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The Proximity Effect: A Comparison of

COMSOL® and Analytic Solutions

( A Possible Means for Accuracy Validation ? )

Chathan Cooke1, Lisa Shatz2,3

1) MIT: RLE, (High Voltage Lab.), Cambridge, MA, USA

2) Suffolk University Electrical Engineering, Boston, MA, USA

3) Benjamin Franklin Institute of Technology, Boston, MA, USA

Thursday, Oct 3, 2019

Newton, MA

d2

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8 Parallel Wire

Magnetic

Flux Density

with

Equal

Total Current

in Each Wire

Non-Equal

Current

Densities

the

Cause:

Proximity

Effect

COMSOL2019

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Outline of Topics

1) Introduction / Goals– Desire: challenging problem with analytic (exact) `solution

2) Selected Problem for “Validation”– Has analytic solution for point and volume values

3) Mathematica® (analytic) and COMSOL® (simulation)– Obtain quantified values

– Compare to analytic solution for accuracy

4) Conclusions

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Goals of “Validation” ProblemAnalytic (Exact) vs Simulation

1) Compatible with Modern Solvers:– Requires Only Common, Expected Capability

• No “Special Physics”

2) Challenging Problem, “Not Simple”– Very high dynamic range of field values

• Severe challenge to numerical algorithms

– 2D: Both radial and angular variations

3) Challenging Post-Processing– For example: bulk electrical properties

• Quantified resistance, added power-loss

Selected: Proximity Effect2D - Quasi-Static EM Problem

Demanding for all three above goals

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Proximity Effect:“External” Currents Influence “Internal” Currents

Induces a redistribution of currents

Single WireEnd View

Uniform Distribution

Multiple WiresNon-Uniform Distribution

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a

2c

edge

wire

center

wire

wire-to-wire gap

= (2c - 2a)

edge

wire

12

CL

Parallel Wire Geometry: 3 Wires

3-ParallelWires

Gap Distance, g,Between Wires

WIreRadius, a

Side View Top View

Angular VariationSurface Currents

c/a =geometryparameter

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Analytic Calculation Method(After Smith 1972)

Kl(q ', z ') = I

2pa( )gl (q ')

gm

(q ') =1+ amp

cos(pq ')p=1

q

å

-¶(A

mz(r,q , z)

¶r=

moI

2pa

æ

èç

ö

ø÷gl (q )

2) ONLY Surface Current Density, K, with angular variation gm:

A = Amz

(r,q, z)z

Assumptions, Definitions:

3) By symmetry only cosine terms for:

4) Magnetic vector potential is z-directed:

(A/m)

1) All wires carry same total current, I: angular variation ofsurface current density

at mth wire:

coefficients, ampgm(q )

gm(q) = gm(p -q)

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Analytic Form for Surface Current Distributions( Series of Integral Equations for )

IntegralEquations

Angle DependentCoefficients

gm

(q ) =1+1

pgl(q ')K

mlq ,q '( )dq '

l=1,l¹m

m

åæ

èçç

ö

ø÷÷

-p

p

ò .

Where:

Kml

q ,q '( ) =1+ 2 c a( )(m- l)cosq - cos(q -q ')

(r 'ml

)2,

gm(q )

normalized by wire-spacing: (c/a)

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Hence, Result:

gm

(q ) =1+ amp

cos(pq )p=1

q

å

Where: summation to q = number of cosine terms to get convergence(typically 6 to 8)

(Calculate via Series Coefficients: )gm(q ) amp

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Solve for the case of c/a = 3, and get:

a11

= .496, a12

= .069,a21

= 0, a22

= .102

gm(q) =1+a

m1cosq +a

m2cos2q

Example Analytic Calculation(Three conductors: n=3, Two terms: q = 2 )

Forwire m

1.0

1.25

0.5

0.75

gm(q )

0 pp

2angle(q)

m=1

m=2

Uniform:No

Proximity-Effect

Smaller CurrentsInside:

Higher CurrentsOutside:

CL (wire-2 in middle)

(wire-1 = wire-3)

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Example Analytic Solution

11-Wires, (c/a) = 2

Uniform:No

Proximity-Effect

Currents Negative:Opposite Direction in Same Wire

0 pp

2angle(q)

0

0.5

1

-0.5

-1.0

1.5

2.0

2.5

gm(q )

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Compare Analytic Solution, Smith (1972)

6-Wires, (c/a) = 1.25

0 pp

2

Analytic via Mathematica® Overlay on Smith (1972)

0

gm(q )

1

2

3

-1

-2

angle(q)

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Solver Simulation Solutions

Via COMSOL ® with AC/DC Module

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Solver Simulation Method - Meshing( COMSOL® + AC/DC Module )

“infinite” boundary

highmesh-density

just insidewire surface

highmesh-densityJust outsidewire surface

Meshing: 3 parallel wires

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Solver Simulation Method – Magnetic Flux Density( COMSOL® + AC/DC Module )

“infinite” boundary

high currentdensityInside

wire surface

high fieldsoutside

wire surface

3 wires, (c/a)= 1.5, skin depth = 6.3µm (freq = 100MHz)

1 ampEachwire

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Expanded View - Meshing( COMSOL® + AC/DC Module )

outsidewire

high mesh densityJust Inside

wire surface

Wiresurface

Ultra Fine Meshing: 3 wires (c/a)= 1.5

insidewire

200 µm

10 µm

skin depth= ~6.3 µm

(5 mesh layers)

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Expanded View – Magnetic Flux Density( COMSOL® + AC/DC Module )

outsidewire

highcurrent density

Insidewire surface

Wiresurface

3 wires, (c/a) = 1.5

insidewire

200 µm

10 µm

skin depth= ~6.3 µm

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Solver Simulation Method – Magnetic Flux Density( COMSOL® + AC/DC Module )

All wires1 amp current

higher fieldsoutside

outer wire surface

20 wires, (c/a) = 2.0, freq = 100MHz (skin depth = 6.3µm)

035 (x 10-5 Tesla)20 1030

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Comparison: Analytic vs COMSOL®

3-Wires, (c/a) = 2: Surface Current Density Distributions

Analytic: Mathematica®

Angle

0 pp

2

Simulation: COMSOL®

Wire 1

Wire 2

Wire 1

Wire 2

Angle

0 pp

2

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Proximity Effect:

2nd Calculation Quantity = Resistance(a bulk volume property)

Added Resistance per Wire:

Rp/Ro

(normalized to skin-effect Ro)

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Calculation of Normalized Proximity

Resistance: Rp/Ro

• Analytic: Use same amp coefficients for :

Rp

Ro

=RT

-NRskin

NRskin

= 1

2amp

2

p=1

q

åæ

èçç

ö

ø÷÷

m=1

N

å

• Simulation: COMSOL® post-process, via:

“Volumetric loss density, electric” function [W/m]

gm(q )

mf.QrhSurface Integration

OverSelected Area

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Calculation of Normalized Proximity Resistance:

Rp/Ro – COMSOL® via: mf.Qrh

“Surface Integration”Over

Area of Each Wire

Example: Rp

Ro

[W3of 20]

=0.22470

0.1289=1.7432

Ref: Single Wire Ro

Rp

Ro=mf .Qrh( )xx-Wiremf .Qrh( )1-Wire

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Comparison: Analytic vs COMSOL®

3-Wire Proximity Loss Factor, (c/a)= 2.0: Rp/Ro

Analytic: Mathematica® Simulation: COMSOL®

MethodRp/Ro Rp/Ro Rp/Ro (ave

outer center of 3-wires)

Theory 0.4986 0.039 0.3455

COMSOL® 0.4968 0.0391 0.3443

Excellent Agreement within 0.5%

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Comparison: Analytic vs COMSOL®N-Wire Average Proximity Loss Factor: (Rp/Ro)ave

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 5 10 15 20 25

Rp/Ro

NumberofParallelWires

ProximityNormalizedAveragePerWireLoss,Rp/Ro

Analytic

COMSOL

Above 10 wires simulations: greater errors due to mesh area outside wires

c/a=2

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3 Wire Mesh Boundary - Example

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3 Wire Mesh Boundary - ExampleMagnetic Flux Density (T)

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Findings from Studies

• Proximity: A Good “Challenge” Analytic Solution Problem

– Proximity Effect for Parallel Wires

• First Solutions 1972, now extended to more wires

– Excellent Agreement; Theory vs Simulated

• Great care to accurately represent problem needed

• Mathematica® Analytic Solutions

– Require adequate terms to converge (troubles for very small spacing)

– Yields both: current distributions and added losses

• COMSOL® Simulation Solutions

– Require very careful meshing for accurate solution

• Large region to external boundary

– Careful post-processing to obtain losses

• Loss best done via mf.Qrh

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Conclusions – Proximity Problem

• Good Agreement: Analytic and Simulations

– Requires careful meshing

• Extra mesh-points in region of rapid field changes

• External boundary needs to be “far” away

– Requires careful number of analytic terms

• Typically 6 to 8 terms is sufficient

• Proximity Effect Results:

– Severity of added resistance increases with number of wires

– Severity of added resistance increases for smaller wire spacing

– Center region wires with many wires less severe change

• Future:

– Might provide a common “calibration” problem

• Could use agreed values as reference

– Possibly a useful means to improve auto meshing

• Try to improve meshing dynamic range

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Example Future Simulation Evaluations

via Proximity

• Mesh Effects

– Quantify accuracy versus mesh density

– Quantify required mesh density versus field gradient

– Quantify “infinite” boundary effects

– Quantify required distance and meshing at “far” boundary region

• Ex: minimum boundary = 5 x largest object dimension

• Post-Processing:

– Improve analysis to quantify losses

– Improve quantification of field gradients

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Thank You

Acknowledge: Very valuable partial support for this work by ProlecGE