Approximate Current on a Wire – A Differential Equation MethodAdam Schreiber, Yuriy Goykhman, Chalmers Butler

Outline

Derivation Solution Method – Solve DE and Iterate Sample Data Discussion Conclusion

Integral Equation

2 ( ') ( ') ' ( ') ( ') ' ( )4 '

h h izh h

d dj k I z K z z dz I z K z z dz E z

k dz dz

Reference Integral Equation Method Break K(z-z’) into it’s real and imaginary parts

KR(z-z’) resembles the delta function

KI(z-z’) resembles sin(x)/x

, ( , )z h h

Justification of KR Approximation

)()(

')'()(')'()(

zzI

dzzzKzIdzzzKzI

R

h

h R

h

h R

-0.5 0 0.5-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

KR(0

-z')

z

Approximation For 0.4 & 0.8 λ( ) ( ') ' ( ) ( )

h

R a RhJ I z K z z dz J I z z

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.22

3

4

5

6

7

8

9

10

11

12

z

J, J

a

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-8

-6

-4

-2

0

2

4

6

8

10

12

J, J

a

z

Differential Equation2

22

( ) ( ) ( ) ( ') ( ') '4 4

( ') ( ') '4 '

hi

R z I

h

h

I

h

d kj k I z z E z I z K z z dz

k dz

d dI z K z z dz

k dz dz

Second Order Differential EquationI(-h)=I(h)=0

Pulse Test

Evaluate the Differential Equation at m points Creates N equations in I(z) Changes intervals from (-h, h) to

(zm - Δ/2, zm + Δ/2)

Triangle Expansion

Replaces I(z) with N unknowns We now have N equations with N unknowns

-h h

1

1 ,( ) ( ) ( )

0,

nz zNn

n n nn

z zI z I z z

elsewhere

Matrix Equation mnmn EIL 0

elsewhere

nmzzk

zz

nmzkz

nmzkz

L

nRnR

nRnR

nRnR

nRnR

mn

,0

,)()(

)()(

1),()(

1),()(

43

43

832

221

43

82

21

43

82

21

2 2

41

11 2

1

( ) ( )

( ) ( ) 2(1 ) ( ) ( )

j k im z m p m

Np k

p m n I m n I m n I m nn

E E z J z

J z j I K z z K z z K z z

Solution Method

Generate tri-diagonal matrix Find I0 with J0 = 0 Generate a new right hand side Find Ip

Repeat the above 2 steps till convergence Compare results with integral equation

data

0.4 Wavelength

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Curr

ent

Integral Equation Method

Imag

Real

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Curr

ent

Differential Equation Method

Imag

Real

0.8 Wavelength

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Cur

rent

Differential Equation Method

Imag

Real

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Curr

ent

Integral Equation Method

Imag

Real

1.0 Wavelength

-0.5 0 0.5-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

Cur

rent

Integral Equation Method

Imag

Real

-0.5 0 0.5-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

Cur

rent

Differential Equation Method

Imag

Real

Error:

0

2

4

6

8

10

12

0.4 0.6 0.8 1.0

Length of Wire in Wavelengths

%E

Lengths without Convergence

Region around 0.5 λ Region around 1.5 λ Odd multiples of 0.5 λ

Reasons for Error

Initial delta function approximation Accuracy of evaluating Almost singular matrices at non-

convergent lengths

R

Future Work

Increase efficiency/speed Extend algorithm to bent/curved wires Improve numerical integration methods

Conclusion

Provides good approximations for current on wires, although not near lengths equal to odd multiples of 0.5 λ

Can be adjusted to improve speed and increase accuracy

Perhaps, method can be fixed for lengths near 0.5 λ

Acknowledgements

Clemson University NSF Dr.Butler Dr. Noneaker & Dr. Xu

Psi R