Method of Moment Solution of SVS-EFIE for 2D Transmission Lines of Complex Cross-Sections

A. Menshov and V. Okhmatovski*

University of Manitoba Department of Electrical and Computer Engineering

Winnipeg, CANADA

May 14th, 2012 SPI 2012, Sorrento, Italy

  Integral Equation Approaches for RL extraction   Volume-IE   Surface-Volume-Surface IE

  Method of Moment for SVS-EFIE   Surface and volume meshing   Matrix structure

  Numerical results   Circular conductor: SVS-EFIE vs. Analytic Solution   Differential pair: SVS-EFIE vs. Volume-IE   Coaxial cable: SVS-EFIE vs. Volume-IE

Outline

SPI 2012, Sorrento, Italy

Traditional Volume-EFIE

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jz (ρ)

σ+ iωµ0 G0 (

ρ, ′ρ ) jz (

′ρ )d ′s

S∫∫ = −Vp.u .l ., 

ρ ∈S

N N

Volume EFIE

G0 (ρ, ′ρ ) = − 1

2πln(| ρ −

′ρ |)Static Green’s function

Volumetric mesh:

Matrix-vector product:

Ez (ρ)+ iωµ0σ G0 (

ρ, ′ρ )Ez (

′ρ )d ′s

S∫∫ = −Vp.u .l ., 

ρ ∈S

Surface-Volume-Surface EFIE

iωµ0 Gσ (

ρ,′ρ )Jz (

′ρ )d′ρ

∂ S∫ +σ (ωµ0 )2 G0(

ρ,′ρ )Gσ (

′ρ ,′′ρ )ds '

S∫∫⎡

⎣⎢

⎤

⎦⎥ Jz (′′ρ )d′′ρ

∂ S∫ =Vp.u.l .

Volume EFIE

Ez (ρ) = −iωµ0 Gσ (

ρ, ′ρ )Jz (

′ρ )d′ρ

∂ S∫ , ρ ∈S

Gσ (ρ, ′ρ ) = i

4 H0(2) kσ

ρ −ρ '( )Full-wave Green’s function

Surface-Volume-Surface EFIE:

ρ ∈∂S.

kσ = ωµ0σ2

(1− i)

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SVS-EFIE: Operators SVS-EFIE in operator form:

Tσ∂ S ,∂ S Jz +σ T0

∂ S ,S TσS ,∂ S Jz = −Vp.u .l .

McN2McMM

(Zσ∂ S ,∂ S +σZ0

∂ S ,S ⋅ZσS ,∂ S ) ⋅I = V

SVS-EFIE in matrix form:

Surface-to-Surface mapping (global impedance operator)

Surface-to-Volume-to-Surface mapping:

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SVS-EFIE: Matrix structure

H0(2) kσ

ρ −ρ '( ) H0

(2) kσρ −ρ '( ) ln(|

ρ −′ρ |)

SVS-EFIE vs. analytic solution Comparison of new model to the analytic solution for ideal circular cross-section

Max Relative error < 2.5%

Simulation Parameters Transmission line Single aluminum conductor,

circular cross-section, r0=0.025m

Surface mesh 200 linear elements

Volumetric mesh 3,620 triangular elements

wire axis0

0.2

0.4

0.6

0.8

1

|j z(!) ||j z|m ax

1Hz

60Hz

1kHz

analytic solution [5]SVS EFIE(3) solution jz (ρ)

jz max

⎛

⎝⎜

⎞

⎠⎟analytic

= J0 (kσ ⋅r)J0 (kσ ⋅r0 )

where: J0 is Bessel function, r – radial coordinate r0 – radius of cross-section

SVS-EFIE vs V-EFIE: Example 1

Relative error1 < 2.7%

Simulation Parameters Transmission line Single aluminum conductor,

hexagonal cross-section, r0=0.025m 2 aluminum conductors of hexagonal cross-section, r0=0.025, centers separated by 0.06m, left – grounded, right – driven 1 V/m

Frequency 60 Hz 60 Hz

Surface mesh 60 linear elements 2×60=120 linear elements

Volumetric mesh 3,697 triangular elements 3,094 triangular elements

1 – with respect to traditional Volume-EFIE solution

0.020.04

0.060.08

00.02

0.040.06

0

0.2

0.4

0.6

0.8

1

x y

|j z( !") ||j z|m ax

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.020.04

0.060.08

0

0.2

0.4

0.6

0.8

x

|j z( !") ||j z|m ax

y

Relative error1 < 1.8%

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R and L for circular conductor

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SVS-EFIE vs V-EFIE: Example 2

0

0.05

0.1

0.020

0.020.04

0.060.3

0.4

0.5

0.6

0.7

0.8

0.9

x y

|j z( !") ||j z|m ax

0

0.05

0.1

0.020

0.020.04

0.060.3

0.4

0.5

0.6

0.7

0.8

0.9

x y

|j z( !") ||j z|m ax

Simula'on  Parameters  Cross-­‐sec(on  approxima(on  

three  hexagons   three  20-­‐sided  polygons  (icosagons)  

Mode   60  Hz,  inner  conductor  is  grounded,  outer  is  driven  1  V/m  

Surface  mesh   360=120  linear  elements   3100=3,000  linear  elements  

Volumetric  mesh   1,660  triangular  elements   1,376  triangular  elements  

Relative error1 < 0.5% Relative error1 < 0.5%

1 – with respect to traditional Volume-EFIE solution SPI 2012, Sorrento, Italy

Resistance for coaxial cable

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Inductance for coaxial cable

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Conclusions

  Surface-Volume-Surface EFIE:   Pros:

  Degrees of freedom are surface based   Degrees of freedom are independent on frequency   Rigorous formulation

  No loss of accuracy from approximations

  No derivatives on the Green’s function   Easily combines with layered media Green’s function

  Highly sparse matrices at high frequencies

  Cons:   Requires both surface and volume meshing

Questions?

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