Method of Moment Solution of SVS-EFIE for 2D Transmission ... · Method of Moment Solution of...
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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
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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?
SPI 2012, Sorrento, Italy