Inductance Computations
Using
Partial Inductance Concepts
Albert Ruehli
November, 2007
Nov., 2007 Slide 1 of 32
Outline
● Partial Inductance Concepts
● Example Macromodels
● Speeding Up Using QR
November, 2007 Slide 2 of32
Inductance Computation
Inductance Definition● General loops with couplings
● Φ flux, I current
V V
II
Φ 2Φ12Φ
1
1
1
2
2
L11 = Φ1/I1 L21 = Φ2/I1
November, 2007 Slide 3 of32
Inductance calculations
Partial Element Equivalent Circuits(PEEC)
● Loops are insufficient for general computations
● How do we model general geometries with loops?
● Many issues like skin-effect models are relevant
1 2
What Is a Partial Inductance?
● Calculate partial inductances of segments
● Building block approach for general geometries
● Circuit analysis to solve general case
November, 2007 Slide 4 of32
Partial Inductance Concepts
Inductance of a Square Loop● Break geometry into segments
● Start: Conventional formulation loop
● Use vector potentialA
Lloop =ΦI
=1I
Z
aΦB·da=
1I
Z
aΦ(∇×A) ·da (1)
Φ
aΦac
B = ∇×A (2)
November, 2007 Slide 5 of32
Partial Inductance Concepts
Lloop =ΦI
=1I
Z
aΦ(∇×A) ·da (3)
Φ
aΦac
Using Stoke’s TheoremZ
aΦ(∇×A) ·da=
I
ℓA·dℓ (4)
Lloop =1I
1ac
Z
ac
I
ℓA·dℓda (5)
November, 2007 Slide 6 of32
Partial Inductance Concepts (Cont.)From last slide
Lloop =1I
1ac
Z
ac
I
ℓA·dℓda
The magnetic vector potential is given by
A =µ
4πIa′c
Z
a′c
I
ℓ′
dℓ′
|r − r ′| ·dℓ′da′
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O
r’ r
| r − r’|
I
Lloop =µ
4π1a′c
1ac
Z
ac
I
ℓ
Z
a′c
I
ℓ′
dℓ′ ·dℓ
|r − r ′|da′da
November, 2007 Slide 7 of32
PEEC Inductance Derivation
Lloop =µ
4π1a′c
1ac
Z
ac
I
ℓ
Z
a′c
I
ℓ′
dℓ′ ·dℓ
|r − r ′|da′da
Φ
aΦac
Lloop =4
∑j=1
4
∑k=1
µ4π
1ak
1a j
Z
a j
Z
ℓ j
Z
ak
Z
ℓk
dℓk ·dℓ j
|r j − rk|dakdaj
November, 2007 Slide 8 of32
PEEC Inductance Calculations
Definition of Partial Inductance
Lp jk =µ
4π1ak
1a j
Z
a j
Z
ℓ j
Z
ak
Z
ℓk
dℓk ·dℓ j
|r j − rk|dakdaj
Lt =4
∑k=1
4
∑j=1
Lp jk (6)
Lp
Lp
Lp
Lp
= ?tL 44
33
22
11
#4
#3
#2
#1
November, 2007 Slide 9 of32
Loop Inductance Calculations
Appy Kirchoff’s Laws for Circuits
● Voltage across partial inductances
● Currents through the partial inductances
V,I
1
I
I
I
I
4V+
3V
++2V
1V
-
-
-
-
+4
3
2
Lp
Lp
Lp
Lp
= ?tL 44
33
22
11
#4
#3
#2
#1
November, 2007 Slide 10 of32
Circuit Analysis using Lps
General Circuit Analysis with Partial Inductances● Can use conventional circuit analysis or Spice
Lp11 Lp12 Lp13 Lp14
Lp21 Lp22 Lp23 Lp24
Lp31 Lp32 Lp33 Lp34
Lp41 Lp42 Lp43 Lp44
sI1
sI2
sI3
sI4
=
V1
V2
V3
V4
V = V1+V2+V3+V4 (7)
I = I1 = I2 = I3 = I4 (8)
L =VsI
=4
∑k=1
4
∑m=1
Lpkm (9)
November, 2007 Slide 11 of32
Skin-Effect Model (VFI)
Skin-effect With(Lp,R)PEEC Model● Break up conductor cross-section into filaments● Compute series resistance for each filament● Compute partial inductance matrix
Conductor Proximity Effect● Interaction of skin-effect in multiple conductors● Strongest between close conductors● Need to solveglobal interaction problem?
Model Simplification (Macromodels)● Different approaches for simpler interaction models● Simplified interaction models for distant conductors● QR model shown below
November, 2007 Slide 12 of32
Volume Filament Skin-Effect Model
One Simple Model for Skin-Effect● Break up conductors into bars (filaments)
● Each bar (filament) has strong coupling to others
● Low frequency skin-effect: need to include resitances
l
t
w S
N
N+11
2N
2
#2#1
Circuit Model for Each Conductor
NN
-
T1I
R
1pL 1N
p
11PL
L
V+
N
1
R
November, 2007 Slide 13 of32
Example L,R Skin-Effect
● Square cross section example
● Conductor spacing is same as size
November, 2007 Slide 14 of32
Simplifications of Lp ComputationsBasic Observations
● Far spaced conductors
● Slow variationofLp with distance
● Very good approximate answer
Lpi j (dmax) < Lpi j (Exact) < Lpi j (dmin)
Lpi j ≈R
c jA·dl j
Ii≃ µo
4πl i l j
r
min
max
d
d
November, 2007 Slide 15 of32
Proximity Macromodel
Simple Coupling Macromodel● Each Coupling: SINGLE Mutual Partial Filament!
● Each Conductor: ONE Self Calculation
● Full Problem:(Np)3 Compute Time
(ω)
(ω)
(ω)
(ω)T
T
2
1I
I 2
1R
R T2
T1
p
p
L
L
21
12p
p
V
V
-
-+
+
November, 2007 Slide 16 of32
Transmission Line Type Model
Example Model● Usual model infinitely long line
● Need to use PEEC model for finite length!
● New insights by example
● Symmetrical Case
● Differential Mode Only
-2
I
1I =
I
I2
1
November, 2007 Slide 17 of32
Transmission Line Type Model
Infinite Line has End Effect Compensation● PEEC model of TL● Simple coupling model● Each section couples along infinite length● Symmetry introduces reduced coupling
Finite Length Line, Corner● Reduced inductance of finite length line● Find error bound between the two models● Can calculate L for examples like corner
November, 2007 Slide 18 of32
TL Model for Infinite Length
• Section both conductors• Compute partial inductances for sections• Section size determines accuracy
2’1’0’-1’
210-1
I
I
2’2’1’1’0’0’
00
-- --
++++
-1 VVVv 210
LpLpLp
LpLpLp 11 22
Lk =Vk−Vk−1
sI= 2
∞
∑m=−∞
(Lpkm−Lpk′m)
Lskm = 2(Lpkm−Lpkm′)
November, 2007 Slide 19 of32
Differential Coupling Decay
• Use section to section couplings
• How fast Does coupling decay?
• Equation for differential couplings
• Distance between sections∆x|k−m|
∆
s
x
Lskm= 0.1∆xq2/|k−m| q := s/[∆x|k−m|]
November, 2007 Slide 20 of32
Differential Coupling DecayExample
SectionN Ls1N Ls1N Approx.
1 0.27765 -
2 0.013375 0.01
3 0.001309 0.00125
5 0.000159 0.0001563
7 4.66e-05 4.6296e-05
9 1.96e-05 1.953e-05
11 1e-05 1e-05
November, 2007 Slide 21 of32
Section based MacromodelsWithout Couplings
Use Section Model
● Can we ignore coupling between sections?
● Check both types of models
● How large is error?
November, 2007 Slide 22 of32
Macromodels Without Couplings
Transmission Line Section Model● Total (coupled) TL
● Table: No coupling between sections results
SectionsF L1−F Ls1−F Macromodel
1 0.27765 0.27765
2 0.582049 0.555299
4 1.196842 1.110598
6 1.813189 1.665898
8 2.429942 2.221197
10 3.046943 2.776496
November, 2007 Slide 23 of32
Semi-Infinite Line Model
Macromodel Applications
● Many practical examples
● Semi Infinite Lines, corner etc.
1’0’-1’
10-1
I
I
1’1’0’0’
00
-- -
+++
-1 VVv 10
LpLp
LpLp 11
November, 2007 Slide 24 of32
Speed up for Partial InductanceCalculations
Can We Use Special Tricks?
● Simplifications examples above
● Stability: Partial inductance matrix is positive definite
● Cannot leave out coupling elements arbitrarily!
● Need other approaches
Mathematically Clean Approach
● Start with partial inductance matrix
● Matrix for subdivided problem is low rank
● Take advantage of low matrix rank
November, 2007 Slide 25 of32
Large Lp Matrix in Block Form
General: Check Coupling factors for
partitioning into blocks● Find many couplings which are weak
● Coupling factorsγ ≤ 0.25,0.5
● Max. inductive coupling:Lc = Lp12/√
Lp11Lp22
● Far coupling factors are very small!
● Can also use distance-size related criteria
● Hence: easy to identify far coupled matrix blocks
November, 2007 Slide 26 of32
Speep-Up for Large Lp Matrices
Example Lp Matrix
● Weak coupling QR evaluation
● QR rank reduction for far couplings
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Far coupling, Low Rank Sections
Local coupling is dense
sI V
November, 2007 Slide 27 of32
Speeding Up of Partial ElementCouplings
Basices of QR Matrix Speed-Up(Gope,Jandhyala)
● Algorithm based on work by Kapur and Long
● Far partial element matrices are dense
● Coefficient values vary slow with distance
● Element matrix vector product is costly
QR Matrix Algorithm
● Far couplings rows/columns have less information
● This is indicated by rankr of matrix
● QR exploits rankr with sampled coupling equations
November, 2007 Slide 28 of32
QR Matrix Compression Scheme
● Thinning of Coupling with QR Algorithm
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= 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
November, 2007 Slide 29 of32
Conventional Partial Inductance 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
November, 2007 Slide 30 of32
QR Compressed Lp Coupling
a
=β
β
β
β
β
β
β
β
b
+
+
+
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
November, 2007 Slide 31 of32
Summary and Conclusions
Inductance Computations
● General, flexible approach● Use for analytical and numerical computations● Partial element computations, Lp matrix
simplifcations● QR matrix vector product speed-up algorithm● QR sampling speeds up matrix element computations● ApproximatelyO(NlogN) solve time● Inductance,quasi-static, full-wave PEEC applications
November, 2007 Slide 32 of32
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