Inductance Computations Using Partial Inductance Concepts/02 - inductance computations... ·...

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


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



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)



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)


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′

��

��

��

��

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


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


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


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


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)


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


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


Example L,R Skin-Effect

● Square cross section example

● Conductor spacing is same as size


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


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

-

-+

+


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


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


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′)


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|]


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


Section based MacromodelsWithout Couplings

Use Section Model

● Can we ignore coupling between sections?

● Check both types of models

● How large is error?


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


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


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


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


Speep-Up for Large Lp Matrices

Example Lp Matrix

● Weak coupling QR evaluation

● QR rank reduction for far couplings

��

��

��

��

Far coupling, Low Rank Sections

Local coupling is dense

sI V


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


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 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


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


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


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