Download FINITE DIFFERENCE TIME DOMAIN METHOD

EEE 431
Computational Methods in
Electrodynamics
Lecture 13
By
Dr. Rasime Uyguroglu
[email protected]
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FINITE DIFFERENCE TIME DOMAIN
METHOD (FDTD)
Numeric Dispersion
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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A dispersion relation gives the relationship
between the frequency and the speed of
propagation.
Consider a plane wave propagating in the
positive z direction in a lossless medium.
In time harmonic form the temporal and spatial
dependence of the wave are given by
e
j ( wt   z )
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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
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Where  is the frequency and  is the
phase constant.
The speed of the wave can be found by
determining how fast a given point on
the wave travels.
Let t   z  =constant
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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Setting this equal to a constant and
differentiating with respect to time gives:
d
d
( wt   z )  (const.)
dt
dt
dz
w
0
dt
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
dz
dt

Where
is the speed of the wave,
propagating in the z-direction.

Therefore the phase velocity yields:
dz w
up 

dt 
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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
This is apparently a function of
frequency, but for a plane wave the
phase constant  is given by w  .
Thus, the phase velocity is:
up 

w
w 

1
 r  o  r o

c
 r r
Where c is the speed of light in free
space.
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)

Note that in the continuous world for a
lossless medium, the phase velocity is
independent of frequency. The
dispersion relationship is
up 
w


c
 r r
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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Since c is a constant, as  r , r for any
given material, all frequencies propagate
at the same speed.
Unfortunately this is not the case in the
discretized FDTD world—different
frequencies have different phase
speeds.
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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Governing Equations
Define the spatial shift-operator S s
And the temporal shift-operator St
Let a fractional superscript represent a
corresponding fractional step.
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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For example:
S s H (k )  H [k  1/ 2]
1/ 2
1/ 2
n
y
St E
1/ 2
St
n 1/ 2
x
E
n
y
(k )  E
n 1/ 2
x
n 1
x
[k ]
(k )  E [k ]
n
x
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)

Using these shift operators the finitedifference version of :
Ex
1 H y

t
 z

Can be written;
St1/ 2  St 1/ 2 n1/ 2
Ss1/ 2  S s1/ 2 n1/ 2
(
) Ex [k ]  (
) H y [k ]
t
z
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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Rather than obtaining an update
equation from this, the goal is to
determine the phase speed for a given
frequency.
Assume that there is a single harmonic
wave propagating such that:
Exn  k   Eo e j ( wnt   k z )
H yn  k   H o e j ( wnt   k z )
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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Where  is the phase constant which exists in
the FDTD grid and Eo , H o are constant
amplitudes.
We will assume that the frequency w
is the same as the one in the continuous world.
Note that one has complete control over the
frequency of excitation, however, one does not
have control over the phase constant, i.e., the
spatial frequency.
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)

Assume, the temporal shift-operator
acting on the electric field:
St
1/ 2
E x  k   Eo e
n
e
e

jwt
2
jwt

2
j [ w ( n 1/ 2) t   k z ]
Eo e j ( wnt   k z )
Ex k 
n
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FINITE DIFFERENCE TIME DOMAIN
METHOD( Numerical Dispersion)

Similarly, the spatial shift-operator acting
on the electric field yields
Ss
1/ 2
E x  k   Eo e
j [ wnt   ( k 1/ 2) z ]
n
e
e
j z
2
j z
2
Eo e
j ( wnt   k z )
Ex k 
n
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)

Thus, for a plane wave, one can equate
the shift operators with multiplication by
an appropriate term:

t
S e

s
S e
j
j
wt
2
z
2
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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Now the scalar equation:

t
(e jwt / 2  e  jwt / 2 ) E x n 1 / 2 [ k ] 
1
 j  z / 2
j  z / 2
n 1 / 2

(e
e
)H y
[k ]
z
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)

Employing Euler’s formula to convert
the complex exponentials to
trigonometric functions:
wt
sin(
) Eo e

2
1
j [ w ( n  ) t   k z ]
t
 z
2
sin(
)Ho e
x
2
1
j [ w ( n  ) t   k z ]
2
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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Canceling the exponential space-time
dependence which is common to both
sides produces:
wt
t
z
sin(
) Eo 
sin(
)Ho
2
z
2
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)

Solving for the ratio of the electric and
magnetic field amplitudes yields:
z
sin( 
)
Eo
t
2

Ho
z sin( w t )
2
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FINITE DIFFERENCE TIME
DOMAIN METHOD (FDTD-ABC’s)
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Another equation relating Eo , H o can be
obtained from;
H y
1 Ex

t
 z
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FINITE DIFFERENCE TIME
DOMAIN METHOD

Expressed in terms of shift operators,
the finite-difference form of the equation:
St1/ 2  St 1/ 2 n
S s1/ 2  S s1/ 2 n
(
) H y [k  1/ 2]  (
) Ex [k  1/ 2]
t
z
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)

As before, assuming plane-wave
propagation, the shift operators can be
replaced with multiplicative equivalents.
The resulting equation is:

t
(e jwt / 2  e  jwt / 2 ) H y n [k  1/ 2] 
1  j z / 2
j z / 2
n
 (e
e
) Ex [k  1/ 2]
z
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FINITE DIFFERENCE TIME
DOMAIN METHOD
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Simplifying and rearranging yields:
t
sin( w )
Eo
z
2

z
Ho
t
sin( 
)
2
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numeral Dispersion)
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Equating
multiplying:
Eo
Ho
equations and cross-
wt
t
2 z
sin

sin (
)
2
2
z
2
2
2
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)

Taking the square root:
wt
t
z
sin(
)
sin(
)
2
2
 z

This equation gives the relation between

w, 
Which is different than the one obtained
for the continuous case.
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numeral Dispersion)
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However, the two equations do agree in
the limit as the discretization gets small.
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FINITE DIFFERENCE TIME DOMAIN
METHOD (numerical Dispersion)
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The first term in the Taylor series
expansion of: sin    for small 
Assume that the spatial and temporal
steps are small enough so that the
arguments of trigonometric functions
are small in the dispersion relation.
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
  w 

From this:

Which is exactly the same as in the continuous
world. However, this is only true when the
discretization goes to zero.
For finite discretization, the phase velocity in
the FDTD grid and in the continuous world
differ.

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FINITE DIFFERENCE TIME DOMAIN
METHOD ( Numerical Dispersion)
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In the continuous world: u p  w

In the FDTD world the same relation
holds u p  w

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FINITE DIFFERENCE TIME
DOMAIN METHOD
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For the one dimensional case a closed
form solution for  is possible.

A similar dispersion relation holds in two
and three dimensions, but there a
closed-form solution is not possible.
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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For a closed form solution:
z 
t
 sin [
sin( w )]
2
t
2
The factor wt   SN 
2
z


Where,
1
ct
S
,   N  z
z
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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Thus
z
2
1
 sin [
 r r
S
sin(
S
N
)]
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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Consider the ratio of the phase velocity
in the grid to the true phase velocity:
z
w/  

  2
u p w /   z
2
up
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)

For the continuous case:
  w   2
c

 o  r o  r 
2

2
 r r 
 r r
N  z
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)

The ratio becomes:
up
up

  r r

1
N  sin [
 r r
S
S
sin( )]
N
This equation is a function of the
material, the Courant number, and the
number points per wavelength.
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FINITE DIFFERENCE TIME DOMAIN
METHOD (Numerical Dispersion)
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