Simulation of Electromagnetic Fields:
The Finite-Difference Time-Domain (FDTD)
Method and Its Applications
Veysel Demir, Ph.D.
demir@ceet.niu.edu
Department of Electrical Engineering, Northern Illinois University, DeKalb, IL 60115
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Veysel Demir
Bachelor of Science, Electrical and Electronics Engineering, Middle East Technical
University, Ankara, Turkey, 1997.
System Analyst and Programmer, Pamukbank, Software Development Department,
Istanbul, Turkey, July 1997 – August 2000.
Master of Science, Electrical Engineering, Syracuse University, Syracuse, NY, 2002.
Doctor of Philosophy, Electrical Engineering, Syracuse University, Syracuse, NY, 2004.
Research Assistant, Sonnet Software, Inc. Liverpool, NY, August 2000 – July 2004.
Visiting research scholar, University of Mississippi, Electrical Engineering Department,
University, MS, July 2004 – Present.
Assistant Professor, Department of Electrical Engineering, Northern Illinois
University, DeKalb,IL, August 2007 – present
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Computational Electromagnetics
 Maxwell’s equations can be given in differential or integral form
Finite-difference
time-domain
(FDTD)
Transmission line
matrix (TLM)
Finite element
method (FEM)
Finite-difference
frequency-domain
(FDFD)
Differential equation methods
Method of Moments
(MoM)
Fast multipole
method (FMM)
Integral equation
methods
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Computational Electromagnetics
 Maxwell’s equations can be given in time domain or frequency domain
Time-domain methods
Finite-difference
time-domain
(FDTD)
Transmission line
matrix (TLM)
Finite element
method (FEM)
Method of Moments
(MoM)
Finite-difference
frequency-domain
(FDFD)
Fast multipole
method (FMM)
Frequency domain methods
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Commercial software packages
 Commercial software packages
Finite element method (FEM)
Method of Moments (MoM)
ADS Momentum
HFSS
Transmission line matrix (TLM)
Finite-difference time-domain (FDTD)
CST Microstripes
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The Finite-Difference Time-Domain Method
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FDTD Books
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Yearly FDTD Publications
 The most popular method in computational electromagnetics
Source: Allen Taflove, “A Perspective on the 40-Year History of FDTD Computational Electrodynamics,”
Applied Computational Electromagnetics Society (ACES) Conference, Miami, Florida, March 15, 2006.
Can be found at http://www.ece.northwestern.edu/ecefaculty/Allen1.html
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Maxwell’s Equations
 The basic set of equations describing the electromagnetic world
 Shows that light is an electromagnetic wave.
Gauss’s law
Gauss’s law for magnetism
Faraday’s law
Ampere’s law
Constitutive relations
∇ ⋅D= ρv
∇ ⋅B= 0
∂B
∇ × E= −
∂t
∂D
∇ × H= J+
∂t
D = ε E ,and B = µ H
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FDTD Overview – Finite Differences
 Represent the derivatives in Maxwell’s curl equations by finite differences
 We use the second-order accurate central difference formula
df ( x)
f ( x + ∆ x) − f ( x − ∆ x)
= f ′ ( x) ≅
dx
2∆ x
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FDTD Overview – Cells
 A three-dimensional problem space is composed of cells
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FDTD Overview – The Yee Cell
 The FDTD (Finite Difference Time Domain) algorithm was first established
by Yee as a three dimensional solution of Maxwell's curl equations.
K. Yee, IEEE Transactions on Antennas and Propagation, May 1966.
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FDTD Overview – Material grid
 A three-dimensional problem space is composed of cells
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FDTD Overview – Updating Equations
 Three scalar equations can be obtained from one vector curl equation.
∂E
ε
= ∇ ×H
∂t
∂ H x ∂ Ey
µx
=
−
∂t
∂z
∂ H y ∂ Ez
µy
=
−
∂t
∂x
∂ H z ∂ Ex
µz
=
−
∂t
∂y
∂ Ez
∂y
∂ Ex
∂z
∂ Ey
∂ Ex ∂ H z
εx
=
−
∂t
∂y
∂ Ey ∂ H x
εy
=
−
∂t
∂z
∂ Ez ∂ H y
εz
=
−
∂t
∂x
µ
∂ Hy
∂z
∂ Hz
∂x
∂ Hx
∂y
∂H
= −∇ × E
∂t
∂x
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FDTD Overview – Updating Equations
 Represent derivatives by finite-differences
∂ Ex ∂ H z ∂ H y
εx
=
−
∂t
∂y
∂z
Exn + 1 (i, j , k ) − Exn (i, j , k )
ε x (i, j , k )
=
∆t
n + 0.5
n + 0.5
H zn + 0.5 (i, j , k ) − H zn + 0.5 (i, j − 1, k ) H y (i, j , k ) − H y (i, j, k − 1)
−
∆y
∆z
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FDTD Overview – Updating Equations
 Represent derivatives by finite-differences
∂ H x ∂ E y ∂ Ez
µx
=
−
∂t
∂z
∂y
H xn + 0.5 (i, j , k ) − H xn − 0.5 (i, j, k )
µ x (i, j , k )
=
∆t
E yn (i, j , k + 1) − E yn + 0.5 (i, j , k ) Ezn (i, j + 1, k ) − Ezn (i, j , k )
−
∆z
∆y
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FDTD Overview – Updating Equations
 Express the future components in terms of the past components
 H zn + 0.5 (i, j , k ) − H zn + 0.5 (i, j − 1, k ) 


∆
y
∆
t


Exn + 1 (i, j , k ) = Exn (i, j , k ) −
ε x (i, j , k )  H yn + 0.5 (i, j , k ) − H yn + 0.5 (i, j , k − 1) 
−

∆z


 E yn (i, j , k + 1) − E yn + 0.5 (i, j, k ) 


∆
t
∆
z


H xn + 0.5 (i, j , k ) = H xn − 0.5 (i, j , k ) +
n
n
µ x (i, j , k )  Ez (i, j + 1, k ) − Ez (i, j , k ) 
−

∆y


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FDTD Overview – Leap-frog Algorithm
Exercise 1D
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Absorbing Boundary Conditions
 The three-dimensional problem space is truncated by absorbing
boundaries
 Most popular absorbing boundary is Perfectly Matched layers (PML)
Exercise 2D PML
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Active and Passive Lumped Elements
 Active and passive lumped elements can be modeled in FDTD
∂E
∇×H= ε
+ σ E+ J
∂t
Voltage source
Current source
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Active and Passive Lumped Elements
Resistor
Capacitor
Inductor
Diode
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Active and Passive Lumped Elements
A diode circuit
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Transformation from Time-Domain to Frequency-Domain
 Results can be obtained for frequency domain using Fourier Transform
A low-pass filter
S11
S22
Exercise 2D object
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Near-Field to Far-field Transformations
An inverted-F antenna
24
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Modeling fine geometries
 It is possible to model fine structures using appropriate formulations
A wire loop antenna
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Incident plane wave
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Scattering Problems
∇ × ( H inc + H scat )
∂
= ε ( Einc + Escat )
∂t
∇ × H inc
∂
= ε 0 Einc
∂t
A dielectric sphere
Exercise 3D PML
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Scattering from a Dielectric Sphere
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Earth / Ionosphere Models in Geophysics
 Snapshots of FDTD-Computed Global Propagation of ELF
Electromagnetic Pulse Generated by Vertical Lightning Strike off South
America Coast
Source: Allen Taflove, “A Perspective on the 40-Year History of FDTD Computational Electrodynamics,”
Applied Computational Electromagnetics Society (ACES) Conference, Miami, Florida, March 15, 2006.
Can be found at http://www.ece.northwestern.edu/ecefaculty/Allen1.html
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Wireless Personal Communications Devices
Source: Allen Taflove, “A Perspective on the 40-Year History of FDTD Computational Electrodynamics,”
Applied Computational Electromagnetics Society (ACES) Conference, Miami, Florida, March 15, 2006.
Can be found at http://www.ece.northwestern.edu/ecefaculty/Allen1.html
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Phantom Head Validation at 1.8 GHz
Source: Allen Taflove, “A Perspective on the 40-Year History of FDTD Computational Electrodynamics,”
Applied Computational Electromagnetics Society (ACES) Conference, Miami, Florida, March 15, 2006.
Can be found at http://www.ece.northwestern.edu/ecefaculty/Allen1.html
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Ultrawideband Microwave Detection of Early-Stage Breast Cancer
Source: Allen Taflove, “A Perspective on the 40-Year History of FDTD Computational Electrodynamics,”
Applied Computational Electromagnetics Society (ACES) Conference, Miami, Florida, March 15, 2006.
Can be found at http://www.ece.northwestern.edu/ecefaculty/Allen1.html
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Focusing Plasmonic Lens
Source: Allen Taflove, “A Perspective on the 40-Year History of FDTD Computational Electrodynamics,”
Applied Computational Electromagnetics Society (ACES) Conference, Miami, Florida, March 15, 2006.
Can be found at http://www.ece.northwestern.edu/ecefaculty/Allen1.html
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Thank You
Exercise 2D PEC
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