1. Computational Electromagnetics
High frequency
rigorous methods IE DE
MoM
FDTD
TLM
field based
current
based
GO/GTD
PO/PTD TD FD VM
FEM

2. Computational Electromagnetics
Electromagnetic problems are mostly described by three methods:
Differential Equations (DE)  Finite difference (FD, FDTD)
Integral Equations (IE)  Method of Moments (MoM)
Minimization of a functional (VM)  Finite Element (FEM)
Theoretical
effort
less
more
Computational
effort
more
less

4. Numerical Differentiation “FINITE DIFFERENCES”

5. Introduction to differentiation
Conventional Calculus
The operation of diff. of a function is a well-defined procedure
The operations highly depend on the form of the function involved
Many different types of rules are needed for different functions
For some complex function it can be very difficult to find closed form solutions
Numerical differentiation
Is a technique for approximating the derivative of functions by employing only arithmetic operations (e.g., addition, subtraction, multiplication, and division)
Commonly known as “finite differences”

6. Taylor Series Problem: For a smooth function f(x),
Given: Values of f(xi) and its derivatives at xi
Find out: Value of f(x) in terms of f(xi), f(xi), f(xi), …. x y
f(x)
f(xi) xi

7. Taylor’s Theorem
If the function f and its n+1 derivatives are continuous on an interval containing xi and x, then the value of the function f at x is given by

8. Finite Difference Approximations of the First Derivative using the Taylor Series (forward difference)
Assume we can expand a function f(x) into a Taylor Series about the point xi+1 x y
f(x)
f(xi) xi
xi+1
f(xi+1) h h

9. Finite Difference Approximations of the First Derivative using the Taylor Series (forward difference)
Assume we can expand a function f(x) into a Taylor Series about the point xi+1
Ignore all of these terms

10. Finite Difference Approximations of the First Derivative using the Taylor Series (forward difference) y
f(x) h xi
xi+1 x
f(xi)
f(xi+1)

11. Finite Difference Approximations of the First Derivative using the forward difference: What is the error?
The first term we ignored is of power h1. This is defined as first order accurate.
First forward
difference

12. Finite Difference Approximations of the First Derivative using the Taylor Series (backward difference)
Assume we can expand a function f(x) into a Taylor Series about the point xi-1 x y
f(x)
f(xi-1)
xi-1 xi
f(xi) h -h

13. Finite Difference Approximations of the First Derivative using the Taylor Series (backward difference)
Ignore all of these terms
First backward
difference

14. Finite Difference Approximations of the First Derivative using the Taylor Series (backward difference) y
f(x) h
xi-1 xi x
f(xi-1)
f(xi)

15. Finite Difference Approximations of the Second Derivative using the Taylor Series (forward difference) y x
f(x)
f(xi) xi
xi+1
f(xi+1) h
xi+2
f(xi+2)
(1)
(2)
(2)-2*
(1)

16. Finite Difference Approximations of the Second Derivative using the Taylor Series (forward difference) y x
f(x)
f(xi) xi
xi+1
f(xi+1) h
xi+2
f(xi+2)
Recursive formula for
any order derivative

17. Higher Order Finite Difference Approximations

18. Centered Difference Approximation
(1)
(2)
(1)-(2)

19. Finite Difference Approximations of the First Derivative using the Taylor Series (central difference) y
f(x) h
xi-1 xi
xi+1 x
f(xi-1)
f(xi+1)
f(xi)

20. Second Derivative Centered Difference Approximation (central difference)
(1)
(2)
(1)+(2)

21. Using Taylor Series Expansions we found the following finite-differences equations
FORWARD DIFFERENCE
BACKWARD DIFFERENCE
CENTRAL DIFFERENCE
CENTRAL DIFFERENCE

22. Forward finite-difference formulas

23. Centered finite difference formulas

24. Finite Difference Approx. Partial Derivatives
Problem: Given a function u(x,y) of two independent variables how do we determine the derivative numerically (or more precisely PARTIAL DERIVATIVES) of u(x,y)

25. Pretty much the same way
STEP #1: Discretize (or sample) U(x,y) on a 2D grid of evenly spaced points in the x-y plane

26. 2D GRID xi xi+1 xi-1 xi+2 yj yj+1 yj-1 yj-2 y axis u(xi,yj) u(xi+1,yj)
x axis

27. ui,j ui+1,j ui-1,j ui,j-1 ui,j+1 SHORT HAND NOTATION i i+1 i-1 i+2 j
y axis i
i+1
i-1
i+2 j
j+1
j-1
j-2
ui,j
ui+1,j
ui-1,j
ui,j-1
ui,j+1
x axis

28. Partial First Derivatives
Problem: FIND
recall:

29. Partial First Derivatives
Problem: FIND Dy
These are central difference formulas
Are these the only formulas
we could use?
Could we use forward or backward
difference formulas? Dx

30. Partial First Derivatives: short hand notation
Problem: FIND Dy Dx

31. Partial Second Derivatives
Problem: FIND
recall:

32. Partial Second Derivatives
Problem: FIND Dy Dx

33. Partial Second Derivatives: short hand notation
Problem: FIND Dy Dx

34. FINITE DIFFERENCE ELECTROSTATICS
Electrostatics deals with voltages and charges that do no vary as a function
of time.
Poisson’s equation
Laplace’s equation
Where, F is the electrical potential (voltage), r is the charge density and
e is the permittivity.

35. FINITE DIFFERENCE ELECTROSTATICS: ExampleFo F1 F2 F3
Find F(x,y) inside the box due to the voltages applied to its boundary. Then
find the electric field strength in the box.

36. Electrostatic Example using FD
Problem: FIND Dy Dx

37. Electrostatic Example using FD
Problem: FIND
If Dx = Dy

38. Electrostatic Example using FD
Problem: FIND
Iterative solution technique:
Discretize domain into a grid of points
Set boundary values to the fixed boundary values
Set all interior nodes to some initial value (guess at it!)
Solve the FD equation at all interior nodes
Go back to step #4 until the solution stops changing
DONE

39. Electrostatic Example using FD
MATLAB CODE EXAMPLE

40. FINITE DIFFERENCE Waveguide TM modes: Example
Where for TM modes
and If
then kz becomes imaginary and the mode does not
propagate.

41. FINITE DIFFERENCE Waveguide TM modes: Example
Goal: Find all permissible values of kt and the corresponding
mode shape (F(x,y)) for that mode.

42. Waveguide Example using FD
Problem: FIND Dy Dx

43. Waveguide Example using FD
Problem: FIND
If Dx = Dy=h

44. Waveguide Example using FD
let
where N is the number of interior nodes (i.e. not on a boundary)
If we now apply the FD equation at all interior nodes we can
form a matrix equation
Where I is the identity matrix

45. Waveguide Example using FD
is an eigenvalue equation usually cast in the form
The eigenvalues will provide the permissible values for the transverse wavenumber
kt and the eigenvectors are the corresponding mode shapes (F(x,y))

46. Waveguide Example using FDh W

47. Waveguide Example using FDh W
Node #1:
Node #2:
Node #3:
Node #4:
Solve using the Matlab “eig” function

48. Waveguide Example using FD
is an eigenvalue equation usually cast in the form
The eigenvalues will provide the permissible values for the transverse wavenumber
kt and the eigenvectors are the corresponding mode shapes (F(x,y))
HOMEWORK: WRITE A MATLAB PROGRAM THAT CALCULATES THE TRANSVERSE WAVENUMBERS AND MODE SHAPES FOR A RECTANGULAR WAVEGUIDE FOR TM MODES

49. Finite Difference Time Domain Method (FDTD)

50. Some big advantages Broadband response with a single excitation.
3D models easily.
Frequency dependent materials accommodated.
Most parameters can be generated e.g.
Scattered fields
antenna patters
RCS
S-parameters
etc…..

51. How does it work?
Based on the 2 Maxwell curl equations in derivative form. These
are linearized by central finite differencing. We only consider
nearest neighbor interactions because all the fields are advanced
temporally in discrete time steps over spatial cells.
ie we sample in space & time
embedding of an antenna
in a FDTD space lattice
(note that the whole volume
is meshed!)

52. closed & open problems
FDTD is especially suitable for computing transients for closed
geometries. When an open geometry is required, such as when
we are dealing with an antenna, we need a boundary condition to
simulate infinity. In this case the FDTD requires an absorbing boundary
condition (ABC) at the grid truncation.
This means there is no reflection from the boundary where the
mesh ends.

53. FDTD: The Basic Algorithm
Maxwell’s Equations in the TIME Domain:

54. Equate Vector Components: Six E and H-Field Equations

55. 2-D Equations: Assume that all fields are uniform in y direction (i. e
2-D Equations: Assume that all fields are uniform in y direction (i.e. d/dy = 0)
2D - TE
2D - TM

56. 1-D Equations: Assume that all fields are uniform in y and x directions (i.e. d/dy =d/dx= 0)
1D - TE
1D - TM

57. Discretize Objects in Space using Cartesian Grid
2D Discretization
3D Discretization z x
1D Discretization

58. Define Locations of Field Components: FDTD Cell called Yee Cell
Finite-Difference
Space is divided into small cells
One Cell: (dx)(dy)(dz)
E and H components are distributed in space around the Yee cell (note: field components are not collocated)
FDTD: Yee, K. S.: Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE Transactions on Antennas Propagation, Vol. AP-14, pp , 1966.

59. Replace Continuous Derivatives with Differences
Derivatives in time and space are approximated as DIFFERENCES

60. Solution then evolves by time-marching difference equations
Time is Discretized
One Time Step: dt
E and H fields are distributed in time
This is called a “leap-frog” scheme.

61. 1-D FDTD
Assuming that field values can only vary in the z-direction (i.e. all spatial derivatives in x and z direction are zero), Maxwell’s Equations reduce to: z Hy Ex
e(z), m(z)

62. 1-D FDTD – Staggered Grid in Space
Interleaving of the Ex and Hy field components in space and time in the 1-D FDTD formulation
Time plane

63. Replace all continuous derivatives with finite differences
1-D FDTD
Replace all continuous derivatives with finite differences

64. Substitution of difference equations in above yields:
1-D FDTD
Substitution of difference equations in above yields:

65. After some simple algebra:
1-D FDTD
After some simple algebra:

66. 1-D FDTD Algorithm – Flow Chart
Start
Compute 1-D Faraday’s FDTD equation: For all nodes n inside the simulation region:
Compute 1-D Ampère-Maxwell’s FDTD equation: For all nodes n inside the simulation region:
Electric current density excitation: For all excitation nodes n:
Boundary condition: For all PEC boundary nodes n: No
Yes
Stop

67. FDTD Solution of 1-D Maxwell’s Equations (EXAMPLE)
FDTD equations
Initial conditions
initial-boundary-value problem
For all values of i
Boundary condition for a perfectly electrically conducting (PEC) material

68. FDTD Solution of the First Two 1-D Scalar Maxwell’s Equations
Excitation pulse: RC2(t) – Time Domain
Amplitude RC2(t)
Excitation pulse: RC2(f) – Frequency Domain
Magntiude |RC2(f)|

69. FDTD Solution of 1-D Maxwell’s Equations

70. FDTD Solution of 1-D Maxwell’s Equations

71. SOME OPEN QUESTIONS??
How do we determine what Dt and Dz should be?
How do we implement real sources?
How do we simulate open boundaries?
How accurate is the solution?

72. Potential Source of Error: Numerical Dispersion
In implementing any numerical technique, it is essential to understand the origin and nature of errors that are introduced as a result of the computational approach.
In the FDTD errors arise from the discretization of the computational space and finite difference approximations to Maxwell’s equations.
One particularly important source of error is call numerical dispersion.
To see this consider the continuous wave equation in ID:
Which has a solution of the form we have looked at before:

73. If we substitute this expression into the wave equation, we get:
From this we see that the wave number is linearly proportional to the frequency.
This is more widely known as the phase velocity, when expressed as np = w/k np=c
To determine how the FDTD effects this relationship, we repeat the same procedure using the difference equations.

74. Let’s now look at the discrete wave equation
Represent the second order difference equation (for both time and space) in terms of:
Doing the same for the time derivative, we get:

75. Now defining the solution from in difference form:
Substituting this into the difference form of the wave equation we get:
Factor out:

76. Factor out:

77. We get:
Group time and space terms and divide both sides by 2
Use Euler’s Identity:
(FDTD)
(continuous)

78. This is a nonlinear equation that represents the relationship between the wave number, and the frequency, w. It is a function of the time step Dt, spatial step Dx and frequency w
This means that as a simulated wave propagate through the solution space it undergoes phase errors because the numerical wave either slows down or accelerates relative to the actual wave propagation in physical space.
To examine the implication we consider 3 special cases:
Case 1
Consider Δt  and Δx  0
In this case, use the first two terms of the Taylor series expansion of terms 
Same as in real space

79. Case 2
Use the relation cΔt = Δx
This is often referred to as the magic time step!
Plugging in we get:
 Implies that
No dispersion
Unfortunately the relation cΔt = Δx is unstable!

80. Case 3
In this case we consider the general case where there is no assumed relation between Δt and Δx.
Therefore we need to solve for a general expression that can tell us the effect of numerical dispersion for a given relationship.
To do this, we solve in terms of the wave number.
Using this equation we can determine the numerical wave number for any relation between Δt and Δx for a given sampling rate.
Substitute in the appropriate parameters and determine the effects of discretizing the computational space.

81. Where we have used k= w/c, use the sampling note ofx
Where we have used k= w/c, use the sampling note of
Now define numerical phase velocity as

82. Over a distance of 10λ, which is in this case in no computational cells
The numerical wave would only propagate cells  phase error of 45.72º
If we repeat this analysis for
Now the phase error at wλ is 11.19º
The conclusion to draw from this is that the larger the solution space, the greater the phase error is going to be.
(MATLAB DEMO)

83. Stability
As stated before, the FDTD method approximates M.E.’s as a set of completed difference equations.
As such this method is useful only when the solution of the difference equations are convergent and stable.
Convergence means that as time stepping continues, the solution of the difference equations asymptomatically approach the solution given by Maxwell’s equations.
On the other hand, stability is basically stated as a set of condition under which the error generated does not grow in an unbounded fashion.

84. In order for the FDTD equations to remain stable the following relationship (called the Courant stability criteria must be met)
This is the stability condition for 1D case
For higher dimensional cases we have the form
,where
D=1, 2, 3 depending on the number of dimensions used
(MATLAB DEMO)

85. So far we have considered a sample 1D FDTD formulation, that physically represents the propagation of a plane wave.
More often than not, EM BVP’s cannot be accurately represented as a 1D problem.
2D FDTD Formulation
In the formulation of the two-dimensional FDTD, one can assume either TE ( electric field is to the plane of incidence, or TM; magnetic field is to the plane of incidence).
This results from the fact that in 2D neither the fields nor the object contain any variations in the z-direction,consequently at

86. So in this context, Maxwell’s Equations can be reduced to:

87. For TM polarization we have:

88. Applying the central difference expression, for TE mode, we get:
For the TM case, we have:

89. Numerical Dispersion 2D case
Using same procedure as for the 1D case we obtain:

90. Numerical Dispersion 2D case
Plot of the normalised phase velocity vs. travelling wave angle.

91. Writing a 2D FDTD Program
Defining physical constants
Define program constants
magnitude of electric field
: total # of time steps
: # of nodes in the x, y direction
If using a pulse source, define Ez and Hx, or we can implement a hard source.
Start time marching nΔt, time value n = 1 : Nt
Increment the spatial index
First apply the absorbing boundary conditions
Apply the ABC along the edges of the computational region

92. (MATLAB DEMO)

93. Yee’s 3D Mesh

94. Transition to 3D
The 1D formulation involved 2 degrees of freedom (one field
quantity E or H and one time variable). In 3D we have 4 degrees
of freedom (3 in space and 1 in time). The following notation is
somewhat standard
where the increment is the lattice space increments and i,j,k are
integers. Any field may be represented as

95. The Yee Cell seen in a more Macroscopic view –
Faraday's and Ampere’s law
Fall 2007

96. So far we have considered the 1D and 2D formulations.
General 3D Formation
So far we have considered the 1D and 2D formulations.
Now we will do the 3D formulation.
Recall that Maxwell’s curl equations are:
(i, j, k)
(i, j, k)

97. For simplicity, consider a linear and isotropic medium.
In Cartesian coordinates, the curl equations reduce to: i j k
/x
/y
/z Ex Ey Ez

98. Also, for Ampere’s Law:
3D Discretization
Given a function f(x, y, z, t) we write it in discrete form as:
f(x, y, z, t) = f(iΔx, jΔy, kΔz, nΔt) =
In general, we let: Δx = Δy = Δz = d

99. 3D formulation These six coupled equations form the basis for the FDTD
algorithm.
3D Discretization
Given a function f(x, y, z, t) we write it in discrete form as:
f(x, y, z, t) = f(iΔx, jΔy, kΔz, nΔt) =
In general, we let: Δx = Δy = Δz = d

100. FDTD Discretization of 3D Curl Equations
Where we have used time interpolation

102. We can simplify these expressions if we set:
And we can define the material constants:

103. Substituting in we get:

104. ABC’s
In general EM analysis of scattering structures often requires the solution of “open region” problems.
As a result of limited computational resources, it becomes necessary to truncate the computational domain in such a way as to make it appear infinite.
This is achieved by enclosing the structure in a suitable output boundary that absorbs all outward traveling waves
ABC

105. In this lecture we will describe some of the more common ABC’s, such as:
Bayliss-Turkel annihilation operators
Enqquist-Majda one way operators
Mur ABC
Liao’s extrapolation method
And the PML, perfectly matched layer
The PML method which was introduced in 1994 by Berenger, represents one of the most significant advances in FDTD development, since it conception I 1966, by Kane Yee.
The PML produces back reflection ~ over a very broad range of incident
PMC Region

106. Early ABC’s
When Yee first introduced the FDTD method, he used PEC boundary conditions.
This technique is not very useful in a general sense.
It wasn’t until the 70’s when several alternative ABC’s were introduced.
However, these early ABC’s suffered from large back reflections, which limited the efficacy of the FDTD method.

107. Extrapolation from Interior Node (Taylor 1969)
If we consider the electric field located on a 2D boundary
Since and are determined in the FDTD calculations, and are therefore known values, we can solve this equation for the exterior node.
This equation is only effective on normally incident waves and degrades rapidly when the incident wave is off-normal.
y, (j)
Wave
x, (i)

108. Extending this method to 2D:
This approach applies to a wide range of incident field angles, but due to the nature of the averaging process often gives rise to significant non-physical back reflections.
j=3
j=2
j=1
i-1 i
i+1

109. Dissapative Medium
In this technique a lossy medium is used to surround the FDTD computational region.
The idea is that as that waves propagate into this medium, they are dissipated before they can undergo back reflection

110. The problem is that there is often an input impedance between the FDTD region and the long media.
The back reflection results from the ratio of electrical-conductivity and magnetic-conductivity parameters, respectively.
Quadratic
Linear

111. Here the are assumed to be free space.
Therefore the input impedance is:
which is not necessarily equal to Zo.
To overcome this the conductivity values can be implemented in a non-uniform fashion.
That is the can have a linear or quadratic profile.
This tends to require relatively thick medium layers - computational requirements!