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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
April, 2013 Slide 2 of 61
OVERVIEW
Motivation
Introduction of PEEC Method
Inductance Computations
Skin-effect, Power Plane Modeling
Summary of Several Issues
April, 2013 Slide 3 of 61
LARGE SET OF EM DESIGN PROBLEMS
Electromagnetic Interference
Electrical Design
RFID Tags
Trains
Power Electronics
Mobile PhonesElectronics
Car
Household
ElectronicsDesktop
Laptop
Large Computes
April, 2013 Slide 4 of 61
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
April, 2013 Slide 5 of 61
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.
April, 2013 Slide 6 of 61
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
April, 2013 Slide 7 of 61
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
April, 2013 Slide 8 of 61
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
April, 2013 Slide 9 of 61
(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
April, 2013 Slide 10 of 61
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!
April, 2013 Slide 11 of 61
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!
April, 2013 Slide 12 of 61
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
April, 2013 Slide 13 of 61
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
April, 2013 Slide 14 of 61
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
April, 2013 Slide 15 of 61
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
April, 2013 Slide 16 of 61
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
April, 2013 Slide 17 of 61
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
April, 2013 Slide 18 of 61
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!
April, 2013 Slide 19 of 61
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
April, 2013 Slide 20 of 61
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
April, 2013 Slide 21 of 61
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
April, 2013 Slide 22 of 61
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
April, 2013 Slide 23 of 61
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
April, 2013 Slide 24 of 61
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
April, 2013 Slide 25 of 61
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
April, 2013 Slide 26 of 61
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
April, 2013 Slide 27 of 61
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
April, 2013 Slide 28 of 61
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
April, 2013 Slide 29 of 61
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
April, 2013 Slide 30 of 61
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
April, 2013 Slide 31 of 61
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
April, 2013 Slide 32 of 61
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
April, 2013 Slide 33 of 61
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
April, 2013 Slide 34 of 61
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
April, 2013 Slide 35 of 61
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
April, 2013 Slide 36 of 61
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
April, 2013 Slide 37 of 61
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
April, 2013 Slide 38 of 61
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
April, 2013 Slide 39 of 61
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
April, 2013 Slide 40 of 61
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
April, 2013 Slide 41 of 61
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
April, 2013 Slide 42 of 61
PC PLANE INDUCTANCE ISSUE
Decoupling capacitors
IC
GND
PWR
April, 2013 Slide 43 of 61
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
April, 2013 Slide 44 of 61
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
April, 2013 Slide 45 of 61
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
April, 2013 Slide 46 of 61
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
April, 2013 Slide 47 of 61
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
April, 2013 Slide 48 of 61
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
April, 2013 Slide 49 of 61
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
April, 2013 Slide 50 of 61
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
April, 2013 Slide 51 of 61
BRIEF TREATMENT OF TWOTOPICS
THREE INTERESTING ISSUES
1. Low frequency electromagneticsolution
2. Speed improvement techniques
April, 2013 Slide 52 of 61
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)
April, 2013 Slide 53 of 61
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
April, 2013 Slide 54 of 61
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
April, 2013 Slide 55 of 61
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
April, 2013 Slide 56 of 61
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
April, 2013 Slide 57 of 61
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
April, 2013 Slide 58 of 61
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
April, 2013 Slide 59 of 61
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
April, 2013 Slide 60 of 61
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
April, 2013 Slide 61 of 61
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