EE757 Numerical Techniques in Electromagnetics Lecture 7
Transcript of EE757 Numerical Techniques in Electromagnetics Lecture 7
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EE757
Numerical Techniques in Electromagnetics
Lecture 7
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Recommended Text
A. Elsherbeni and V. Demir, The Finite Difference Time
Domain Method for Electromagnetics with MATLAB
Simulations, ACES Series on Computational
Electromagnetics and Engineering, SciTech Publishing
Inc. an Imprint of the IET, Second Edition, Edison, NJ,
2015
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FDTD: an Introduction
the FDTD method is based on simple formulations that do
not require complex Green’s functions
it solves the problem in time, it can provide frequency-
domain responses over a wide band using the Fourier
transform
finite difference approximations are used to approximate
time and space derivatives in Maxwell’s differential
equations
unlike TLM, the electric and magnetic fields are staggered
in both time and space
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Derivative Approximations
In 1966, Yee came up with a set of finite-difference
equations for the time-dependent Maxwell’s curl equations
system
these equations employ the second-order accurate central
difference formulas
for a function f(x), its first-order derivative with respect to x
is given by the second-order accurate formula:
The same formula is applied to approximate time derivatives
( / 2) ( / 2)( )
f x x f x xf x
x
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The 3D FDTD Case
the domain is discretized into a grid of
cells
( )x y zN N N
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Yee’s Cell
electric field vector components are placed at the centers of
the edges of the Yee’s cell
magnetic field vector components are placed at the centers of
the faces of the Yee’s cell
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Spatial Field Notation
( , , ) (( 0.5) ,( 1) ,( 1) ),xE i j k i x j y k z
( , , ) (( 1) ,( 0.5) ,( 1) ),yE i j k i x j y k z
( , , ) (( 1) ,( 1) ,( 0.5) ),zE i j k i x j y k z
( , , ) (( 1) ,( 0.5) ,( 0.5) ),xH i j k i x j y k z
( , , ) (( 0.5) ,( 1) ,( 0.5) ),yH i j k i x j y k z
( , , ) (( 0.5) ,( 0.5) ,( 1) ).zH i j k i x j y k z
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Temporal Field Staggering
for a time-sampling period , the electric field components
are sampled at time instants
the magnetic field components are sampled at time instants
0, ,2 , , ,t t n t t
1 1 1,(1 ) , , ( ) ,
2 2 2t t n t
Example: the z-component of the electric located at
at time instant
is
the y-component of a magnetic field vector positioned at
and sampled at time instant
is
(( 1) ,( 1) ,( 0.5) )i x j y k z n t
( , , )n
zE i j k
(( 0.5) ,( 1) ,( 0.5) )i x j y k z 1
( )2
n t
1
2 ( , , )n
yH i j k
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Material Interpolation
material properties are indexed in the same way as the field
components
interpolation of material properties is a must. Why?
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Discretization Example
consider the Maxwell’s equation
1= ,
y ex zx x ix
x
HE HE J
t y z
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Discretization Example (Cont’d)
1 11 2 2
1 1
2 2
1 1
2 2
( , , ) ( , , ) ( , , ) ( , 1, )1=
( , , )
( , , ) ( , , 1)1
( , , )
( , , ) 1( , , ) ( , , ).
( , , ) ( , , )
n nn n
x x z z
x
n n
y y
x
en n
xx ix
x x
E i j k E i j k H i j k H i j k
t i j k y
H i j k H i j k
i j k z
i j kE i j k J i j k
i j k i j k
the derivatives are approximated by central finite
differences at the position of and at the time
instant
( , , )xE i j k
( 0.5)n t
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Discretization Example (Cont’d)
eliminating , we can write
1
1 1
2 2
1 1
2 2
2 ( , , ) ( , , )( , , ) = ( , , )
2 ( , , ) ( , , )
2( , , ) ( , 1, )
2 ( , , ) ( , , )
2( , , ) ( , , 1)
2 ( , , ) ( , , )
2
en nx xx xe
x x
n n
z ze
x x
n n
y ye
x x
i j k t i j kE i j k E i j k
i j k t i j k
tH i j k H i j k
i j k t i j k y
tH i j k H i j k
i j k t i j k z
1
2 ( , , ).2 ( , , ) ( , , )
n
ixe
x x
tJ i j k
i j k t i j k
1
2 ( , , )n
xE i j k
electric field at time (n+1)t is evaluated in terms of electric
field at time instant nt and magnetic fields at time instant
(n+0.5)t
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Second Discretization Example
consider the Maxwell’s differential equation
1= ,
yx z
x
EH E
t z y
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Second Discretization Example (Cont’d)
discretizing this equation at the location of Hx(i,j,k) and at
a time instant nt, we have
1 1
2 2( , , ) ( , , )=
( , , 1) ( , , ) ( , 1, ) ( , , )1
( , , )
n n
x x
n n n ny y z z
x
H i j k H i j k
t
E i j k E i j k E i j k E i j k
i j k z y
reorganizing, we get the update equation
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Second Discretization Example (Cont’d)
1 1
2 2( , , ) = ( , , )
( , , 1) ( , , )
.( , , ) ( , 1, ) ( , , )
n n
x x
n n
y y
n nx z z
H i j k H i j k
E i j k E i j k
t z
i j k E i j k E i j k
y
the magnetic field at time (n+0.5)t is evaluated in terms of
magnetic field at time instant (n-0.5)t and the electric fields at
time instant nt
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The Electric Field Update Equations 1
1 11 1
2 22 2
1
2
( , , ) = ( , , ) ( , , )
( , , ) ( , , 1)( , , ) ( , 1, )
( , , ) ,
( , , )
n n
x exe x
n nn n
y yz z
exh
n
ix
E i j k C i j k E i j k
H i j k H i j kH i j k H i j k
C i j k y z
J i j k
1
1 1 1 1
2 2 2 2
1
2
( , , ) = ( , , ) ( , , )
( , , ) ( , , 1) ( , , ) ( 1, , )
( , , ) ,
( , , )
n n
y eye y
n n n n
x x z z
eyh
n
iy
E i j k C i j k E i j k
H i j k H i j k H i j k H i j k
C i j k z x
J i j k
1
1 1 1 1
2 2 2 2
1
2
( , , ) = ( , , ) ( , , )
( , , ) ( 1, , ) ( , , ) ( , 1, )
( , , ) ,
( , , )
n n
z eze z
n n n n
y y x x
ezh
n
iz
E i j k C i j k E i j k
H i j k H i j k H i j k H i j k
C i j k x y
J i j k
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The Electric Field Update Equations (Cont’d)
2 ( , , ) ( , , )( , , ) = ,
2 ( , , ) ( , , )
2( , , ) = , , , .
2 ( , , ) ( , , )
e
m meme e
m m
emh e
m m
i j k t i j kC i j k
i j k t i j k
tC i j k m x y z
i j k t i j k
update coefficients are calculated and stored before hand
for all electric field components
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Magnetic Field Update Equations
1 1
2 2( , , ) = ( , , )
( , , 1) ( , , ) ( , 1, ) ( , , )( , , ) ,
n n
x x
n n n ny y z z
hxe
H i j k H i j k
E i j k E i j k E i j k E i j kC i j k
z y
1 1
2 2( , , ) = ( , , )
( , , 1) ( , , )( 1, , ) ( , , )( , , ) ,
n n
y y
n nn n
x xz zhye
H i j k H i j k
E i j k E i j kE i j k E i j kC i j k
x z
1 1
2 2( , , ) = ( , , )
( 1, , ) ( , , )( , 1, ) ( , , )( , , ) ,
n n
z z
n nn ny yx x
hze
H i j k H i j k
E i j k E i j kE i j k E i j kC i j k
y x
(2.24)
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Magnetic Field Update Equations (Cont’d)
( , , ) = , , , .( , , )
hme
m
tC i j k m x y z
i j k
the magnetic field update coefficients are calculated
beforehand
all coefficients are calculated using interpolated
material properties for accurate results!
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Algorithm Steps