Lumped-element modelling of RF-MEMS - unibo.itreggiani/mems_lecture_021106_parte_II.pdf ·...

ARCES -Advanced Research Centre on Electronic Systems

University of Bologna - Italy

Thursday, 2nd November 2006

Lumped-element modellingof RF-MEMS

Roberto Gaddi

Email: [email protected]

ARCES - University of Bologna

Strutture MEMS in un transceiver a radio frequenza

Potenziale applicativo dei MEMS include la sostituzionedi componenti tradizionali


Applicazione MEMS in sistemi wireless

Possibili elementi in un transceiver adatti a realizzazionein tecnologia MEMS:

Induttori ad alto fattore di qualità integratiCapacitori variabiliInterruttori accoppiati in DC o ACMicro-risonatori, per filtri o tank per oscillatori locali

Caratteristiche favorevoli dei nuovi componenti MEMS permettono di rivedere anche la descrizione del sistema a livello di architetturaIn particolare: selezione di canale e riconfigurabilità…


Componenti passivi ad alto fattore di qualità

Induttori integrati isolati dal substrato semiconduttivohanno perdite ridotte ⇒ maggiore fattore di idealità (Q)


Interruttori MEMS accoppiati in DC o AC-RF

Deformazioni di strutture conduttive, tramite trasduzione elettrostatica o magnetica, si utilizza per implementareinterruttori


Capacitori variabili integrati

Tramite strutture deformabili e trasduzione elettrostaticaè possibile realizzare capacitori variabili MEMS integrati


Esempio: ricevitore a banco di switch


Simulation approaches for MEMS

System Level

Sub-system / Circuit Level

Device / Physical LevelTOP-DOWN

BOTTOM-UP

• System modeling• Behavioral analysis of complete MEMS devices

• Reduced order modeling• Electrical equivalent• Lumped elements• Modified nodal analysis

• 3D modeling• FEM / FVM / BEM field solvers• Coupled domains


Sub-system / Circuit Level Modeling

This modeling level involves:Terminal characteristics description of a sub-systemMultiple physical domains phenomena and quantitiesHierarchy compatible model complexity

Reduced-order modelling approach:Starts from exact continuous 3D modelling; space discretisation and reduction of mechanical degrees of freedom are appliedUsually requires expertise and intuition to avoid loss of significant device behaviour descriptionLately some automated model reduction tools are available also from commercial CAD toolsSeems more appropriate to a bottom-up design methodology…


Generalized Kirkhoffian networks

Kirkhoffian network theory is applicable to diverse energy domains, provided that:

Flow (through) and difference (across) quantities can be identified, with relationships between them given as implicit/explicit equations or differential equations depending only on terminal quantities and internal states. Conservation laws apply:Zero sum of across quantity along a closed network loopZero sum of through quantity into a node or network cut-set

Physical domain Flow quantity Difference quantityElectrical Current Voltage

Mechanical-trans Force Velocity / Displ.

Mechanical-rot Torque Ang. Velocity / Displ.

Pneumatic Volume Flow PressureThermal Heat Flow Temperature


Motivations for Lumped element modelling

Simplified device representationExpressible with equivalent circuit model (compatibility with spice-like circuit simulators)Exploitation of powerful know-how on circuits simulation: well established methods for small-signal and large signal analysisInterface with electronic circuits is straightforward: key aspect of MEMS devices is coupled physical domains simulations


Power flow

Two components can exchange energy:

A BPAB

PBA

Power flow is energy flow per unit time:

PAB=r12 PBA=r2

2

Net Power Flow from A to B is:PNet=r1

2-r22=(r1+r2)(r1-r2)=e(t)·f(t)

i.e. a product of two real numbers…


Conjugate power variables

A pair of real quantities whose product yields the net power flow between two elements

They are generally defined as:effort e(t) and flow f(t)From integration in time we also have:

generalized displacementgeneralized momentum

so that p*f and e*q are both Energies

( ) ( )00

)( tqdttftqt

t+= ∫

( ) ( )00

)( tpdttetpt

t+= ∫


Example: mechanical power variables

Effort will be a Force, Flow will be Velocity:Power = Force * VelocityEnergy = Force * Displacement

= Momentum * VelocityNote: in n-dimensional space we have scalar products

of vectors…

viscFr

vr


Conjugate power variables


Generalized port definition

Port: a pair of terminals on a circuit element that must carry the same through quantity (e.g. current) within the element, while sustaining a given across quantity between the same terminals.The net power flow in the element is an appropriately signed product between through and across quantity.Electrical convention is:effort = across variableflow = through variable.

+

-

e(t)

f(t)

f(t)

+

-

e(t)

f(t)

f(t)


Variable-Assignment Conventions

Depending on physical domain, different conventions can be appliedAppropriate power flow definition must be adopted for each variable conventionThe e→V always associates potential energy to the energy stored in capacitors


Electrical convention: generalized sources

In general, either flow or effort sources can be introduced as boundary conditionActive elements supplying power to the network (negative net power flow)


The generalized resistor

Direct dependence between through and across quantities at its terminals:

e = R(f)R must go through the originCan be linear or non-linear

Dissipative behaviour if R is in I and II quadrants(positive net power flow)


The generalized capacitor

Direct relation between effortand displacement:

e = Φ(q)Φ is a well behaved functiongoing through the originElement associated to potential energy storage:

( ) ∫ ⋅=1

01

q

dqeqW

( ) ∫ ⋅=1

01

*e

deqeW


The electrical capacitor

A parallel plate capacitor, ideal, is defined by the following:

The energy stored in the capacitor is:CQV =

CQdqqVQW

Q

2)()(

2

0

== ∫


The Hooke’s Law spring

The force (flow) F is related to the displacement x by a linear function:

F=kxThe stored energy in the spring as displaced by x1 is:

Same potential energy as electrical capacitance: the two elements are equivalent (e→V convention)

21

01 2

1)()(1

kxdxxFxWx

== ∫k

C 1↔


The generalized inertance

Electrical inductance, generalizedas the function relation between the flow f and the momentum p

f=Ψ(p)Ψ must go through the origin of the f-p planeIt is related to stored kinetic energy:

( ) ∫ ⋅=1

01

p

dpfpW


The electrical inductance

Inductance definition:

If we were to define an electrical momentum it would be:

So we recognize the inertance definition:

The stored kinetic energy and co-energy are:

0

0

)( idtLvti

dtdiLv

t

t∫ +=→⋅=

0

0

)( e

t

te pvdttp ∫ +=

( )0)(1)( ee ptpL

ti −=

211

*21

1 21)(

2)( LiiW

LppW ==


The inertial mass

In the mechanical domain we have:p=mv

The inertance definition is:The stored kinetic energy and co-energy are:

Hence, adopting the e→V convention, an inertial massand an inductance are equivalent:

mpv =

211

*21

1 21)(

2)( mvvW

mppW ==

mL ↔


Equivalent networks: connections

Two possible situations will lead to connection of elements to form networks

1. Shared flow and displacement: series connection2. Shared effort: parallel connectionExample: mass-spring-dashpot system:

All three elementsshare the samedisplacement…


Mass-spring-dashpot example

Series connection of the three equivalent circuit elements


Kirchhoff’s Laws

Connectivity between circuit elements is based on generalized Kirchhoff’s lawsTheir validity is maintained across diverse physical domains

Kirchhoff’s Current Law (KCL): The sum of all currents (flows) entering a node is zero

Kirchhoff’s Voltage Law (KVL): The oriented sum of all voltages (efforts) around any closed path is zero


Spring-mass-dashpot example

The KVL expresses the resultant force principle as follows:-F+ek+em+eb=0

The total applied force has three resulting components: the inertia of the mass, the stiffness of the spring and the damping of the dashpot.


Lumped element electrical equivalence

Different energy domains can have formally identical constituent relationships (implicit/explicit or differential)

extFxkxBxM =⋅+⋅+⋅ &&& ext

t

idvLR

vvC =⋅++⋅ ∫∞−

τ1&

geometry parameters

mech. model abstraction

energy domain equivalence:

force ↔ currentvelocity ↔ voltage

electrical simulation

NO DIRECT LINK WITH DESIGN PARAMETERS


Higher level electrical equivalent approach

Equivalent electrical network modelling is suitable to small-signal analysis of generalised dumped resonatorsElectrical equivalent extraction quickly looses track of geometrical and mechanical design parameters


MEMS component library in Cadence®


Prediction of beams eigenfrequencies

F1=173.9KHz

F2=1.088MHz

F3=3.039MHz


Resonance modes of a composite device (1)

res1=110kHz res2=225kHzres3=275kHz res4=350kHz


Resonance modes of a composite device (2)

Small-signal ac simulation of the device with a punctual force stim.

res1=109kHzres2=204kHzres3=278kHzres4=347kHz


Complete MEMS example: tunable capacitor

MEMS varactor with T-shaped spring suspensionsParasitic extraction from RF characterisation or electromagnetic simulations should be performed for accurate RF modellingHere only access resistancedue to finite conductivity of beams is accounted for


MEMS Varactor top-down design (1)

Critical specs for varactor as tuning element within an electronic circuit are: tuning ratio (Cmax/Cmin), nominal capacitance (Cnom) and pull-in voltage (VPI)

Vbias

Z-pos

Vbias

Z-posAll geometrical parameters are available for design: MEMS design tool based on Spectre simulatorParametric static (DC) simulations quickly allow for Pull-in voltage design



The tuning ratio is technology defined

A sweep from 200x200μm2 to 400x400μm2, at f=1.8GHz and bias voltage VNOM

ox

oxairox

ttg

CC +

=ε

min

max

Total plate area A and nominal voltage VNOM define the capacitance value Cnom

Small signal (ac) analysisperformed at given frequency and sweeping Aleads quickly to the desired nominal capacitance

AA ~=



Spring beams dimensions control the overall spring constant k, e.g. the pull-in voltageAccess resistance also depends on beams W/LPossible trade-off: tuning range vs. resistive losses

⇓ width: ⇑ tuning range ⇓ Q factor


Varactor transient behaviour

Transient simulationscan give insight to response time to VBIAS

Spectre® simulator does not show any convergence issues, even with added electronicsBoth electrical and mechanical quantities can be observed


Varactor insertion within an LC tank

Typical application can be the tuning element within an LC tank for an RF voltage controlled oscillator (VCO)LC network includes two varactors that provide isolation from controlling voltage


Mixed-domain complete VCO simulation

Differential VCO: CMOS technology from UMC, 0.18μm channel lengthModel library based on BSIM3 modelSpectre achieves convergence in transient analysisPeriodic-steady-state (PSS) simulation for noise analysis still have issues…

time

Vout

time

Vout

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