Post on 18-Jan-2018
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EEE 431Computational Methods in
ElectrodynamicsLecture 13
ByDr. Rasime Uyguroglu
Rasime.uyguroglu@emu.edu.tr
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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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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
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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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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FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material)
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.