Modeling

ENERGY-BASED MACRO MODELS

ENERGY-BASED MACRO MODELS

• Overview

• Generalized variables

• Models of transducers

ENERGY-BASED MACRO MODELS

• Capacitive accelerometer

ENERGY-BASED MACRO MODELS

SourceSenturia:MicrosystemDesign

SourceSenturia:MicrosystemDesign

Microsystems

• Microsystems are low-power devices in greatly confined spaces

RESULT: Strong coupling of physical quantities between the components of a microsystem

• In addition to component modelling and simulation also system level simulation using macromodels is needed

Macromodel

• Also called lumped parameter model

• Analytical, not numerical (FEM)

• Exhbits correct dependencies on device geometry and constitutive properties

• Covers dynamic and quasi-static behavior

• Simple to use, e.g. network analogy

• Easy to connect to system level simulators

• Agrees with 3D multiphysics simulation

Generalized variables

Generalized variables - duality

Generalized variables – through and across

Models of transducers

• Plate capacitance

• Inductance

AC

dε=

AL

dµ=

Model of angular capacitor

Model of angular capacitor

Models of transducers – physical diagrams

Model of a capacitor

Models of transducers

Models of transducers

Magnetic levitation

Magnetic levitation

MAGNETIC LEVITATION

Magnetic levitation

Magnetic levitation

Mechanical modelmMz F Mg= −

0 or lim

m

m mz

W F z

W W dWF F

z z dz→

∆ = ∆∆ ∆

= = =∆ ∆

Virtual work principle

Force2

2

( )

2

( )

2m

L z iW

dW dL z iF

dz dz

=

= =

Electrical model

( ( ( )) )

( ( )) ( ( ))

= ( ( ))

dE Ri L z t i

dtdi dL z t

Ri L z t idt dt

di dL dzi L z t i

dt dz dt

= +

= + +

+ +

Theoretically2

0 2

ANL

zµ=

From measurements

A = Cross-sectional areaN = Number of turns

42.1 100.1 (H)L

z

−⋅= −

Magnetic levitation

Force-gap characteristic

Operating point is always unstable. Feedback control is needed.

State-space model

mFz g

Mdi E R i dL dz

idt L L L dz dt

= −

= − −

1

2

3

vertical position

vertical velocity

current in the field coil

x z

x z

x i

===

Define the states

2

421 1 2

22

1

2

(H); 0.1; 2.1 10

m

dLF i

dzk

L k k kz

dL k

dz z

−

=

= − = = ⋅

=

State-space model

1 2

232

2 21

2 323 3 2 3 3 2

1

42

2

1

2.1 10

m

x x

F xkx g g

M M x

x xE R dL E R kx x x x x

L L L dz L L L x

k −

=

= − = −

= − − = − −

= ⋅

Magnetic levitation

Model of a capacitor

Model of dependent resistorPolynomial form P(x) = P0(1+f(x))

0

(1 ( ))V

g xR

+

0

1 1(1 ( ))

( )g x

P x P= +

V=iR(x)=iR0(1+f(x)) = V0 + V0 f(x)

Model of dependent resistor

0

(1 ( ))V

i g xR

= +

Alternate form

Model of dependent resistor

i = i0 + i0 g(x) = i0 + (V/R0) g(x)

Model of temperature dependent resistor

R(T) = R0 (1 +αΤ)

which can be written as

V12=i R0

Model of temperature dependent resistor

Model of dependent capacitor

q = CV = C0 (1 + f(x))V = C0 V + C0 Vf(x)

= q0 +q0 f(x)

Model of dependent capacitor

Example – capacitive system

Example – capacitive system• Electrical

0 0 0

0

1 1 1( ) ( ) ( )

( ( )) ( ( )) ( ( ))

( )

t t tpdV d d d

i d i d i ddt dt C x t dt C x t C x t dt

A A d dC x C

d x d d x d x

α α α α α α

ε ε

= = +

= = =− − −

∫ ∫ ∫

0( )) ( ( ))x t C x t

=

Example – capacitive system

Example – capacitive system

Domains of simulation

Gyroscope – mechanical model

Gyroscope – mechanical model

SOLENOID

LAGRANGE’S EQUATION

MICROPHONE

MICROMACHINED MICROPHONE

MICROPHONE - EQUIVALENT CIRCUIT

MICROPHONE – ELECTRICAL SYSTEM

MICROPHONE – MECHANICAL SYSTEM

ELECTROMECHANICAL SYSTEM WITH CAPACITIVE COUPLING

ENERGY EXPRESSIONS

RESULTING MODEL

VHF free-free beam high-Q micromechanical resonator

Miniaturized communication devices

Resonator beam structure