· 1. be creative, exploit any opportunity to reduce order: do not neglect “details”, e.g....

© Daniel IOANAutumn School on MOR, Terschelling-NL, Sept. 21–25, 2009

Parametric Reduced Order Models for Passive Integrated

Components Coupled with their EM Environment

Daniel Ioan and LMN team

Politehnica University of Bucharest, Romania, Laboratorul de Modele Numerice [email protected]


Outline

A IntroductionB Problem formulationC Numerical approachD Order Reduction and VariabilityE Domain PartitioningF Numerical examplesG Conclusions www.codestar.imec.be

www.chameleon-rf.orgwww.comson.orgwww.lmn.pub.ro/neda

http://www.codestar.imec.be/

http://www.chameleon-rf.org/

http://www.comson.org/


A. INTRODUCTION

• My talk is mainly not about solving !

• But it is about formulating problems

• If the problem is well formulated it is half solved !


European Technology Platform in Nano-electonics

Strategic Research Agenda

1960 1980 2000 2020 2040 2060

10μm

1μm

100nm

10nm

1nm

2. More than Moore

1.More Moore ( RF, HV, MS)

4. Beyond CMOS

For more details see www.eniac.eu

3. EDA

0.2MHz

0.2GHz

4GHz

60GHz

200GHz


Real life complexity

zoom

Technology variability

EM coupling between blocks


Numerical approaches used to compute EM field

Idealized geometry models

BEM or FEM mesh

Can not handle the complexity of real designs !


Proposed modeling approach

.

Discretemodel:FIT

Reducedmodel –Kirchhoffeqs.

Continuousmodel–Maxwell eqs.and b.c.

• EM field problem for passive components after Domain Partitioning:- Maxwell equations with- appropriate boundary conditions

for EM coupling modeling

• After discretization (not solving!) non-compact model is generated

• After reduction by MOR an equivalent parametric reduced circuit is synthesized

PDE

DAE

ODE

Model extraction: from Maxwell to Kirchhoff

LAE

In frequency domain


Outline

A IntroductionB Problem formulation: hooks and EMCEC Numerical approach: FIT for EMCED Order Reduction: ALROM, VariabilityE Domain PartitioningF Numerical examplesG Conclusions


EM coupling with environment

Electric terminals(metal traces)

Electric and magnetic virtual connectors (hooks): windows for capacitive and inductive coupling

Capacitive, inductive and conductive couplings:

Sio2 layer

Electric current

Magnetic fieldElectric field

Hooks: connectors and terminals


Requirements for boundary conditions

• Modeled domain should interact with the Environment (the rest of IC)

• Interaction can be: Electric or/and Magnetic• Interaction is bidirectional

Modeled domain EM Environment


Requirements for boundary conditions

• Modeled domain should interact with the Environment (the rest of IC)

• Interaction can be: Electric or/and Magnetic• Interaction is bidirectional• Usually the environment is modeled as a circuit, thus b.c.

should be compatible with it

Modeled domain Circuit model of

Env.

Only boundary condition for Maxwell PDE we know which comply these requirements is EMCE (El-Mg. Circuit Element b.c.).

See for instance: Bossavit, A. (2000). Most general “non-local" boundary conditions for the Maxwell equations in a bounded region. COMPEL, 19(2), 239{245).


EMCE Boundary conditions

(4) "kSP U∈∀( ) 0Hn =× P,t

( ) 0=P,tcurl Hn

(3) 'kSP U∈∀

(1) 0SP∈∀

( ) 0En =× P,t

( ) 0=P,tcurl En

(2) 0SP∈∀

• No Magnetic flux outside magnetic term.:

• No Electric current outside electric term.:

• Electric scalar potential is constant over each electric terminal

• Magnetic scalar potential is constant over each magnetic connectorElectric

terminals

.These boundary conditions allow the EM field-circuit coupling, hence the compatibility and interconnection with an external circuit

Magnetic terminals

kk SSS ′′−′−Σ= UU0


Field-circuit coupling

Passive component or its reducedmodel

Electric Environment:Electric circuit which contains R, L, C, controlled sources, transistors, etc.

Magneticenvironment:a magnetic circuit which contains Rm and controlled sources

( ) ∫Γ′

=k

dtik rH

( ) ∫′

=kC

rEdtvk

∫Γ ′′

=k

)( rEdtkϕ&

( ) ∫′′

=kC

k dtu rH

For electric terminals:

For magnetic terminals:

( ) ∑∑−

=

−

=

+=1"

1

1'

1

n

k

kk

n

kkk dt

duivP ϕPower P

Power P


Consequences of the EMCE b.c.• Lemma 1 - of the current and flux conservation (KC/FL):

The sum of all terminal currents and the sum of all magnetic fluxes are always zero for any EMCE:

• Lemma 2 - of voltages (KVL): The sum of electric/magnetic voltages over any closed loop which does not contains magnetic/electric terminals is zero:

• Lemma 3 of the power transferred by the EMCE terminals: The electromagnetic power transferred outside to inside of EMCE is:

,in

kk 0

1=∑

′

=

,in

kk 0

1=∑

′

=

01

=∑′′

=

n

jjϕ

( )∑ ∑−′

=

−′′

=

+=1

1

1

1

n

k

n

j

kkkk dt

duivP

ϕ

,vk

k 0=∑Γ∈

,uk

k 0=∑Γ∈


Other consequences• Uniqueness theorem for the field in the linear EMCE: The fundamental field problem

associated to an EMCE with Maxwell’s equations, EMCE boundary conditions, known electric/magnetic voltage or current/flux for any electric/magnetic terminal and initial conditions for B and D has a unique field solution: E(M,t), D(M,t), B(M,t), H(M,t), J(M,t),ρ(M,t), for M in EMCE and t>0.

• Consequently, the responses of the EMCE (currents/fluxes and electric/magnetic voltages or all electric/magnetic terminal) are unique, real functions well defined, for t≥0.

• Linearity/superposition theorem (operational form of the input-output relation in the case of a hybrid-controlled, linear EMCE): The Laplace transform of the output signals of a linear EMCE are linear combinations of input signal in zero initial conditions. The coefficients of that combination are functions of complex frequency s:

W(s) = H(s) X(s),where H(s) = [Hjk(s)] is the hybrid matrix, element of C (n'+n"-2) x (n'+n"-2) . The matrix H is the

transfer function of the EMCE and it has the following structure:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

==

i

i

i

i

o

)s(

ΦUIV

HHHH

HHHH

HHHH

HHHH

UΦVI

HXWo

o

o

44434241

34333231

24232221

14131211EMCE is a dynamical (LTI) system with infinite state space but finite number of I/O

Model extraction goal: to find a finite approximation, represented by rational functions in the frequency domain


Alexandru Timotin (1925-2007)

• Timotin’s theorem for the power transferred to MEMCE:

MEMCEMultiple connected Electro-Magnetic

Circuit Element

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )∑∑

∑∑

∫

==

−

=

−

=

ΣΣ

−

+Φ

+

=×≡

q

sSS

q

sSS

"n

jm

j'n

hkk

inttt

tftetfte

tudt

dtitu

sntp

j

1

0

1

0

1

1

1

1

δhe

Timotin Al, Passive EM element of circuit,Rev. Roum. Sci Techn. – Electrotech. et Energ. 21, 2, 347-342, 1971


Outline



C. NUMERICAL APPROACH

Finite Integration Technique (FIT) is a numerical method to solve field problems, based on spatial discretization “without shape functions” intensively studied by Th. Weiland et al. since 1977

Principles of Finite Integrals Technique (FIT):

• dual staggered orthogonal grids, (Yee type = “complex of dual Cartesian cells”), suitable for our Manhattan geometry;

• global variables as DOFs: voltages and fluxes on grid elements, and not local field components;

• global form of field equations (neither differential form - FDM, nor weak-variational form - FEM, nor integral equations - BEM/VIE).


FIT principles

• Each mesh cell complies the global field equations


Maxwell Grid Equations

• No discretization errors in MGE fundamental equations• They are metric-free, sparse, mimetic and conservative DAE,

without spurious modes

dtdd

td

tdiv

dtd

dd

dt

d

divt

curl

dtdv

d

dt

d

divt

curl

qDiAJJ

qDψ

ψiuC'

ρAD

ADJrH

ρD

DJH

D

C

AB

ABr

B

BE

−=⇒∂∂

−=⇒∂∂

−=⇒

⎪⎩

⎪⎨⎧

=⇒

+=⇒

⎪⎩

⎪⎨

⎧

=⇒∂∂

+=⇒

⎪⎩

⎪⎨⎧

=⇒∂∂

+=

⎪⎩

⎪⎨⎧

=⇒

−=⇒

⎪⎩

⎪⎨

⎧

=⇒∂∂

−=⇒

⎪⎩

⎪⎨⎧

=⇒∂∂

−=

∫∫∫∫∫

∫∫∫∫∫∫∫∫

∫∫∫∫∫

v

v

)(

0'00

ρρ

ϕ

ϕΕ


Hodge operators

• They are metric-dependent and they hold the discretization error.

• Classical FIT (MGE+Hodge) must be improved and adapted, in order to achieve the requirements of the nowadays designers.

We did it with the ALROM technology

⎪⎩

⎪⎨

⎧

=⇒==⇒==⇒=

ϕνψε

σ

νm

ε

σ

MuBHvMED

vMiEJdescribe material behavior:

⎪⎩

⎪⎨

⎧

===

iiiii

iiiii

iiii

ltwMltwMtwlM

//

)/( 0

σεμ

σ

ε

μ


Electric and magnetic terminals on the FIT grids

An electric terminal on the electric grid.

TERMINAL = union of elementary grid faces

A magnetic terminal on the magnetic grid and itsshadow on the electric grid


Structure of C, G matricesFaraday’s law for inner electric loops;

Ampere’s law for magnetic loops;

Faraday’s law for electric loops on the boundary;

Current conservation for nodes on the boundary (not ET);

Current conservation for electric terminals;

Flux conservation for magnetic terminals;

Voltages of electric terminals – sums along paths to ground;

Magnetic voltages of the magnetic terminals.

On inner loops

On the boundary000

0=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ ′−+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡uv

BBG

uv

GC e

m

e

dtd

First implementation of the EMCE boundary conditions in FIT !


C

( )⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

−−

−

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡−

=

−−

−

gndmTMmMTMM

E

MTMm

ETE

Sl

Sl

iT

TM

TE

Sl

i

m

RRSGPP

SGSG

GB

GBBB

C

CC

CG

000

0000000000

,

00000000000000000000

1

21

GC

( )⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

−−

−

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡−

=

−−

−

gndmTMmMTMM

E

MTMm

ETE

Sl

Sl

iT

TM

TE

Sl

i

m

RRSGPP

SGSG

GB

GBBB

C

CC

CG

000

0000000000

,

00000000000000000000

1

21

GC

G

u’m v’e um ve veb

Faraday’s law for inner electric loops;








dtdϕ Σ ve


C

( )⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

−−

−

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡−

=

−−

−

gndmTMmMTMM

E

MTMm

ETE

Sl

Sl

iT

TM

TE

Sl

i

m

RRSGPP

SGSG

GB

GBBB

C

CC

CG

000

0000000000

,

00000000000000000000

1

21

GC

( )⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

−−

−

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡−

=

−−

−

gndmTMmMTMM

E

MTMm

ETE

Sl

Sl

iT

TM

TE

Sl

i

m

RRSGPP

SGSG

GB

GBBB

C

CC

CG

000

0000000000

,

00000000000000000000

1

21

GC

G

u’m v’e um vev’eb veb








dtdψ iΣum


C

( )⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

−−

−

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡−

=

−−

−

gndmTMmMTMM

E

MTMm

ETE

Sl

Sl

iT

TM

TE

Sl

i

m

RRSGPP

SGSG

GB

GBBB

C

CC

CG

000

0000000000

,

00000000000000000000

1

21

GC

G








Σ veb


C

( )⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

−−

−

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡−

=

−−

−

gndmTMmMTMM

E

MTMm

ETE

Sl

Sl

iT

TM

TE

Sl

i

m

RRSGPP

SGSG

GB

GBBB

C

CC

CG

000

0000000000

,

00000000000000000000

1

21

GC

G







Cap.current

Conductioncurrent


C

( )⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

−−

−

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡−

=

−−

−

gndmTMmMTMM

E

MTMm

ETE

Sl

Sl

iT

TM

TE

Sl

i

m

RRSGPP

SGSG

GB

GBBB

C

CC

CG

000

0000000000

,

00000000000000000000

1

21

GC

G






Cap.current

Conductioncurrent

y=[it, φt]x’: x:


C

( )⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

−−

−

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡−

=

−−

−

gndmTMmMTMM

E

MTMm

ETE

Sl

Sl

iT

TM

TE

Sl

i

m

RRSGPP

SGSG

GB

GBBB

C

CC

CG

000

0000000000

,

00000000000000000000

1

21

GC

G





Magn. Flux

y=[it, φt]


C

( )⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

−−

−

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡−

=

−−

−

gndmTMmMTMM

E

MTMm

ETE

Sl

Sl

iT

TM

TE

Sl

i

m

RRSGPP

SGSG

GB

GBBB

C

CC

CG

000

0000000000

,

00000000000000000000

1

21

GC

G





C

( )⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

−−

−

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡−

=

−−

−

gndmTMmMTMM

E

MTMm

ETE

Sl

Sl

iT

TM

TE

Sl

i

m

RRSGPP

SGSG

GB

GBBB

C

CC

CG

000

0000000000

,

00000000000000000000

1

21

GC

G



y=[it, φt]


Structure of C, G matrices

( )⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

−−

−

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡−

=

−−

−

gndmTMmMTMM

E

MTMm

ETE

Sl

Sl

iT

TM

TE

Sl

i

m

RRSGPP

SGSG

GB

GBBB

C

CC

CG

000

0000000000

,

00000000000000000000

1

21

GC

T

tLBLxyBuGxxC ===+ ,,

dd

• “red” part of C is diagonal and of G is sym (inner faces equations)• conductances G, capacitances C, magnetic reluctances/permeances Rm =1/Gm• topological matrices B, S, P• Semi-sate equations: • are the eqs, of the FIT equivalent circuit ⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

yuu

x e

m


MIMO State Space Model based on FIT

000

0=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ ′−+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡uv

BBG

uv

GC e

m

e

dtdFIT equations:

with boundary conditions

in the frequency domain

State variables: electric and magnetic voltages of grid edge

Electric/magnetic terminals may be excited in current/flux or in voltage:

⎪⎩

⎪⎨⎧

=

=+

Lxy

zxGxCdtd

Current

VoltageMagn. Circ.Flux

.

(sC+G)x = z

y = Hz

Voltage

Current

Flux

Magn. Circ.


The graph of electric circuit The graph of magnetic circuit

Magneto-Electric Equivalent Circuits (MEEC)

Graphs E and M circuits = dual FIT grids


Hodge’s operators ⇒ Constitutive branch relations

dtd

mm

uGieMD

RuhMb

uCieMd

′=′′⇔=

ϕ=′⇔=

=′⇔=

σ

μ

ε

Typical branch in electric circuit Typical branch in magnetic circuit

iR

Ci ′′

i′u′

u

∑ϕ

=dt

de k mR ∑=θ ki

mumu′

ϕ

Branches of MEEC –Magneto-Electric Equivalent Circuit


Magnetic sub-circuit

Summation sub-circuit

Derivative sub-circuit

• The SPICE equivalent circuit consists in four mutual coupled sub-circuits (VPEC)

• The SPICE equivalent circuit has linear complexity w.r.t #FIT nodes

• Although, the number of DOFs is still large. MOR is a must !

Equivalent SPICE circuit of FIT modelElectric sub-circuit


Outline



D. MOR - What is Order Reduction

Large system

e.g. >10 000

DoFs

Small system

e.g. <100

DOFs

Essentially same I/O relation

Discretemodel:FIT DAE eqs.

Reducedmodel –Kirchhoffeqs.

Continuousmodel–Maxwell eqs.and b.c.

Apriori ROM (discretization)

Reduction on the fly Aposteriori ROM and model realization

Pre-grid Final grid


Aproiri Reduction Order Methods

Examples of apriori order reduction techniques:

• Optimal truncation of model domain (see ALROM)

• Cell homogenization - CellHo

• EQS+MS in Si, (LL)FW in SiO2, MQS in metal, ES+MS in air

• Local-integral equations for field vectors, Fourier transform, TL

• EMCE boundary conditions, DD with EM hooks

Any pre-processing for an effective discretization:

• Geometric approximations of the model domain

• Simplification of material behaviour

• Appropriate equations (field regime) in each sub-domain

• Field problem (re)formulation: equations, quantities

• Boundary and interface condition


Order Reduction on the fly

Examples of such techniques:

• Hierarchical structuring

• FIT, dFIT, dELOB

• Yee type for Manhattan geometries, local adapted grids

• Frequency dependent Hodge operators, FredHo for skin effect

• Algebraic Sparsified (ASPEEC), Hierarchical Substrate Struct. (HSS)

• Identification of optimal hooks

Any technique to generate a discrete model with reduce number of DoFs:

• Domain Decomposition

• Numeric method for discretization

• Appropriate grid or mesh

• Macro(cells)-models

• Equation sparsification

• Terminals reduction


Aposteriori Reduction Order Methods and model realization

Examples of aposteriori order reduction techniques:

• Krylov type, e.g. PvL, PRIMA

• Truncate balance realizations (TBR)

• Iterative Vector Fitting (VF)

• Differential Equation Macromodel (DEM) in time domain and Direct Stamping Macromodel (DSM) in frequency domain

• Parametric pmTBA

Any post-processing to generate a reduced circuit model:

• State space projection methods

• Truncate SS systems realizations

• Interpolation or fitting in the frequency domain

• Spice circuit synthesis

• Parametric Model Order Reduction


Variability Modeling

)()()(

21

21

21

P

P

P

pppLLpppCCpppRR

L

L

L

===

Extractor

Linear system becomes parametric!Lxy

BuxpGxpC

=

=+ )()(dtd

[ ]Pppp L21=p Parameter Vector

Geometric params Pppp L21


State equations and their derivatives extracted with FIT

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡−

=

0 0 00 0 0 0 0000000

TECC

CG

Sl

i

m

C

[ ][ ] signalsoutput i

signals,input v

variable,state yvu

t

t

e

m

==

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

yu

x ⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

=

0 0 0

0 000000

1

E

ETE

Sl

Sl

iT

PSG

GBGBB

G⎪⎩

⎪⎨⎧

=

=+

Lxy

BuxGxCdtd

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

=

0 0 00 0 0 0 0000000

TE

Sl

i

m

k

CC

CG

dpd

&

&

&

&

C

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡−

=

00 00 00 000000000

TE

Sl

i

k

GG

G

dpd

&

&

&

G

EEC case:

⎪⎩

⎪⎨

⎧

===

cbai

cbai

cbam

pppGpppCpppG

///0

σεμ


Validity range of first order Taylor Series expansion2

12

2

0

20

1 )(2 pp

yy

p εξε∂∂

=

2

1011

2D

typt <Δ⇒<ε

By “reversing” circuit function 1/y(p) (interchanging I/O signals)

rDytp

20

12<Δ⇒

1) The validity of the first order TS expansion depends on the value of the second order sensitivity.

2) The validity range can be increased by reverting the circuit function, i.e. by simply changing the excitation type of terminals


Variability analysis in the frequency domain: AFT

Adjoint Field Technique – AFT applied to the Finite Integral Techniques - FIT is used to handle the parameter variability. It provides accurate and low cost gradient information.

• AFT: only two field problems are solved (direct and adjoint one), regardless the number of params. No overhead for sensitivity evaluation

• The small variations model is based on results of sensitivity analysis

• The models are represented as parametric compact SPICE circuits.

• First order methods (direct and reverse) suitable to handle technology variability


Controlled sources in the original and adjoint circuits

pn

pn

φ

Rm

Gm

i

Cm

+1

-jω

a. Original circuit

Gm

i

Cm

φ

Rm

+1

-jω

b. Adjoint circuit


Principle: reduction have to be applied as early as possible !Steps of the Algorithm:

• Domain decomposition

• 3D Grid (mesh) calibration with dFIT

• Virtual Boundary Calibration with dELOB

• 3D Frequency Analysis by AFS

• Length Extension (TL)

• Extr. of par. red. model by VF

• Integration of compact parasitic extracted model into designand standard/variability (e.g. Monte Carlo) SPICE simulation

All Levels Reduced Order Modeling

On the fly order reduction

Apriori order reduction

Aposteriori order reduction

Mod

el e

xtra

ctio

n


Steps of the Algorithm for accurate Modeling and Simulation of Interconnect and Passives:

• Critical interconnects identification and domain decomposition

• 2D Grid (mesh) calibration

• Virtual Boundary Calibration

• 2D Frequency Analysis

• Length Extension (TL)

• Reduced Par. Model DEM

• Integration of compact parasitic extracted model intodesign and standard SPICE simulation

ALROM approach for interconnects

On the fly order reduction

Apriori order reduction

Aposteriori order reduction

Mod

el e

xtra

ctio

n


Frequency vs time domain MOR

• Advantages of MOR in the frequency domain:– very low order of the reduced model– adjoin technique ideal for the first order parametric

models extraction• Drawbacks:

– passivity enforcement– no error control– dangerous behavior: when work all competitors are

smashed, but if do not they fail deplorably

• Conclusion: – there is not a perfect approach


Outline



Typical partitioning of ICs

Air

< λ/10 = 500μ

Substrate

Environment

Long inter-connects (TL)

hooks

hooks

active comp.

passive comp.

Environment

EMCE is the best interface condition


First attend in the hook identification:chess-board regular structure

Interface between subdomain

Magnetic and electric connectors = windows for EM interaction

• EM field is perturbed if the number of connectors is too low

• The interface became transparent when each connector contains one grid-node


Air

SiO2

Si

Al

Connectors (hooks)

GND

Results with regular structured hooks

Relative error of the extracted capacitance matrix vs number of hooks

Impedance error

• Conform grid

• Non-conform grid

TL2 Test problem


A simple heuristic approach On the interface between devices and Si substrate:• Electric connectors under conductive path (capacitive coupling) • Magnetic connectors in rest (inductive coupling)

U2 Test case

Interface

CHRF201 Test case

Interface


Importance of magnetic hooks

L2 test case:

• Full (no DD)

• EL hooks

•MG hooks

• EM hooks


Optimal placement of hooks

• In order to identify the hooks, nodes on interface have to be merged in a minimal number of clusters, so that the approx. error is kept at acceptable level

• Thus, the hooks identification is formulated as a discrete optimization problem (with continuous restriction: accepted error, which can be evaluated by Adjoint Field Technique – param. Sens.)

• Since the E and M connectors can not be overlapping, the hook reduction is similar to the strategic game of GO• Being a NP problem, heuristic solutions (soft computing) can not be competed.

http://upload.wikimedia.org/wikipedia/en/0/08/Go-Equipment-Narrow-Black.png


Domain Decomposition versus Partitioning (DD vs DP)

• The hooks technique has practical importance when their number is low (e.g. <10-100)

• In this case the extracted models are reduced (by using: frequency dependent circuit functions Y, state matrices ABCD, or reduced order Spice circuits) and then interconnected in the global model of IC. Thus the hierarchical structure is preserved

• Unlike DD, which is basically an iterative process, the proposed approach we call Domain Partitioning (DP) is a “direct” one

• The challenge to reduce the number of hooks has to be accepted, otherwise, the EM field modeling in nowadays RF-ICs is insolvable


Outline


www.chameleon-rf.org

http://www.chameleon-rf.org/


Domain Partitioning: passive components

Air

< λ/10 = 500μ

Substrate

Environment


hooks

hooks

active comp.

passive comp.

Environment


ELMGE+M

Frequency characteristics with several kinds of hooks

CHRF201 benchmark - experimentally validated


CHRF-201 Y parameters

•ROM ord 10

• ROM ord 15

•Measured


Air

< λ/10 = 500μ

Substrate

Environment


hooks

hooks

active comp.

passive comp.

Environment

Domain Partitioning: Substrate treatement


Hierarchical Structured Substrate

Substrate – Level 3

Substrate – Level 2cells


Top contacts contacts on Level 2

Substrate – Level 3



Top contacts contacts on Level 2

• Si substrate is structured in virtual layer with increasing thickness• Layers are structured in rectangular super-elements/cells • Each cell is an EMCE with 10 terminals• Each cell generates a macro-model with 45 lumped elements.


Hierarchical multilevel sparsification of substrate

Level 0: top contacts

Level 1 (internal) contacts

Level 2contacts

Level 3 contact

Layer 1 cells

Layer 2 cells

Layer 3 cells

Level 0 -Top contacts

Level 1Clusters

Level 2Clusters

Vertical cross section

Upper view

(clusters)


Hierarchical sparse substrate model

Contacts on level n-1

Contacts at level n

Contacts at level n+1

Backbone tree

Equivalent circuit/stamp of the standard cell from layer n

• Optimal contacts identification, by Fourier analysis

• With appropriate DP, only one cell have to be modeled, by field solving

• It is a sparse circuit with tree back-bone


Application of the hierarchical sparsification – Test case TL2

Air

SiO2

Si

Connectors (hooks)

Al

0.093sec.Tin

22495451554515519695

42.4sec.Tin

21395097550975561495

HS

DD

=

⎥⎦

⎤⎢⎣

⎡−

−=

=

⎥⎦

⎤⎢⎣

⎡−

−=

m/pF....

C

m/pF....

C

HS

DD

The field solving is reduced to Laplace equation in a standard cell with 10 terminals


Air

< λ/10 = 500μ

Substrate

Environment


hooks

hooks

active comp.

passive comp.

Environment

Domain partitioning: Interconnects


Typical interconnect configuration

Si

SiO2metal

Modeled with ALROM-TL


Codestar benchmark #27• Nodes of initial mesh =

2 866 441• Initial no. of DOFs =

17,198,646• Reduced computational

domain: 200μ × 46.588μ× 17.74μ;

• Order of reduced model = 10

• CPU time for model extraction = 161 s

• Rel.err. = 5.0 %


Compact models #27q = 2 q = 6 q = 10


ROM error vs order q


Measurements vs simulation of ROM for q=10

* Rel.err (sim,red) =

1.3 %* Stable

* cir2sys =>non-minimal =>some more toolsfrom robust control systemare used

* Rel.err (mas,red) = 5.5 %


Outline



A – Introduction: coherent, “intelligent” approach

B - Problem formulation: EMCE boundary conditions

C - Numerical Approach: FIT (SS, MEEC)D – Domain partitioning: EM hooks, HSSE – Order reduction: ALROM+HooksF – Variability analysis: AFT, First order rational models

Conclusions. Main novelties


1. Be creative, exploit any opportunity to reduce order: do not neglect “details”, e.g. boundary conditions, magnetic coupling or type of terminal excitations

2. Keep it simple, but not primitive, “the brute force” approach is easy to be implemented, but it is seldom the best. Discretisation is the crucial step

3. Apply order reduction as soon as possible, in any modeling step, e.g. apriori and “on the fly” MOR

4. There is not a perfect MOR method, each category of systems (e.g. interconnects, substrate, passives) has its suitable approach

General recommendations


More readings

• Daniel Ioan, Irina Munteanu and Gabriela Ciuprina, Adjoint Field Technique Applied in Optimal Design of a Nonlinear Inductor, IEEE TRANSACTIONS ON MAGNETICS, VOL 34, NO. 5, SEPTEMBER 1998, 2849 [Ioan98-IEEE-MAG-AFT.pdf]

• Daniel Ioan and Irina Munteanu, Missing Link Rediscovered: The Electromagnetic Circuit Element Concept, JSAEM Studies in Applied Electromagnetics and Mechanics, vol. 8, oct. 1999, pp. 302{320, ISSN 1343-2869 [Ioan99-JSAEM-MissingLnk.pdf]

• Irina Munteanu and Daniel Ioan, A Survey on Parameter Extraction Techniques for Coupling Electromagnetic Devicesto Electric Circuits, Published in LNSCE (Lecture Notes in Computational Science and Engineering), Springer Verlag, Vol. 18, 2001, pp.337-358 [Munteanu01-LNCSE-Springer.pdf]

• D. Ioan, M.Radulescu,G.Ciuprina - Fast Extraction of Static Electric Parameters with Accuracy Control”, in Scientific Computing in Electrical Engineering (W.H.A. Schilders et alEds), Springer, 2004, pp.248-256 [Ioan04SCEE.pdf]

• Daniel Ioan, Marius Radulescu FDTD cell homogenization based on dual FIT – PIERS Abstracts - Progress in Electrical Engineering Research, March 28-31, Pisa 2004, Italy [Ioan04-Piers2004_LMN1.doc]

• Daniel Ioan, Marius Piper FIT Models with Frequency Dependent Hodge Operators for HF Effects in Metallic Conductors - PIERS Abstracts - Progress in Electrical Engineering Research, March 28-31, Pisa 2004, Italy [Ioan04-Piers2004_LMN2.doc]

• Daniel Ioan, Catalin Ciobotaru Equivalent Circuits of Linear Order for Electromagnetic Field Problems - PIERS Abstarcts - Progress in Electrical Engineering Research, March 28-31, Pisa 2004, Italy [Ioan04-Piers2004_LMN3.doc]

• Daniel IOAN, Gabriela CIUPRINA, Reduced Order Models at All Levels for CODESTAR problems - PIERS Abstarcts -Progress in Electrical Engineering Research, March 28-31,Pisa 2004, Italy [Ioan04-Piers2004_LMN3.doc]

• Daniel Ioan, Gabriela Ciuprina, M. Radulescu and M. Piper – Algebraic Sparsified Partial Equivalent Circuit (ASPEEC). Scientific Computing in Electrical Engeneering - SCEE 2004, Digest Book, Sept. 5-9, 2004, Capo D’Orlando Italy [scee04lmn_ioan.pdf]

• Daniel Ioan, Gabriela Ciuprina, Marius Radulescu, Ehrenfried Seebacher – Compat modeling and fast simulation of on-chip interconnect lines, Compumag 2005 Seyshan, 25-19 June, 2005 [Ioan05-COMPUMAG.pdf]

• Daniel Ioan, G. Ciuprina, M. Radulescu, Absorbing Boundary onditions for Compact Modeling of On-chip Passive structures, ISEF 2005 International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering, Baiona, Spain, Sep. 15-17, 2005, Best Paper Award [Ioan05_ISEF-Digest.doc] [Ioan05-isef.pdf]

• Daniel Ioan, Gabriela Ciuprina and Marius Radulescu, Theorems of Parameter Variations Applied for the Extraction of Compact Models of On-Chip Passive Structures, IEEE ISSCS Iasi, 14-15 July, 2005 [Ioan05-ISSCSIasi.pdf]

• P Meuris, G Ciuprina, E Seebacher , High-frequency simulations and compact models compared with measurements for passive on-chip -INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS Int. J. Numer. Model. 2005; 18:189–201 [MeurisCiuprinaSeebacher04INMJ.pdf]

• D. Ioan, G. Ciuprina, M. Radulescu, and E. Seebacher –"Compact Modeling and Fast Simulation of On-Chip Interconnect Lines", IEEE Transactions of Magnetics. Volume 42 Number 4, April 2006, pp 547-550 [Ioan06-IEEEMAG.pdf]

• Daniel Ioan, Gabriela Ciuprina, Marius Radulescu –"ABSORBING BOUNDARY CONDITIONS FOR COMPACT MODELING OF ON-CHIP PASSIVE STRUCTURES" COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 25, No. 3, 652-659, 2006. [Ioan06-Compel25(3).pdf].


More readings• Seebacher, D. D. Zutter, J. Maubach - "COMPACT

MODELING OF PASSIVE ON CHIP COMPONENTS" - Revue Roum. Sci. Tech. - Electrotechn. et Energ., Romainian Academy, Tome 51, No.3. pag. 309-312, 2006 [IoanCiuprinaRevue05.doc]

• Ioan,D., G. Ciuprina and M. Radulescu, “Algebraic Sparsefied Partial Equivalent Electric Circuit - ASPEEC,” in Scientific Computing in Electrical Engineering, Vol. 9, (Anile, A.M.; Alì, G.; Mascali, G., Eds.), 45-50, Springer, 2006 [scee04lmn_ioan.pdf]

• G. Ciuprina, Course on Model Order Reduction, Introduction to Matlab ROM workbench, Hands-on experience with ROM workbench, Centre for Analysis, Scientific Computing and Applications (CASA), Eindhoven University of Technology, April 10-12, 2006 [Ioan06-CEFC.pdf]

• G. Ciuprina, D. Ioan, D. Mihalache "Reduced Order Electromagnetic Models based on dual Finite Integrals Technique", Scientific Computing in Electrical Engineering, Vol. 10 (Ciuprna G., Ioan D., Eds), Springer, 2007 [Ciuprina07-Springer.pdf]

• Daniel Ioan, Gabriela Ciuprina – “Parametric Models for Electromagnetic Field Systems Related to Passive Integrated Components” PIERS 2007, Beijing, China, 26-30 March, 2007 [Ioan07-PIERS.doc]

• D. Ioan, WHA Schielders, G. Ciuprina – “Parametric-Circuit Models for Electromagnetic Field Systems Related to Passive Integrated Components” Compumag 2007, Aachen, Germany, June 24-28, 2007, IEEE Trans. on MAG [Ioan07-COMPUMAG.doc]

• Daniel Ioan, Gabriela Ciuprina, Sebastian Kula, Reduced Order Models for HF Interconnect over Lossy Semiconductor Substrate, SPI 07, 11th IEEE WORKSHOP ON SIGNAL PROPAGATION ON INTERCONNECTS, May 13-16, 2007, Ruta di Camogli (Genova), Italy [Ioan07-SPI.pdf]

• Daniel Ioan, Wil Schilders, Gabriela Ciuprina, Nick van der Meijs, Wim Schoenmaker, MODELS FOR INTEGRATED COMPONENTS COUPLED WITH THEIR EM ENVIRONMENT, ISEF 2007 - XIII International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering Prague, Czech Republic, September 13-15, 2007. Accepted for publication in Compel. [Ioan-ISEF2007_2page_digest[1].doc], [Ioan-ISEF2007.pdf]

• Gabriela Ciuprina, Daniel Ioan, Dragos Niculae, Jorge Fernandez Villena, L. Miguel Silveira Parametric models based on sensitivity analysis for passive components, ISEF 2007 - XIII International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering Prague, Czech Republic, September 13-15, 2007 [Ciuprina_ISEF2007_2page_digest]

• Ioan, D., Ciuprina, G.: Reduced order models of on-chip passive components and interconnects, workbench and test structures. in (W.H.A. Schilders, H.A. van der Vorst, J. Rommes, Eds). Model Order Reduction: Theory, Research Aspects and Applications, Springer series on Mathematics in Industry, Springer-Verlag 13, 447–467 (2008)

• Stefanescu, A., Ioan, D., Ciuprina, G.: Parametric models of transmission lines based on first ordersensitivities. in (J. Roos., L. Costa, Eds). Scientific Computing in Electrical Engineering, Springer, ECMI series (2009).

• Ciuprina, G., Ioan, D., Mihalache, D., Stefanescu, A.: The electromagnetic circuit element - the key of modelling em coupled integrated components. Revue Roumanie des Sciences Techniques, Serie Electrotechnique et energetique 54(1), 37–46 (2009)

· 1. be creative, exploit any opportunity to reduce order: do not neglect “details”, e.g....

Documents