <span class="plainlinks">[https://doi.org/10.1007/3-540-27909-1_11 Oberwolfach Benchmark Collection]</span>, Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.</ref>
+<ref name="hildenbrand2005">Hildenbrand J., Bechtold T., J. Wöllenstein, <span class="plainlinks">[https://doi.org/10.1007/3-540-27909-1_14 Microhotplate Gas Sensor]</span>. In: Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol 45: 333-336, 2005.</ref>
+<ref name="wollenstein03">J. Wöllenstein, H. Böttner, J.A. Pláza, C. Carné, Y. Min, H.L. Tuller, <span class="plainlinks">[https://doi.org/10.1016/S0925-4005(03)00218-1 A novel single chip thin film metal oxide array]</span>, Sensors and Actuators B: Chemical 93 (1-3): 350--355, 2003.</ref>
+<ref name="hildenbrand03">J. Hildenbrand, <span class="plainlinks">[https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/GasSensor-Hildenbrand03.pdf Simulation and Characterisation of a Gas sensor and Preparation for Model Order Reduction]</span>, Diploma Thesis, University of Freiburg, Germany, 2003.</ref>
+<ref name="bechthold04">T. Bechtold, J. Hildenbrand, J. Wöllenstein, J. G. Korvink, <span class="plainlinks">[https://doi.org/10.1109/ESIME.2004.1304049 Model Order Reduction of 3D Electro-Thermal Model for a Novel, Micromachined Hotplate Gas Sensor]</span>, Proceedings of 5th International Conference on Thermal and Mechanical Simulation and Experiments in Microelectronics and Microsystems, EUROSIME2004, May 10-12, 2004, Brussels, Belgium: 263--267, 2004.</ref>
+<ref name="bechthold05">T. Bechtold, "<span class="plainlinks">[https://www.freidok.uni-freiburg.de/volltexte/1914/ Model Order Reduction of Electro-Thermal MEMS]</span>", PhD thesis, Department of Microsystems Engineering, University of Freiburg, 2005.</ref>
+</references>
+==Contact==
+'' [[User:Feng|Lihong Feng]] ''
+[https://www.jade-hs.de/team/tamara-bechtold/ Tamara Bechtold]