1 eee 431 computational methods in electrodynamics lecture 13 by dr. rasime uyguroglu

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EEE 431Computational Methods in

ElectrodynamicsLecture 13

ByDr. Rasime Uyguroglu

[email protected]

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FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

Lossy Material

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) When a material has a finite

conductivity , a conduction current term is added to Ampere’s Law (different than the source term). Thus:

EXH Et

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) As before assuming the x component

of E and the variation only in the z direction:

yxx

HEE

t z

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) This equation can be expanded about a

point : to find FDTD update equation.

However, when loss is present, the undifferentiated electric field appears on the left side of the equation.

1(( , ( ) )2

k z n t

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) With the assumption that the electric field

point is , there is no electric field at this space-time point.

This problem can be solved by using the average of the electric field to the either side of the desired point:

11/ 2 [ ] [ ][ ]

2

n nn x xx

E k E kE k

1( , ( ))2

k z n t

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) Thus a suitable discretization of

Ampere’s Law when loss is present is:

1/ 2 1/ 21 1 [ 1/ 2] [ 1/ 2][ ] [ ] [ ] [ ]2

n nn n n nyx x x x

H k H kE k E k E k E kt z

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) As before this can be solved for ,

which the present field, in term of purely past fields. The result is:

1[ ]nxE k

1 1/ 2 1/ 21

2[ ] [ ] ( [ 1/ 2] [ 1/ 2])1 1

2 2

n n n nx x y y

t tzE k E k H k H k

t t

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) Now, consider the Faraday’s Law:

When, no magnetic loss is assumed

HXEt

( 0)m

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) Now considering that there is ,

varying with z direction:yH

y xH Et z

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) Discretized form:

1/ 2 1/ 2( 1/ 2) [ 1/ 2] ( 1) [ ]n n n ny y x x

H k H k E k E kt z

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) Solving for Yields the following update equation:

1/ 2[ 1/ 2]nyH k

1/ 2[ 1/ 2] ( ( 1) ( ))n n ny x x

tH k E k E kz

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) Consider a lossy dielectric half-space

which starts at node 100. The relative permittivity is 9 as before. However there is a also an electric loss present such that :

0.012t

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) A previously employed simple ABC

does not work at the right edge of the grid. It can be removed. But the left side of the grid can be terminated by using the same ABC as before.

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) The magnetic field update remained the

same. That us observe the pulse propagation for different time steps. (i.e. multiples of 10)

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)

0 20 40 60 80 100 120 140 160 180 200-2

-1

0

1

2

3

4x 10

-4

Ex

Space (spatial index)

time step 1

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0 20 40 60 80 100 120 140 160 180 200-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03E

xtime step 10

Space index

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0 20 40 60 80 100 120 140 160 180 200-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Space Steps

Time Step=20E

z

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0 20 40 60 80 100 120 140 160 180 200-0.2

0

0.2

0.4

0.6

0.8

1

1.2

space steps

time step 30E

x

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0 20 40 60 80 100 120 140 160 180 200-0.2

0

0.2

0.4

0.6

0.8

1

1.2time step 40

Ex

space steps

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0 20 40 60 80 100 120 140 160 180 200-0.2

0

0.2

0.4

0.6

0.8

1

1.2time step 50

Ex

Space Steps

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0 20 40 60 80 100 120 140 160 180 200-0.2

0

0.2

0.4

0.6

0.8

1

1.2E

xtime step 60

space step index

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0 20 40 60 80 100 120 140 160 180 200-0.2

0

0.2

0.4

0.6

0.8

1

space step index

Ex

time step 70

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0 20 40 60 80 100 120 140 160 180 200-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

space step index

Ex

time step 80

25


0 20 40 60 80 100 120 140 160 180 200-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

space step index

Ex

time step 90

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0 20 40 60 80 100 120 140 160 180 200-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4time step 100

Ex

space steps index

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0 20 40 60 80 100 120 140 160 180 200-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1time step 60,70,80,90,100

space steps

Ex

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0 20 40 60 80 100 120 140 160 180 200-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

space steps index

time step 110E

x

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0 20 40 60 80 100 120 140 160 180 200-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3time step 150

Ex

space step index

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0 20 40 60 80 100 120 140 160 180 200-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2time step 170

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FINITE DIFFERENCE TIME DOMAIN METHOD ( Lossy Material)

0 20 40 60 80 100 120 140 160 180 200-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2time step 190

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0 20 40 60 80 100 120 140 160 180 200-0.1

-0.05

0

0.05

0.1

0.15time step 200

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0 20 40 60 80 100 120 140 160 180 200-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14time step 210

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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Dielectric)

0 20 40 60 80 100 120 140 160 180 200-14

-12

-10

-8

-6

-4

-2

0

2x 10

-3 time step 440

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FINITE DIFFERENCE TIME DOMAIN METHOD (TFSF-Lossy Material) Conclusion: The pulse decays as it propagates in the

lossy region and eventually decays to a rather negligible value.

The lack of an ABC at the right side of the grid is not really a concern in this particular instance.

1 eee 431 computational methods in electrodynamics lecture 13 by dr. rasime uyguroglu

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