inductance inﬂuence in high speed design and evolution of ...peec model with skin-effect...

Inductance Influence in

High Speed Design

and Evolution of (PEEC)

Partial Element Equivalent Circuit

Techniques

Albert E. Ruehli

EMC Dept., Missouri U. of Science and TechnologyEmeritus IBM Research, Yorktown

April, 2013 Slide 1 of 61

ACKNOWLEDGMENTS

Contributions to this work by:

Prof. Giulio Antonini

Universita degli Studi dell’Aquila

L’Aquila, Italy

Prof. Lijun Jiang

The University of Hong Kong

Hong Kong

Many authors: papers, theses and dissertations


OVERVIEW

Motivation

Introduction of PEEC Method

Inductance Computations

Skin-effect, Power Plane Modeling

Summary of Several Issues


LARGE SET OF EM DESIGN PROBLEMS

Electromagnetic Interference

Electrical Design

RFID Tags

Trains

Power Electronics

Mobile PhonesElectronics

Car

Household

ElectronicsDesktop

Laptop

Large Computes


ELECTRICAL AUTOMOTIVE DESIGN

Up to 100 Motors and Printed Ckt. Boards Up to 500 sensors and actuators

Cable harness: total of 3 km length!

WLAN, TV, radio, GPS, mobile phone

More than 15 highly sensitive antennas

Source:Troscher@CST


MOST IMPORTANT EARLYDEVELOPMENTS

Foundations of All Our Work 1850 Kirchhoff KCL and KVL: I cannot

think of anything I use more in my dailywork.

1857 Maxwell’s Equations:Richard Feynman: The most significantevent of the nineteenth century is Maxwell’sdiscovery of the law of electrodynamics.


FORMULATION USED IN EM SOLVERS1960’s

Zienkiewicz, Cheung, Finite Elements inElectromagnetics,(FE), 1965

K. S. Yee, FDTD, Differential Eq. (DE), 1966 R. F. Harrington, MoM, Integral Eq. (IE), 1967

1970’s P-B. Johns, TLM, Circuit (DE), 1971 A. E. Ruehli, PEEC, Circuit (IE), 1972 T. Weiland, FIT (DE), 1977

1980-1990’s G. Mur, Absorb. Boundary Condition (DE), 1981 V. Rokhlin, Multipole (IE), 1985 J. Fang, Hybrid solvers (IE,DE,Ckt), 1993 Kapur, Long, SVD(QR) decomposition (IE), 1996


PARTIAL ELEMENT EQUIVALENTCIRCUIT (PEEC), 1972

Integral Equation for Total Electric Field (pcEFIE)

KVL: v=∫

E ·dl

Ei(r, t) =J(r, t)

σ+µ

∫v′

G(r, r ′)∂J(r′, td)

∂tdv′

+∇ε0

∫v′

G(r , r′)q(r′, td)dv′

PEEC Circuit Model Element Computation

KVL Loop: Voltage = R I + s Lp I + Q/C RHS Term 1: Resistance RHS Term 2: Partial Inductance RHS Term 3: Coefficient of Potential


PEEC MODEL FOR A CONDUCTINGSHEET

J y

xJ

Currents are flowing in volume cells in the x,y,zdirections; Uniform in each cross-section cell

1Φ 2Φ 3Φ Φ4

Q1

Q Ν

Charges are on the surfaces of the conductors


(Lp,P,R,τ)PEEC EQUIVALENT CIRCUIT

PEEC Equivalent Ckt. For Two Basic Loops Model for MNA (Modified Nodal Analysis) Coupled Partial Inductances and Capacitances Example: 3 Node Discretization of “Metal Stick”

iL1

iL2

v1

v2 v

3


WHEN DID SI,PI, NI START?

SI = Signal Integrity, PI = Power Integrity,

NI = Noise Integrity 1972 CAD definition of an interconnect: L=0,

R=0, C=(maybe)

Limited interconnect modeling used in general

Mainframes: 1967, IBM 360/91: 60 nsec cycle

IBM PC: 1981 4.77 MHz, 210 nsec cycle

Session on SI/PI modeling, 1981 ISCASAttendance: 7 people!


HOW INDUCTANCE BECAME A KEYISSUE

First Ground Bounce (∆I ) Noise Work,

1974 Ground and VDD noise problems were first

important issue

Clear need for inductance calculations

Example: ∆V = L∆I/∆t

∆ I = 10 A, ∆ t = 5ns, L = 1 nH,Result: ∆ V =2 V !

L Problem of great concern for mainframe powerand ground - PEEC full wave? Not needed!


END 1960’s: INDUCTANCE WORK

Inductance Formulation for Interconnects Inductance computation example

General circuit analysis applies to PEEC!

Small example illustration, no branching

Lp Lp

Lp

Lp

44 L = ?

33 22

11


SMALL PEEC EXAMPLE LOOP L

Lp Lp

Lp

Lp

44 L = ?

33 22

11

L

L p11

p22

L p33

L p44L

Simple circuit analysis yields:

Lloop= Lp11+Lp33−2Lp13+Lp22+Lp44−2Lp24


COMPUTATION OF PARTIALINDUCTANCES

l

a

Example: Evaluation of Partial Inductances

“Near” and “Far” Coefficients; High AccuracyRequired!

Lpkm=µ

4πakam

∫ak

∫am

∫lk

∫lm

dlk · ¯dlm| rk− rm |

dakdam

Lpkm=µlklm

4πRkm


ISSUES WITH PARTIAL INDUCTANCES

Practical, Accurate Partial Inductance

Computation is Challenging Practical approach: combination of analytical

and numerical techniques

Need accuracy for cells with very large aspectratios

Important: Accuracy, speed, non-orthogonalshapes

Today: Good contributions by many researchers


PARTIAL L ACCURACY ISSUE

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10

−6

Length [m]

Lp

13/l [

H/m

]

Round conductorFinite thicknessZero thickness

0 0.5 1 1.5 2 2.5 31.6

1.8

2

2.2

2.4

2.6

2.8

3x 10

−7

Length [m]

(Lp

11−

Lp

13)/

l [H

/m]

Round conductorFinite thicknessZero thickness

Lp11 Partial inductance Lp11 - Lp12 Difference


IMPACT OF AVAILABILITY OF EMSOLVERS

1970 - 1990 Key Tools: SPICE circuit solver, L, C

calculators Education: Mostly conventional courses in

electromagnetics Needed: Design engineers need to be

specialists with much practical insight

1990 - Today Tools: Full wave 3D EM solvers are mandatory

for high-end designs today Education: Today EM education required


MODELING IN TODAY’S ENVIRONMENT

2010

Miniaturization leads to speed-up (today some)

PC processor: Example Intel, clock: 3.6 GHz

Mainframe: IBM Z196 Clock rate: 5.2 GHz

190 to 280 picoseconds for several transitions!

Signal frequency content up to 40 GHz

Today: Need full-wave - more that inductance!


BLOCK-BASED PEEC CIRCUIT-EMSOLVER IMPLEMENTATION

.option cap = yes

+ delay = yes

+ ind = yes+ res = yes

3D Capacitances(Lp)PEEC . (Lp,R)PEEC

Low Impedance Power Supply Modeling

Signal Propagation Modeling

(Lp,C,R)PEEC

(Lp,P,R,τ) PEEC

GENERAL FULL WAVE PEEC MODEL


IMPORTANT SOLVER FEATURES

Practical PEEC Models Must Include

Skin-Effect and Dielectric Loss Models! Dielectric loss model not presented here

Several types of skin-effect models considered

1D model,Volume Filament (VFI) and GlobalSurface Impedance (3D-GSI) models

Ruehli,Antonini,Jiang, “Skin-effect loss modelsfor time and frequency domain PEEC solver”,IEEE Proceedings, Febr. 2013


WHY 3D SKIN-EFFECT MODELS?

MODELING REQUIREMENTS

3D Discontinuities: Connectors, vias, cornermodels, ground plane holes

Skin-effect distorts responses in time andfrequency domain

Important for 2D transmission line problemsGood 2D skin-effect models available today

Challenge: Efficient 3D skin-effect models whichwork well


APPROACHES USED IN PEEC

DIFFERENT MODEL ISSUES

Established Volume Filament Model (3D-VFI)provides reliable solutions - works well

However, would like model with fewer globalcouplings than VFI

New model presented is extension of 2D GlobalSurface Impedance (GSI) work Coperich,Cangellaris


PEEC MODEL WITH SKIN-EFFECT

EQUIVALENT CIRCUIT FOR EACH PEEC CELL External model is standard PEEC Interface is zero thickness partial inductance Zs is the impedance of skin-effect model

+l +

1Ic2Ic

Is 2I s 1

1

p22

1Lv

111

e

e e

e

e

LpI

11ep

1I i

Zs I


THE CONVENTIONAL 1D SKIN-EFFECTMODEL

1D BASED MODEL WORKS ONLY FOR VERY

HIGH FREQUENCIES!

Example for conventional 1D skin-effect model

Skin-depth penetration must be small

δ <<

ZsRegion 1

Dielectric

Region 2

Conductor


1D VFI SKIN-EFFECT MODEL

WIDELY USED MODEL FOR TL CONDUCTORS

Each filament is a partial inductance Coupled partial inductances, high accuracy

needed!

W

L

T

NN

-

T1I

R

1pL 1N

p

11PL

L

V+

N

1

R


MESHED CONDUCTING BLOCK FOR3D-VFI PEEC CIRCUIT

J y

xJ

Currents are flowing in volume cells in the x,y,z directions

z

xy

pLR

All branches consist of the series resistor R and coupled partial Lp


GSI COUPLING TO THIN CONDUCTORSKIN-EFFECT MODEL

Model 2 region: GSI models for different shapes

Coupled to skin-effect model through surface

GSI skin-effect sandwich for thin conductors

z

yx

Top surface

Bottom surface

Skin−Effect

Model

Lp

Lp


GSI MODEL WITH INTERNALMACROMODEL

1D Example of Skin-Effect Circuit Model

Physics-based macromodel for currentdistribution

We solve differential equation with circuit

Finite thickness: Do not have exponentialthickness current decay for low frequencies

E = xEx H = yHy

∂Ex

∂z=−µ

∂Hy

∂t

∂Hy

∂z=−σEx


PHYSICS BASED MACROMODEL

x

y

z

∆x

∆y

I1

d

A

BS

U

2R

1R

R

RK

K−1

0.5

0.5 L

L11L0.5

K−1K−1L0.5

AC

S U


COMPARISON OF LADDER AND1D-EXP MODEL

10−3

10−2

10−1

100

101

102

10−4

10−3

10−2

10−1

100

Frequency [GHz]

Interna

l induc

tance

[nH]

PM−GSI1DExp

10−3

10−2

10−1

100

101

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency [GHz]

Interna

l resis

tance

[Ω]

PM−GSI1DExp


INTERNAL MODEL EXTENSION

EXTENSION TO 3D CONDUCTORS

General GSI: Coupling between ALL surfacecells on conductor

Subdivide non-uniform in all 3 directions

Efficient: small cells towards all surfaces

y

z

x


MULTIPLE CELLS ON ALL SURFACES

Cross-section GSI model for internal conductorThe cross-section surface nodes are coupled to all

external PEEC circuit cells

+l +l

PEEC Model

GSI Impedance Model

1Ic2Ic

Is 2Is 1

1

p22

1Lv1

11

e

e e

e

e

LpI

11ep

12Ic

Is 21

p22

Lv1

e e

e

e

LpI 22 2


NUMERICAL RESULTS

MODELING OF CHALLENGING PROBLEMS

Problem difficulty depends on geometry,cross-section shape, etc.

Current path can vary greatly with frequency ortime

Excitations, contacts are different for non-circuitmodels


HORSE SHOE PROBLEM

x

y

z

d

x w

ly

l

wcWide Conductor + Corners Problem

Narrow conductor is simple problem Wide conductor W = 10 µm, d = 6µm , Yℓ = 50 µm


RESULTS FOR HShoe PROBLEM

Horse-Shoe problem 6 µ thick and 10 µ wide

10−2

100

102

0.5

1

1.5

2

2.5

3

3.5x 10

−4

Frequency [GHz]

R [k

Ω]

3D−VFI3D−GSI

10−2

100

102

2

2.5

3

3.5

4

4.5

5x 10

−5

Frequency [GHz]

L [µ

H]

3D−VFI3D−GSI

Left: Inductance Right: Resistance


L-SHAPED CONDUCTOR PROBLEM

x

y

z

zg

z

y

x

w

w

L,R = ?

l

Short

gy

xg

w

d

wx

Non-Trivial 3D Skin-Effect Problem

Low frequency: minimum resistance current flow

High frequency: minimum inductance result


RESULTS FOR L-SHAPED CONDUCTORPROBLEM

Conductor and groundplane are 5 µ thick

10−4

10−2

100

1.2

1.4

1.6

1.8

2

2.2

2.4

x 10−4

Frequency [GHz]

Indu

ctan

ce [µ

H]

3D−VFIThin−GSI

10−4

10−2

100

102

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

−4

Frequency [GHz]

Res

ista

nce

[kοh

ms]

3D−VFIThin−GSI

Left: Inductance Right: Resistance


LOSSY TRANSMISSION LINE VFIMODEL TEST EXAMPLE

PEEC model used cells with largest aspect ratio

1 : 4750

I s

R L = 50 Ohms

L = 50 mm

T = 1

W = 20

S = 20

µm

µm

µm


LOSSY TRANSMISSION LINE TESTEXAMPLE

0 0.5 1 1.5 2 2.5 3 3.5

x 10−9

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Time [s]

Volta

ge [V

]

5 cm transmission line waveform comparison

PEECPowerSpice

Waveform comparisons for 3D PEEC and2D transmission Line model, Spice solvers


SUMMARY ON SKIN-EFFECT MODELS

Skin-Effect Models to Choose From

Volume Filament Model (VFI) model

Circuit-oriented form of 3D-GSI skin-effect model

Thin conductor GSI volume model efficient

State of PEEC Skin-Effect Research

Improvement of compute time for solutions withskin-effect

Skin-effect paper published Proc. IEEE withsurface IE model comparison


PPP POWER PLANE MODELING

FAST POWER PLANE PEEC MODEL

Currents are equal and opposite in planes(reduces unknowns)

Inductive coupling currents decay fast for planepairs

Far coupling can be totally ignored; introducesparsity in coupling

Li, Ruehli, Fan, “Accurate and efficientcomputation of power plane pair inductances”,Missouri Univ. S+T, EPEPS, 2012, Tempe, AZ


PC PLANE INDUCTANCE ISSUE

Decoupling capacitors

IC

GND

PWR


DIFFERENTIAL CURRENT MODEL

EARLY WORK ON TRANSMISSION LINES

Reduce currents:– opposite in the planes

Unknowns are plane voltage differences

Reduce unknowns by factor of almost 2!

−− −−

Lp 000−1

Lp 111

+

Lp

2

+++

v−1

Lp0’0’

0’−1’Lp

1’1’1’

Lp2’2’

2’

V0 V1 V2

22

I

I


FAST COUPLING DELAY

New differential Ls and fast approximation

Lskm= 2(Lpkm−Lpkm′) Lskm= 0.1∆xq2/|k−m|

10−1

100

101

102

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Section Distance,mm

Norm

alize

d to

par

tial−

self

indu

ctan

ce


DIFFERENTIAL PLANE MODEL

AT ANY NODE OF THE PLANE WE CAN APPLY

Observation point, where we inject a unit current

Between any node pair can insert short orL-macromodel

Holes in planes: remove inductive cell pairs

−

−

1−

N1

N2

N3

Is

−I2x

I

Iy2

Ix1

I

Ix1

y

Iy

1

x

2y

y

4N

Ix2


MNA MATRIX FORMULATION

Modified Nodal Matrix (MNA) with added shorts

=

A

A 0Ib

Vn

0

Is

I V

T xLs

yLss

s

−

−

SHORTS ETC EQUATIONS sh sh

Example: 50000 Nodes and 500 Shorts


LARGE NUMBER OF PORTS

Realistic boards have many contacts

0 100 200 300 400 500 60020

40

60

80

100

Contact Number

Tim

e, s

ec

0 10 20 30 40 500

100

200

300

400

500

600

Contact Number

Tim

e, s

ec

Left: PPP model Right: Zpp model


NUMERICAL RESULTS

POWER PLANE RESULTS USING PPP

10cm x 10cm plane, 0.2mm plane spacing

Uniform: 0.5mm x 0.5mm mesh

Sub-mesh: 3mm x 3mm areas: 0.5mm mesh,1mm general mesh, 4 GBytes memory

Sparsification of matrix used in both cases

Solution Details No. Unknowns Induct. [pH] Comp. Time [s]

Uniform mesh 120762 350.57 6384

Sub-mesh 30548 351.12 87


PLANE INDUCTANCE MAP

5cm x 5cm Planes, Plane to Plane spacing 0.2mm

0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

mm

mm

100

200

300

400

500

600

700

800

Dark blue:0.1 nH; Light blue:0.3 nH; Green:0.4 nH; Yel-low:0.55 nH; Red:0.7 nH


SUMMARY ON PPP MODELS

SPECIAL PEEC MODEL FOR PC PLANE

Parallel plane (PPP) model, for differentialcurrents

Exploit fast decay of differential coupling

New time saving mesh reduction method

FUTURE PLANS

More levels in mesh reduction

Inclusion of decoupling capacitors ESC

Multi-layer models


BRIEF TREATMENT OF TWOTOPICS

THREE INTERESTING ISSUES

1. Low frequency electromagneticsolution

2. Speed improvement techniques


1. LOW FREQUENCY SOLUTIONACCURACY PROBLEM

INTEGRAL EQUATION (IE) FORMULATION

USING MOMENTS METHOD SOLUTION Good low frequency solution is difficult

100

105

Real Current Low−Frequency Instability

Frequency (Hz)

C u r

r e n

t (A)


LOW FREQUENCY SOLUTION

MANY IMPORTANT APPLICATIONS

dc solution is very important for VLSI systems

Need to model long strings time domain signal

Low frequency solution needed for powerengineering

CONVENTIONAL FIXES

Local star-delta solution for IE-MoM solutions

Implementation is complicated; done at the localmeshing level


PEEC DC TO DAYLIGHT SOLUTION

Formulation with Modified Nodal Analysis (MNA)

Current-only (Moments) formulations becomesingular

The PEEC method separates the capacitanceand inductance path

The MNA circuit formulations provide currentand voltage (potentials) unknowns

sP−1 −Aℓ

−ATℓ −(Zs(s)+sLp)

Φn(s)

I ℓ(s)

=

Ai I i(s)

0


2. SPEED-UP METHOD USING QR

Faster Solution of Large Problems PEEC matrix has dense areas and a large

number of far couplings

Apply QR Matrix Based Algorithm Issues : Full matrix operations can be costly Observation: Coefficients vary slowly with

distance Hence: Far couplings rows/columns contain less

real information This is indicated by rank r of matrix Solution: QR find rank r and tries to compress

matrix to use only relevant information


QR MATRIX COMPRESSION SCHEME

Thinning of Coupling with QR Algorithm

= m

r

r

pp

m

Q

R

Operations:mp vs. r (p + m)

Qk =

[

Ak−k−1

∑i=1

RikQi)

]

/Rkk

Rik = QiTAk ; k= 1· · · r


CONVENTIONAL PARTIALINDUCTANCE COUPLING

+ + +

++++

++++

+Lp11

Lp

Lp

Lp

22

Lp

Lp

Lp

33

44

55

66

77

I

I

I

I

1Lps 15 I1

sLp I47 4

2

3

4


QR COMPRESSED Lp COUPLING

a

=β

β

β

β

β

β

β

β

b

+

+

+

Source: Gope, Ruehli, Jandhyala

Lp

V 5

V

2I

I3

I

I

I

11

44

Lp 33

Lp

Lp22

1

4I

V 6

7

Lp

Lp

Lp

6a

7a

5a 5b

Lp 6b

Lp 7b

s

s

s

s

Lp

a1 a2 a3 a4

b1 b2 b3 b4

55Lp

66Lp

77Lp

V

V

V

5

6

7

I

I

I

I

1

2

3

4

QR

CCCSLp


SUMMARY OF QR MATRIXSPARSIFICATION

Feasibility shown for partial inductances

QR applies to inductance, capacitancequasi-static, full-wave solvers

Similar approaches in use today: ACA, SVD,H-matrix

Use the new sparser matrix in iterative, etc.speed-up techniques


SUMMARY

SUMMARY OF PEEC OVERVIEW Shown as many important aspects as possible PEEC is very flexible, allows flexible building

block solution Structure stamped matrix circuit solver approach

- very flexible solution!

APPLICATION OF PEEC TECHNIQUES Increased use in power engineering 3D modeling in SI/PI/NI Recent research: Efficient power plane modeling


inductance inﬂuence in high speed design and evolution of ...peec model with skin-effect...

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