1 lezione1 introduzione a femlab

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ARCES – Advanced Reserch Center on Electronic SystemsDEIS - Dipartimento di Elettronica Informatica e Sistemistica

UNIVERSITA’ DI BOLOGNA

[email protected]

SIMULAZIONI DI

MULTIFISICHE CON FEMLABELEMENTI DI BASE

Bruno Iafelice

http://www.micro.deis.unibo.it/~iafelice/solid_state_course

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Advanced Research Center on Electronic SystemsUniversity of Bologna

FASI DI PROGETTO

Scelta dellaTecnologia

Progetto

Modellizzazione eSimulazione

Test e Misura

fine


Physic Simulation Solvers

qMAGIC

qMAFIA

qHFSS

qARGUS

qFEMLAB (Comsol Multiphysics …now!)

q…

a full list is available at: http://www.emclab.umr.edu/csoft.html

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ELF/MAGIC

q http://www.elf.co.jp/eng/eng_page.htmlq for 3D nonlinear magneto-dynamic analysisq for simulating a system composed of magnetic bodies, magnets,

currents, conductors, and chargesq strong in evaluating the magnetic moment in material, magnetic

field in space, magnetic forces/torques on magnetic bodies and currents, and Maxwell's stress

q completed calculation without difficulty even if the system contains nonlinear materials

q easy system to re-model every time to arrange changes q Arbitrary time-varying current and a time-varying applied field

are available in a modelq Magnetic heads, Electromagnets, Various motors, Permanent

magnets, Superconducting magnets, Electromagnetic sensors


M.A.F.I.A. 4

q http://www.cst.de/Content/Products/MAFIA/Overview.aspxq The MAFIA 4 software package is a multi-purpose ECAD

system designed to solve all kinds of electromagnetic problemsq Its applications include most of today's problems in the

simulation of electromagnetic fields, ranging from static to thehighest frequencies, even including space charge fields of free moving charges.

q The tool MAFIA 4 is one of the most advanced and sophisticated codes in the area and offers the broadest range ofapplications.

q MAFIA 4 is based on the Finite Integration methodq MAFIA 4 is easy to useq MAFIA 4 can be run on most UNIX workstations and on PCs

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HFSS 9.0

q http://www.ansoft.com/products/hf/hfss/index.cfmq Ansoft HFSS is based on the finite element method (FEM)q automatic adaptive mesh generation and refinement, tangential

vector finite elementsq Application fields:Ø RF and microwave componentsØ Antenna, array, and feed structuresØ High-frequency ICsØ High-speed packagesØ High-speed or RF PCBs


COMSOL MULTIPHYSICS (Femlab 3.2)

COMSOL Multiphysics is a modeling package for the simulation of any physical process you can describe with partial differential equations (PDEs).

It features state-of-the-art solvers that address complex problems quickly and accurately, while its intuitive structure is designed to provide ease of use and flexibility.

COMSOL Multiphysics provides a friendly, fast and versatile environment for multiphysics modeling.

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COMSOL MULTIPHYSICS : KEY FEATURES

q Fast, interactive, and user-friendly graphical user interface for all steps of the modeling process

q Powerful direct and iterative solvers q Linear and nonlinear stationary, timedependent, and eigenvalue analyses of

models q Total freedom in the specification of physical properties, whether as analytical

expressions or functions q Unlimited multiphysics capabilities for coupling all types of physics, even on

domains in different space dimensions q General formulations for quick and easy modeling of arbitrary systems of PDEsq CAD tools for solid modeling in 1D, 2D and 3D q Triangular, quadrilateral, tetrahedral, brick, and prism meshes using fully

automatic and adaptive mesh generation q Extensive model libraries that document and demonstrate more than 100 solved

examples q Parametric solver for efficient solution of highly nonlinear models q Interactive postprocessing and visualization q Report generator for documenting models q 64-bit platform support for large-scale computations q Smooth interface to MATLAB


FEMLAB (Comsol Multiphysics)

Oltre alle caratteristiche dei tool concorrenti,Femlab permette anche:

q Simulazioni multifisicheq Vasta gamma di modelli simulativi gia’ impostatiq Creare modelli simulativi ex-novo da parte

dell’utenteq Post-processingq Creare scriptq Uso delle librerie di MATLABq … ed altro ancora

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Problematiche fisiche diverse

011

02

2

20

=

+∇−∇+

∂∂

qptp

c ρρ

Analisi acustiche Analisi di diffusione

( ) 0=−∇−∇+∂∂

RcDtc

Analisi termiche

( ) 0=−∇∇−∂∂

QTktT

cρ

Analisi fluido-dinamiche

( )( ) ( )

=∇

=∇+∇⋅+∇+∇⋅∇−∂∂

0u

Fpuuuutu T ρηρ

Analisi elettro-magnetiche

=⋅∇=⋅∇∂∂

−=×∇∂∂

+=×∇

0,

,

BDtB

EtD

JH

ρ

Analisi strutturali

Kuctu

=∇⋅∇−∂∂

2

2

ρ

courtesy Comsol (www.comsol.com)


Analisi in ambito multi-fisico

q Possibilità di accoppiare più aspetti fisicicontemporaneamente.


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Accoppiamento degli aspetti fisici

qSequenziale.

A ⇒ B

(Variabili caratteristiche)

qAccoppiamento pieno.

A ⇔ B

(Variabili caratteristiche, proprietà costitutive)



FEMLAB Evolution

q Toolbox basato su Matlab [1,2]

q Software indipendente dalla corrente versione 3.0q 1D e 2D, dalla nuova versione 3D su tutti i modelliq 2005 Femlabà Comsol Multiphysics [1] fonte www.comsol.com

[2] fonte www.mathworks.com


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COMSOL MULTIPHYSICS 3.2 ModulesNew Module

q Sono presenti modelli simulativi per diversi ambiti: dall’elettronica allachimica, dalla fisica alla meccanica

q Si puo’ usare il motore di Femlab per simulare modelli definiti dall’utente

Reaction Engineering Module


CHEMICAL ENGINEERING Module

q The Chemical Engineering Module has been developed specifically for the modeling of reactors, filtration and separation units, heat exchangers and other equipment found in the chemical industry. It deals with the couplings of fluid flow, diffusion and reaction processes, as well as heat transport couplings often found in systems of interest to the chemical engineer.

q The foundation of this module has been established on the theories described in the classic works Transport Phenomena by Bird, Stewart and Lightfoot plus Elements in Chemical Reaction Engineering by H. Scott Fogler.

q This module offers the best of both worlds when it comes to modeling: ready-to-use applications for the most common couplings between different phenomena; and free equation-based modeling where you input equations much in the same way as you would write them with pen and paper.

q Key FeaturesØ Interactive interface tailored for chemical engineering Ø Predefined applications and equations for momentum,

heat and material balances fully coupled to chemical reactions Ø Postprocessing and visualizationØ Model library with fully documented chemical-engineering examples

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ELECTROMAGNETICS Module

q This module specializes in component design in virtually all areas where engineers need electromagnetic field simulations–from statics and quasistaticsto microwaves and photonics. The static, transient and frequency domain analyses allow for material properties that are complex-valued, anisotropic, frequency or time-dependent.

q Simulation outputs are electromagnetic field distributions, resistance, inductance, capacitance and S-parameters.

q The Electromagnetics Module works smoothly with COMSOL Multiphysics and the other discipline-specific modules to provide comprehensive multiphysicsmodeling where electromagnetic fields interact with other physical phenomena. This includes resistive and dielectric heating, electromagnetic fluids, and electromechanical devices such as MEMS.

q Application AreasØ Low Frequency

• Devices Modeled : Magnets, Electrical Motors, Power Generators, Coils, Capacitors, Insulators, Electromagnetic shielding, MEMS

Ø Radio Frequency• Radio Components, Antennas, Transmission Lines,

Microwave WaveguidesØ Photonics

• Photonic Waveguides, Photonic Crystals, Semiconductor Lasers

Balanced Patch Antenna


STRUCTURAL MECHANICS Module

q This module is specializedin the analysis ofcomponents andsubsystems where it isnecessary to evaluatestructural deformations.It also contains specialapplication modes for themodeling of shells.

q Application modes in this module solvestationary and dynamic models, and let youperform eigenfrequency, parametric, quasi-staticand frequency-response analyses. 3D solid as wellas 2D plane stress, plane strain and axisymmetricanalyses allow the specification of elastoplastic andhyperelastic material laws as well as large deformations.

q The Structural Mechanics Module works in tandem with COMSOLMultiphysics and the other discipline-specific modules to couple structuralanalysis to any multiphysics phenomenon.

Structural deformation ofa blood-vessel stent

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EARTH SCIENCE Module

q This module is well suited for studies such as oil and gas flow in porous media, the modeling of groundwater flow, and the spread of pollution through the earth.

q A variety of specialized interfaces are available for easy application of the Richards and Navier-Stokes equations, Darcy’s law, and Brinkman’s extension of Darcy’s law. In addition, the module handles the transport and reaction of solutes as well as heat transport in porous media.


HEAT TRANSFER Module

q Problems involving any combination of conduction, convection, and radiation are solved with the Heat Transfer Module.

q It finds extensive use in systems that involve the generation and flow of heat in any form.

q A variety of specialized modeling interfaces are available for different formulations and applications such as surface-to-surface radiation, non-isothermal flow, heat transfer in structures made of thin layers and shells, and heat transfer in biological tissue.

q This is particularly relevant to applications such as thermal management in the electronics industry, thermal processing and manufacturing, and medical technology and bioengineering.

Isotherm surfaces in a body cooled by a coolant flow through a channel.

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MEMS Module

q The MEMS Module addresses design issues that arise in the micro-world. It models physical phenomena in actuators and sensors plusmicrofluidic and small piezoelectric devices. Most MEMS applications are multiphysics by their verynature and usually include electromagnetic-structural, thermal-structural, fluid-structure(FSI), or electromagnetic-fluid interactions.To this end, the MEMS module providesequations and settings optimized forthe single- and coupled-physicsmodeling that these interactionsmay require.

q The module includes analyses inthe stationary and transientdomains as well as eigenfrequency,parametric, quasi-static and frequency-response analyses.

Mixing from a microfluidic micromixer


REACTION ENGINEERING Module

q The COMSOL Reaction Engineering Lab is an innovative tool for modeling and simulating chemical systems. Just enter chemical reaction formulas and have the material and energy balances set up automatically!

q This one-of-a-kind package generates the chemical kinetics of a reaction network automatically as you enter the reaction formulas.

q In addition, a library of predefined expressions for thermodynamic and transport properties completes the physical description of the reacting system. The governing equations of the problem are organized into material and energy balances, providing a structure as well as an overview.

q The Reaction Engineering Lab is the perfect environment for virtual experiments with chemical kinetics. When you arrive at a relevant kinetic model, simply pass the equations to the Chemical Engineering Module and evaluate the chemistry in the specific geometry of your application.

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New Products

qCOMSOL ScriptTM

qCAD Import Module


CAD IMPORT Moduleq This utility greatly simplifies the transition from geometric

designs created with specialized CAD tools to mathematical modeling in COMSOL Multiphysics.

q The CAD Import Module is based on Parasolid® geometry kernel and includes ACIS® to support the SAT® format. In addition to the native Parasolid and SAT formats, the CAD Import Module also supports the STEP and IGES file formats. The CAD Import Module works with optional add-ons dedicated to a variety of CAD formats including: Pro/ENGINEER®, CATIA® V5, CATIA® V4, VDA-FS, and Autodesk Inventor®.

q Another highlight of the CAD Import Module is the live synchronization between COMSOL Multiphysics and the SolidWorks® CAD environment. This feature allows for the real-time updating of a COMSOL Multiphysics geometry immediately following any changes to the design in SolidWorks.

VDA-FS (.vda)CAD Import ModuleVDA-FS Import Module

Pro/ENGINEER®Pro/E (.prt, .asm)CAD Import ModulePro/E Import Module

Autodesk Inventor®Inventor (.ipt)CAD Import ModuleInventor Import Module

CATIA® V5CATIA V5 (.catpart)CAD Import ModuleCATIA V5 Import Module

CATIA® V4CATIA V4 (.model)CAD Import ModuleCATIA V4 Import Module

SolidWorks®, Solid Edge®, NX™

STEP (.stp), IGES (.igs), SAT (.sat) and Parasolid (.x_t)

COMSOL Multiphysics

CAD Import Module

STL (.stl), VRML (.wrl), DXF (2D) (.dxf), GDS (2D) (.gds) and NASTRAN

COMSOL Multiphysics™

Format Native to CAD Software

FormatsPrerequisitesProduct

The three steps for importing a CAD geometry using the CAD Import Module. The geometry exists in a CAD program such as SolidWorks®, and it is imported as a geometry into COMSOL Multiphysics before undergoing meshing.

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COMSOL Script

COMSOL Script — command-line modeling

q COMSOL Script™ can function both as a standalone technical programming language and as the ideal complement to COMSOL Multiphysics™ modeling. With over 500 commands, it provides everything required for robust modeling: data acquisition, matrix calculations, advanced postprocessing and graphical abilities.

q COMSOL Script M-files are compatible with both COMSOL Multiphysics and MATLAB.


COMSOL Mesh Import

q NASTRAN format

q COMSOL native text format (.mphtxt)

q Linear and non/linear meshes

q Automatic boundary and subdomain recognition based onØ material data Ø geometric shapes

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ELECTROMAGNETICS Module


Mathematical Model:

Maxwell’s Equations

Maxwell-Ampere’s law

Faraday’s law

Gauss’ law, electric

Gauss’ law, magnetic

Equation of Continuity

1:st Order PDEs

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Mathematical Model

Magnetic Vector Potential, A

Electric Potential, V

Potentials


Mathematical Model

Constitutive Relations

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Mathematical Model

Electrostatics, unknown V:

Quasi-statics, unknown H:

ρεε =−∇⋅∇ )( 0 PVr

0))(( 01

0 =×−−×∇×∇+∂

∂ − HvJHH e

rr

tµµσ

µµ

2:nd Order PDEs


Mathematical Model

2:nd Order PDEsQuasi-statics, unknown V and A:

Electromagnetic Waves, unknown E or H:

e

e

JAvAA

JAvA

=∇++×∇×−×∇×∇++

=−∇++×∇×−+⋅∇−−− Vjj

Vjj

rrr

rr

)()()()(

0))()()((

011

002

002

εωεσσµµεεωωσ

εωεσσεεωωσ

0HH

0EE

=−×∇−×∇

=−−×∇×∇

−

−

rr

rr

kj

jk

µωε

σε

ωεσ

εµ

20

1

0

0

20

1

))((

)()(

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q V = Vb,i

q A = Ab,i

with i = 1, 2, 3, … n for n geometry boundarieswhere Vb and Ab are the values at the geometry

boundaries

Mathematical Model: Boundary conditions (1)


Mathematical Model: Boundary conditions (2)

V+

V-

Vb=V+continuityVb=V-

continuity

continuity

Example: plane faces capacitor

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q er = er,i

q µr = µr,i

q s = s i

with i = 1, 2, 3, … n for n geometry subdomains

Mathematical Model: Subdomain conditions (1)


Mathematical Model: Subdomain conditions (2)

metal

metal

dielectric

Example: plane faces capacitor

s metal emetal

s metal emetal

s dielectric edielectric

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Impostare una simulazione


Impostare una simulazione

1. Cosa simulare?2. Quali fenomeni fisici simulare?3. Quale modello usare?4. Implementare un nuovo modello

matematico/circuitale?5. Cosa si può trascurare?6. Quale geometria simulare?7. 2D o 3D

8. … SIMULARE

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Quale geometria simulare? (1)

q E’ fondamentale scegliere cosa simulare. La scelta della geometria determina i tempi di simulazione

q Sulla base della struttura fisica il simulatore costruisce una maglia. In ogni nodo della maglia durante il processing verrà risolto il problema matematico

q Una struttura complessa con molte variazioni delle caratteristiche dei materiali (e,µ e s ) in poco spazio, necessita di una maglia molto fitta


Quale geometria simulare? (2)

V+

V-

Vb=V+continuityVb=V-

Esempio: condensatore a facce piane

s dielectric edielectric

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2D o 3D?

q Da 2D a 3D cresce la complessità della risoluzione (complessità della maglia), e quindi i tempi di simulazione

q Esistono problemi bi-dimensionali?q E’ preferibile fare prima una simulazione 2D

per capire Cosa simulare e Cosa trascuraree poi una in 3D

q Femlab permette di ottenere una struttura 3D a partire da una 2D


Femlab Simulation Steps

q Draw modeq Boundary modeq Subdomain modeq Mesh modeq Solve modeq Post-processing mode

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SENSORE IMPEDENZIOMETRICOper

CARATTERIZZAZIONE DI CAMPIONI BIOLOGICI

Bruno Iafelice


Introduzione

Definizione: I sensori impedenziometrici sfruttano la variazione dell’effetto capacitivo, resistivo o di entrambi del sistema in esame per rilevare un evento biologico al suo interno

Applicazioni: caratterizzare un campione biologico (ad esempio tessuti sani vs. tessuti malati), monitorizzare la crescita cellulare, calcolare la densita’ cellulare in una soluzione, …

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Esempio: crescita cellulare

La crescita cellulare si compone di diverse fasi (graficoqualitativo):

Morte

Cre

scita

loga

ritm

ica

Plateau

Tenpo [giorni]

Con

cent

razi

one

[cel

lule

/ml]

Inoculo0 2 4 6 8 10


Sensore

Il sensore puo’ esser fatto come un condensatore a facce piane dove il dielettrico e’ una soluzione contente il campione cellulare:

La presenza delle cellule, e le loro dimensioni influiscono sulle caratteristiche del dielettrico, quindi sull’impedenza Z.

Z

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Campione: sferoide

Per semplificare le simulazioni in Femlab consideriamo un unico campione biologico detto sferoide [1,2] :

Z

[1] Per sferoide intendiamo un aggregato cellulare. Esistono diversi metodi per realizzarlo, ad esempio aggregando con una centrifuga cellule precedentemente coltivate. Inoltre lo sferoide ben emula un campione di tessuto prelevato da paziente.

[2] H. Thielecke, A. Mack and A. Robitzki, “A multicellular spheroid-based sensor for anti-cancer therapeutics”, Biosensors & Bioelectronics, vol. 16 (2001), pp. 261–269


Modello circuitale di una soluzione elettrolitica

La soluzione elettrolitica in cui è sospeso il campione biologico è descrivibile in termini circuitali attraverso il parallelo di una resistenza e di una capacità:

q l distanza tra gli elettrodiq S superficie degli elettrodi

Sl

Rsol σ1

=

lS

C solsol εε0=

q s conducibilità dell‘elettrolitaq e0 permettività del vuotoq esol permettività dell’elettrolita

La presenza del campione biologico può esser caratterizzato attraverso la variazione della Rsol e Csol.

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Sensing del campione

qLa presenza dello sferoide (s sample esample) in sospensione tra gli elettrodi altera le caratteristiche del sistema liquido (s liq eliq):

s * = s *(s liq , s sample)e* = e*(eliq , esample)

qLa variazione di s e e si manifesta come variazione dell’effetto resistivo e di quello capacitivo del sistema tra gli elettrodi


Interfaccia metallo-liquido

Impedenza di interfaccia

Zinterfaccia

Impedenza dell’elettrolita

Bisogna inoltre considerare il contributo all’impedenza del sistema dovuto all’interfaccia metallo-liquido dell’elettrodo:

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Fenomeni fisici: legame con la frequenza

qAl crescere della frequenza del segnale applicato, è possibile trascurare gli effetti dell’interfacciaqIPOTESI SEMPLIFICATIVA:

misuriamo l’impedenza del sistema ad una frequenza tale da trascurare gli effetti delle interfacceqIl sistema è approssimabile al solo bulk


Introduzione

Vediamo come simulare in Femlab il sensore, e come sfruttare il risultato delle simulazioni per calcolarel’impedenza associata al sistema

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Disegno della struttura (Draw mode)

q Per semplicita’ partiamo da una struttura 2D destinando quella 3D solo a raffinamenti successivi

q Individuiamo cosa simulare

Semplificare lastruttura

Ridurre i tempidi computazione

sferoide

Liquido disospensione

elettrodi

Liquido disospensione

sferoide

elettrodi


( )t

V∂∂

−=∇•∇−ρ

σ

V⋅∇−=E

Modello matematico

EJ s=

t∂∂

−=•∇ρ

J

Data l’equazione di continuità

e la legge di Ohm

si ha

Ma definendo V il potenziale elettrico

si ha

quindi un’equazione di Poisson:

( )t

s∂∂

−=•∇ρ

E

( ) fuc =∇•∇−

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Disegno della struttura (Draw mode)

qNella finestra Model Navigator scegliere 2D e il modello Poisson’s equation disponibile come sottomodello nella sezione ClassicalPDEs

qDisegniamo la struttura come due cerchi concentrici (200 e 100 µm di diametro)

qGli elettrodi li descriveremo imponendo le condizioni al contorno del cerchio più grande


Femlab Classical PDEs model

fauuuuctu

da =+∇+−+∇⋅∇− βγα∂∂

)(

=

′+=+−+∇⋅rhu

hgquuucn µγα )(

inside subdomain

on subdomain boundary

Poisson’s equation model:

fuc =∇⋅∇− )(

it implies that:

0,, =µγα

==+∇⋅

rhugquucn )(

on subdomain boundary

in subdomain

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Boundary mode

Condizione al contorno di Dirichlet o di Neumann:Ø agli elettrodi si sceglierà la condizione di Dirichlet,

discontinuità dovuta alla presenza del potenziale applicatoØ per gli altri lati si sceglierà la condizione di Neumann

garantendo la continuità del campo elettrico

==+∇⋅

rhugquucn )(

gquucn =+∇⋅ )( Condizione al contornodi Neumann

Condizione al contornodi Dirichlet


Boundary mode

Condizione al contornodi Neumann


q, g = 0

• q, g = 0• h = 1• r = 0 (potenziale)


• q, g = 0• h = 1• r = (potenziale)2

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q LiquidoØ f = 0 (sorgenti di campo interni ai sottodomini)Ø c = s liq + j*2*p*freq*e0*eliq (conducibilita’ complessa)

q SferoideØ f = 0Ø c = s sample + j*2*p*freq*e0*esample

q conØ freq = 1 MHz (frequenza)Ø s sample = 10-15 S/m conducibilita’ sferoideØ s liq = 10-3 S/m conducibilita’ liquido di sospensioneØ e0 = 8.854e-12Ø esample = 2.5Ø eliq = 78.5

Subdomain mode


Cell Model (Saccharomyces cerevisiae yeast cell)

Esercitazioni di Sensori a stato Solido LS

2.5 µm0.4 µm

0.008 µm

cytoplasms = 0.55 S/me = 55

membranes = 10-6 S/me = 8

walls = 0.01 S/me = 55 smembrane depends on the conductivity

of the suspension solution

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Mesh mode


Calcolo dell’impedenza (1)

( )∫Ω

Ω=

dP

VZ

eT

2

21

ω

2

21

EmeP σ=

La resistenza e la capacita’ del sistema sono calcolabili come segue. Sia Pe la densita’ di potenza ohmica, esprimibile come:

Integrando la Pe sul volume O in cui si risolve il problema matematico si ottiene:

dove ZT(w) e’ l’impedenza CpRp del sistema e V la differenza di potenziale applicata.

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Calcolo dell’impedenza (2)Da ZT si possono ricavare la resistenza e la capacita’ con le seguenti formule:



22V

dWC

e∫ΩΩ

=

202

1Eεε meW =

La capacita’ del sistema e’ anche calcolabile col seguente metodo. Sia Wela densita’ di energia ohmica e O il volume in cui si risolve il problema matematico, si ha:

dove V e’ la differenza di potenziale applicata. Questo metodo deve dare lo stesso valore della Csol calcolata col metodo precedente e puo’ essere usato per verifica.

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qAggiungiamo dunque tra le variabili cheFemlab dovra calcolare anche We e Pe

qSelezioniamo dunque la voce del menu’ Optionsà Add/Edit Expressions....

qUsiamo ux e uy per indicare E essendo u la variabile che usa Femlab per indicare ilpotenziale e ux terminologia per indicare la derivata parziale di u rispetto a xqAllora |E|2 sara’ ux*conj(ux)+uy*conj(uy)


Solve mode : u (Potenziale)

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Post-processing mode : E (campo elettrico)


Post-processing mode

qTramite la funzione Subdomain Integration, applicata ai due sottodomini si calcolano gliintegrali di volume di We e Pe e quindi C e R

qIl risultato appare nella finestra informativapresente in basso nella finestra di Femlab

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Esercitazioniq Presso LAB1, edificio Aule Nuove, piano terra

q PRIMA ESERCITAZIONEMartedì 21 febbraio ore 14,00 – 16,00

q SECONDA ESERCITAZIONEMartedì 28 febbraio ore 14,00 – 16,00

q Le esercitazioni sono parte integrante del corso di Sensori a Stato Solido LS, quindi obbligatorie

q Munirsi di floppy disk e della presente dispensa (scaricabile dal sito-web delle esercitazioni)

1 lezione1 introduzione a femlab

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