Fdtd

FDTD: An Introduction

Dr. Rakhesh Singh Kshetrimayum

2/25/2015FDTD by R. S. Kshetrimayum1

Dr. Rakhesh Singh Kshetrimayum


� Background on commercial softwares and Who’s Who’s in CEM

(a) EMSS software FEKO� Originated from Dr. Ulrich Jacobus.� Dr. Francs Meyer (a student of Prof. D. B. Davidson, a


� Dr. Francs Meyer (a student of Prof. D. B. Davidson, a Professor at University of Stellenbosch, South Africa) is one of the main person behind this company


(b) IE3D by Zeland Software Inc.� Founded by Dr. Jian-Xiong Zheng� Did PhD under Prof. D. C. Chang (President Emeritus of

NYU Poly)(c) REMCOM XFDTD is a product of Penn. State University


(c) REMCOM XFDTD is a product of Penn. State University� Founders are H. Scott Langdon, Dr. Raymond Lubbers and

Dr. Christopher Penny


(d) James C. Rautio is founder of SONNET� He did his PhD from Syracuse university, USA under Prof. R.

F. Harrington (considered as originator of MoM in EM area)(e) Dr. WenhuaYu who was a member of Prof. Raj Mittra group

at Penn. State University, USA founded GEMS software


at Penn. State University, USA founded GEMS software� And list goes on


� FDTD is one of the very popular techniques � in Computational electromagnetics (CEM)

� What is CEM?� In CEM

� Try to solve complex EM problems


� Try to solve complex EM problems� which do not have tractable analytical solutions � Using numerical methods employing computers


Why CEM?� Possible to simulate a device/experiment/phenomenon any

number of times � Try to achieve the best or optimal result before actually

doing the experiments (preserving resources)


doing the experiments (preserving resources)� Some experiments are

� dangerous to perform like lightning strike on aircraft or � not possible to conduct because of limited facilities/resources


� How to learn CEM (including FDTD)?� Can be learnt by doing it� Once you learn the basic concepts� You should write or code simple programs in any language� Always start with simple problems to gain confidence


� Always start with simple problems to gain confidence� and try for more difficult problems later


� Many CEM methods are available� MoM� FEM� FDTD

� FDTD (our choice)


� FDTD (our choice)� Easy to implement� Can see fields and waves in real time (time domain techniques)

� Expected outcome of this talk � Understand fundamentals of FDTD� Will explain and give 1-D, 2-D and 3-D FDTD programs� Can start your research works right way


� Let us start with 4 Maxwell’s equations

2. 0B∇ • =r

ε

ρvE =•∇r

.1


3.D

H Jt

∂∇× = +

∂

rr r

4.B

Et

∂∇× = −

∂

urur

Maxwell curl equations (why?)

FDTD: An Introduction� What is FDTD?� Finite Difference Time Domain� Basically, we replace spatial & time derivatives in the two

Maxwell’s curl equations � by central finite difference approximation


� What is central finite difference approximation of derivatives?� Consider a function f(x), its derivative at x0 from central finite

difference approximation is given by

0 0

'0

0

( ) 2 2( )

x xf x f x

df xf x

dx x

∆ ∆ + − −

= ≅∆


� The time-dependent and source free ( ) Maxwell’s curl equations in a medium with ε=ε0εr and µ = µ0 µr are

0J =r

0 0

1 1;

r r

E HH E

t tε ε µ µ

∂ ∂= ∇× = − ∇×

∂ ∂

r rr r

3 equations 3 equations


� For Cartesian coordinate systems, � expanding the curl equations, � equating the vector components,

� we have 6 equations


� Expanding the first vector curl equation

� we get 3 equations

0

1

r

EH

t ε ε

∂= ∇×

∂

rr


0

0

0

1

1

1

yx z

r

y x z

r

y xz

r

HE H

t y z

E H H

t z x

H HE

t x y

ε ε

ε ε

ε ε

∂ ∂ ∂= −

∂ ∂ ∂

∂ ∂ ∂ = −

∂ ∂ ∂

∂ ∂∂= −

∂ ∂ ∂


� Expanding the second vector curl equation

� We get 3 equations

0

1

r

HE

t µ µ

∂= − ∇×

∂

rr

1 EH E∂ ∂ ∂


0

0

0

1

1

1

yx z

r

y xz

r

yxz

r

EH E

t z y

H EE

t x z

EEH

t y x

µ µ

µ µ

µ µ

∂ ∂ ∂= −

∂ ∂ ∂

∂ ∂∂ = −

∂ ∂ ∂

∂ ∂∂= −

∂ ∂ ∂


As we know that FDTD is a time-domain solverThe question is how do we solve those 6 equations above?1. 1-D FDTD update equations

� For 1-D case (a major simplification), we can consider (a) Linearly polarized wave along x-axis


� (a) Linearly polarized wave along x-axis � exciting an electric field which has Ex only (Ey = Ez = 0)

� (b) propagation along z-axis� no variation in the x-y plane, i.e. ∂/∂x = 0 and ∂/∂y = 0


� Then, the above 6 equations� reduce to 2 equations for 1-D FDTD

0

1 yx

r

HE

t zε ε

∂ ∂= −

∂ ∂


� We want to solve these equations at different locations and time in solution space

� Hence we need to discretize both in time and space

0

1y x

r

H E

t zµ µ

∂ ∂ = −

∂ ∂


2 3 4 5

∆z

1

space discretization size

running index k


Fig. 1(a) Discretizations in space

Fig. 1(b) Discretization in time

∆t

1 2 3 4 5

time discretization size running index n


� Notations:� n is the time index and � k is the spatial index,

� which indexes � Locate positions z = k∆z


� Locate positions z = k∆z� Usually ∆z is taken as 10-15 per wavelength� gives accurate

results � times t = n∆t and� ∆t (dependent on ∆z) chosen from Courant stability criterion � gives stable FDTD solution (will be discussed later)


� for the first equation:

� the central difference approximations for both the temporal and spatial derivatives are obtained at

0

1 yx

r

HE

t zε ε

∂ ∂= −

∂ ∂


and spatial derivatives are obtained at � z = k∆z, (space discretization size and index)� t = n∆t (time discretization size and index)

0

1 1 1 1, , , ,

12 2 2 2x x y y

r

E k n E k n H k n H k n

t zε ε

+ − − + − −

= −∆ ∆


� for the second equation:

� the central difference approximations for both the temporal and spatial derivatives are obtained at

� at (z + ∆z/2, t + ∆t/2)

0

1y x

r

H E

t zµ µ

∂ ∂ = −

∂ ∂


� at (z + ∆z/2, t + ∆t/2) � (increment all the time and space steps by 1/2):

0

1 1 1 1, 1 , 1, ,

12 2 2 2y y x x

r

H k n H k n E k n E k n

t zµ µ

+ + − + + + − +

= −∆ ∆


� The above equations can be rearranged as a pair of ‘computer update equations’, � which can be repeatedly updated in loop, � to obtain the next time values of Ex (k, n+1/2) and Hy (k + ½.

n+1)


n+1) � from the previous time values as follows

0

1 1 1 1 1, , , ,

2 2 2 2x x y y

r

tE k n E k n H k n H k n

zε ε

∆ + = − − + − − ∆

0

1 1 1 1 1, 1 , 1, ,

2 2 2 2y y x x

r

tH k n H k n E k n E k n

zµ µ

∆ + + = + − + + − + ∆


k-2 k-1 k k+1 k+22/1−n

xE

nH


Fig. 2 Interleaving of E and H fields

k-2 k-1 k k+1 k+2

k-5/2 k-3/2 K-1/2 K+1/2K+3/2K+5/22/1+n

xE

n

yH


� Interleaving of the E and H fields in space and time in the FDTD formulation

� to calculate Ex(k), the neighbouring values of Hy at k-1/2 and k+1/2 of the previous time instant are needed

� Similarly, to calculate Hy(k+1/2), for instance, the


� Similarly, to calculate Hy(k+1/2), for instance, the neighbouring values of Ex at k and k+1 of the previous time instant are needed


� In the above equations, � ε0 and µ0 differ by several orders of magnitude, � Ex and Hy will differ by several orders of magnitude

� Numerical error is minimized by making the following change of variables as


change of variables as

xx EEµ

ε='


� Hence,

' '1 1 1 1 1 1 1, , , ,

2 2 2 2x x y y


zεε ε

µ µ

∆ + = − − + − − ∆


' '

' '

1 1 1 1 1, , , ,

2 2 2 2

1 1 1 1, ,

2 2

x x y y

x x y


z

tE k n E k n H k

z

ε

µ ε

µε

∆ ⇒ + = − − + − − ∆

∆ ⇒ + = − − +

∆

' '

1, ,

2 2

1 1 1 1 1, , , ,

2 2 2 2

y

x x y y

n H k n


zµε

− −

∆ ⇒ + = − + − − + ∆


� Similarly,

' '1 1 1 1 1 1, 1 , 1, ,

2 2 2 2

1 1 1 1 1

y y x x


z

t

µε

µ

∆ + + = + − + + − + ∆

∆


' '1 1 1 1 1, 1 , 1, ,

2 2 2 2

1 1 1, 1 ,

2 2

y y x x

y y


z

tH k n H k n E

z

µε

µε

∆ ⇒ + + = + − + + − + ∆

∆ ⇒ + + = + +

∆

' '1 1, 1,

2 2x xk n E k n

+ − + +


Courant stability criteria:

� Courant stability criteria dictates the � relationship between the time increment ∆t with respect to

space increment ∆z� in order to have a stable FDTD solution of the electromagnetic


� in order to have a stable FDTD solution of the electromagnetic problems

� In isotropic media, an electromagnetic wave propagates a distance of one cell in time ∆t = ∆z/vp, � where vp is the phase velocity


� This limits the maximum time step� This equation implies that an EM wave cannot be allowed

� to move more than a space cell during a time step� Otherwise, FDTD will start diverging� If we choose ∆t > ∆z/v ,


� If we choose ∆t > ∆z/vp, � the distance moved by the EM wave over the time interval ∆t

will be more than ∆z, � the EM wave will leave out the next node/cell and � FDTD cells are not causally interconnected and

� hence, it leads to instability


cell


y∆

x∆Fig. 3 (a) 2-D discretizations

x

y


∆

2

∆ Propagation at θ=450


Plane wave front

∆

Propagation at θ=00

Fig. 3 (b) Plane wave front propagation along θ=00 andθ=450


� Propagation at θ=450

� Wavefront jumps from one row of nodes to the next row, � the spacing between the consecutive rows of nodes is

2

∆


� Similarly for 3-D case2

3

∆


� In the case of a 2-D simulation, � we have to allow for the propagation in the diagonal

direction, � which brings the time requirement to ∆t = ∆z/√2vp

� Obviously, three-dimensional simulation requires ∆t =


� Obviously, three-dimensional simulation requires ∆t = ∆z/√3vp

� We will use in all our simulations a time step ∆t of

2p

zt

v

∆∆ =

⋅


� where vp is the phase velocity, � which satisfies the requirements in 1-D, 2-D and 3-D for all

media (√2≈1.414<√3≈1.7321<2)� In 3D case, it may be more appropriate to modify the above

stability criteria as


stability criteria as

� The above factor now can be simplified as

( )min , ,

2 p

x y zt

v

∆ ∆ ∆∆ =

⋅

1 1

2 2

p

p

p

v zt tv

z z v zεµ

⋅ ∆∆ ∆= ⋅ = =

∆ ∆ ⋅ ⋅ ∆

FDTD: An Introduction� Making use of this in the above two equations, � we obtain the following equations

' '1 1 1 1 1, , , ,

2 2 2 2 2x x y yE k n E k n H k n H k n

+ = − + − − +

' '1 1 1 1 1, 1 , , 1,H k n H k n E k n E k n

+ + = + + + − + +


� Hence the computer update equations are� ex[k] = ex[k] + (0.5/er(k))*( hy[k-1] - hy[k] )� hy[k] = hy[k] + 0.5*( ex[k] - ex[k+1] )

' '1 1 1 1 1, 1 , , 1,

2 2 2 2 2y y x xH k n H k n E k n E k n

+ + = + + + − + +

FDTD: An Introduction� k+1/2 and k-1/2 are replaced by k or k-1� Note that the n or n + 1/2 or n −1/2 in the superscripts do not

appear� FDTD Simulation of Gaussian pulse propagation in free space (fdtd_1d_1.m)


(fdtd_1d_1.m)� FDTD Simulation of Gaussian pulse hitting a dielectric medium (fdtd1_die.m)

� From theoretical analysis of EM wave hitting a dielectric surface

21

21

12

12

εε

εε

ηη

ηη

+

−=

+

−=Γ

21

1

12

222

εε

ε

ηη

ητ

+=

+=


� For ε1=1.0 and ε2=4.0, we have � FDTD simulation of Absorbing Boundary Condition (fdtd_1d_3.m)

� In calculating the E field, � we need to know the surrounding H values;

3

2;

3

1=−=Γ τ


� we need to know the surrounding H values; � this is the fundamental assumption in FDTD

� At the edge of the problem space, � we will not have the value to one side, � but we know there are no sources outside the problem space


� The wave travels ∆z/2 (=c0 · ∆ t) distance in one time step, � so it takes two time steps for a wave front to cross one cell

� Suppose we are looking for a boundary condition at the end where k=1

� Now if we write the E field at k=1 as


� Now if we write the E field at k=1 as � Ex (1,n) = Ex (2, n-2), � then the fields at the edge will not reflect

� This condition must be applied at both ends


2. FDTD Solution to Maxwell’s equations in 2-D Space

� In deriving 2-D FDTD formulation, we choose between one of two groups of three vectors each:

� (a) Transverse magnetic (TMz) mode, which is composed of Ez, Hx, and Hy or


Ez, Hx, and Hy or � (b) Transverse electric (TEz) mode, which is composed of Ex,

Ey, and Hz


� Unlike time-harmonic guided waves, � none of the fields vary with z so that � there is no propagation in the z-direction

� But in general propagation along x- or y- directions or both is possible


� FDTD simulation of TM mode wave propagation (fdtd_2d_TM_1.m )

� Expanding the Maxwell’s curl equations with � Ex = 0, Ey = 0, Hz = 0 and ∂/∂z = 0,

� we obtain, � 3 equations from the 6 equations


� 3 equations as follows

1 y xzH HE

t x yε

∂ ∂∂= −

∂ ∂ ∂

1H E ∂ ∂


1

1

x z

y z

H E

t y

H E

t x

µ

µ

∂ ∂=

∂ ∂

∂ ∂ = −

∂ ∂


� Central finite difference approximations of the above 3 equations are as follows

1 1, , , ,

2 2

1 1 1 1

z zE i j n E i j n

t

+ − −

∆

1 y xzH HE

t x yε

∂ ∂∂= −

∂ ∂ ∂


� n+1/2�n+1 & j�j+1/2

0

1 1 1 1, , , , , , , ,

1 2 2 2 2y y x x

r

H i j n H i j n H i j n H i j n

x yε ε

+ − − + − −

= −∆ ∆

0

1 1 1 1, , 1 , , , 1, , ,

12 2 2 2x x z z

r

H i j n H i j n E i j n E i j n

t yµ µ

+ + − + + + − +

= −∆ ∆


n+1/2�n+1 & i�i+1/2

� Rearranging the above equations, we can write the final 3 update equations’ expressions as

0

1 1 1 1, , 1 , , 1, , , ,

12 2 2 2y x z z

r

H i j n H i j n E i j n E i j n

t xµ µ

+ + − + + + − +

=∆ ∆


update equations’ expressions as

0

1, ,

2

1 1 1 1, , , , , , , ,

1 2 2 2 2, ,

2

z

y y x x

z

r

E i j n

H i j n H i j n H i j n H i j nt

E i j nx yε ε

+

+ − − + − − ∆ = − + −

∆ ∆


0

1 1, 1, , ,

1 1 2 2, , 1 , ,

2 2

z z

x x

r

E i j n E i j nt

H i j n H i j nyµ µ

+ + − + ∆ + + = + −

∆


0

1 11, , , ,

1 1 2 2, , 1 , ,

2 2

z z

y x

r

E i j n E i j nt

H i j n H i j nxµ µ

+ + − + ∆ + + = + +

∆


� Computer update equations are (k+1/2 and k-1/2 are 0

1, ,

2

1 1 1 1, , , , , , , ,

1 2 2 2 2, ,

2

z

y y x x

z

r

E i j n

H i j n H i j n H i j n H i j nt

E i j nx yε ε

+

+ − − + − − ∆ = − + −

∆ ∆


� Computer update equations are (k+1/2 and k-1/2 are replaced by k or k-1)

� ez(i,j)=ez(i,j)+{dt/(eps_0*eps_r)}*[{Hy(i,j)-Hy(i-1,j)}/dx-{Hx(i,j)-Hx(i,j-1)}/dy]

� Hx(i,j) = Hx(i,j)-{dt/(mu_0*dy)}*{Ez(i,j+1)-Ez(i,j)}� Hy(i,j) = Hy(i,j)+{dt/(mu_0*dx)}*{Ez(i+1,j)-Ez(i,j)}


� Some new MATLAB commands:� image: IMAGE(C) displays matrix C as an image� imagesc: data is scaled to use the full color map� Colorbar: appends a vertical color scale to the current axis

Colormap: a matrix may have any number of rows but it


� Colormap: a matrix may have any number of rows but it must have exactly 3 columns

� Each row is interpreted as a color, with the first element specifying the intensity of red, the second gree and the third blue

� Color intensity is specified on the interval of 0.0 to 1.0


� For example, [0 0 0] is black� [1 1 1] is white� [1 0 0] is red� [0.5 0.5 0.5] is gray

[127/255 1 212/255] is aquamarine


� [127/255 1 212/255] is aquamarine


FDTD simulation of TM wave hitting a dielectric surface (right side), source at the center (2-D) (fdtd_2d_TM_2.m)

� Modify the above program as follows:� Specify the position of the dielectric surface (id=..,jd=..)� Choose a value of eps_r=10.0..


� Choose a value of eps_r=10.0..� Modify the constant value C1 accordingly� Run the program� In free space wave is propagating freely and � on the other side reflection and transmission of the wave at

the interface


� Wave propagates more slowly on the medium on the right and � the pulse length is shorter

� FDTD simulation of TM wave propagation due to multiple sources (fdtd_2d_TM_multi_source.m)


(fdtd_2d_TM_multi_source.m)3. 3-D FDTD

� From two Maxwell’s curl equations, � we need to consider all 6 equations for 3-D case

0 0

1 1;

r r

E HH E

t tε ε µ µ

∂ ∂= ∇× = − ∇×

∂ ∂

r rr r


Yee’s FDTD grid

� Electric fields are calculated at “integer” time-steps � and magnetic field at “half-integer” time-steps

� Electric field components are placed at mid-points of the corresponding edges


corresponding edges� E.g. � Ex is placed at midpoints of edges oriented along x-direction� Ex is on half-grid in x and n the integer grids in y & z


� The magnetic field components are � placed at the centers of the faces of the cubes and � oriented normal to the faces

� E.g.� Hx components are placed at the centers of the faces in the


� Hx components are placed at the centers of the faces in the yz-plane

� Hence Hx is � on the integer grid in x and � on the half-grid in y & z


2/25/2015FDTD by R. S. Kshetrimayum50Fig. 4 Yee’s FDTD grid


( ) ( )

( ) ( )

( ) ( )

∆

−+−++−

∆

−+−++

=∆

+−+

++

++

+

z

rqpHrqpH

y

rqpHrqpH

t

rqpErqpE

n

y

n

y

n

z

n

z

r

n

x

n

x

2/1,,2/12/1,,2/11

,2/1,2/1,2/1,2/11

,,2/1,,2/1

2/12/1

2/12/1

0

1

εε

εε

∂

∂−

∂

∂=

∂

∂

z

H

y

H

t

E yz

r

x

0

1

εε


∆zr 0εε

( ) ( )

( ) ( )

( ) ( )

∆

+−−++−

∆

−+−++

=∆

+−+

++

++

+

z

rqpHrqpH

z

rqpHrqpH

t

rqpErqpE

n

z

n

z

r

n

x

n

x

r

n

y

n

y

,2/1,2/1,2/1,2/11

2/1,2/1,2/1,2/1,1

,2/1,,2/1,

2/12/1

0

2/12/1

0

1

εε

εε

∂

∂−

∂

∂=

∂

∂

x

H

z

H

t

Ezx

r

y

0

1

εε


( ) ( )

( ) ( )

( ) ( )

∆

+−−++−

∆

+−−++

=∆

+−+

++

++

+

y

rqpHrqpH

x

rqpHrqpH

t

rqpErqpE

n

x

n

x

n

y

n

y

r

n

z

n

z

2/1,2/1,2/1,2/1,1

2/1,,2/12/1,,2/11

2/1,,2/1,,

2/12/1

2/12/1

0

1

εε

εε

∂

∂−

∂

∂=

∂

∂

y

H

x

H

t

E xy

r

z

0

1

εε


∆

−yr 0

εε

( ) ( )

( ) ( )

( ) ( )

∆

+−++−

∆

+−++

=∆

++−++ −+

y

rqpErqpE

z

rqpErqpE

t

rqpHrqpH

n

z

n

z

n

y

n

y

n

x

n

x

2/1,,2/1,1,1

,2/1,1,2/1,1

2/1,2/1,2/1,2/1,

0

0

2/12/1

µ

µ

∂

∂−

∂

∂−=

∂

∂

z

E

y

E

t

H yzx

0

1

µ


( ) ( )

( ) ( )

( ) ( )

∆

+−++−

∆

+−++

=∆

++−++ −+

z

rqpErqpE

x

rqpErqpE

t

rqpHrqpH

n

x

n

x

n

z

n

z

n

y

n

y

,,2/11,,2/11

2/1,,2/1,,11

2/1,,2/12/1,,2/1

0

2/12/1

µ

µ

∂

∂−

∂

∂−=

∂

∂

x

E

z

E

t

Hzxy

0

1

µ


∆z

0µ

( ) ( )

( ) ( )

( ) ( )

∆

+−++−

∆

+−++

=∆

++−++ −+

x

rqpErqpE

y

rqpErqpE

t

rqpHrqpH

n

y

n

y

n

x

n

x

n

z

n

z

,2/1,,2/1,11

,,2/1,2/1,2/11

,2/1,2/1,2/1,2/1

0

0

2/12/1

µ

µ

∂

∂−

∂

∂−=

∂

∂

y

E

x

E

t

H xyz

0

1

µ


� FDTD simulation of cubical cavity (fdtd_3D_demo.m)

� We will use FDTD to find � the resonant frequencies of an air-filled cubical cavity with

metal walls� Peaks in the frequency response indicate


� Peaks in the frequency response indicate � the presence of resonant modes in the cavity

� An initial Hz field excites only � the TE modes

� The first four eigenfrequencies� TE101, TE011/TE201, TE111 and TE102 modes


4. Perfectly Matched Layers

� Berenger’s PML is an artificial material � that is theoretically designed to create no reflections

regardless of the � frequency,


� frequency, � polarization and � angle of incidence of a plane wave upon its interface


� PML could be analyzed in stretched coordinate systems� Source-free Maxwell’s equations

( ) ( )HE

EjHHjE

ss

ss

=•∇=•∇

=×∇−=×∇

;0;0

;;rr

rrrr

µε

ωεωµ


� Consider plane waves

( ) ( )

( ) ( ) ( ) zzsz

yysy

xxsx

zyx

s

ss

∂

∂+

∂

∂+

∂

∂=∇

1ˆ

1ˆ

1ˆ

zkykxkrk

eHHeEE

zyx

rkjrkj

++=•

== •−•−

rr

rrrr rrrr

00 ;


� Substituting these into Maxwell’s equations, we have,

( ) ( )

ˆˆˆ

;0;0

;;

++=

=•=•

−=×=×

zyxs

ss

ss

s

kz

s

ky

s

kxk

HkEk

EHkHEk

r

rrrr

rrrrrr

µε

εωµ


� Solution to the above equation is given by

222

+

+

=•

y

z

y

y

x

xss

zyx

s

s

k

s

k

s

kkk

sss

rr

θφθφθ cos,sinsin,cossin zzyyxx kskkskksk ===


� If sx is a complex number with a negative imaginary part, � the wave will be attenuated in the x direction

� Wave impedance is independent of coordinate stretching

( )0,1,

'''cossincossin '''

>≥== −−−−sseee

xjssjkxjksxjk xx φθφθ


� Wave impedance is independent of coordinate stretching

ε

µωµη ===

kH

Err

r

FDTD: An Introduction'''

,1 jsssss yzx −===

==

'''

,1:4

jssss

scorners

yx

z

−==

=

y


Object

Fig. 5 Computational domain truncated using the conductor backed PMLs

''',1 jsssss yzx −===

'''

,1

jsss

ss

x

zy

−=

=='''

,1

jsss

ss

x

zy

−=

==x


( ) ( ) ( )

( ) ( ) ωµωµ

ωµ

syy

sxx

zyxs

HjEyys

HjExxs

HjEzzs

Eyys

Exxs

E

−=×∂

∂−=×

∂

∂

−=×∂

∂+×

∂

∂+×

∂

∂=×∇

;ˆ1

;ˆ1

;ˆ1

ˆ1

ˆ1

rrrr

rrrrr


( )

ωε

σ

ωε

σ

ωε

σ

ωµ

zz

y

yx

x

szsysxszz

yx

jsjsjsgChoo

HHHHHjEzzs

ysxs

−=−=−=

++=−=×∂

∂

∂∂

1,1,1sin

;;ˆ1 rrrrrr


( )

( )

( ) szzsz

szz

sy

ysy

syy

sxxsx

sxx

Ht

HHsjEz

z

Ht

HHsjEy

y

Ht

HHjjEx

x

rr

rr

rr

rr

rr

rr

ωε

µσµωµ

ωε

µσµωµ

ωε

µσµ

ωε

σωµ

−∂

∂−=−=×

∂

∂

−∂

∂−=−=×

∂

∂

−∂

∂−=

−−=×

∂

∂

ˆ

ˆ

1ˆ


( )

( )

( )

( ) szzsz

syy

sy

sxxsx

szszz

Et

EHz

z

Et

EHy

y

Et

EHx

x

tz

rr

r

rr

r

rr

r

σε

σε

σε

ωε

+∂

∂=×

∂

∂

+∂

∂=×

∂

∂

+∂

∂=×

∂

∂

∂∂

ˆ

ˆ

ˆ


5. References1. S. D. Gedney, Introduction to the Finite-Difference Time-

Domain Method for Electromagnetics, Morgan & Claypool, 2011

2. R. Garg, Analytical and Computational Methods in Electromagnetics, Artech House, 2008


Electromagnetics, Artech House, 20083. A. Taflove and S. C. Hagness, Computational

electrodynamics the Finite-Difference Time-Domain Method, Artech House, 2005

4. T. Rylander, P. Ingelstrom and A. Bondeson, Computational Electromagnetics, Springer, 2013


5. J. M. Jin, Theory and Computation of Electromagnetic Fields, IEEE Press, 2010

6. A. Elsherbeni and V. Demir, The Finite-Difference Time-Domain Method for Electromagnetics with MATLAB Simulations, Scitech Publishng, 2009


Simulations, Scitech Publishng, 20097. D. B. Davidson, Computational Electromagnetics for RF

and Microwave Engineering, Cambridge University Press, 2011

8. U. S. Inan and R. A. Marshall, Numerical ElectromagneticsThe FDTD Method, Cambridge University Press, 2011

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