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Computational Electromagnetics: The Finite-Difference Time-Domain Stephen Gedney Department of Electrical Engineering University of Kentucky EE699 Fall 2003 © Copyright, Stephen Gedney

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Page 1: Computational Electromagnetics: The Finite-Difference …optics.sgu.ru/~ulianov/Students/Books/Applied_Optics/Gedney... · EE-699 S. Gedney Introduction 1-2 Computational Electromagnetics

Computational Electromagnetics: The Finite-Difference Time-Domain

Stephen Gedney Department of Electrical Engineering

University of Kentucky

EE699

Fall 2003

© Copyright, Stephen Gedney

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EE-699 S. Gedney

Introduction 1-2

Computational Electromagnetics via The Finite-Difference Time-Domain Method

• FDTD is One of the Most Popular Computational Techniques of Current Date for

Simulation of Electromagnetic Phenomona • Based on 2nd Order Accurate Central Difference Approximations in Space and

Time of Maxwell's Eqns. • Provides a Direct Solution of Time-Dependent EM Fields in a Volumetric Region • Advantages over Moment Method or Finite Element Method:

- Inhomogeneous Media is Inherently Modeled - Lossy, Non-Linear and Anisotropic Media Supported - Boundary Conditions are Inherently Represented - Broad Band Response Obtained with Single Simulations - Robustness and Ease of Implementation - Much Broader Range of Direct Applicability

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EE-699 S. Gedney

Introduction 1-3

• Some Disadvantages: - Accuracy

- The standard Yee-algorithm is second order accurate - Error is generally dispersive - Error accumulates as the wave propagates through the mesh - Large problems require finer meshing!

- Computationally Intensive - requires 3D orthogonal gridding - staircase approximation of curved surfaces - Time stepping is a function of the grid dimension - smallest detail in the geometry can thus determine grid size (Yee-algorithm)

and the time step

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EE-699 S. Gedney

Introduction 1-4

• Application areas of FDTD:

- EM Scattering - EMC - Microwave Circuits - Non-linear device modeling - Antenna modeling & Design - Geo-sensing - Photonic device modeling - Optical gratings, periodic structures -

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EE-699 S. Gedney

Introduction 1-5

Applications

Optical Interferance Due to Two PEC Spheres (Computed using LLNL Tsar-Lite)

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EE-699 S. Gedney

Introduction 1-6

Balanced Antipodal Vivaldi Antenna (REMCOM X-FDTD) Source: http://www.remcom.com/html/cad-vivaldi.html

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EE-699 S. Gedney

Introduction 1-7

Electromagnetic Interaction with a Helicoptor (REMCOM X-FDTD) Source: http://www.remcom.com/html/cad-helicopter.html

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EE-699 S. Gedney

Introduction 1-8

Microwave Circuit Device Modeling (Simulated Using UK/JPL PGY Simulator)

Vertical Electric Field Beneath the Substrate of a

3 dB Gysel Power Divider

Vertical Electric Field Beneath the Substrate of a

16-Element Phased Array

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EE-699 S. Gedney

Introduction 1-9

Micro-Ring Resonator (OptiFDTD) Source: http://www.optiwave.com

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EE699 S. Gedney

1D Solution of the Wave Equation 2-1

1.1 One-Dimensional Scalar Wave Equation

∂ 2 f∂t 2 = c2 ∂ 2 f

∂x2

• Assume an infinite homogeneous space • The wave equation supports the general solutions

f (x, t) = F1(x − ct) + F2(x + ct) • Represents superposition of forward and backward traveling waves

c = 1µε

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EE699 S. Gedney

1D Solution of the Wave Equation 2-2

Taylor's series expansion of f(x,t) about the point x to the point x ± ∆x at a fixed time t

f (x + ∆x,t) = f (x,t) +∂f (x,t)

∂x∆x +

∂ 2 f (x,t)∂x 2

∆x2

2+ ∂ 3 f (x,t)

∂x 3∆x3

6+ ∂ 4 f (x,t)

∂x4∆x 4

24+ ...

f (x − ∆x,t) = f (x,t) −∂f (x,t)

∂x∆x +

∂ 2 f (x,t )∂x2

∆x2

2− ∂ 3 f (x,t)

∂x3∆x3

6+ ∂ 4 f (x,t)

∂x4∆x 4

24+ ...

Adding these two expressions

f (x + ∆x,t) + f (x − ∆x,t) = 2 f (x,t) +∂ 2 f (x,t)

∂x2 ∆x2

+ ∂ 4 f (x,t)∂x 4

∆x4

12+ ...

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EE699 S. Gedney

1D Solution of the Wave Equation 2-3

∂ 2 f (x,t)

∂x2 =f (x + ∆x,t) − 2 f (x,t) + f (x − ∆x,t )

(∆x)2⎡ ⎣ ⎢

⎤ ⎦ ⎥ + O (∆x)2[ ]

This expression is commonly referred to as the CENTRAL DIFFERENCE approximation

of the second-order derivative. It is said to be second-order accurate since the error is of order (∆x)2. Namely, the error decays as (∆x)2 with decreasing values of ∆x. It is noted that the error is also dependent on the smoothness of f(x,t). Similarly,

∂ 2 f (x,t)∂t 2 =

f (x,t + ∆t) − 2 f (x,t) + f (x,t − ∆t)(∆t)2

⎡ ⎣ ⎢

⎤ ⎦ ⎥ + O (∆t)2[ ]

Combining to Form a Discrete Wave Equation

f (x,t + ∆t )− 2 f (x,t) + f (x,t − ∆t)(∆t)2

⎡ ⎣ ⎢

⎤ ⎦ ⎥ + O (∆t)2[ ]=

c2 f (x + ∆x,t) − 2 f (x,t) + f (x − ∆x,t)(∆x)2

⎡ ⎣ ⎢

⎤ ⎦ ⎥ + O (∆x)2[ ]⎧

⎨ ⎩

⎫ ⎬ ⎭

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EE699 S. Gedney

1D Solution of the Wave Equation 2-4

DISCRETE SPACE

Let xi = i∆x,tn = n∆t, andf i

n = f (xi , tn )

Discrete Wave Equation:

f i

n+1 − 2 f in + f i

n−1

(∆t)2

⎣ ⎢ ⎤

⎦ ⎥ + O (∆t)2[ ]= c2 fi+1n − 2 f i

n + f i−1n

(∆x )2

⎣ ⎢ ⎤

⎦ ⎥ + O (∆x)2[ ]⎧ ⎨ ⎩

⎫ ⎬ ⎭

Explicit Time-Marching Solution for f:

f in+1 = (c∆t)2 f i+1

n − 2 f in + f i−1

n

(∆x)2

⎣ ⎢ ⎤

⎦ ⎥ + 2 f in − f i

n−1 + O (∆x)2[ ]+ O (∆t)2[ ]

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EE699 S. Gedney

1D Solution of the Wave Equation 2-5

• Given the Initial Conditions: 1 0,i if f−

• Given the Boundary Conditions: 0 ,x

n nNf f

• Solve for 1,ni if + ∀

x

t

x∆

t∆

i

nnif

1-D FDTD Stencil

1nif +1

nif −

1nif+

1nif−

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EE699 S. Gedney

1D Solution of the Wave Equation 2-6

DISCRETIZATION

The Stability and Accuracy of the Explicit Solution will be Dependent on the values of ∆x and ∆t.

MAGIC TIME STEP: ∆x = c ∆t. Discrete Wave Equation Using the Magic Time Step:

f in+1 = f i+1

n − 2 f in + f i−1

n[ ]+ 2 f in − f i

n−1

= f i+1n + f i−1

n − fin−1[ ]

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EE699 S. Gedney

1D Solution of the Wave Equation 2-7

Analysis:

Let f i

n = F1(i∆x − cn∆t), and gin = G2 (i∆x + cn∆t)

and ∆t = ∆x/c

f in+1 = F1((i − n −1)∆x ) = f i−1

n ,

gin+1 = G2 ((i + n +1)∆x) = gi+1

n !

This is the exact solution! In fact, for this choice of discretization parameters, the discrete solution is exact! This choice of ∆t is referred to as the magic time step. Unfortunately, we only have this luxury for one-dimensional problems with homogeneous material regions, however, in the following discussions, it will provide useful insight into understanding the discrete solution.

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EE699 S. Gedney

1D Solution of the Wave Equation 2-8

Dispersion for 1D Wave Equation • Dispersion: Variation of the propagating velocity as a function of frequency. Time-Harmonic Plane Wave Analysis Let

f (x, t) = e j(ωt−kx)

From the wave equation (∂ 2 f∂t 2 = c2 ∂ 2 f

∂x2 )

−ω2e j(ωt−kx ) = −c2k2e j(ωt−kx ) or

k = ω / c c = phase velocity of the wave = vp, where

vp = ω / k Group velocity (velocity of energy transport):

vg =∂ω∂k

= 1/∂k∂ω

= c

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EE699 S. Gedney

1D Solution of the Wave Equation 2-9

Numerical Dispersion for 1D Wave Equation • Discrete Space

f (x,t) = fin = e j(ωn∆t − ˜ k i∆x)

Explicit Update Equation:

f in+1 = (c∆t)2 f i+1

n − 2 f in + f i−1

n

∆x2

⎣ ⎢ ⎤

⎦ ⎥ + 2 f in − f i

n−1

Plugging the discrete form of f(x,t) into the update expression:

e j(ω (n+1)∆t− ˜ k i∆x) = (c∆t)2 e j(ωn∆t− ˜ k (i+1)∆x) − 2e j(ωn∆t− ˜ k i∆x) + e j(ωn∆t− ˜ k (i−1)∆x)

∆x2

⎣ ⎢

⎦ ⎥

+ 2e j(ωn∆t− ˜ k i∆x) − e j(ω (n−1)∆t− ˜ k i∆x)

Factor out the term e j(ωn∆t− ˜ k i∆x) , and grouping terms, this reduces to: e jω∆t + e− jω∆t

2=

(c∆t)2

∆x2e j ˜ k ∆x + e− j˜ k ∆x

2−1

⎣ ⎢

⎦ ⎥ +1

Dispersion Relationship of the Discrete Space:

cos(ω∆t) =(c∆t)2

∆x2 cos( ˜ k ∆x) −1[ ]+1

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EE699 S. Gedney

1D Solution of the Wave Equation 2-10

Numerical Phase Velocity: 1. Assume: ∆t → 0, ∆x → 0. Dispersion Relationship becomes

1−(ω∆t)2

2≅

(c∆t)2

∆x2 1−( ˜ k ∆x)2

2−1

⎣ ⎢ ⎤

⎦ ⎥ +1

which reduces to: (ω∆t)2

2≅

(c∆t)2

∆x2( ˜ k ∆x)2

2⇒ ω2 = c2 ˜ k 2 ⇒ ˜ k = ±

ωc

2. Magic Time Step: c∆t = ∆x

cos(ω∆t) = 1⋅ cos( ˜ k ∆x) −1[ ]+1= cos( ˜ k ∆x)

∴ω∆t = ± ˜ k ∆x → ˜ k = ±ω / c

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EE699 S. Gedney

1D Solution of the Wave Equation 2-11

3. Numerical Dispersion:

˜ k =1

∆xcos−1 ∆x2

(c∆t)2 cos(ω∆t )−1[ ]+1⎧ ⎨ ⎩

⎫ ⎬ ⎭

vp = ω / ˜ k = 2πc

cos−1 ∆x2

(c∆t)2 cos(ω∆t) −1[ ]+1⎧ ⎨ ⎩

⎫ ⎬ ⎭

∆xλo

Example: ∆x = λo / 10 and c∆t = ∆x / 2. Then

˜ k = 1∆x

cos−1 1+ 4 cos( 2π∆x2λ o

) −1[ ] = 0.63642∆x

= 6.3642 / λ o

vp = ω / ˜ k = 0.9873c

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EE699 S. Gedney

1D Solution of the Wave Equation 2-12

Example: c∆t = ∆x / 2

0

0.2

0.4

0.6

0.8

1

1.2

0.001 0.01 0.1 1

c∆ t/ ∆ x = 1/2

vp/c

∆x/ λo

vp = 0.98726 c

vp = 0.99689 c

( )Phase Error = 360 360 3601p

p

wv ck k cwk vc

ω

λ λ λ λ

⎛ ⎞⎟⎜ ⎟−⎜ ⎟⎜ ⎛ ⎞⎟⎟⎜− ° ° °⎝ ⎠ ⎟⎜ ⎟⋅ = ⋅ = − ⋅⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠

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EE699 S. Gedney

1D Solution of the Wave Equation 2-13

Calculate the Phase Error over 10 λo when c∆t = ∆x / 2 and: 1. ∆x = 0.1 λo, vp = 0.98726 c

Phase Error: 10 [1/0.98726-1] 360o = 46.45o

2. ∆x = 0.05 λo, vp = 0.99689 c

Phase Error: 10[1/0.99689-1] 360o = 11.23o

Phase Error Reduces by a Factor of 4!

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EE699 S. Gedney

1D Solution of the Wave Equation 2-14

STABILITY • Stability of the Formulation Implies that the Eigenvalues of the linear update expression

are within the unit circle. Starting with:

f in+1 = (c∆t)2 f i+1

n − 2 f in + f i−1

n

∆x2

⎣ ⎢ ⎤

⎦ ⎥ + 2 f in − f i

n−1

Rewrite this as:

f n+1 = 2 A[ ]f n − f n−1

where [A] is a matrix. Stability requires that the energy is bounded. That is:

21

2 1n

n

f

f

+

≤ .

To analyze the stability problem, we will analyze the eigenspectrum of the difference operator and determine the relationship of t∆ and x∆ that leads to bounded energy within the system.

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EE699 S. Gedney

1D Solution of the Wave Equation 2-15

STABILITY ANALYSIS The difference operator is re-written in matrix format:

21 1

1 12

( ) 2 2n n n n n ni i i i i i

c tf f f f f fx

+ −+ −

∆ ⎡ ⎤= − + + −⎣ ⎦∆ 1 12[ ] [ ]n n nf A f I f+ −⇒ = − (i)

where,

2

2

( )[ ] [ ] [ ]2c tA I L

x∆

= −∆ ,

[I] is the Identity Matrix, and [L] is a tridiagonal Matrix and is defined as:

[ ] [0,0, ,0, 1,2, 1,0, ,0,0]i throw

L − = − −

[L] is Positive Semi-Definite with eigenvalues 0 ≤ λ ≤ 4. Currently, our update operator is in the form of a 2nd Order Difference Operator.

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EE699 S. Gedney

1D Solution of the Wave Equation 2-16

Reducing 1st Order Difference Eqn. to 2nd Order To study stability, we will reduce the 2nd order difference equation into a 1st order expression. To this end, let

yn+1 = f n+1

f n⎡ ⎣ ⎢

⎤ ⎦ ⎥ , yn = f n

f n−1⎡ ⎣ ⎢

⎤ ⎦ ⎥ .

Then, (i) can be rewritten as a 1st order difference equation

1 2[ ] [ ][ ] ; [ ]

[ ] 0n n A I

M MI

+ −⎡ ⎤= = ⎢ ⎥

⎣ ⎦y y

Each iteration of the difference operator is simply a matrix product. Therefore, given the initial condition oy , we can define:

[ ]n n oM=y y

Stability of the first-order equation thus requires

limn→∞

[M]n < K

where K is a constant that is finite. This will be satisfied if ρ([M]) ≤ 1, that is, if the eigenvalues of M are within or on the unit circle in the complex plane.

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EE699 S. Gedney

1D Solution of the Wave Equation 2-17

Define (λ, x1) to be an eigensolution of [A], or [ ] 0A x xλ− = and ( )[ , ] ,Tx y ξ to be an eigen solution of [M]:

[M] xy

⎡ ⎣

⎤ ⎦ = ξ x

y⎡ ⎣

⎤ ⎦ .

Assume the eigenvalues of [ ]A are real and distinct, such that [ ]A is diagonalizable. Then 1[ ] [ ][ ][ ]A P D P −=

where [P] is the matrix of eigenvectors of [A] and [D] is a diagonal matrix containing the eigenvalues of [A]. Subsequently,

[P] 00 [P]

⎡ ⎣ ⎢

⎤ ⎦ ⎥

2[D] −[I][I] 0

⎡ ⎣

⎤ ⎦

[P]−1 00 [P]−1

⎡ ⎣ ⎢

⎤ ⎦ ⎥

xy

⎡ ⎣

⎤ ⎦ = ξ x

y⎡ ⎣

⎤ ⎦

Transforming

2[D] −[I ][I ] 0

⎡ ⎣

⎤ ⎦

′ x ′ y

⎡ ⎣ ⎢

⎤ ⎦ ⎥ = ξ ′ x

′ y ⎡ ⎣ ⎢

⎤ ⎦ ⎥ , where

1

1

[ ] 00 [ ]

x PP

′ ⎡ ⎤⎡ ⎤ ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥′⎣ ⎦ ⎣ ⎦⎣ ⎦

xy y

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EE699 S. Gedney

1D Solution of the Wave Equation 2-18

Stability Criterion of Explicit Method This is then reduced to 2 x 2 matrices for each eigenvalue λ

2 11 0

x xy y

λξ

′ ′⎡ ⎤− ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥′ ′⎣ ⎦ ⎣ ⎦⎣ ⎦

The eigenvalues ξ can then be easily derived from the characteristic equation ξ 2 − 2λξ +1 = 0

which has solutions ξ = λ ± λ2 −1. Stability requires that |ξ| ≤ 1, which is true if |λ| < 1, or, if ρ([A]) ≤ 1.

The spectral radius of [A] is determined from the generalized eigenvalue equation [ ] 0A x xλ− =

or 2

2

( )[ ] [ ] [ ] 02c tI L x I x

⎧ ⎫∆− − =⎨ ⎬∆⎩ ⎭

This can be rewritten as:

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EE699 S. Gedney

1D Solution of the Wave Equation 2-19

( )( )

2

22[ ] 1 [ ]

o

xL x I xc t

λ ∆= −

or [L]x = κ[ I]x Therefore,

( )( )

( )22

2 2

21 12

o

o

c txxc t

κ λ λ κ∆∆

= − ⇒ = −∆∆

Stability requires 1λ ≤ . Also, κ is the eigenvalues of [L], and by definition: 0 <κ ≤ 4 Therefore, stability requires:

( )2

2 22

oc tx

κ∆

≤∆

or,

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EE699 S. Gedney

1D Solution of the Wave Equation 2-20

( )

2

24o

xc t

κ ∆≤

This inequality must satisfy all eigenvalues κ . Consequently, choosing the largest value κ = 4, leads to:

( )

2

21 oo

x c t xc t∆

≤ ⇒ ∆ ≤ ∆∆

Thus, for stability, we must have that:

1. The eigenvalues of A are real and distinct (inherently true) 2. The space and time steps satisfy the Courant Stability Limit:

xtc

∆∆ ≤

If this inequality is not satisfied, the energy in the system will grow unboundedly.

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EE699 S. Gedney

1D Solution of the Wave Equation 2-21

An Implicit 1-Dimensional Formulation with Unconditional Stability ∂ 2 f∂t 2 = c2 ∂ 2 f

∂x2

Using the Newmark-Beta Time-Integration Scheme, we can approximate the wave equation in the discrete space as

∆x2

(co∆t)2 f in+1 − 2 f i

n + f in−1( )= β f i+1

n+1 − 2 f in+1 + f i−1

n+1( )+ (1− 2β ) f i+1

n − 2 f in + f i−1

n( )+ β f i+1

n−1 − 2 f in−1 + f i−1

n−1( )

⇒ [A] f n+1 = 2[B] f n − [C] f n−1

or

f n+1 = 2[A]−1[B] f n − [A]−1[C] f n−1

where,

[A] = [C] = β[L]+ ∆x2

(co∆t)2 [I],

[B] = 2β−12 [L]+ ∆x2

(co∆t)2 [ I]

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EE699 S. Gedney

1D Solution of the Wave Equation 2-22

[I] is the Identity Matrix, and [L] is a tridiagonal Matrix and is defined as:

[L]i−th

row= [0,0, ,0,−1,2,−1, 0, ,0,0]

[L] is Positive Semi-Definite with eigenvalues 0 ≤ λ ≤ 4. With [A]= [C], the implicit update expression is

f n+1 = 2[A]−1[B] f n − f n−1 (i) This scheme is implicit since it requires the inversion of A. For the one-dimensional problem, A is tri-diagonal, and can be factorized in O(N) operations. Question: What is the range of stability for this implicit operator as a function of β, c∆t, and ∆x?

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EE699 S. Gedney

1D Solution of the Wave Equation 2-23

Reducing 1st Order Difference Eqn. to 2nd Order Again, reduce the 2nd order differ-ence equation into a 1st order expression by letting:

yn+1 = f n+1

f n⎡ ⎣ ⎢

⎤ ⎦ ⎥ , yn = f n

f n−1⎡ ⎣ ⎢

⎤ ⎦ ⎥ .

Then, (i) can be rewritten as a 1st order difference equation

yn+1 = [M]yn ; [M] = 2[A]−1[B] −[I ][ I] 0

⎡ ⎣ ⎢

⎤ ⎦ ⎥

Stability of the first-order equation requires

limn→∞

[M]n < K

where K is a constant that is finite. This will be satisfied if ρ([M]) ≤ 1.

Define (λ, x1) to be an eigensolution of [A]-1[B], or [B]x − λ[A]x = 0 The eigenspectrum of [M] is found from

[M] xy

⎡ ⎣

⎤ ⎦ = ξ x

y⎡ ⎣

⎤ ⎦ .

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EE699 S. Gedney

1D Solution of the Wave Equation 2-24

Assume that the eigenvalues of [A]-1[B] are real and distinct. Then, it is diagonalizable:

1 1[ ] [ ] [ ][ ][ ]A B P D P− −= [D] is a diagonal matrix containing the eigenvalues of [A]-1[B]. Subsequently,

[P] 00 [P]

⎡ ⎣ ⎢

⎤ ⎦ ⎥

2[D] −[I][I] 0

⎡ ⎣

⎤ ⎦

[P]−1 00 [P]−1

⎡ ⎣ ⎢

⎤ ⎦ ⎥

xy

⎡ ⎣

⎤ ⎦ = ξ x

y⎡ ⎣

⎤ ⎦

Let: 1

1

[ ] 00 [ ]

PP

′ ⎡ ⎤⎡ ⎤ ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥′⎣ ⎦ ⎣ ⎦⎣ ⎦

x xy y

Leads to:

2[D] −[I ][I ] 0

⎡ ⎣

⎤ ⎦

′ x ′ y

⎡ ⎣ ⎢

⎤ ⎦ ⎥ = ξ ′ x

′ y ⎡ ⎣ ⎢

⎤ ⎦ ⎥

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EE699 S. Gedney

1D Solution of the Wave Equation 2-25

Stability Criterion of Explicit Method This is then reduced to 2 x 2 matrices for each eigenvalue λ

2λ −11 0

⎡ ⎣ ⎢

⎤ ⎦ ⎥

′ x ′ y

⎡ ⎣ ⎢

⎤ ⎦ ⎥ = ξ ′ x

′ y ⎡ ⎣ ⎢

⎤ ⎦ ⎥

The eigenvalues ξ can then be easily derived from the characteristic equation ξ 2 − 2λξ +1 = 0

which has solutions ξ = λ ± λ2 −1. Stability requires that |ξ| ≤ 1, which is true if |λ| < 1, or, if ρ([A]-1[B]) ≤ 1.

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EE699 S. Gedney

1D Solution of the Wave Equation 2-26

The spectral radius of [A]-1[B] is determined from the generalized eigenvalue equation [B]x − λ[A]x = 0

or 2β−1

2 [L]+ ∆x2

(co∆t)2 [ I] x − λ β[L] + ∆x2

(co∆t)2 [ I] = 0

[L]x =λ −1

2β−12 − λβ

∆x2

(co∆t)2 [ I]x

or [L]x = κ[ I]x However, by definition, 0 <κ ≤ 4. Finally, solving for λ,

λ =∆x2

(c∆t )2 − κ 1−2β2( )

∆x2

(c∆t)2 + κβ

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EE699 S. Gedney

1D Solution of the Wave Equation 2-27

Regions of Stability Stability requires |λ| ≤ 1 1. β = 0 (Central Difference Approximation)

λ =∆x2

(c∆t )2 − κ2( )

∆x2

(c∆t)2

=1− κ2( )(c∆t)2

∆x2

For |λ |≤1; κ2( )(c∆t)2

∆x2 ≤ 2

∵κmax = 4, → (c∆t)2

∆x2 ≤1, or ∆t ≤ ∆xc

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EE699 S. Gedney

1D Solution of the Wave Equation 2-28

2. β > 0.

λ =1− κ 1−2β

2( )(c∆t)2

∆x2

1+ κβ (c∆t)2

∆x2

If |λ| ≤ 1, 00 4 and tκ ∆ >£ £ " then we can write this as: ( )1 2

211

β αλ

β α

--=

+ ◊

where 2

2( )c t

xα κ ∆

∆= For this to have a magnitude greater than one for 0 α£ £ •

1−2β

2( )≤ β , or

β ≥ 14

In this instance, the formulation is unconditionally stable! Namely, |λ| ≤ 1 for all time steps.

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EE699 S. Gedney

1D Solution of the Wave Equation 2-29

Question: What are practical values for β and t∆ ? o 1/ 4β = - the Newmark scheme is equivalent to 2nd order trapezoidal integration

rule. This is the most accurate.

( ) ( )

( )

( )

21 1 1 1 1

1 12

1 1

1 1 11 1

12 2( ) 4

1 22

1 24

n n n n n ni i i i i i

o

n n ni i i

n n ni i i

x f f f f f fc t

f f f

f f f

∆∆

+ - + + ++ -

+ -

- - -+ -

- + = - +

+ - +

+ - +

o 1/ 2β = - the Newmark scheme is equivalent to a 1st order Simpson integration rule:

( ) ( )

( )

21 1 1 1 1

1 12

1 1 11 1

12 2( ) 2

1 22

n n n n n ni i i i i i

o

n n ni i i

x f f f f f fc t

f f f

∆∆

+ - + + ++ -

- - -+ -

- + = - +

+ - +

o Typically, you should choose 1/ 4β = for 2nd order accuracy. o Next, ∆t must still be small enough to minimize dispersion error, and to maintain the

proper sampling rate. o A good choice is ∆t ≤ λo/20/c, where, λo is the wavelength, of the highest frequency

within the Fourier spectrum of the propagating signal.

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-1

Transmission-Line Equations

( , ) ( , )( , ) ,V x t I x tRI x t Lx t

∂ ∂∂ ∂

= − −

∂I(x, t)∂x

= −GV(x, t) − C∂V (x, t)

∂t.

C – Capacitance per unit length L– Inductance per unit length R – Resistance per unit length G – Conductance per unit length Scalar Wave Equations – Transmission Line

∂ 2V(x, t)∂x 2 = LC

∂ 2V (x, t)∂t 2 + (RC + GL)

∂V(x, t)∂t

+ RGV(x, t)

∂ 2 I(x, t)∂x2 = LC

∂ 2I(x, t)∂t2 + (RC + GL)

∂I(x,t )∂t

+ RGI(x, t)

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-2

Lossless Transmission Lines (R = G = 0)

Lossless Transmission Line Equations:

( , ) ( , ) ( , ) ( , ), .V x t I x t I x t V x tL Cx t x t

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

= − = −

Taylor's series expansion function f(x,t)

f (x + ∆x 2, t) = f (x, t )+∂f (x,t )

∂x∆x2

11!

+∂ 2 f (x, t)

∂x2∆x2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

2 12!

+∂ 3 f (x, t)

∂x 3∆x2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

3 13!

+∂ 4 f (x, t)

∂x 4∆x2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

4 14!

+ ...

Similarly,

f (x − ∆x 2 , t) = f (x, t) −∂f (x, t)

∂x∆x2

+∂ 2 f (x, t)

∂x 2∆x2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

2 12!

−∂ 3 f (x, t)

∂x3∆x2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

3 13!

+∂ 4 f (x, t )

∂x4∆x2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

4 14!

+ ...

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-3

Subtracting

f (x + ∆x 2, t) − f (x + ∆x 2, t) =∂f (x, t)

∂x∆x +

124

∂ 3 f (x, t)∂x3 ∆x( )3 + ...

∂f (x, t)

∂x=

f (x + ∆x 2, t )− f (x − ∆x 2, t)∆x

⎡ ⎣ ⎢

⎤ ⎦ ⎥ + O (∆x)2[ ]

∂f (x, t)

∂t=

f (x, t + ∆t 2) − f (x, t − ∆t 2)∆t

⎡ ⎣ ⎢

⎤ ⎦ ⎥ + O (∆t)2[ ]

Note that second-order accuracy is obtained at (x,t) by differencing two discrete points centered by (x,t). Hence, this is called a central difference approximation.

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-4

Discrete Transmission Line Equations:

Ii+1/2n − Ii−1/2

n

∆x= −C

∂Vin

∂t Second-order accurate iff V is evaluated at xi and I is evaluated at / 2ix x± ∆ .

Ii+1/2

n − Ii−1/2n

∆x= −C

Vin +1/2 − Vi

n−1/ 2

∆t . Second-order accurate iff I is evaluated at nt and V is evaluated at / 2nt t± ∆ . Observing the equation, we can write a recursive relationship to compute

1/2niV + given the

previous values of V and I:

Vin+1/ 2 = Vi

n−1/2 −∆tC

Ii+1/2n − Ii−1/ 2

n

∆x

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-5

FULLY EXPLICIT TIME-MARCHING SOLUTION OF THE DISCRETE LOSSLESS TRANSMISSION LINE EQUATIONS

Vin+1/ 2 = Vi

n−1/2 −∆tC

Ii+1/2n − Ii−1/ 2

n

∆x

Ii+1/2n+1 = Ii+1/2

n −∆tL

Vi+1n +1/2 − Vi

n+1/ 2

∆x

o Explicit update expression based on 1D coupled eqns o V and I are staggered in both space and time.

o Given the initial condition of V and I along the entire line length,V and I can be time advanced

o We still need to guarantee stability and understand the error properties

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-6

STABILITY OF 1D TX-LINE EQUATIONS

1/ 2 1/ 21 1

1/ 2 1/ 2

n nn n i ii i

V VtI IL x

+ ++ +

+ +

−∆= −

∆, 1/ 2 1/ 2 1/ 2 1/ 2

n nn n i i

i iI ItV V

C x+ − + −−∆

= −∆

Plug in the expressions for V into the first equation:

1/ 2 1/ 23/ 2 1/ 2 1/ 2 1/ 21

11/ 2 1/ 2

n n n nn ni i i i

i in ni i

I I I It tV Vt C x C xI IL x

− −+ + + −+

++ +

− −∆ ∆− − +∆ ∆ ∆= −

Noting that:

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

n n n ni i i i

L xV V I It

− − −+ + +

∆ ⎡ ⎤− = − −⎣ ⎦∆

( )21 1

1/ 2 1/ 2 1/ 2 3/ 2 1/ 2 1/ 222 2n n n n n ni i i i i i

c tI I I I I I

x+ −

+ + + + + −

∆⎡ ⎤= − − − +⎣ ⎦∆

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-7

This can again be repeated for V:

( )21/ 2 1/ 2 3/ 2 1/ 2 1/ 2 1/ 2

1 122 2n n n n n ni i i i i i

c tV V V V V V

x+ − − − − −

+ −

∆⎡ ⎤= − − − +⎣ ⎦∆

These equations are identical to those derived via the 2nd-order wave equation using central difference approximations! (with the exception of a shifted time or space index.)

Stability and Dispersion:

o Same as One-Dim. Scalar Wave Equation o Stability of the coupled 1D difference operators satisfies the Courant limit:

xtc

∆∆ ≤ , where

1cLC

= is the wave speed.

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-8

Example 1.

Assume a uniform transmission-line with C = 100.0 pf/m, and L = 250.0 nH/m (Z = 50 Ω, and c = 2 x 108 m/s). The transmission-line has a length of 1 m, and is terminated by a short circuit at each end. The spatial discretization ∆x is 0.1 m. ! Initial condition, a rectangular pulse with amplitude of 1 V –

Given 1/2niV − and 1/2

niI + . Find 1/2n

iV + if xtc

∆∆ =

.

Solution:

At xt LC xc

∆∆ = = ∆

1/ 2 1/ 2

1/ 2 1/ 2n n n n

i i i iLV V I IC

+ −+ −⎡ ⎤= − −⎣ ⎦ , ( )1 1/ 2 1/ 2

1/ 2 1/ 2 1n n n ni i i i

CI I V VL

+ + ++ + += − −

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-9

V in−1/ 2

Ii +1/ 2n

150

A

1V

1V

i = 1 2 3 4 5 6 7 8 9 10

i = 1 2 3 4 5 6 7 8 9 10

i = 1 2 3 4 5 6 7 8 9

V in+1/ 2

where , Vin+1/ 2 = Vi

n −1/ 2 − 50( Ii+1/ 2n − Ii −1/ 2

n ) Substituting 1

1/2niI ++ and 1

1/2niI +− , leads to: 1/ 2 1/ 2 1/ 2 3/ 2

1 1n n n n

i i i iV V V V+ − − −+ −= + −

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-10

Lossy Transmission Lines

Transmission Line Equations: ∂I(x, t)

∂x= −GV(x, t) − C

∂V (x, t)∂t

∂V(x, t)∂x

= −RI(x, t )− L∂I(x, t)

∂t Discrete Transmission Line Equations:

Ii+1/2n − Ii−1/2

n

∆x= −GVi

n − CVi

n +1/2 − Vin−1/ 2

∆t 1/ 2 1/ 2 1

1/ 21 1/ 2 1/ 21/ 2

n n n nni i i ii

V V I IRI Lx t∆ ∆

+ + +++ + ++

- -= - -

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-11

Averaging Loss Terms in Time for 2nd Order Accuracy Ii+1/2

n − Ii−1/2n

∆x= −G

Vin +1/2 + Vi

n−1/ 2

2− C

Vin+1/2 − Vi

n −1/2

∆t Vi+1

n+1/ 2 −Vin+1/2

∆x= −R

Ii+1/2n+1 + Ii+1/2

n

2− L

Ii +1/ 2n +1 − Ii+1/2

n

∆t

Vin+1/ 2 =

C∆t

−G2

C∆t

+G2

⎢ ⎢ ⎢

⎥ ⎥ ⎥ Vi

n−1/ 2 −1

C∆t

+G2

⎢ ⎢ ⎢

⎥ ⎥ ⎥

Ii+1/ 2n − Ii−1/2

n

∆x

Ii+1/2n+1 =

L∆t

−R2

L∆t

+R2

⎢ ⎢ ⎢

⎥ ⎥ ⎥ Ii+1/2

n −1

L∆t

+R2

⎢ ⎢ ⎢

⎥ ⎥ ⎥

Vin+1/ 2 − Vi

n+1/2

∆x

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-12

Boundary Conditions: Loads and Sources

Rg

RLv(t)

z = 0 z = d

Constraints must be introduced to properly terminate the mesh since nV cannot be updated at the terminating load

1/ 2 1/ 2 1n n

n n i ii i

I ItV VC x

+ − −−∆= −

and 1/ 2 1/ 2

1 1n n

n n i ii i

V VtI IL x

+ ++ + −∆

= −∆

.

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-13

To update 1V or , NV we need to have 1/2I− and 1/2NI + . Unfortunately, neither of these quantities are in our solution space. As a result, an additional constraint must be introduced in order to update the voltages at the endpoints of the grid. Such constraints must preserve stability and accuracy in the solution.

0V 1V1/ 2I 3/ 2I NV1NV − 1/ 2NI −

x∆

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-14

Short and Open Circuit Loads 1. RL = 0 and Rg = 0

1/ 2 1/ 21 ( ), 0;n n

N nV v n t V+ += ∆ = ∀

2 Rg = 0, and RL = ∞?

V1

n+1/ 2 = v(n∆t), IN −1n+1 = 0; ∀n

[Note: ∆x = L / (N −1.5) ]

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-15

Arbitrary Loads

Poor First-Order Approximation:

1/ 21 ( )n n

N N L L oV I R R Z+-= £

• ∆t/2 displacement in time and ∆x/2 displacement in space • Unconditionally Unstable if RL > Zo (∴ IN −1

n +1 = VN −1n+1/2 / RL )

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-16

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

50 Ω Uniform Transmission LineExcited by a 2 V Voltage Source

Zg = 50 Ω, Z

L = 50 Ω (Magic Time Step)

V: t = .50773E-08 sV: t = 0.15232E-07s

V: t = 0.30464E-07 s

V: t = 0.40619E-07 s

Line

Vol

tage

(V)

x (meters)

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-17

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

50 Ω Uniform Transmission LineFirst Order Boundary Condition

Zg = 50 Ω, Z

L = 25 Ω (Magic Time Step)

V: t = 0.50773E-08 s

V: t = 0.15232E-07 s

V: t = 0.30464E-07s

V: t = 0.40619E-07s

Line

Vol

tage

(V)

x (meters)

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-18

Equivalent Distributed Circuit Model of a Uniform Transmission Line

L L

V i+ 2V i+ 3

Ii +2Ii +1IiV i V i+1

L∆x L∆x

C∆x C∆x C∆x

L∆x

C∆x

∆x

LC∆x2

C∆ x2

C∆ x2

C∆x2

C∆ x2

L∆xL∆x

∆ x

V 1 V 2 V 3

I1 I2

At the source end, we need to account for the affects of the discrete current source.

∂I(x, t)∂x

= −GV(x, t) − C∂V (x, t)

∂t− Is (x, t)

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-19

Second-Order Accurate Source and Load Models for the Finite-Length

Transmission Line

LC∆x2

C∆ x2

C∆ x2

C∆x2

L∆xL∆x

∆x

V 1 V 2

I1V NV N −1

IN−1

RLRg

V g

Rg

Applying KCL at Node 1:

1 11 2

n n ng n

g g

V x V VI CR t R

∆ ∂= + +

1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2

1 12 2 2

n n n n n nng g i i i i

g g

V V V V V VI CR x x t R x

+ − + − + −+ − += + +

∆ ∆ ∆ ∆

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-20

Finally,

V1n+1/ 2 =

RgC∆x2∆t

−12

RgC∆x2∆t

+12

⎢ ⎢ ⎢

⎥ ⎥ ⎥ V1

n −1/ 2 −1

RgC∆x2∆t

+12

⎢ ⎢ ⎢

⎥ ⎥ ⎥

RgI1n −

Vgn+1/2 + Vg

n −1/2

2

⎧ ⎨ ⎩

⎫ ⎬ ⎭

Similarly, at the load end:

VNn+1/ 2 =

RLC∆x2∆t

−12

RLC∆x2∆t

+12

⎢ ⎢ ⎢

⎥ ⎥ ⎥ VN

n −1/2 +1

RLC∆x2∆t

+12

⎢ ⎢ ⎢

⎥ ⎥ ⎥

RL IN −1n

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-21

EXAMPLE Steady State Response of a Uniform Transmission Line Excited by A Unit Step

Function

Rg

RLv(t)

z = 0 z = d C = 100.0 pf/m, L = 250 nH/m, d = 10 m

8150 , 2 10 /o pLZ v m sC LC

Ω= = = = ¥

transit time = / 50nspd v = Assume a source voltage ( ) ( )2v t u t= . Example 1: 50 , 50 , Example 2: 25 150

Example 2: 100 20g L g L

g L

Z Z Z

Z

Ω Ω Ω Ζ Ω

Ω Ζ Ω

= = = , =

= , =

probe the near and far end voltages.

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-22

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 2 10 -8 4 10 -8 6 10 -8 8 10 -8 1 10 -7 1.2 10 -7 1.4 10 -7

Uniform 50 Ω Tx Line (Lossless)2 V Unit Step Excitation, t

rise = 2 ns

Zg = 50 Ω, Z

L = 50 Ω

Source End (V)

Load End (V)Volts

seconds

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-23

-0.5

0

0.5

1

1.5

2

2.5

0 100 1 10-7 2 10-7 3 10-7 4 10-7 5 10-7

50 Ω Uniform Transmission LineExcited by 2 V Unit Step with 2 ns Rist Time

Zg = 25 Ω, Z

L = 150 Ω (Magic Time Step)

Source End (V)

Load End (V)

Volts

seconds

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EE699 S. Gedney

FDTD Solution of the Tx-Line Equations 3-24

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 10-7 2 10-7 3 10-7 4 10-7 5 10-7

50 Ω Uniform Transmission LineExcited by 2 V Unit Step with 2 ns Rist Time

Zg = 100 Ω, Z

L = 20 Ω (Magic Time Step)

Source End (V)

Load End (V)Vo

lts

seconds

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EE699 S. Gedney

Gedney, EE699 - FDTD Method for Electromagnetics 3-25

Reactive Load Models for the Finite-Length Transmission Line

C∆ x2

C∆ x2

L∆x

∆ x

V NV N−1

IN−1

RL

G∆x2

G∆x2

CL

Parallel Capacitance/Resistance

From KCL at node N:

IN −1n = VN

n G∆x2

+1

RL

⎝ ⎜

⎠ ⎟ +

C∆x2

∂VNn

∂t+ CL

∂VNn

∂t

Central difference the time derivatives, and using linear interpolation of VN

n

IN −1n =

Vin+1/2 + Vi

n−1/2

2G∆x

2+

1RL

⎝ ⎜

⎠ ⎟ +

C∆x2

+ CL⎛ ⎝ ⎜

⎞ ⎠ ⎟

Vin+1/2 − Vi

n−1/2

∆t

Finally,

VNn+1/2 =

C∆x + 2CL

2∆t−

12

G∆x2

+1RL

⎝ ⎜

⎠ ⎟

C∆x + 2CL

2∆t+

12

G∆x2

+1RL

⎝ ⎜

⎠ ⎟

⎢ ⎢ ⎢ ⎢ ⎢

⎥ ⎥ ⎥ ⎥ ⎥

VNn−1/2 +

1C∆x + 2CL

2∆t+

12

G∆x2

+1

RL

⎝ ⎜

⎠ ⎟

⎢ ⎢ ⎢ ⎢ ⎢

⎥ ⎥ ⎥ ⎥ ⎥

IN −1n

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EE699 S. Gedney

Gedney, EE699 - FDTD Method for Electromagnetics 3-26

EXAMPLE Steady State Response of a Uniform Transmission Line Excited by A Trapezoidal

Pulse (trise = tfall = 250 ps, tdur = 250 ps)

Rg

v(t)

z = 0 z = d

RL CL

C = 100.0 pf/m, L = .250 µH/m (Zo = 50 Ω)d = 0.5 m Case 1: RL = 50 Ω, CL = 0.1 pf, Rg = 50 Ω

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5

2.5 ns

4.0625 ns

Volts

x (m) Load EndSource End

-0.04

-0.02

0

0.02

0.04

0.15 0.2 0.25 0.3 0.35 0.4

4.0625 ns

Volts

x (m) Load EndSource End

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EE699 S. Gedney

Gedney, EE699 - FDTD Method for Electromagnetics 3-27

EXAMPLE 2 Trapezoidal Pulse (trise = tfall = 200 ps, tdur = 250 ps)

Rg

v(t)

z = 0 z = d

RL CL

C = 100.0 pf/m, L = .250 µH/m (Zo = 50 Ω)d = 0.5 m Case 2: RL = 150 Ω, CL = 5 pf, Rg = 50 Ω

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5

t = 2.5 nst = 2.5 ns (CL = 0)

Volts

x (m)

-0.2

0

0.2

0.4

0.6

0.8

0 0.1 0.2 0.3 0.4 0.5

t = 2.75 nst = 2.75 ns (CL = 0)

Volts

x (m)

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EE699 S. Gedney

Gedney, EE699 - FDTD Method for Electromagnetics 3-28

-0.4

-0.2

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.4 0.5

t = 3.125 nst = 3.125 ns (CL = 0)

Volts

x (m)

-0.3

-0.2

-0.1

0

0.1

0.2

0.30.4

0.5

0 0.1 0.2 0.3 0.4 0.5

t = 3.4375nst = 3.4375 ns (CL = 0)

Volts

x (m)

-0.3

-0.2

-0.1

0

0.1

0.2

0.30.4

0.5

0 0.1 0.2 0.3 0.4 0.5

t = 4.375 nst = 4.375 ns (CL = 0)

Volts

x (m)

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EE699 S. Gedney

Gedney, EE699 - FDTD Method for Electromagnetics 3-29

Inductive Load Models

∆ x

V N

RLC∆x

L∆x2

G∆xIN−1 IN

LL

L∆x2

R∆x2

R∆x2

Series Inductance/Resistance

Typical Model for Wire Bond to load, or via. From KVL at current node N:

VNn+1/ 2 = IN

n+1/2 RL +R∆x

2⎛ ⎝ ⎜ ⎞

⎠ ⎟ +

L∆x2

∂INn+1/2

∂t+ LL

∂INn+1/2

∂t

Central difference the time derivatives, and using linear interpolation of IN

n+1/2

VNn+1/ 2 =

INn+1 + IN

n

2RL +

R∆x2

⎛ ⎝ ⎜ ⎞

⎠ ⎟ +

L∆x2

+ LL⎛ ⎝ ⎜ ⎞

⎠ ⎟ IN

n+1 − INn

∆t

Finally,

INn+1 =

L∆x + 2LL

2∆t−

R∆x + 2RL

4L∆x + 2LL

2∆t+

R∆x + 2RL

4

⎢ ⎢ ⎢

⎥ ⎥ ⎥ IN

n +1

L∆x + 2LL

2∆t+

R∆x + 2RL

4

⎢ ⎢ ⎢

⎥ ⎥ ⎥ VN

n+1/ 2

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EE699 S. Gedney

Gedney, EE699 - FDTD Method for Electromagnetics 3-30

Inductive Load at a Voltage Node

NV

LR

LL

1NI -

2C x∆

L x∆

( )Li t2

G x∆

KCL: ( ) ( ) ( ) ( )1 2 2N

N N L

V tC x G xI t V t i tt

∆ ∆-

∂= + +

∂ KVL: ( ) ( ) ( )L

N L L L

i tV t i t R L

t∂

= +∂

Discretize in time:

(1) 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2

1 2 2 2 2

n n n n n nn N N N N L LN

V V V VC x G x I IIt

∆ ∆∆

+ - + - + -

-- + += + +

(2) 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2

2 2

n n n n n nN N L L L L

L LV V I I I IR L

t∆

+ - + - + -+ + -= +

Where, the load current is expressed at the same time as the load voltage. This is due to the fact that they are co-located in space, and there is no time-delay between them.

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EE699 S. Gedney

Gedney, EE699 - FDTD Method for Electromagnetics 3-31

(1) and (2) lead to a coupled set of equations. This can be expressed in matrix form as: 1 12 2

11 12 2

1 12 4 2 2 4 2

01 12 2 2 2

n n nN N N

n nL LL L

C x G x C x G xV Vt t I

L R L RI I

t t

∆ ∆ ∆ ∆∆ ∆

∆ ∆

+ -

-

+ -

Ê ˆ Ê ˆÊ ˆ Ê ˆÊ ˆ Ê ˆ+ + - +Á ˜ Á ˜Á ˜ Á ˜Ë ¯ Ë ¯ Ê ˆÁ ˜ Á ˜= -Á ˜ Á ˜ Á ˜Á ˜ Á ˜ Ë ¯Ê ˆ Ê ˆÁ ˜ Á ˜+ - + + - -Ë ¯ Ë ¯Á ˜ Á ˜Á ˜ Á ˜Ë ¯ Ë ¯Ë ¯ Ë ¯

which can be written in reduced notation as:

1 12 2

n n nPx Qx y+ -= +

Where, P and Q are 2 2× matrices. Therefore, at each time step, we can solve:

11 12

12 212

nn nN n

n

L

V

Ix P Qx y

++ --

+

Ê ˆÊ ˆÁ ˜ = = +Á ˜Á ˜ Ë ¯

Ë ¯

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EE699 S. Gedney

Gedney, EE699 - FDTD Method for Electromagnetics 3-32

EXAMPLE 3 Trapezoidal Pulse (trise = tfall = 15 ps, tdur = 150 ps)

C = 100.0 pf/m, L = .250 µH/m (Zo = 50 Ω), d = 0.1 m RL = 10 Ω, LL = 10 nf, Rg = 50 Ω, .001 , / 5x m t x v ps∆ = ∆ = ∆ =

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.02 0.04 0.06 0.08 0.1

t=0.5 nst = 0.5 ns (L = 0)

Volts

x (m)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.02 0.04 0.06 0.08 0.1

t=0.625 nst = 0.625 ns (L = 0)

Volts

x (m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.02 0.04 0.06 0.08 0.1

t = 0.75 pst = 0.75 ns (L = 0)

Volts

x (m)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.02 0.04 0.06 0.08 0.1

t = 1 nsL = 1 ns (L = 0)

Volts

x (m)

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EE699 S. Gedney

Gedney, EE699 - FDTD Method for Electromagnetics 3-33

Example of a Lossy Transmission Line Rg

v(t)

z = 0 z = d

RL CL

C = 100.0 pf/m, L = .250 µH/m, d = 0.5 m G = 0.01 s/m, R = 10 Ω/m (Zo, vp are dispersive)

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

0.275 ns1.25 ns2.5 ns

Volts

x (m)

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EE699 S. Gedney

Gedney, EE699 - FDTD Method for Electromagnetics 3-34

NON-UNIFORM TRANSMISSION LINES

Rg

v(t)

z = 0 z = dz = d1

L1, C1 L2, C2 ZL

Line 1: Z1 = L1C1

, v1 = 1L1C1

Line 2: Z2 = L2C2

, v2 = 1L2C2

Impact: a. Lines have different impedances b. Lines have different phase velocities (affects ∆t) c. Differential Operators invalid at interface Assume L1 = L2:

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EE699 S. Gedney

Gedney, EE699 - FDTD Method for Electromagnetics 3-35

L1, C1 L1, C2

V I Vi-1 I

∆x

i-1i-2i-2 Vi Ii IVi+1 i+1

At node Vi:

∂I(x, t)∂x

= −C∂V(x, t)

∂t

V is continuous, however, C is not! Therefore, ∂I(x, t)

∂x not defined at interface

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EE699 S. Gedney

Gedney, EE699 - FDTD Method for Electromagnetics 3-36

Split into two regions and approximate:

L1, C1 L1, C2

Vi-1 I

∆ x

i-1 Vi Ii Vi+1

Ii'+Ii'

/2

Ii'

− − Ii−1

∆x / 2

n

= −C1Vi

n+1/2 − Vin−1/2

∆t;

Ii − Ii'+

∆x / 2

n

= −C2Vi

n+1/2 − Vin−1/ 2

∆t

Adding the two and enforcing current continuity:

Iin − Ii−1

n

∆x= −

C1 + C2

2⎛ ⎝ ⎜ ⎞

⎠ ⎟ Vi

n+1/2 − Vin−1/ 2

∆t

Average Value of the Capacitance per unit length. Similarly, for discontinuity in the line inductance:

Vin+1/ 2 −Vi−1

n+1/2

∆x= −

L1 + L2

2⎛ ⎝ ⎜ ⎞

⎠ ⎟ Ii

n+1 − Iin

∆t

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EE699 S. Gedney

Gedney, EE699 - FDTD Method for Electromagnetics 3-37

An alternate derivation is presented via a distributed circuit model:

iV1iI -

1

2C x∆

1L x∆1iV -

1

2C x∆

1iV +iI

2

2C x∆

1L x∆iV

2

2C x∆

The interface is represented by the interface of the cascaded pi-networks. Applying KCL at the interface:

1 21 2 2

nn nii i

VC x C xI It

∆ ∆-

∂Ê ˆ= + +Á ˜Ë ¯ ∂

Applying a central difference approximation for the time derivative: Ii

n − Ii−1n

∆x= −

C1 + C2

2⎛ ⎝ ⎜ ⎞

⎠ ⎟ Vi

n+1/2 − Vin−1/ 2

∆t Which is identical to that derived previously. Note that this is easily extendable to lossy lines. Conclusion: FDTD approximation of a step discontinuity is approximated by a linear approximation. Also, discontinuity in C and L are separated by ½ of a space cell.

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-1

FDTD Solution of Maxwell's Equations Faraday's Law:

imp

impS C S

B E Mt

B dS E d M dSt

∂∂∂∂

= −∇× −

⋅ = − ⋅ − ⋅∫∫ ∫ ∫∫

impE Jσ− − Ampere's Law:

imp

impS C S S

D H E Jt

D dS H d E dS J dSt

∂ σ∂∂ σ∂

= ∇× − −

⋅ = ⋅ − ⋅ − ⋅∫∫ ∫ ∫∫ ∫∫

Gauss' Laws:

0

0S V S

D B

D dS dV B dS

ρ

ρ

∇ ⋅ = ∇ ⋅ =

⋅ = ⋅ =∫∫ ∫∫∫ ∫∫

Constitutive Relations: , ,D E B H J Eε µ σ= = =

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-2

Maxwell's Curl Equations From Faraday's and Ampere's Laws:

y yx xz zx

y yx xz zy

y yx xz zz

E HB DE H Jt y z t y z

B DE HE H Jt z x t z x

E HE HB D Jt x y t x y

∂ ∂∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂

∂ ∂∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂

− = − = − −

− = − = − −

− = − = − −

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-3

2-Dimensional Form of Maxwell's Equations: 1. TMz (Hz = 0, Ez = Ez(x,y)).

yx xz zz

y z

HB HE D Jt y t x y

B Et x

∂∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂

∂ ∂∂ ∂

− = = − −

=

2. TEz (Ez = 0, Hz = Hz(x,y)).

yx xz zx

y zy

ED EH BJt y t x y

D H Jt x

∂∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂∂ ∂

= − − = −

= − −

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-4

1-Dimensional Form of Maxwell's Equations:

1. TMz (Hz = 0, Ez = Ez(y)).

or,1 1 1

x xz zz

x xz zz

B HE D Jt y t y

H HE E Jt y t y

∂ ∂∂ ∂∂ ∂ ∂ ∂

∂ ∂∂ ∂∂ µ ∂ ∂ ε ∂ ε

− = = −

= − = − −

2. TEz (Ez = 0, Hz = Hz(y)).

or,1 1 1

x xz zx

x xz zx

D EH BJt y t y

E EH HJt y t y

∂ ∂∂ ∂∂ ∂ ∂ ∂

∂ ∂∂ ∂∂ ε ∂ ε ∂ µ ∂

= − − = −

= − =

Can Model 1-Dimensional Maxwell's Equations in a manner identical to the 1-Dimensional Transmission Line Equations.

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-5

The Yee Algorithm in 2D, TMz Case Maxwell’s Equations TMz (Hz = 0, Ez = Ez(x,y)).

yx xz zz

y z

HB HE D Jt y t x y

B Et x

∂∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂∂ ∂

− = = − −

− = −

Discrete Space

Let

, ,

, ,

, and

( , )

i j

nn

i j i j n

x i x y j y

t n t

f f x y t

= ∆ = ∆

= ∆

=

Use Central Differences

( ) ( )1 1 1 12 2 2 2, , , ,2 2( , , ) ( , , ),

n n n ni j i j i j i jf f f ff x y t f x y tO x O y

x x y y∂ ∂

∂ ∂+ − + −− −

= + ∆ = + ∆∆ ∆

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-6

Staggered Grid:

Hy i,

j +1 2

Ezi+1

2, j+12

Hxi+1

2, j

i

j

zE is located at center of grid cells xH is located at the center of horizontal edges yH is located at the center of vertical edges

x z

y z

y xzz

H Et y

H Et x

H HE Jt x y

∂ ∂µ∂ ∂∂ ∂µ∂ ∂

∂ ∂∂ε∂ ∂ ∂

− =

− = −

= − −

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2-Dimensional FDTD Analysis 4-7

Discrete coupled 1st-order equations for linear, homogeneous, isotropic medium:

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

1/ 2 1/ 2i j i j i j i j

n n n nx x z zH H E E

t yµ + + + + + −

+ −− −− =

∆ ∆

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

1/ 2 1/ 2i j i j i j i j

n n n ny y z zH H E E

t xµ + + + + − +

+ −− −− = −

∆ ∆

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

1 1/ 2 1/ 2i j i j i j i j

n n n nz z y yE E H H

t xε + + + + + + +

+ + +− −=

∆ ∆

( )1/ 2, 1/ 2 1/ 2, 1/ 21/ 2, 1 1/ 2,

11/ 2 1/ 2

2i j i ji j i j

n nn nz zx x E EH H

yσ + + + ++ + +

++ + +−− −

Constitutive relations:

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

1 1i j i j

n nz zD Eε+ + + +

+ += , 1/ 2, 1/ 2,

1/ 2 1/ 2i j i j

n nx xB Hµ+ +

+ += , 1/ 2 , 1/ 2

1/ 2 1/ 2i j i j

n ny yB Hµ

+ +

+ +=

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-8

Fully Explicit Time-Marching Solution of the Discretized Maxwell’s Equations in 2-D (TMz)

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

1/ 2, 1/ 2,

1/ 2 1/ 2 i j i j

i j i j

n nz zn n

x x

E EtH Hyµ

+ + + −

+ +

+ −−∆

= −∆

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

, 1/ 2 , 1/ 2

1/ 2 1/ 2 i j i j

i j i j

n nz zn n

y y

E EtH Hxµ

+ + − +

+ +

+ −−∆

= +∆

1, 1/ 2 , 1/ 2

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

1/ 2 1/ 21 1 1

1 1i j i j

i j i j

n ny yn n

z z

H HtE Ex

ξξ ξ ε

+ + +

+ + + +

+ ++

−− ∆= +

+ + ∆

1/ 2, 1 1/ 2,

1/ 2 1/ 21

1i j i j

n nx xH Ht

yξ ε+ + +

+ +−∆−

+ ∆

where

2tσξε∆

=

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-9

Stability for the 2-D TMz Finite Difference Equations • Stability ensures that the energy is bounded within our system. That is, for all n:

21

2 1n

n

f

f

+

≤ .

where here f represents the electric and magnetic fields. Solving the Stability Problem:

Again, we will derive the relationships between ( ),x y∆ ∆ and t∆ that is sufficient to ensure that the energy is bounded.

This will be done in combination of a Von-Neumann analysis and an

eigenvalues problem.

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-10

Von Neumann Stability Analysis

• Due to the regularity of the grid, a Von Neumann stability analysis can be used. To this end, the grid is assumed infinite in extent, and regularly spaced.

• As done for the dispersion analysis, assume a TMz Plane Wave Solution in the discrete space:

( )

( )

( )

, 0

, 0

, 0

x y

i j

x y

i j

x y

i j

j k i x k j yn nx x

j k i x k j yn ny y

j k i x k j yn nz z

H H e

H H e

E E e

∆ + ∆

∆ + ∆

∆ + ∆

=

=

=

• The plane wave expansion is then inserted into the update equations on page 4-7. Note, that we will coincide the electric and magnetic fields in time.

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-11

• Looking first at the Hx update:

( )( ) ( )( )

( ) ( )( ) ( ) ( )( )

1 11 12 22 2

0 0

1 1 1 12 2 2 2

0

x y x y

x y x y

j k i x k j y j k i x k j yn nx x

j k i x k j y j k i x k j ynz

H e H e

t E e eyµ

+ ∆ + ∆ + ∆ + ∆+ −

+ ∆ + + ∆ + ∆ + − ∆

= −

∆ ⎛ ⎞−⎜ ⎟⎝ ⎠∆

• Canceling exponentials, and applying Euler’s law: 1 12 2

0 0 02 sin

2n n ynx x z

k ytH H j Eyµ

+ − ⎛ ⎞∆∆= − ⎜ ⎟⎜ ⎟∆ ⎝ ⎠

• This is repeated for the other updates, leading to:

( )

1 1 1 12 2 2 2

0 0 0 0 0 0

1 12 2

0 0 0 0

2 , 2

2

n n n nn nx x z y y z

n n n nz z y x

t tH H j YE H H j XE

tE E j XH YH

µ µ

ε

+ − + −

+ −

∆ ∆= − = +

∆= + −

where

sin sin2 2,

y xk y k x

Y Xy x

⎛ ⎞∆ ⎛ ⎞∆⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠= =∆ ∆

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-12

• These equations can be combined in a linear form as:

[ ]1 12 22n nnx j A x I x+ −= − + (ii)

• where

[ ]

12

0

1 12 2

0

12

0

0 0

, 0 0

0

nx

n ny

nz

YH

Xx H A c t

E Y X

η

ηη η

+

+ +

+

⎡ ⎤⎢ ⎥⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= = ∆ −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ −⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦

and

µηε

=

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-13

• Equation (ii) can be reduced to a first order equation by introducing the vectors 121

212

,nn

n nnn

xxx x

++

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= =y y

and then expressing it as 12

2 [ ] [ ][ ] ; [ ][ ] 0

n n j A IM MI

+ ⎡ ⎤⎢ ⎥⎣ ⎦

−= =y y .

• Stability of the first-order equation requires ρ([M]) < 1, where ρ([M]) is the spectral radius of M. Following the arguments from Set 2 of the notes, it can be shown that if if ρ([A]) < 1, then it will follow that ρ([M]) < 1.

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-14

[ ]

0 0

0 0

0

Y

XA c t

Y X

η

ηη η

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= ∆ −⎢ ⎥⎢ ⎥

−⎢ ⎥⎢ ⎥⎣ ⎦

• The eigenvalues of [A] can be determined analytically as:

2 2 2 20, ,A c t X Y c t X Yλ = ∆ + − ∆ + • We are interested in the extremum of Aλ . Thus, the maximum value of Aλ for all

possible values of and x yk k is

2 2max

1 1A c t

x yλ = ∆ +

∆ ∆

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-15

Stability of the Explicit 2-Dimensional Formulation

• If ρ([A]) < 1, thenmax

1Aλ < . Hence, this leads to the stability limit for the two-dimensional explicit update equations:

2 2

11 1

tc

x y

∆ <+

∆ ∆

• Note, that under the conditions x y∆ = ∆ = ∆ , this is expressed as

2t

c∆

∆ <

• It is noted that the stability limit excludes the limiting value / 2t c∆ = ∆ . This is because this limit is the weak stability limit. In fact, due to numerical rounding error, at the weak stability limit, the simulation may go unstable.

• It will be interesting to observe in the following studies that the dispersive error of the discrete approximation is actually minimized by choosing the time step as close as possible to this limit. Thus, typically one would choose the time step to be at least 99 % of the stability limit.

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-16

Study of TMz Wave Propagation in an Unbounded Medium Time-Harmonic Plane Wave Analysis – Continuous Space Let

( )( , , ) x yj t k x k yf x y t e ω − −=

From the wave equation (2 2 2

22 2 2

f f fct x y

∂ ∂ ∂∂ ∂ ∂

⎛ ⎞= +⎜ ⎟

⎝ ⎠)

( )( ) ( )2 2 2 2x y x yj t k x k y j t k x k yx ye c k k eω ωω − − − −− = − +

This leads to the dispersion relationship: 2

2 2 22x yk k k

+ = =

c = phase velocity of the wave = vp, where /pv kω=

Group velocity (velocity of energy transport):

1/gkv c

k∂ω ∂∂ ∂ω

= = =

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-17

Numerical Dispersion for Maxwell’s Equations in 2D • Assume a TMz Plane Wave Solution for the discrete difference equations:

( )

( )

( )

, 0

, 0

, 0

x y

i j

x y

i j

x y

i j

j k i x k j y n tnz z

j k i x k j y n tnx x

j k i x k j y n tny y

E E e

H H e

H H e

ω

ω

ω

∆ + ∆ − ∆

∆ + ∆ − ∆

∆ + ∆ − ∆

=

=

=

where, ω is a constant, and we have defined xk and yk as the numerical wave-numbers that will be solved for. Plugging the discrete form of the plane wave solution into the update expression for Hx:

0

0sin sin

2 2z yo

x

E k ytHt y

µ ω ⎛ ⎞∆∆⎛ ⎞ = ⎜ ⎟⎜ ⎟ ⎜ ⎟∆ ∆⎝ ⎠ ⎝ ⎠

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-18

Repeat for Hy and Ez gives:

0

0

0

1sin 0 sin2 2

10 sin sin 02 2

1 1sin sin sin2 2 2

yo

x

o xy

z

y x o

k ytt y

Hk xt H

t xE

k y k x ty x t

µ ω

µ ω

ε ω

⎡ ⎤⎛ ⎞∆∆⎛ ⎞ −⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟∆ ∆⎝ ⎠⎢ ⎥⎝ ⎠ ⎡ ⎤⎢ ⎥⎢ ⎥⎛ ⎞∆∆⎛ ⎞⎢ ⎥ =⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥∆ ∆⎝ ⎠ ⎝ ⎠ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎛ ⎞∆ ⎛ ⎞∆ ∆⎛ ⎞−⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟∆ ∆ ∆ ⎝ ⎠⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-19

Dispersion Relationship of the Discrete Space:

2 2 22 2 2 2

1 1 1sin sin sin 02 2 2

yx k yk xtc t x y

ω ⎛ ⎞∆⎛ ⎞∆∆⎛ ⎞ − − =⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟∆ ∆ ∆⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Numerical Phase Velocity: 1. Assume: 0, 0t x∆ → ∆ → . Dispersion Relationship becomes

22 2

2 0x yk kcω

− − =

2. Magic Time Step?

Try c t x y∆ = ∆ = ∆ for plane wave propagation at an angle α with respect to the x axis:

2 22 2 2 2

22 2

1 1 cossin sin2 2

1 sinsin 02

t tc t c t

tc t

ω ω α

ω α

∆ ∆⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟∆ ∆⎝ ⎠ ⎝ ⎠∆⎛ ⎞− ≠⎜ ⎟∆ ⎝ ⎠

3. Numerical Dispersion: In general, one should choose( ),20

x y λ∆ ∆ ≤ .

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-20

CHARACTERISTICS OF NUMERICAL DISPERSION OF 2-DIM. GRID DISPERSION RELATIONSHIP IN DISCRETE SPACE:

2 2 22 2 2 2

1 1 1sin sin sin 02 2 2

yx k yk xtc t x y

ω ⎛ ⎞∆⎛ ⎞∆∆⎛ ⎞ − − =⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟∆ ∆ ∆⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Let,

cos( ), sin( )x yk k k kθ θ= = Then,

2 2 22 2 2 2

1 1 cos( ) 1 sin( )sin sin sin2 2 2

t k x k yc t x y

ω θ θ⎛ ⎞ ⎛ ⎞∆ ∆ ∆⎛ ⎞ = +⎜ ⎟ ⎜ ⎟⎜ ⎟∆ ∆ ∆⎝ ⎠ ⎝ ⎠ ⎝ ⎠

k is solved with a non-linear solver. Apply Newton's method 1 ( ) / ( )i i i ik k f k f k+ ′= −

( ) ( )

2 2 22 2 2 2

1

cos( ) sin( )1 1 1sin sin sin2 2 2

cos( ) sin( )sin cos( ) sin sin( )2 2

i i

i i

i i

k x k y tx y c t

k kk x k y

x y

θ θ ω

θ θθ θ+

⎛ ⎞ ⎛ ⎞∆ ∆ ∆⎛ ⎞+ −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∆ ∆ ∆ ⎝ ⎠⎝ ⎠ ⎝ ⎠= −∆ + ∆

∆ ∆

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2-Dimensional FDTD Analysis 4-21

Numerical Dispersion - Examples

• Numerical Dispersion: Numerical Phase Velocity: ∆t = 0.9 C.L.

0.95

0.96

0.97

0.98

0.99

1

0 15 30 45 60 75 90

∆x = ∆y = 0.2 λ

∆x = ∆y = 0.1 λ

∆x = ∆y = 0.05 λ

v p/c

o

θ • vp < co

• Anisotropic Behavior

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-22

Numerical Phase Velocity: ∆t = 0.1 C.L.

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 15 30 45 60 75 90

∆x = ∆y = 0.2 λ

∆x = ∆y = 0.1 λ

∆x = ∆y = 0.05 λv p

/co

θ

Note that the error increases as the time step decreases! We find that the dispersion error is minimized at the stability limit.

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-23

Numerical Phase Velocity: ∆t = 0.9 C.L., dx ≠ dy

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 15 30 45 60 75 90

∆x = 0.2 λ ∆y = 0.1 λ∆x = 0.1 λ ∆y = 0.05 λ∆x = 0.1 λ ∆y = 0.025 λ∆x = 0.1 λ ∆y = 0.01 λ

v p/c

o

θ

• vp < co

• Anisotropic Behavior • Accuracy of wave propagating along y-direction increases. • Accuracy of wave propagating along the x-direction actually decreases!

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-24

Numerical Implementation (TMz - Polarization)

In Discrete Homogeneous Space:

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

1/ 2, 1/ 2,

1/ 2 1/ 2 i j i j

i j i j

n nz zn n

x xo

E EtH Hyµ

+ + + −

+ +

+ −−∆

= −∆

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

, 1/ 2 , 1/ 2

1/ 2 1/ 2 i j i j

i j i j

n nz zn n

y yo

E EtH Hxµ

+ + − +

+ +

+ −−∆

= +∆

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

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

1/ 2 1/ 2 1/ 2 1/ 21 i j i j i j i j

i j i j

n n n ny y x xn n

z zo o

H H H Ht tE Ex yε ε

+ + + + + +

+ + + +

+ + + ++

− −∆ ∆= + −

∆ ∆

Hy i,

j +1 2

Ezi+1

2, j+12

Hxi+1

2, j

i

j

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-25

High-Level Language Implementation (FORTRAN) real hx(maxi,maxj), hy(maxi,maxj) real ez(maxi,maxj) co = 2.998e8 dtoco = 0.95/(sqrt(1./dx**2+1./dy**2) eta = 120*Pi c1 = dtoco/eta/dy c2 = dtoco/eta/dx c3 = dtoco*eta/dx c4 = dtoco*eta/dy do it = 1,maxits do 10 j = 1,ny do 10 i = 1,nx-1 hx(i,j) = hx(i,j)-c1*(ez(i,j)-ez(i,j-1)) 10 continue do 20 j = 1,ny-1 do 20 i = 1,nx hy(i,j) = hy(i,j)+c2*(ez(i,j)-ez(i-1,j)) 20 continue do 30 j = 1,ny-1 do 30 i = 1,nx-1 ez(i,j) = ez(i,j)+c3*(hy(i+1,j)-hy(i,j))-c4*(hx(i,j+1)-hx(i,j)) 30 continue enddo

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2-Dimensional FDTD Analysis 4-26

Dirichlet and Neumann Boundary Conditions • Are the Explicit Update Operators Well-Posed as Stated on the Exterior Boundaries? (i.e., i = 1; i = nx; j = 1; j = ny). • PMC Boundary:

Ht = 0, and 1 0tEn∂

µ ∂=

Dirchlet boundary condition: hx(i,1) = 0, hx(i,ny) = 0 hy(1,j) = 0, hy(nx,j) = 0

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2-Dimensional FDTD Analysis 4-27

• PEC Boundary

Ez = 0, and 1 0tHn∂

ε ∂= , ˆ 0n H⋅ =

Cannot Strictly Enforce as Neumann Boundary Condition on Ht Use Image Theory: Let i = 1 be a PEC wall, then ez(0,j) = - ez(1,j). Therefore, update becomes: do 25 j = 1,ny-1 hy(1,j) = hy(1,j)+2*c2*ez(1,j) 25 continue • Exterior Radiation Conditions: -Will Discuss in Set 7 of notes

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-28

1 m

1 m

Source

PEC

Statistics: ∆x = 0.01 m, ∆y = 0.01 m, nx = 101, ny = 101, ∆t = 2.1213203E-11 10,000 time iterations

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-29

Source (Soft Source): • Gaussian Profile

2( )2

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

n t totw

i j i j

n nz zE E y

∆ −

+ + + +

⎛ ⎞−⎜ ⎟⎝ ⎠= + ∆

-60

-40

-20

0

20

40

60

0 1 10-8 2 10-8 3 10-8 4 10-8 5 10-8 6 10-8 7 10-8

Time-Dependent Signal in PEC Cavity Excited by TM

z Source

Ez(15,15)

V/m

t (s)

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-30

Fourier transform the time-dependent data using a FFT:

0

2 104

4 104

6 104

8 104

1 105

1.2 105

1.4 105

1.6 105

0 0.2 0.4 0.6 0.8 1

Fourier Spectrum of Sampled Electric Field inPEC Bound Cavity Excited by TM

z Wave

TMz (TE

x) mode

V/m

f (GHz)

TE11

TE21

TE3,1

TE32

Cavity Mode Approx. (FDTD) Exact

TM11 212.2 MHz 212.1 MHz TM21 334.7 MHz 335.4 MHz TM3,1 474.2 MHz 474.3 MHz TM3,2 541.2 MHz 540.8 MHz

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-31

0

2 104

4 104

6 104

8 104

1 105

1.2 105

1.4 105

0 0.2 0.4 0.6 0.8 1

Fourier Spectrum of Sampled Electric Field inPMC Bound Cavity Excited by TM

z Wave

TMz (PMC Walls)

V/m

f (GHz)

TM10

TM20

TM30

TM40

TM50

Cavity Mode Approx. (FDTD) Exact

TM10 150.8 MHz 150.0 MHz TM20 300.5 MHz 300.0 MHz TM30 450.2 MHz 450.0 MHz TM40 599.5 MHz 600.0 MHz TM50 749.1 MHz 750.0 MHz

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-32

TEz POLARIZATION - DISCRETE SOLUTION

yx xz z

y z

ED EH Bt y t x y

D Ht x

∂∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂∂ ∂

= − = −

= −

i

j

Hzi+1/2, j+1/2n+1

Exi+1/2, jn+1/2

E yi,

j+1 /

2n+

1/2

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-33

i

j

Hzi+1/2, j+1/2n+1

Exi+1/2, jn+1/2

E yi,

j+1 /

2n+

1/2

In Discrete Homogeneous Space:

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

1/ 2, 1/ 2,

1/ 2 1/ 2 i j i j

i j i j

n nz zn n

x xo

H HtE Eyε

+ + + −

+ +

+ −−∆

= +∆

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

, 1/ 2 , 1/ 2

1/ 2 1/ 2 i j i j

i j i j

n nz zn n

y yo

H HtE Exε

+ + − +

+ +

+ −−∆

= −∆

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

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

1/ 2 1/ 2 1/ 2 1/ 21 i j i j i j i j

i j i j

n n n ny y x xn n

z zo o

E E E Et tH Hx yµ µ

+ + + + + +

+ + + +

+ + + ++

− −∆ ∆= − +

∆ ∆

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-34

High-Level Language Implementation (FORTRAN) real ex(maxi,maxj), ey(maxi,maxj) real hz(maxi,maxj) co = 2.998e8 dtoco = 0.98/(sqrt(1./dx**2+1./dy**2) eta = 120*Pi c1 = dtoco*eta/dy c2 = dtoco*eta/dx c3 = dtoco/eta/dx c4 = dtoco/eta/dy do it = 1,maxits do 10 j = 1,ny do 10 i = 1,nx-1 ex(i,j) = ex(i,j)+c1*(hz(i,j)-hz(i,j-1)) 10 continue do 20 j = 1,ny-1 do 20 i = 1,nx ey(i,j) = ey(i,j)-c2*(hz(i,j)-hz(i-1,j)) 20 continue do 30 j = 1,ny-1 do 30 i = 1,nx-1 hz(i,j) = hz(i,j)-c3*(ey(i+1,j)-ey(i,j))+c4*(ex(i,j+1)-ex(i,j)) 30 continue enddo

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-35

Dirichlet and Neumann Boundary Conditions for TEz Polarization

• The Explicit Update Operators are Ill-Posed as Stated on the Exterior Boundaries (i.e., i = 1; i = nx; j = 1; j = ny). • PEC Boundary:

Et = 0, and 1 0tHn∂

ε ∂=

Dirchlet boundary condition: ex(i,1) = 0, ex(i,ny) = 0 ey(1,j) = 0, ey(nx,j) = 0

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-36

• PMC Boundary

Hz = 0, and 1 0tEn∂

µ ∂=

Cannot Strictly Enforce as Neumann Boundary Condition Use Image Theory: Let i = 1 be a PMC wall, then hz(-1,j) = - hz(1,j). Therefore, update for PMC boundary becomes: do 25 j = 1,ny-1 ey(1,j) = ey(1,j)-2*c2*hz(1,j) 25 continue

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-37

1 m

1 m

Source

PEC

Statistics: ∆x = 0.01 m, ∆y = 0.01 m, nx = 101, ny = 101, ∆t = 2.1213203E-11 10,000 time iterations Source (Soft Source): • Gaussian Profile

2( )2

, 1/ 2 , 1/ 2exp ; 1, 1

n t totw

i j i j

n ny yE E y j ny

∆ −

+ +

⎛ ⎞−⎜ ⎟⎝ ⎠= + ∆ = −

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-38

0

200

400

600

800

1000

0 0.2 0.4 0.6 0.8 1

Fourier Spectrum of Sampled Electric Field inPEC Bound Cavity Excited by TE

z Wave

TEz (PEC Boundaries)

V/m

f (GHz)

TE10

TE20

TE30 TE

40

TE50

Parallel Plate Mode

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EE699 S. Gedney

2-Dimensional FDTD Analysis 4-39

Cavity Mode Approx. (FDTD) Exact

TE10 150.0 MHz 150.0 MHz TE20 299.92 MHz 300.0 MHz TE30 449.97 MHz 450.0 MHz TE40 599.94 MHz 600.0 MHz TE50 749.53 MHz 750.0 MHz TE60 899.17 MHz 900 MHz

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-1

THREE-DIMENSIONAL FDTD SOLUTION OF MAXWELL'S EQUATIONS - THE YEE ALGORITHM

• Maxwell's Curl Equations in Cartesian Coordinates:

y yx xz zx

y yx xz zy

y yx xz zz

E HB DE H Jt y z t y z

B DE HE H Jt z x t z x

E HE HB D Jt x y t x y

∂ ∂∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂

∂ ∂∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂

∂ ∂∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂

− = − = − −

− = − = − −

− = − = − −

• Mapping the Curl Equations into the Discrete Space

Define a regular orthogonal grid. A function f(x,y,z,t) evaluated at the vertices of the grid is defined in the discrete coordinate space as

f(x,y,z,t) = f(i∆x,j∆y,k∆z,n∆t) = , ,

ni j kf

⇒Central Difference Approximation of Derivatives:

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-2

yx z EB E

t y z∂∂ ∂

∂ ∂ ∂− = − ⇒

1 12 2

1 1 1 1 1 1, , , , , , 1 , ,2 2 2 2 2 2

1 1, 1, , ,2 2

1 1

1

i j k i j k i j k i j k

i j k i j k

n n n nx x y y

n nz z

B B E Et z

E Ey

+ + + + + + +

+ + +

+ −⎡ ⎤ ⎡ ⎤− = − −⎢ ⎥ ⎢ ⎥∆ ∆⎣ ⎦ ⎣ ⎦⎡ ⎤−⎢ ⎥∆ ⎣ ⎦

yx zx

HD H Jt y z

∂∂ ∂∂ ∂ ∂

= − − ⇒

1 12 2

1 1 1 1 1 1, , , , , , , ,2 2 2 2 2 2

1 1 12 2 2

1 1 1 1 1, , , , , ,2 2 2 2 2

11 1

1

i j k i j k i j k i j k

i j k i j k i j k

n nn nx x z z

n n ny y x

D D H Ht y

H H Jz

+ + + + + −

+ + + − +

− −−

− − −

⎡ ⎤ ⎡ ⎤− = − −⎢ ⎥ ⎢ ⎥∆ ∆⎣ ⎦ ⎣ ⎦

⎡ ⎤− −⎢ ⎥∆ ⎣ ⎦

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-3

YEE-CELL (PRIMARY LATTICE CELL)

(i,j,k) Ey

Ey

Ex

Ex Ez

Ez

Hz

Hz

Hx

Hx

Hy Hy

Ex

Ex

Ey

Ey

Ez Ez

x

yz

• Assumptions: 1) Discrete Vector Electric Fields are parallel to and are constant along the edges of the

Primary Lattice. 2) Discrete Vector Magnetic Fields are normal to and constant throughout each face of

the Primary Lattice. 3) Discrete Vector Magnetic Fields are parallel to and are constant along the edges of the

Secondary Lattice. 4) Discrete Vector Electric Fields are normal to and constant throughout each face of the

Secondary Lattice.

• Secondary Lattice edges connect the cell centers of the Primary Lattice.

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-4

DERIVATION OF YEE-ALGORITHM FROM MAXWELL'S EQUATIONS IN INTEGRAL FORM

Hxm ,n+ 1

2, p+ 12

l− 12

Eym ,n+ 1

2, p

l

Eym ,n+ 1

2, p+1

l

Ezm ,n, p+ 1

2

l Ezm ,n+1, p+ 1

2

l

∆y

∆z

Let ˆdS xdydz= . Then, within the discrete space:

1 12 2

1 1 1 1 1 1, , , , , , , 1,2 2 2 2 2 2

1 1, , 1 , ,2 2

i j k i j k i j k i j k

i j k i j k

n n n nx x y z

n ny z

z y B B E y E zt

E y E z

+ + + + + + +

+ + +

+ −∆ ∆ ⎡ ⎤ ⎡− = − ∆ + ∆⎢ ⎥ ⎢∆ ⎣ ⎦ ⎣⎤− ∆ − ∆ ⎥⎦

S C

B dS E dt

∂∂

⋅ = − ⋅∫∫ ∫

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-5

1 12 2

1 1 1 1 1 1, , , , , , , 1,2 2 2 2 2 2

1 1, , 1 , ,2 2

i j k i j k i j k i j k

i j k i j k

n n n nx x y z

n ny z

z y B B E y E zt

E y E z

+ + + + + + +

+ + +

+ −∆ ∆ ⎡ ⎤ ⎡− = − ∆ + ∆⎢ ⎥ ⎢∆ ⎣ ⎦ ⎣⎤− ∆ − ∆ ⎥⎦

Multiply this expression by 1/∆z∆y leads to

1 12 2

1 1 1 1 1 1, , , , , , 1 , ,2 2 2 2 2 2

1 1, 1, , ,2 2

i j k i j k i j k i j k

i j k i j k

n n n nx x y y

n nz z

tB B E Ez

t E Ey

+ + + + + + +

+ + +

+ − ∆ ⎡ ⎤= + − −⎢ ⎥∆ ⎣ ⎦∆ ⎡ ⎤−⎢ ⎥∆ ⎣ ⎦

Similarly, from a Secondary Lattice Face:

1 12 2

1 1 1 1 1 1, , , , , , , ,2 2 2 2 2 2

1 1 12 2 2

1 1 1 1 1, , , , , ,2 2 2 2 2

1

i j k i j k i j k i j k

i j k i j k i j k

n nn nx x z z

n n ny y x

tD D H Hy

t H H Jz

+ + + + + −

+ + + − +

− −−

− − −

∆ ⎡ ⎤= + − −⎢ ⎥∆ ⎣ ⎦∆ ⎡ ⎤− −⎢ ⎥∆ ⎣ ⎦

Similar approximations for the remaining components of B and D are also derived.

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-6

STABILITY • The study of the one-dimensional problem demonstrated that the explicit update scheme was

conditionally stable. The stability of the three-dimensional scheme is also conditionally stable. Stability is again analyzed here using a Von Neumann scheme. Specifically, all the fields will be expanded into a Fourier Plane Wave Spectrum, which can be thought of as a plane wave expansion, e.g. x y zjk x jk y jk z

oE E e− − −= . • Following the analogy of the 1D problem, The 6 update equations are written as a coupled

2nd order difference operator: [ ] 1

21 2 nn nx j A x x++ = +

nxnyn

n znxnynz

EEE

xHHH

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

[ ]

0 0 0 00 0 0 00 0 0 00 / / 0 0 0/ 0 / 0 0 0/ / 0 0 0 0

z y

z x

y x

z y

z x

y x

S SS S

S SA

S SS SS S

η ηη η

η ηη η

η ηη η

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥−

= ⎢ ⎥−⎢ ⎥⎢ ⎥−⎢ ⎥−⎢ ⎥⎣ ⎦

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-7

where:

( ) ( ) ( )22 2sinsin sin, ,

yx zk yk x k z

x y z

c tc t c tS S S

x y z

∆∆ ∆∆∆ ∆= = =

∆ ∆ ∆

• From the previous analysis, it is sufficient to show that a stable solution requires |λA| < 1

(strict stability). • Computing the eigenvalues using Maple™, shows there are two sets of degenerate

eigenvalues: 2 2 20,A x y zS S Sλ = ± + +

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-8

STABILITY CONTINUED • Note that stability requires |λA| < 1 for all Fourier Harmonics. Therefore, one needs to

choose the maximum eigenvalues for all kx, ky, and kz. To this end,

max( ) , max( ) , max( )x y zc t c t c tS S S

x y z∆ ∆ ∆

= = =∆ ∆ ∆

• Enforcing |λA| < 1, this leads to the strict stability limit:

2 2 2

1 11 1 1

tc

x y z

∆ <+ +

∆ ∆ ∆

• The CFL (Courant-Friedrichs-Lewy) number is expressed as:

2 2 2

1 1 1 1CFL tcx y z

= ∆ + + <∆ ∆ ∆

• Thus, the CFL number must always be less than unity for strict stability. For the sake of

numerical dispersion, one typically wants to run the simulation with as large a CFL number as possible. This will be demonstrated when studying dispersion error.

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-9

GAUSS' LAWS • For the Yee-algorithm to be well-posed, the approximate fields must also satisfy Gauss'

Laws. If the fields do not satisfy Gauss' Laws, then spurious charge will result, leading to an erroneous solution.

• It will be shown that Gauss' Laws are satisfied in a global sense, by treating each unit cell individually.

• Electric Flux: - Gauss' Law

charge free media0

S V

D ds dVρ⋅ = =∫∫ ∫∫∫

- Differentiate with respect to time:

0S

D dst

∂∂

⋅ =∫∫

- Integrate over a single Yee-cell, and superimpose faces 6 6 ?

1 10

i i

ii iS S C

D ds D ds H dt t

∂ ∂∂ ∂= =

⋅ = ⋅ = ⋅ =∑ ∑∫∫ ∫∫ ∫

- The line integrals of the magnetic fields around the contours bounding the cell faces will cancel out along each edge. Therefore, the right-hand-side is zero.

- Since this is true for all time-dependent fields in a source free media, then we can conclude for the discrete fields that Gauss' law is satisfied over each Yee-cell.

0Yee cellS

D ds−

⋅ =∫∫

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-10

NUMERICAL DISPERSION • Due to the 2nd-Order Nature of the Algorithm, the Discrete Solution is Inherently

Dispersive. A Harmonic Analysis based on the Discrete Operators can Again be Performed, as Discussed in Section 4. This Leads to the Dispersion Relationship:

2 2 22 2 2 2

22

1 1 1sin sin sin2 2 2

1 sin2

yx

z

k yk xtc t x y

k zz

ω ⎛ ⎞∆⎛ ⎞∆∆⎛ ⎞ = + ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟∆ ∆ ∆⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞∆

+ ⎜ ⎟∆ ⎝ ⎠

• Limit ∆t, ∆x, ∆y, ∆z → 0:

222 2

2 2 2 2 2

( )( )( ) ( )yx zk yk xt k zc t x y zω ∆∆∆ ∆

= + +∆ ∆ ∆ ∆

or

2 2 2 2x y zk k k k= + +

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-11

DISPERSION RELATIONSHIP FOR THE DISCRETE SPACE: Plane Wave Expansion of the field components, e.g.,

( cos sin sin sin cos )j k x k y k zz oE E e φ θ φ θ θ− + +=

Leads to the Dispersion Relationship:

2 2 2 22 2 2 2

1 1 1 1sin sin sin sin 02 2 2 2

yx zk yk x k z tx y z c t

ω⎛ ⎞∆⎛ ⎞ ⎛ ⎞∆ ∆ ∆⎛ ⎞+ + − =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟∆ ∆ ∆ ∆ ⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠

Let,

cos sin , sin sin , cosx y zk k k k k kφ θ φ θ θ= = = Then solve for k using a non-linear solver. Again, use Newton's method [Numerical Recipes]:

1( )( )

ii i

i

f kk kf k+ = −′

where the initial condition is assumed to be: 0 2 /k π λ= . This method converges in a few

iterations. • From k the numerical phase velocity is estimated ( /pv kω= ) and the phase error is

estimated as a function of φ and θ.

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-12

NUMERICAL DISPERSION 3-DIMENSIONS

00.5

11.5

22.5

33.5

4

0

30

6090

120

210

240270

300

330

Dispersion vs. CFL Number

( θ = 90 o, ∆x = ∆y = ∆z = 0.1 λ )

CFL = 0.5

CFL = 0.99

Phas

e E

rror

per

λ

(deg

rees

)

00.5

11.5

22.5

33.5

0

30

6090

120

210

240270

300

330

Dispersion vs. Cell Size( θ = 90 o, CFL = 0.99)

∆x= ∆y= ∆z=0.1 λ∆x= ∆y= ∆z=0.05 λ∆x=0.1, ∆y= ∆z=0.05 λ

Phas

e E

rror

per

λ

(deg

rees

)

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-13

NUMERICAL DISPERSION 3-DIMENSIONS

00.10.20.30.40.50.60.70.8

0

30

6090

120

210

240270

300

330

Dispersion vs. Cell Size (Zoomed In)( θ = 90 o, CFL = 0.99)

∆x= ∆y= ∆z=0.1 λ∆x= ∆y= ∆z=0.05 λ∆x=0.1, ∆y= ∆z=0.05 λ

Phas

e E

rror

per

λ

(deg

rees

)

• The Numerical Grid is a Dispersive and Anisotropic Medium. • Cell sizes should be on the order of λ/15 (~1o error/λ) to λ/20 (~0.5o error/λ ). • The CFL # should be as large as possible (0.999). • Square meshes minimize anisotropy and phase error. • This study assumes smooth boundaries. When geometric singularities are present in a

model (e.g., the edge of a microstrip line) the fields typically become singular in their near vicinity. As a result, the spatial frequency spectrum of the fields in these regions are much higher, and much finer discretization will be needed (recall the error is (3)f∝ .)

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-14

NUMERICAL DISPERSION 3-DIMENSIONS

0.988

0.99

0.992

0.994

0.996

0.998

1

0 30 60 90 120 150 180

∆x = ∆y = ∆z = 0.1 λo CFL = 0.99 φ = 0o

φ = 30o

φ = 45o

φ = 60o

φ = 90o

v p / c o

θ 0.997

0.9975

0.998

0.9985

0.999

0.9995

1

0 30 60 90 120 150 180

∆x = ∆y = ∆z = 0.05 λo CFL = 0.99 φ = 0o

φ = 45o

φ = 60o

v p / c o

θ

0.996

0.9965

0.997

0.9975

0.998

0.9985

0.999

0 30 60 90 120 150 180

∆x = ∆y = ∆z = 0.05 λo CFL = 0.5 φ = 0o

φ = 45o

φ = 60o

v p / c o

θ 0.985

0.99

0.995

1

0 30 60 90 120 150 180

∆x = ∆y = 0.1 λo ∆z = 0.05 λ

o CFL = 0.99

φ = 0o

φ = 45o

φ = 60o

φ = 90o

v p / c o

θ

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-15

Numerical Implementation

x

yz

∆x

∆z ∆y

(0,0,0) Ey

Ey

Ex

Ex Ez

Ez

Hz

Hz

Hx

Hx

Hy Hy

Ex

Ex

Ey

Ey

Ez Ez

(∆x, ∆y, ∆z)

• Data Array Storage:

12

12

12

1 12 2

1 12 2

( , , ) (( ) ,( 1) ,( 1) )

( , , ) (( 1) ,( ) ,( 1) )

( , , ) (( 1) ,( 1) ,( ) )

( , , ) (( 1) ,( ) ,( ) )

( , , ) (( ) ,( 1) ,( ) )

x x

y y

z z

x x

y y

z

E i j k E i x j y k z

E i j k E i x j y k z

E i j k E i x j y k z

H i j k H i x j y k z

H i j k H i x j y k z

H

= − ∆ − ∆ − ∆

= − ∆ − ∆ − ∆

= − ∆ − ∆ − ∆

= − ∆ − ∆ − ∆

= − ∆ − ∆ − ∆1 12 2( , , ) (( ) ,( ) ,( 1) )zi j k H i x j y k z= − ∆ − ∆ − ∆

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-16

YEE-SCHEME

∴Can Express Faraday's Law in Discrete Form as:

1 12 2

, , , , , , 1 , , , 1, , ,i j k i j k i j k i j k i j k i j k

n n n n n nx x y y z z

t tH H E E E Ez yµ µ+ +

+ − ∆ ∆⎡ ⎤ ⎡ ⎤= + − − −⎣ ⎦ ⎣ ⎦∆ ∆

1 12 2

, , , , 1, , , , , , 1 , ,i j k i j k i j k i j k i j k i j k

n n n n n ny y z z x x

t tH H E E E Ex zµ µ+ +

+ − ∆ ∆⎡ ⎤ ⎡ ⎤= + − − −⎣ ⎦ ⎣ ⎦∆ ∆ 1 12 2

, , , , , 1, , , 1, , , ,i j k i j k i j k i j k i j k i j k

n n n n n nz z x x y y

t tH H E E E Ey xµ µ+ +

+ − ∆ ∆⎡ ⎤ ⎡ ⎤= + − − −⎣ ⎦ ⎣ ⎦∆ ∆

And can Express Ampere's Law in Discrete Form as:

1 1 1 12 2 2 2

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

1i j k i j k i j k i j k i j k i j k

n n n nn nx x y y z z

t tE E H H H Hz yε ε− −

+ + + ++ ∆ ∆⎡ ⎤ ⎡ ⎤= − − + −⎣ ⎦ ⎣ ⎦∆ ∆

1 1 1 12 2 2 2

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

1i j k i j k i j k i j k i j k i j k

n n n nn ny y z z x x

t tE E H H H Hx zε ε− −

+ + + ++ ∆ ∆⎡ ⎤ ⎡ ⎤= − − + −⎣ ⎦ ⎣ ⎦∆ ∆

1 1 1 12 2 2 2

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

1i j k i j k i j k i j k i j k i j k

n n n nn nz z x x y y

t tE E H H H Hy xε ε− −

+ + + ++ ∆ ∆⎡ ⎤ ⎡ ⎤= − − + −⎣ ⎦ ⎣ ⎦∆ ∆

Updates can be implemented using triple-vector loops for each field component. The bounds of the loops will be determined by the boundary conditions at the extremities of the lattice.

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-17

Treating Inhomogeneous Media

y yx xz zx y

y xzz

H DD HH HE Et y z t z x

H HD Et x y

∂ ∂∂ ∂∂ ∂σ σ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂∂ σ∂ ∂ ∂

+ = − + = −

+ = −

• Consider a Piece wise Homogeneous Medium. Let the Interface of this medium coincide

with an edge of the Primary Lattice. • The four regions surrounding the lattice are defined by the profile εi , σi, µo (i = 1,2,3,4)

ε

1,σ1 ε2 ,σ2

ε

3, σ3

ε

4 , σ4

S1 S2

S3

S4

E ⋅ ˆ n

H1

H2

H3

H4

H12

H23

H34

H41

Secondary cell face within an inhomogeneous medium.

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-18

4 4

1 1i i i

i ii iS S C

E ds E ds H dt

∂ ε σ∂= =

⎛ ⎞ ⎛ ⎞⋅ + ⋅ = ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∑ ∑∫∫ ∫∫ ∫

• Si is the surface of each sub-area that is bound by the contour Ci • Perform the integrals over each Si and the bounding Ci. For example, in region 1 leads to

1 1 1 12 2 2 2

1 11 1 1 1

1 1

1 1 12 12 41 41 4 4

ˆ ˆ2

n n n n

n n n n

E E E EndA ndAt

H H H H

ε σ+ +

+ + + +

⎛ ⎞ ⎛ ⎞− +⋅ + ⋅ =⎜ ⎟ ⎜ ⎟∆⎝ ⎠ ⎝ ⎠

⎡ ⎤⋅ − ⋅ − ⋅ + ⋅⎣ ⎦

• 1

nE is the electric field in region 1, n is the unit normal to the planar surface

• Repeat for Remaining Three Regions. Then Enforce the Continuity of the Tangential Electric Field:

1 2 3 4ˆ ˆ ˆ ˆn n n nE n E n E n E n⋅ = ⋅ = ⋅ = ⋅ • Adding the Results from the Four Regions:

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-19

1 1 1 12 2 2 2

11 1

1 2 3 4

11 1

1 2 3 4

1 1 2 2 3 3 4 4

ˆ( )

ˆ( )2

n n

n n

n n n n

E E n dAt

E E n dA

H H H H

ε ε ε ε

σ σ σ σ

+

+

+ + + +

⎛ ⎞−⋅ + + + +⎜ ⎟∆⎝ ⎠

⎛ ⎞+⋅ + + + =⎜ ⎟

⎝ ⎠⎡ ⎤⋅ + ⋅ + ⋅ + ⋅⎣ ⎦

• Since dA = A/4, this can be interpreted as an effective permittivity that is the average of the

4 surrounding regions:

1 2 3 4

1 2 3 4

( )4

( )4

ave

ave

ε ε ε εε

σ σ σ σσ

+ + +=

+ + +=

• Linear Interpolation of the Media. In the Discrete Space, the Media is not Truly Piece Wise

Constant. Rather, it is Linearly Varying across a Discontinuity • Second-Order Accurate Representation of an Inhomogeneous Interface.

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-20

Yee's Scheme in General Inhomogeneous Media

1 12 2

, , , , , , 1 , ,

, 1, , ,

, ,

, ,

i j k i j k i j k i j k

i j k i j k

n n n nx x y y

i j k

n nz z

i j k

tH H E Ez

t E Ey

µ

µ

+

+

+ − ∆ ⎡ ⎤= + −⎣ ⎦∆

∆ ⎡ ⎤− −⎣ ⎦∆

1 12 2

, , , , , , , , 1

1 12 2

, , , 1,

1

, ,

, ,

i j k i j k i j k i j k

i j k i j k

n nn nx x y y

i j k

n nz z

i j k

tE E H Hz

t H Hy

ε

ε

+ ++

+ +

∆ ⎡ ⎤= − −⎣ ⎦∆

∆ ⎡ ⎤+ −⎣ ⎦∆

• Compute average ε for each edge of the primary lattice. Then store εi,j,k • Compute average for µ for each edge of the secondary lattice and store µ i,j,k

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-21

3D-FDTD Main FDTD Routine:

program fdtd ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Three-dimensional FDTD simulator based on a regular c orthogonal grid. ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

include 'fdtd.inc' all arrays defined in common call input() Read input data call fieldinit() initialize fields to zero do n = l,maxts

call injsrce(n) source condition

call hupdate() update h-field over all space

call eupdate() update e-field over all space

call outerboundary() update e-fields on exterior boundaries (ABC, PEC/PMC) call epec() zero out e-fields on PECs call output(n) write data to file

enddo end

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-22

Include File (fdtd.inc):

parameter(maxi=400,maxj=400,maxk=400, maxs=100) real ex(maxi,maxj,maxk),ey(maxi,maxj,maxk),ez(maxi,maxj,maxk) real hx(maxi,maxj,maxk),hy(maxi,maxj,maxk),hz(maxi,maxj,maxk) integer is1(maxs),js1(maxs),ks1(maxs) integer is2(maxs),js2(maxs),ks2(maxs) common /efields/ ex,ey,ez common /hfields/ hx,hy,hz common /pec/ nsurf,is1,js1,ks2,is2,js2,ks2

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-23

H-Field Update

subroutine hupdate()

include 'fdtd. inc' C

cx = co*dt/dx cy = co*dt/dy cz = co*dt/dz

do 10 k = l,nk-l do 10 j = l,ny-l do 10 i = l,nx hx(i,j,k) = hx(i,j,k)+cz*(ey(i,j,k+l)-ey(i,j,k))

1 -cy*(ez(i,j+l,k)-ez(i,j,k)) 10 continue

do 20 k = l,nk-l do 20 j = l,ny do 20 i = l,nx-l hy(i,j,k) = hy(i,j,k)+cx*(ez(i+l,j,k)-ez(i,j,k))

1 -cz*(ex(i,j,k+l)-ex(i,j,k)) 20 continue

do 30 k = l,nk do 30 j = l,ny-l do 30 i = 1,nx-1 hz(i,j,k) = hz(i,j,k)+cy*(ex(i,j+l,k)-ex(i,j,k))

1 -cx*(ey(i+l,j,k)-ey(i,j,k)) 30 continue

return end

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-24

E-Field Update (Lossless, Inhomogeneous) subroutine eupdate()

include 'fdtd. inc'

C cx = co*dt/dx cy = co*dt/dy cz = co*dt/dz

do 10 k = 2,nk-l exclude exterior bdries at k=1 & k =nk do 10 j = 2,ny-l exclude exterior bdries at j=1 & j =nj do 10 i = l,nx-1 ex(i,j,k) = ex(i,j,k)-cz*(hy(i,j,k)-hy(i,j,k-1))

1 +cy*(hz(i,j,k)-hz(i,j-1,k))/eps(iepx(i,j,k)) 10 continue C

do 20 k = 2,nk-l do 20 j = l,ny-1 do 20 i = 2,nx-l ey(i,j,k) = ey(i,j,k)-cx*(hz(i,j,k)-hz(i-1,j,k))

1 +cz*(hx(i,j,k)-hx(i,j,k-1))/eps(iepy(i,j,k)) 20 continue C

do 30 k = l,nk-1 do 30 j = 2,ny-l do 30 i = 2,nx-1 ez(i,j,k) = ez(i,j,k)-cy*(hx(i,j,k)-hx(i,j-1,k))

1 +cx*(hy(i,j,k)-hy(i-1,j,k))/eps(iepz(i,j,k)) 30 continue

return end

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EE699 S. Gedney

3-Dimensional FDTD Analysis 5-25

More Efficient Implementation of the Explicit Update Loops • The number of floating-point operations can be further reduced by weighting the fields by

their edge length, and normalize the magnetic field by oη namely: , ,

/ , / , /x x y y z z

x x o y y o z z o

e xE e yE e zE

h xH h yH h zHη η η

= ∆ = ∆ = ∆

= ∆ = ∆ = ∆

• Secondly, define eps(i) = /( )o rc t x y zε∆ ∆ ∆ ∆ The update loop can be rewritting as:

do 10 k = 2,nk-l do 10 j = 2,ny-l do 10 i = l,nx-1 ex(i,j,k) = ex(i,j,k)-(hy(i,j,k)-hy(i,j,k-1)

1 -hz(i,j,k)+hz(i,j-1,k))*eps(iepx(i,j,k)) 10 continue

• This has reduced the number of Floating point operations from 39 N to 33 N (~20 %

speedup). Furthermore, we have eliminated a floating point division. • Further speedup can be made on vector-based and RISC processor based computers via

more advanced steps such as combining the three field update loops into a single loop, etc.†.

† S. D. Gedney, "Finite-difference time-domain analysis of microwave circuit devices on high performance vector/parallel computers,"

IEEE Trans. on MTT, vol. 43, pp. 2510-2514, Oct. 1995.

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EE699 S. Gedney

Source Modeling 6-1

SOURCE EXCITATIONS • We have discussed 1D, 2D, and 3D FDTD Algorithms - Central Difference Approximations - Discrete Explicit Update Operations - Natural Boundary Conditions (PEC, PMC) - Inhomogeneous Media • Need to Introduce Physical Source Models to Excite the Fields for Accurate Full

Wave Analysis • Choice of Source is Problem Dependent - Current Source (Jx, Jy, Jz, Mx, My, or Mz) - Voltage Source - Discrete Source Models (Thévenin or Norton) - Hard Source (eigen mode, fictitious voltage source) - Soft Source (analogous to current source) - Controlled Sources (non-linear device modeling) - Plane Wave Source

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EE699 S. Gedney

Source Modeling 6-2

CURRENT SOURCES • ELECTRIC CURRENT SOURCE - Example: 2D, TMz Polarization

∂Bx

∂t=

∂Ez

∂y∂Dz

∂t=

∂Hy

∂x−

∂Hx

∂y− Jz

∂By

∂t= ∂Ez

∂x

- Discrete Form:

−Bxi+1/2, j

n+1/ 2 − Bxi+1/2, jn−1/2

∆t=

Ezi+1/2, j+1/2n − Ezi+1/2, j−1/2

n

∆y

−Byi, j+1/2

n+1/ 2 − Byi, j+1/2n−1/2

∆t= −

Ezi+1/2, j+1/2n − Ezi−1/2, j+1/2

n

∆x

Dzi+1/2, j+1/2n+1 − Dzi+1/2, j+1/2

n

∆t=

Hyi+1, j+1/2n+1/ 2 − Hyi, j+1/2

n+1/2

∆x

−Hxi +1/ 2, j +1

n+1/ 2 − Hxi +1/2, j

n+1/ 2

∆y−

Jzi+1/2, j+1/2

n + Jzi+1/2, j+1/2

n+1( )2

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EE699 S. Gedney

Source Modeling 6-3

- Faraday's Law Remains Unchanged from Previous Analysis

- Ampere's Law leads to the explicit Update Expression:

Ezi+1/2, j+1/2

n+1 = Ezi+1/2, j+1/2

n +∆tε

Hyi +1, j +1/ 2

n+1/2 − Hyi,j +1/ 2

n−1/ 2

∆x−

Hxi +1/ 2, j +1

n+1/2 − Hxi +1/2, j

n−1/ 2

∆y

⎣ ⎢ ⎢

−Jzi+1/ 2,j +1/ 2

n+1 + Jzi+1/ 2,j +1/ 2

n

2

⎦ ⎥ ⎥

• Applications: Current Sheet Excitation, Known Current Source

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EE699 S. Gedney

Source Modeling 6-4

CURRENT SOURCES

• MAGNETIC CURRENT SOURCE - TEz Polarization (Mz) - Dual of TMz Electric Current Source (Jz) - TMz Polarization (Mx, My) • MAGNETIC CURRENT SOURCE - Example: 2D, TMz Polarization

, yx z zx y

BB E EM Mt y t x

∂∂ ∂ ∂∂ ∂ ∂ ∂

- = + = +

- Discrete Form:

Hxi+1/2, j

n+1/2 = Hxi +1/ 2, j

n−1/2 −∆tµ

Ezi+1/2, j+1/2

n − Ezi+1/2, j−1/2

n

∆y+

Mxi+1/ 2, j

n+1/2 + Mxi+1/2, j

n−1/ 2

2

⎣ ⎢ ⎢

⎦ ⎥ ⎥

Hyi, j +1/ 2

n+1/2 = Hyi, j +1/ 2

n−1/2 +∆tµ

Ezi +1/ 2, j +1/2

n − Ezi −1/2, j+1/2

n

∆x+

Myi, j +1/ 2

n+1/2 + Myi, j +1/ 2

n−1/2

2

⎣ ⎢ ⎢

⎦ ⎥ ⎥

• Observation - Magnetic Current Sources are Applied Along Secondary Grid Edges - Electric Current Sources are Applied Along Primary Grid Edges

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EE699 S. Gedney

Source Modeling 6-5

TIME SIGNATURE OF SOURCES • Initial Condition of Fields - E and H must be Specified Throughout Lattice - Must satisfy Maxwell's Equations throughout the entire space - Typically, Fields are Initialized to Zero • At zero, source should be zero (if fields init. to zero), and smooth (~ zero derivative). • Typical Sources

1. CW response (ramped): f (t) =sin(2πft)exp(−(t − to )2 / tw

2 )), t ≤ to

sin(2πft), t > to

⎧ ⎨ ⎩

-1.5

-1

-0.5

0

0.5

1

1.5

0 200 400 600 800 1000

gaus(t)sin(2πft)gaus(t)

gaus

(t)s

in(2

πft

)

t (ps)

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EE699 S. Gedney

Source Modeling 6-6

Broad Band Pulses with DC Component 2. Gaussian: f (t) = exp(−(t − to )2 / tw

2 )) • Spectral Response:

2 2( ) exp( ( ) ))exp( 2 )w w

o

F f t f tj ft

π ππ

= −

⋅ −

Also a Gaussian Pulse centered at the Origin! -3 dB Band Width f-3db = 1 πtw

Broad Band Pulses with no DC Component 3. Gaussian Derivative:

2 2 2 22( ) exp( ( ) / )) ( )exp( ( ) / ))w o w o o ww

f t t t t t t t t t tt t

∂∂

-= - - = - - -

• Spectral Response: 2 2 2( ) exp( ( ) ) exp( 2 ) 2w w oF f t f t j ft j fπ π π π= - ◊ - ◊

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400

Gau

ssia

n Pu

lse

t (ps)

tw

= 225 ps

to = 3 t

w

-1

-0.5

0

0.5

1

0 100 200 300 400

t w d

/dt(

Gau

ssia

n Pu

lse)

t (ps)

tw

= 225 ps

to = 3 t

w

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EE699 S. Gedney

Source Modeling 6-7

Broad Band Pulses with No DC (Continued)

• Modulated Gaussian: 2 2( ) exp( ( ) / )) sin(2 )o w mf t t t t f tπ= - - ◊

• Spectral Response:

2 2( ) exp( ( ( )) / )) exp( 2( ) )m w m ow

F f j f f t j f f fttπ

π= - - ◊ - -

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0 100 200 300 400

Mod

ulat

ed G

auss

ian

t (ps)

tw

= 225 pst

o = 3 t

wf

m = 3/t

w

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EE699 S. Gedney

Source Modeling 6-8

VOLTAGE SOURCES Discrete Voltage Source:

V

i

PEC

PEC

VEy

Ex

2

1 j1

j2

• Line Voltage:

V21 = −

E ⋅d

1

2∫ = − Eyi, j

∆yj = j1

j2−1∑

∴Eyi,j= − V21

∆y( j2 − j1)

• Hard Source:

Eyi, jn = −

V21n

∆y( j2 − j1); j1 ≤ j < j2

• Soft Source:

, ,

21 ; 1 2( 2 1)i j i j

nn ny y

VE E j j jy j j∆

= - £ <-

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EE699 S. Gedney

Source Modeling 6-9

DISCRETE SOURCE MODELS WITH LUMPED LOADS • Ampere's Law

H ⋅ d ∫ =

D ∂t

⋅ d s ∫ + σ

E ⋅d s ∫ +

J ⋅d s ∫

• Total Current

I =

D ∂t

⋅ d s ∫ + σ

E ⋅d s ∫ +

J ⋅d s ∫

or, I = Id + Ic + Il

Id = displacement current, Ic = conduction current, Il = impressed current, or the lumped current.

VOC

2

1

RTH

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EE699 S. Gedney

Source Modeling 6-10

Expressing:

V21 = −

E ⋅d

1

2

• Express Lumped Current Using a Voltage/Current Relationship

Il = f (V21 ) = f

E ⋅ d

1

2

∫⎛ ⎝ ⎜

⎞ ⎠ ⎟

• For Understanding, Let Voc = 0 (Resistive Load Only). Thus, Il = V21/RlTH.

VOC

2

1

RTH

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EE699 S. Gedney

Source Modeling 6-11

LUMPED RESISTANCE • Relationship for Lumped Resistance Il = V21/Rl. • Evaluate Ampere's Law over a Secondary Lattice Face:

H ⋅ d

∫ =

D ∂t

⋅ d s ∫ + σ

E ⋅d s ∫ +

J ⋅d s ∫

1 11 1 1 1 1, , , ,2 22 2 2 2 2

1 1 1 1 1 1 1 1, , , , , , , ,2 2 2 2 2 2 2 2

1

i j k i j k

i j k m n p i j k i j k

n nx x

n n n n nz z y y l

D DH H z H H y y z I

t∆ ∆ ∆ ∆

∆+ +

+ + + - + + + -

-

- - - - --È ˘Ê ˆ Ê ˆ- - - = +Á ˜ Á ˜Í ˙Ë ¯ Ë ¯Î ˚

• From Voltage Current Relationship:

1 12 2

1 1 1 11 , , , , , , , ,2 2 2 221, ,2

1

2i j k i j k i j k i j k

i j k

n n n nl x x x

nl

V E x E ExI

R R R+ + + +

+

− − −

−∆ +

∆= = =

• Inserting this Back into Ampere's Law (Discrete) Leads to:

1 1 1 12 2 2 21 1 1 1 1 1 1 1, , , , , , , ,2 2 2 2 2 2 2 2

1 1 1 1, , , , , , , ,2 2 2 2

1 1

2

i j k m n p i j k i j k

i j k i j k i j k i j k

n n n nz z y y

n n n nx x x x

H H z H H y

E E E Exy z

t Rε

+ + + − + + + −

+ + + +

− − − −

− −

⎡ ⎤⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎢ ⎥− ∆ − − ∆ =⎟ ⎟⎜ ⎜⎟ ⎟⎢ ⎥⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠⎣ ⎦− +

∆∆ ∆ +∆

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EE699 S. Gedney

Source Modeling 6-12

• Explicit Update Expression:

Exi +1

2, j,k

n =

ε∆t

−∆x

∆y∆z2Rε∆t

+∆x

∆y∆z2R

Exi +1

2, j,k

n−1 + 1ε∆t

+∆x

∆y∆z2R

Hzi+1

2 ,j +12,k

n− 12 − Hz

m +12,n −1

2,p

n− 12

⎛ ⎝ ⎜

⎞ ⎠ ⎟ / ∆y − Hy

i+12, j,k+1

2

n− 12 − Hy

i+12, j,k−1

2

n− 12

⎛ ⎝ ⎜

⎞ ⎠ ⎟ / ∆z

⎡ ⎣ ⎢

⎤ ⎦ ⎥

• Comparing with Previous Update Expressions For Lossy Media - Locally Lossy Media with σeff =

∆x∆y∆zR

(s/m)

- Expression is Stable (local C. L. is less restrictive than lossless case) - Lumped Resistance Has an Effective Width of Secondary Grid Face, and Effective Length of Grid Edge

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EE699 S. Gedney

Source Modeling 6-13

DISCRETE VOLTAGE SOURCE • KVL:

V12 = IlRTH - Voc. –> Il = (V12 + Voc)/RTH • In the Discrete Space:

12

1 1 11 , , , , , ,2 2 221, ,2

1 2

2i j k i j k i j k

i j k

nn nx x oc

nl

TH

E E x VI

R+ + +

+

−−

⎛ ⎞⎟⎜ + ∆ +⎟⎜ ⎟⎜⎝ ⎠=

• Applying this Discrete Form to Ampere's Law Leads to:

( ) ( )

1 1, , , ,2 2

12

11 1 1 1 , ,22 2 2 2

1 1 1 1 1 1 1 1, , , , , , , ,2 2 2 2 2 2 2 2

1 12

2 2

/ /

i j k i j k

i j k

i j k m n p i j k i j k

n nTHx x

TH TH

noc

n n n nz z y y

TH

xt y z RE Ex xt y z R t y z R

VH H y H H z

y zR

ε

ε ε+ +

+

+ + + − + + + −

− − − −

∆−∆ ∆ ∆= + ⋅∆ ∆+ +∆ ∆ ∆ ∆ ∆ ∆

⎡ ⎤⎢ ⎥⎢ ⎥− ∆ − − ∆ −⎢ ⎥∆ ∆⎢ ⎥⎣ ⎦

• Implement as a Soft Source (to maintain Vectorization). • Represents a Physical Source with Internal Resistance.

VOC

2

1

RTH

Il

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EE699 S. Gedney

Source Modeling 6-14

TOTAL FIELD SCATTERED FIELD FORMULATIONS (PLANE WAVE)

• Plane Wave Excitation Poses Challenge Since It Cannot Be Easily Generated By Finite

Sources, as Posed Previously, in a Finite Volume • Number of Techniques Have Been Presented To Excite Plane Wave. Most are Based on

a Decomposition of the Total Field into Scattered and Incident Fields, i.e.,

E total =

E inc +

E scattered ,

H total =

H inc +

H scattered

where, the incident fields satisfy Maxwell’s equations for a homogeneous free space:

,inc inc

inc inco oH EE Ht t

µ ε∂ ∂∇× =− ∇× =∂ ∂

• Popular Techniques:

1. Superimpose Scattered Field and Incident Fields Throughout the Space, where incident field is known.

- problem: Have to include non-linear numerical dispersion in incident field

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EE699 S. Gedney

Source Modeling 6-15

2. Solve for the Scattered Field Only. - Incident Field Arises from Boundary Conditions! - surface sources for perfectly conductor boundaries - volume sources for dielectric/magnetic medium

E scattered

PEC

n ×

E scat . = − ˆ n ×

E inc

3. Solve for Total Field in the Intererior Region of the Mesh, including all inhomogeneities, and Compute Scattered Field at the Extremeties of the Mesh. Source is Excited at the Interface Between the Two Regions

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EE699 S. Gedney

Source Modeling 6-16

TOTAL FIELD - SCATTERED FIELD FORMULATION

• Plane Wave Excitated at the Interface Between the Total Field Region and the Scattered Field Region:

E scattered

PEC

E total n ×

E total = 0

• At the Interface, make use of the fact that

E total =

E inc +

E scattered ,

H total =

H inc +

H scattered to transition between the scattered and total field regions • Subsequently, the incident field is excited about the surface defining the interface

between the two regions

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EE699 S. Gedney

Source Modeling 6-17

One-Dimensional Example

Ez1scat Ez2

scat Ez3tot Ez4

tot Ez5tot Ez6

tot

Hy5totHy4

totHy3totHy2

scatHy1scat

• Updates in Total Field Region:

Ezitotn+1

= Ezitotn

+∆t

ε∆xHyi

totn+1

2− Hyi−1

totn+1

2⎛

⎝ ⎜

⎠ ⎟

• Updates in Scattered Field Region:

Eziscatn+1

= Eziscatn

+∆t

ε∆xHyi

scatn+1

2− Hyi−1

scatn+1

2⎛

⎝ ⎜

⎠ ⎟

• Update at the Interface

Ez3totn+1

= Ez3totn

+∆t

ε∆xHy3

totn+1

2− Hy2

scatn+1

2⎛

⎝ ⎜

⎠ ⎟ −

∆tε∆x

Hy2inc

n+12⎛

⎝ ⎜

⎠ ⎟

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EE699 S. Gedney

Source Modeling 6-18

• Similarly:

Hy2scat

n+12

= Hy2scat

n−12

+∆t

µ∆xEz3

totn− Ez2

scatn⎛ ⎝ ⎜ ⎞

⎠ ⎟ −

∆tµ∆x

Ez3incn⎛

⎝ ⎜ ⎞

⎠ ⎟

• Incident fields satisfy independently Maxwell’s equations for a homogeneous free

space. For example, solution for a monochromatic plane wave propagating in the x-direction:

( )( )

12

2

3

1 52 2cos ( ) ( ) /

cos (3 )

n

n

incy zo

incz zo

H E n t x

E E n t x

ω β η

ω β

+

=− + ∆ − ∆

= ∆ − ∆

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EE699 S. Gedney

Source Modeling 6-19

Two-Dimensions (TMz)

i

j

Eztot

Ezscat

Eztot

Ezscat

Hxtot

Hxscat

Hyscat

Hytot

Total Field Region:

Hxi+1/2, jtotn+1/ 2

= Hxi+1/ 2, jtotn−1/2

−∆t

µ i+1/ 2, j

Ezi+1/2, j +1/2totn − Ezi+1/2, j−1/2

totn

∆y

Hyi,j +1/2totn+1/ 2

= Hyi, j+1/ 2totn−1/2

+∆t

µi, j+1/2

Ezi+1/2, j +1/2totn − Ezi−1/2, j+1/2

totn

∆x

1/ 2 1/ 2 1/ 2 1/ 2

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

1/ 2 , 1/ 2 1/ 2, 1/ 21/ 2, 1/ 2 1/ 2, 1/ 2

1 1 11 1 1

n n n n

n n i j i j i j i j

i j i j

tot tot tot toty y x xtot tot

z zi j i j

H H H Ht tE Ex y

ξξ ξ ε ξ ε

∆ ∆∆ ∆

+ + + +

+ + + + + + +

+ + + ++ + + +

- --= + -+ + +

where, ξ =σ i+1/2, j+1/2 ∆t2εi+1/ 2, j +1/2

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EE699 S. Gedney

Source Modeling 6-20

i

j

Ezi+1

2, j+ 12

tot Ezi+ 3

2, j+ 12

tot

Hyi , j+1

2

scat

Hxi+ 3

2, j+1

scat

Hxi+ 3

2, j

tot

Total Field/Scattered Field Interface:

1/ 2 1/ 2 1/ 2 1/ 2 1/ 2

1 2, 1/ 2 1, 1/ 2 3/ 2, 1 3/ 2, 3 / 2, 1

3/ 2, 1/ 2 3/ 2, 1/ 23/ 2, 1/ 2 3/ 2, 1/ 2

1 1 11 1 1

n n n n n

n n i j i j i j i j i j

i j i j

tot tot scat tot incy y x x xtot tot

z zi j i j

H H H H Ht tE Ex y y

ξξ ξ ε ξ ε

∆ ∆∆ ∆ ∆

+ + + + +

+ + + + + + + + + +

+ + + ++ + + +

È ˘- -- Í ˙= + - ++ + + Í ˙Î ˚

Corner:

Ezi+1/2, j+1/2totn+1

=1− ξ1+ ξ

Ezi+1/ 2,j +1/2totn

+1

1+ ξ∆t

εi+1/2, j +1/ 2

Hyi +1,j +1/2totn+1/2

− Hyi, j +1/2scatn+1/ 2

∆x

−1

1+ ξ∆t

ε i+1/ 2, j+1/2

Hxi+1/2, j+1scatn+1/2

− Hxi+1/2, jtotn+1/2

∆y+

Hxi+1/2, j+1incn+1/ 2

∆y+

Hyi, j+1/2scatn+1/2

∆x

⎢ ⎢

⎥ ⎥

H-Field Update:

Hxi+3/2, j+1scatn+1/ 2

= Hxi+3/2, j+1scatn−1/2

−∆t

µi+3/2, j

Ezi+3/ 2,j +3/2scatn

− Ezi+3/2, j+1/ 2totn

+ Ezi+3/2, j+1/2incn

∆y

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EE699 S. Gedney

Source Modeling 6-21

FORTRAN IMPLEMENTATION 2D (TMz) E-Field Update: do 10 jj=2,ny-1 do 10 ii=2,nx-1 ez(ii,jj)=ca(ii,jj)*ez(ii,jj) 1 + cb(ii,jj)*(hy(ii+1,jj)-hy(ii,jj)+hx(ii,jj)-hx(ii,jj+1)) 10 continue • Field is updated everywhere using the same update loop • Preserves Vectorization • Incident Field is Added in as a Soft Source do 16 ii=i0,i1 c total-front face ez(ii,j0)=ez(ii,j0)+ cb(ii,j0)*hxinc(ii,j0,float(nt)-.5) c total-back face ez(ii,j1)=ez(ii,j1)- cb(ii,j1)*hxinc(ii,(j1+1),float(nt)-.5) 16 continue

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EE699 S. Gedney

Source Modeling 6-22

do 17 jj=j0,j1 * total-left face ez(i0,jj)=ez(i0,jj)- cb(i0,jj)*hyinc(i0,jj,float(nt)-.5) c total-right face ez(i1,jj)=ez(i1,jj)+ cb(i1,jj)*hyinc((i1+1),jj,float(nt)-.5) 17 continue • hxinc, hyinc are functions

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EE699 S. Gedney

Source Modeling 6-23

H-Field Update: * update hx-fields * do 20 jj=1,ny do 20 ii=1,nx-1 hx(ii,jj)=dax(ii,jj)*hx(ii,jj) + dbx(ii,jj)*(ez(ii,jj-1)-ez(ii,jj)) 20 continue * update hy-fields * do 30 jj = 1,ny-1 do 30 ii=1,nx hy(ii,jj)=day(ii,jj)*hy(ii,jj) + dby(ii,jj)*(ez(ii,jj)-ez(ii-1,jj)) 350 continue • Again, Field is updated using a unified update loop • Incident Field is added in as a Soft Source

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EE699 S. Gedney

Source Modeling 6-24

do 40 ii=i0,i1 * scatt-front face hx(ii,j0)=hx(ii,j0)+ dbx(ii,j0)*ezinc(ii,j0,float(nt)) * scatt-back face hx(ii,(j1+1))=hx(ii,(j1+1))-dbx(ii,(j1+1))*ezinc(ii,j1,float(nt)) 40 continue do 50 jj=j0,j1 * scatt-left face hy(i0,jj)=hy(i0,jj)- dby(i0,jj)*ezinc(i0,jj,float(nt)) * scatt-right face hy((i1+1),jj)=hy((i1+1),jj)+dby((i1+1),jj)*ezinc(i1,jj,float(nt)) 50 continue

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EE699 S. Gedney

Absorbing Boundary Conditions 7-1

ABSORBING BOUNDARY CONDITIONS • We have discussed 2D, and 3D FDTD Algorithms For the Analysis of Cavity Problems • Desirable to Analyze Electromagnetic Interaction in Unbounded Media • Have to Truncate the Mesh in a Manner that the Exterior Boundary Effective Looks

Like an Unbounded Medium • Essentially Need to Match the Exterior Region to the Interior Region • Number of Techniques Can be Used to Acheive This: - Exact Radiation Boundary Condition Based on Green's Functions and a Surface or

Volume Convolutional Integral in Time and Space - Local Radiation Boundary Condition, or Absorbing Boundary Condition Based on a

Differential Operator, or Psuedo-Differential Operator - Absorbing Media that is Matched to the Interior Region and Rapidly Attenuates

Electromagnetic Energy • Desirables: - Reflectionless Boundary over Broad Spectrum, and for All Angles of Incidence and

Polarizations - Minimal in Computational Cost and Memory Requirements

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EE699 S. Gedney

Absorbing Boundary Conditions 7-2

ENGQUIST-MAJDA ONE-WAY WAVE EQUATION • Engquist-Majda one-way wave equation is based on enforcing the wave equation on the

fields near the exterior boundary • In 2D, each field component must satisfy the scalar wave equation. Let U represent an

individual field component: 2 2 2

2 2 2 2

1 0U U Ux y c t

∂ ∂ ∂∂ ∂ ∂

+ − =

where c is the wave phase velocity. • Introducing the Operator L

2 2 22 2 2

2 2 2 2 2

1 1x y tL D D D

x y c t c∂ ∂ ∂∂ ∂ ∂

= + − = + −

• The Wave Equation is then written in operator form as L U = 0, and L is factored as: 0LU L L U+ −= = , where:

2 21 ; 1 ; yt tx x

t

DD DL D S L D S Sc c D c

− +≡ − − ≡ + − =

• Engquist and Majda showed that the application of the wave function L- to the wave function U absorbs a plane wave impinging on a boundary at any angle θ! Thus,

0L U− = [B. Engquist and A. Majda, "Absorbing boundary conditions for the numerical simulation of waves", Mathematics of Computation, vol. 31, 1977, pp. 629-651]

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EE699 S. Gedney

Absorbing Boundary Conditions 7-3

• The difficulty with the Engquist-Majda boundary operator is that it is non-linear due to the square root.

21 0tx

DD S Uc

⎡ ⎤− − =⎢ ⎥⎣ ⎦

• Have to derive an approximate form, or a pseudo-differential operator that can be Implemented numerically.

• First-order Approximation based on a Taylor-series expansion: 21 1S− ≈

• Enguist-Majda Becomes: 1 0U U

x c t∂ ∂∂ ∂

− =

• If we knew the phase-velocity exactly relative to the x-direction, this is exact! e.g., at the x = 0 Boundary:

( )( , )1 1( ) ( ) 0

x y

x x x

j x y j to

xx

p p p

U x y U e eU U jj U j Ux v t v v

β β ω

β∂ ∂ ωβ ω∂ ∂ ω

+=

− = − = − =

• Difficulty: This requires a non-dispersive wave with a single angle of incidence. This is not practical for typical problems, such as scattering problems, which can have arbitrary angles of incidence on the exterior boundary.

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EE699 S. Gedney

Absorbing Boundary Conditions 7-4

• Second-Order Taylor-Series Expansion: 2 211 1

2S S− ≈ −

• This leads to: 2

2112 2

yt tx x

t

cDD DL D S Dc c D

− ⎛ ⎞≅ − − ≅ − +⎜ ⎟⎝ ⎠

• Introducing the ABC Operator: 2 22

0 02 2

y yt tx t x

t

cD cDD DL U D U D D Uc D c

− ⎛ ⎞ ⎛ ⎞≅ − + = ⇒ − + =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

which becomes: 2 2 2

2 2

1 0 ( 0)2

U U c U xx t c t y

∂ ∂ ∂∂ ∂ ∂ ∂

− + = =

• Similarly for the Other Boundaries: 2 2 2 2 2 2

2 2 2 2

2 2 2

2 2

1 10 ( ), 0 ( 0)2 2

1 0 ( )2

U U c U U U c Ux h yx t c t y y t c t xU U c U y h

y t c t x

∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂ ∂

+ − = = − + = =

+ − = =

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Absorbing Boundary Conditions 7-5

MUR ABSORBING BOUNDARY OPERATOR • Based on the Enquist-Majda Formulation • Second-order accurate • Standard ABC of the 80’s and early 90’s • E-M ABC is based on mixed derivatives in space and time. Mur implemented these in a

second-order accurate manner.

E y1 , jE y 2 , j

x = 0 Boundary:

3 3, , 2, 1, 2, 1,2 2

32

1 1 1 1 1 12

,

1 12 2

j j j j j j

n nn n n n ny y y y y yy

j

E E E E E EEx t t x x t x x

∂ ∂∂∂ ∂ ∂ ∂

+ −+ + − −⎛ ⎞ ⎛ ⎞− −⎜ ⎟= − = −⎜ ⎟⎜ ⎟⎜ ⎟∆ ∆ ∆ ∆⎜ ⎟ ⎝ ⎠⎝ ⎠

2, 1 2, 2, 1 1, 1 1, 1, 1

32

2

2 2 2,

2 212

j j j j j j

n n n n n n ny y y y y yy

j

E E E E E EEy y y

∂∂

+ − + −⎛ ⎞− + − +

= −⎜ ⎟⎜ ⎟∆ ∆⎝ ⎠

2, 2, 2, 1, 1, 1,

32

1 1 1 12

2 2 2,

2 212

j j j j j j

n n n n n n ny y y y y yy

j

E E E E E EEt t t

∂∂

+ − + −⎛ ⎞− + − += −⎜ ⎟⎜ ⎟∆ ∆⎝ ⎠

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EE699 S. Gedney

Absorbing Boundary Conditions 7-6

• Assembling the terms, this leads to an update expression for the boundary field:

1, 1, 2, 1, 1, 2,

1, 1 1, 1, 1 2, 1 2, 2, 1

1 1 1 1

2

2

2( ) ( )

( ) ( 2 2 )2( ) ( )

j j j j j j

j j j j j j

n n n n n ny y y y y y

n n n n n ny y y y y y

c t x xE E E E E Ec t x c t x

c t x E E E E E Ey c t x + − + −

+ − + −∆ − ∆ ∆= − + + + +

∆ + ∆ ∆ + ∆∆ ∆

+ − + + − +∆ ∆ + ∆

• This is known as the 2nd-Order Mur ABC • The first-order Mur is similarly derived from the Taylor Series Expansion 21 1S− ≈ ,

and evaluating at the center point between i = 1 and i = 2, leading to:

1, 1, 2, 1, 1, 2,

1 1 1 1 2( ) ( )j j j j j j

n n n n n ny y y y y y

c t x xE E E E E Ec t x c t x

+ − + −∆ − ∆ ∆= − + + + +

∆ + ∆ ∆ + ∆

• Second-Order Mur leads to a Second-Order Approximation of the Difference Equation.

However, the Difference Equation only provides a 2-term T.S. approximation based on a normally incident plane wave (i.e., with phase velocity c).

• First-Order Mur is a second-order accurate approximation of a 1-term T.S.,

approximation of the wave annihilator

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EE699 S. Gedney

Absorbing Boundary Conditions 7-7

• Approximating the Reflection Coefficient Based on the First and Second-Order Mur ABCs

• Let:

( cos sin )

( cos sin ) ( cos sin )

,

Re

inc j t kx kyy

total j t kx ky j t kx kyy

E e

E e

ω φ φ

ω φ φ ω φ φ

+ −

+ − − −

=

→ = +

• First-Order Mur: E-field satisfies:

1 0y yE Ex c t

∂ ∂∂ ∂

− = 1cos 1cos 1

R φφ

−⇒ =

+

• Second-Order Mur: E-field satisfies:

2 2 2

2 2

1 0 ( 0)2

y y yE E Ec xx t c t y

∂ ∂ ∂∂ ∂ ∂ ∂

− + = =

21

22 21

2

cos 1 sincos 1 sin

R φ φφ φ

− +⇒ =

+ −

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EE699 S. Gedney

Absorbing Boundary Conditions 7-8

10-9

10-7

10-5

0.001

0.1

0 15 30 45 60 75 90

Reflection Error of 1st & 2nd Order Mur ABC

R1

R2

Ref

lect

ion

Err

or

φ (degrees)

• Note that the Error is Based on Exact Differential Operators. The discrete Operators Approximate this (2nd-Order Accurate).

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EE699 S. Gedney

Absorbing Boundary Conditions 7-9

OTHER ABC OPERATORS • The Mur ABC annihilates a normally incident wave highly accurately. However, for

obliquely incident waves, large reflections can occur.

• Other boundary operators have been introduced to annihilate waves at multiple angles. This produces a higher levels of absorption over multiple angles.

• At least 2 schemes have been proposed with this intention - Trefethen-Halpern Generalized ABC - Higdon Boundary Operator.

Trefethen-Halpern Approximation • Returning to the Approximation of 21 S− . Let,

2 221 oS p p S− ≅ +

2 2 2

22 2 0opU U Up cx t c t y

∂ ∂ ∂∂ ∂ ∂ ∂

⇒ − − =

be a polynomial approximation. Then the coefficients can be determined by an interpolation technique such as a Padé, least-square, or Chebyshev approximation with the goal of interpolating 21 S− over the interval [-1,1].

(see Tables 7.1, 7.2, pg. 162 of Text). • Higher Order polynomial approximations:

22 2

22

1 o

o

p p SSq q S

+− ≅

+

3 3 3 32

2 22 2 3 2 0oo

pU U U Uq q c p cx t x y c t t y

∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

⇒ + − − =

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EE699 S. Gedney

Absorbing Boundary Conditions 7-10

HIGDON BOUNDARY OPERATOR

• Consider the Wave Impinging on the Exterior Boundary to be a Linear Superposition of Plane Waves

( , , ) ( cos sin )i i ii

U x y t f ct x yφ φ= + ±∑

• Higdon Proposed the Annihilator function [Math. Comp., vol. 49, pp. 65-90, 1987]:

1

cos ( , , ) 0n

ii

c U x y tt x

∂ ∂φ∂ ∂=

⎡ ⎤⎛ ⎞− =⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∏

• This operator is exact at each of the n angles iφ . • First-Order Higdon (n = 1):

1cos ( , , ) 0c U x y tt x

∂ ∂φ∂ ∂

⎛ ⎞− =⎜ ⎟⎝ ⎠

• Actually Equivalent to the First-Order Mur, with

1cospv c φ=

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EE699 S. Gedney

Absorbing Boundary Conditions 7-11

10-6

10-5

0.0001

0.001

0.01

0.1

1

0 15 30 45 60 75 90

First-Order Higdon ABC

R1

R1 (φ

1 = 10o)

R1 (φ

1 = 27o)

Ref

lect

ion

Err

or

φ (degrees)

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EE699 S. Gedney

Absorbing Boundary Conditions 7-12

• Second-Order Higdon (n = 1):

cosφ1∂∂t

− c∂∂x

⎛ ⎝ ⎜ ⎞

⎠ ⎟ cosφ2

∂∂t

− c∂∂x

⎛ ⎝ ⎜ ⎞

⎠ ⎟ U(x, y,t) = 0

• Equivalent to Trefethen-Halpern ABC, with po, and p2 chosen to annihilate at angles φ1

and φ2. - Multiplying through leads to

2 2 21 2

2 21 2 1 2

1 1 cos cos 0(cos cos ) (cos cos )

U U c Ux t c t y

∂ φ φ ∂ ∂∂ ∂ φ φ ∂ φ φ ∂

+− − =

+ +

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EE699 S. Gedney

Absorbing Boundary Conditions 7-13

10-10

10-8

10-6

0.0001

0.01

1

0 15 30 45 60 75 90

Second-Order Higdon Boundary Condition

R2

R2 (φ

1 = 10, φ = 27)

R2 (φ

1 = 22.5, φ = 67.5)

Ref

lect

ion

Err

or

φ (degrees)

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EE699 S. Gedney

Absorbing Boundary Conditions 7-14

• Implementing as a Difference Operator

1 2

( , , ) 0 (x=h)cos cos

c c U x y tt x t x

∂ ∂ ∂ ∂∂ φ ∂ ∂ φ ∂

⎛ ⎞⎛ ⎞+ + =⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

• Discretizing the Time and Space Derivatives at a half cell, as done previously with the

Mur Boundary Operator, leads to: UM

n = −(γ1 + γ 2 )UM−1n − γ1γ 2UM−2

n + (γ1 + γ 2 )UMn−1

+ (2 + 2γ1γ 2 )UM−1n−1 + (γ 2 + γ1)UM−2

n−1

− γ1γ 2UMn−2 − (γ 2 + γ1)UM−1

n−2 −UM−2n−2

where, 1 ,1 cos

ii i

i i

c tx

ργ ρρ φ

− ∆= =

+ ∆

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EE699 S. Gedney

Absorbing Boundary Conditions 7-15

• Advantages of Higdon Boundary Operator • Normal Derivative Only. • This is absolutely important for corners of the lattice, and for material inomogeneities • Same Degree of Accuracy as 2nd Order Mur with added flexibilityof broadening the

absorption band • Disadvantage

• Can be Unstable in Late Time. Principally due to rounding error and the fact that the operator is weakly causal. Can be eradicated by adding a damping term:

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EE699 S. Gedney

Absorbing Boundary Conditions 7-16

• Modified Higdon Boundary Operator†

11 2

( , , ) 0 (x=h)cos cos

c c U x y tt x t x

∂ ∂ ∂ ∂α∂ φ ∂ ∂ φ ∂

⎛ ⎞⎛ ⎞+ + + =⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

• This leads to the Difference Operator:

11 2 1 1 2 2 1 2

1 11 2 1 2 1 2

2 2 21 2 2 1 1 2

( ) ( )

(( 1) 2 ) ( )

( )

n n n nM M M M

n nM M

n n nM M M

U U U U

U U

U U U

γ γ γ γ γ γ

β γ γ βγ γ

γ γ βγ γ β

−− −

− −− −

− − −− −

= − + − + +

+ + + + +

− − + −

where,

1 1

1 1, ,1 (1 ) cos 1 (1 )

i ii i

i i i

c tx x x

ρ ργ ρ βρ α φ ρ α

− +∆= = =

+ + ∆ ∆ + + ∆

• Typically, 1 xα ∆ = 0.1 is chosen. This provides a stable formulation while maintaining

the absorptive properties. † V. Betz and R. Mittra, "Comparison and evaluation of boundary conditions for the absorption of guided waves in an FDTD

simulation," IEEE Microwave and Guided Wave Letters, vol. 2, pp. 499-501, December 1992.

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EE699 S. Gedney

Absorbing Boundary Conditions 7-17

IMPLEMENTATION • Example of update of Ex on the y = 0 plane alp = 0.1 r1 = vi(1)*dt/dy r2 = vi(2)*dt/dy g1 = (1.-r1)/(1.+r1*(1.+alp)) g2 = (1.-r2)/(1.+r2) bet = (1.+r1)/(1.+r1*(1.+alp)) c1 = g1+g2 c2 = g1*g2 c3 = bet*g2+g1 c4 = bet+1.+2.*c2

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EE699 S. Gedney

Absorbing Boundary Conditions 7-18

c j = 1 plane c c If r1 = 0, then assume j = 1 plane is a PEC wall. Else Update c if(r1 .ne. 0.) then c j = 1 do 10 k = 2,nk-1 do 10 i = 1,npx-1 sv1 = ex(i,j,k) ex(i,j,k) = c4*exy2(i,k,1,1)-bet*exy3(i,k,2,1)+c1*ex(i,j,k) 1 -c1*ex(i,j+1,k)-c3*(exy2(i,k,2,1)-exy3(i,k,1,1)) 2 -c2*(exy1(i,k,1)+ex(i,j+2,k)) exy1(i,k,1) = sv1 10 continue

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EE699 S. Gedney

Absorbing Boundary Conditions 7-19

c c Back store Field Information c do 108 k = 2,nk-1 do 108 i = 1,npx-1 exy2(i,k,2,1) = exy2(i,k,1,1) exy3(i,k,2,1) = exy3(i,k,1,1) 108 continue c do 110 k = 2,nk-1 do 110 i = 1,npx-1 exy2(i,k,1,1) = ex(i,2,k) exy3(i,k,1,1) = ex(i,3,k) 110 continue c

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EE699 S. Gedney

Absorbing Boundary Conditions 7-20

PERFECTLY MATCHED LAYER ABC

• Absorbing Material Boundary Condition' • To Be Effective, Must: - Be thin (only a few lattice cells in thickness) - Be Effective in the Near Field of Source or Scatterer - Reflectionless to All Impinging Waves (Polarization, Angle) - Reflectionless over a Broad-Band

AbsorbingMaterial Medium

FDTD Space

• Perfectly Matched Layer (PML) Material Medium First Introduced by J. P. Berenger Has Been Found to Meet all of these Criterion [J.-P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," Journal of Computational Physics, October 1994]

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EE699 S. Gedney

Absorbing Boundary Conditions 7-21

Dispersionless Medium • Consider a plane wave incident from a free space (x < 0) onto a material half space (x >

0) • The material half space is assumed to have both a magnetic conductivity σ* and an

electric conductivity σ.

Incident Plane Wave

Lossy Medium

Reflected Wave

y

x • TEz polarized uniform plane wave incident upon this interface ˆ

i ix yj x j yinc

oH zH e β β− −= , propagating with angle θi relative to the normal x-axis.

• The fields in the two regions are then posed as:

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EE699 S. Gedney

Absorbing Boundary Conditions 7-22

( )1 1

21

2 21

ˆ (1 )( 0)

ˆ ˆ(1 ) (1 )

i iix yx

i i iii iy x yxx x

j x j yj xo

j x j yj x j xo

H zH e ex

E x e y e H e

β ββ

β β βββ βωε ωε

− −

− −

⎫= + Γ ⎪ <⎬= − + Γ + − Γ ⎪⎭

( ) ( )2 22 2

2

2 1 1

ˆ( 0)

ˆ ˆ

t tx y

t t tty x yx

j j

j x j yo

j x j yo

H zH ex

E x y H eσ σωε ωε

β β

β β ββ

ωε ωε

τ

τ

− −

− −

+ +

⎫=⎪

>⎬⎛ ⎞= − + ⎪⎜ ⎟⎝ ⎠ ⎭

• From the dispersion relationships:

1

1

cos( 0)

sin

i ixi iy

kx

k

β θ

β θ

⎫= ⎪ <⎬= ⎪⎭

2 22

2 2

*1 1 ( 0)t tx yk x

j jσ σβ βωε ωµ

⎫⎛ ⎞⎛ ⎞ ⎪= + + − >⎬⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎪⎭

• where i i ik ω ε µ= (i = 1,2). • Enforcing the continuity of the tangential fields across the boundary interface (x = 0),

1 sint i iy y kβ β θ= = , and:

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EE699 S. Gedney

Absorbing Boundary Conditions 7-23

( )

( )

2

2

1 2

1 2

1

1

; 1

i tx x

j

i tx x

j

σωε

σωε

β βωε ωε

β βωε ωε

τ+

+

−Γ = = + Γ

+

• If the wave is normally incident (θi = 0): 1 2

1 2

η ηη η

−Γ =

+

where

( )( )

2

2

*21

1 21 2

1,

1j

j

σωµ

σωε

µµη ηε ε

+= =

+

Subsequently, if µ2 = µ1, ε2 = ε1, and

1 1

*σ σµ ε

=

then, Γ = 0! • Also,

1 1 11

1tx k k j

jσβ σηωε

⎛ ⎞= + = −⎜ ⎟

⎝ ⎠.

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EE699 S. Gedney

Absorbing Boundary Conditions 7-24

• The real part of txβ is the propagating component of the wave

• The imaginary part is the attenuative component. • Thus, the wave speed in medium 2 is identical to that in medium 1. Furthermore, despite

being a lossy medium, the wave is dispersionless – i.e., the wave speed is frequency independent for an axially propagating wave. Hence, this type of medium has been quantified as a dispersionless medium.

• From (7.1.b), the resultant fields in the dispersionless medium excited by a normally incident wave are expressed as:

1 1

1 1

2

2 1

ˆ

ˆ

jk x xo

jk x xo

H zH e e

E y H e e

ση

σηη

− −

− −

=

=.

• In summary, it is seen that given a medium with magnetic and electric conductivity

defined by 1 1

*σ σµ ε

= the medium will be matched across a planar boundary for all

normally incident waves. Furthermore, the wave propagating in the dispersionless medium has the same propagation characteristics as the incident wave, but will attenuate along the normal direction.

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EE699 S. Gedney

Absorbing Boundary Conditions 7-25

Berenger's PML

TM-Pol. Plane Wave PML Medium

Reflected Wave

x

z • The Dispersionless medium is matched only for normal incidence. J. P. Berenger was

the first to introduce a technique that is matched for all angles of incidence and polarizations. This method is based on a mathematical step known as "Field splitting". This technique is now described.

• Consider again, a TEz Polarized Incident Wave • Within the lossy medium, Maxwell's curl equations are expressed in their time-harmonic

form as:

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EE699 S. Gedney

Absorbing Boundary Conditions 7-26

2 21 , 1 ,z zx y

o o

H Hj E j Ej y j xσ ∂ σ ∂ωε ωεωε ∂ ωε ∂

⎛ ⎞ ⎛ ⎞+ = + = −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

2*1 .yx

zo

EEj Hj y x

∂∂σωµωµ ∂ ∂

⎛ ⎞+ = −⎜ ⎟

⎝ ⎠

• It is noted that the electric and magnetic conductivities in (7.9.a) and (7.9.b) are intentionally normalized by the relative permittivity and permeability, respectively.

• These equations are then modified by splitting the transverse magnetic field into two orthogonal terms:

z zx zyH H H= + such that Faraday's Law is rewritten as:

*

2 1 ,yxzx

o

Ej H

j x∂σωµ

ωµ ∂⎛ ⎞

+ = −⎜ ⎟⎝ ⎠

*

2 1 .y xzy

o

Ej Hj yσ ∂ωµωµ ∂

⎛ ⎞+ =⎜ ⎟⎜ ⎟

⎝ ⎠

• Introduce the variables:

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EE699 S. Gedney

Absorbing Boundary Conditions 7-27

**1 , 1 ; ( , )k k

k ko o

s s k x yj jσ σωε ωµ

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪= + = + =⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭

• The split field equations above are then rewritten as:

*2 ,y

x zx

Ej s H

x∂

ωµ∂

= −

*2 .x

y zyEj s Hy

∂ωµ∂

=

• Then, from Ampere's law:

( )2 ,zx zy

y x

H Hj s E

y∂

ωε∂

+=

( )2

zx zyx y

H Hj s E

x∂

ωε∂

+= −

• From these equations, the wave equation can be derived:

( )22 2 *

1 1zx zx zy

x x

H H Hs x s x

∂ ∂ω µ ε∂ ∂

− = − +

( )22 2 *

1 1zy zx zy

y y

H H Hs y s y

∂ ∂ω µ ε∂ ∂

− = − +

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EE699 S. Gedney

Absorbing Boundary Conditions 7-28

• Adding these together and leads the representative wave equation: 2

2 2* *

1 1 1 1 0z z zx x y y

H H Hs x s x s y s y

∂ ∂ ∂ ∂ ω µ ε∂ ∂ ∂ ∂

+ + =

• This wave equation supports the solutions: * *

x x x y y yj s s x j s s yz oH H e β βτ − −=

with the dispersion relationship: 2 2 2 2

2 2 2x y kβ β ω µ ε+ = =

• Then, from Ampere's law: * *

*

2

x x x y y yj s s x j s s yy yx o

y

sE H e

sβ ββ

τωε

− −= −

* **

2

x x x y y yj s s x j s s yx xy o

x

sE H es

β ββτωε

− −=

• The object of this exercise is to determine the characteristics of *, , ,x x ys s s and *ys such

that the boundary interface is reflectionless.

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EE699 S. Gedney

Absorbing Boundary Conditions 7-29

• Despite the field splitting, the tangential fields must still be continuous across the x = 0 interface.

• Assuming a plane wave is assumed to be incident on this hypothetical material medium, the reflection and transmission coefficient are derived by enforcing the continuity of the tangential fields at x = 0, leading to:

*

1 2

*

1 2

; 1

ix x x

x

ix x x

x

ss

ss

β βωε ωε

τβ β

ωε ωε

−Γ = = + Γ

+

and 1 sini iy y kβ β θ= = .

• From the phase match condition, we see that sy = *ys = 1.

• It is observed from the dispersion relationship that if ε2 = ε1, then ix xβ β= .

• Further, it is seen that if sx = *xs (or

*x x

o o

σ σε µ= ), then Γ = 0 for all incident angles θI!

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EE699 S. Gedney

Absorbing Boundary Conditions 7-30

• The resultant TEz field transmitted into the absorbing medium is then:

1 1 cosi i i i ix x y x y rjs x j y j x j y x

z o oH H e H e eβ β β β ση ε θ− − − − −= =

1 1

1 1

cos1

cos1

sin ,

cos .

i i ix y r

i i ix y r

j x j y xix o

j x j y xiy o

E H e e

E H e e

β β ση ε θ

β β ση ε θ

η θ

η θ

− − −

− − −

= −

=

• Thus, it is observed that the purely transmitted field propagates with the same wave speed as the impinging field, while attenuating along the normal direction with an attenuation constant of 1 1 cos i

rση ε θ . • Unlike the dispersionless material in the previous section, this property applies to all

angles incidence. • This exercise can be repeated for the dual polarization splitting z zx zyE E E= + . It is found

that if µ2 = µ1, and sx = *xs , then i

x xβ β= and Γ = 0 for all iθ . • In summary, if a wave impinges on a half space of this perfectly matched absorbing

material, it will be perfectly transmitted for all polarizations, all frequencies, and all angles of incidence. Furthermore, the transmitted wave will propagate with the same wave speed as the incident wave, have the same characteristic impedance as the incident wave, but will attenuate along the normal direction. Finally, this hypothetical absorbing material can be used within the framework of a discrete numerical method such as FDTD to terminate all outward traveling waves.

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EE699 S. Gedney

Absorbing Boundary Conditions 7-31

FDTD Implementation of PML • PML Equations (TEy - 2 Dimemsional)

( ) ( )1 ( ), 1 ( )y x

o oo r x zx zy o r y zx zyj y j xj E H H j E H Hσ σ∂ ∂ωε ∂ ωε ∂ωε ε ωε ε+ = + + = − +

( ) ( )1 , 1 yx

o oo r zx y o r zy xj x j yj H E j H Eσσ ∂ ∂ωε ∂ ωε ∂ωµ µ ωµ µ− + = − + = −

• Time-Dependent PML Equations (TEy - 2 Dimemsional)

( ), ( )yxo r y r x zx zy o r x r y zx zyy x

EE E H H E H Ht t

∂ ∂∂ ∂

∂∂ε ε σ ε ε ε σ ε∂ ∂

+ = + + = − +

2 2, zyzxo r x o r zx y o r y o r zy xx y

HH H E H Et t

∂ ∂∂ ∂

∂∂µ µ σ η µ µ µ σ η µ∂ ∂

+ = − + =

• Discrete Form Derived Using Central Difference Approximations. For example:

1 1 1 12 2 2 2

1 1 1 1 1 1 1 1 1 1, , , , , , , , , , , ,2 2 2 2 2 2 2 2 2 2

1

12

22i j k i j k i j k i j k i j k i j k

yo

n n n nn n rx x zx zx zy zy

y o zo

ytE E H H H H

tt

σεε

σ ε σε+ + + + + − + + + −

+ + + ++− ∆ ⎛ ⎞∆= − − + −⎜ ⎟

⎝ ⎠++∆∆

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EE699 S. Gedney

Absorbing Boundary Conditions 7-32

Theoretical Performance of the PML

• PML Backed by PEC Wall: 0yσ =

( ) 2 cosx odR e σ η θθ −= • In practice, the abrupt transition between the discrete PML space and the discrete FDTD

space can lead to a large reflection. One way to circumvent this is to scale the conductivity spatially along the normal axis:

max

mo

x xm

x xd

σ σ−

=

This results in a reflection Coefficient: ( ) 2 cos /( 1)x od m

mR e σ η θθ − +=

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EE699 S. Gedney

Absorbing Boundary Conditions 7-33

THREE-DIMENSIONAL PML • PML is easily represented in three-dimensions following the previous analogies:

( )( )( )( )( )( )

( )

( )

( )

( )

( )

( )

o r y r xy zx zyt y

o r z r xz yx yzt z

o r z r yz xy xzt z

o r x r yx zx zyt x

o r x r zx yx yzt x

o r y r zy xy xzt y

E H H

E H H

E H H

E H H

E H H

E H H

∂ ∂∂ ∂

∂ ∂∂ ∂

∂ ∂∂ ∂

∂ ∂∂ ∂

∂ ∂∂ ∂

∂ ∂∂ ∂

ε ε σ ε

ε ε σ ε

ε ε σ ε

ε ε σ ε

ε ε σ ε

ε ε σ ε

+ = +

+ = − +

+ = +

+ = − +

+ = +

+ = − +

( )( )( )( )( )( )

2

2

2

2

2

2

( )

( )

( )

( )

( )

( )

o r y o r xy zx zyt y

o r z o r xz yx yzt z

o r z o r yz xy xzt z

o r x o r yx zx zyt x

o r x o r zx yx yzt x

o r y o r zy xy xzt y

H E E

H E E

H E E

H E E

H E E

H E E

∂ ∂∂ ∂

∂ ∂∂ ∂

∂ ∂∂ ∂

∂ ∂∂ ∂

∂ ∂∂ ∂

∂ ∂∂ ∂

µ µ σ η µ

µ µ σ η µ

µ µ σ η µ

µ µ σ η µ

µ µ σ η µ

µ µ σ η µ

− + = +

− + = − +

− + = +

− + = − +

− + = +

− + = − +

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EE699 S. Gedney

Absorbing Boundary Conditions 7-34

UNIAXIAL PERFECTLY MATCHED LAYER ABSORBING MEDIUM • Berenger’s PML Provides an Efficient and Highly Accurate Means for Truncating the

FDTD Lattice. • Limitations of Berenger's Method: + Based on a Non-Maxwellian Formulation. + No Physical Insight for Application to General Problems. + Results in a Field Splitting - Additional Memory and CPU Time. • There are Alternate PML Based on a Uniaxial Anisotropic Media that also provides

Perfect Absorption - but based directly on Maxwell's Equations.†,†† † Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary

condition," IEEE Transactions on Antennas and Propagation , vol. 43, pp. 1460-1463, December 1995. †† S. D. Gedney, "An Anisotropic PML Absorbing Media for FDTD Simulation of Fields in Lossy Dispersive Media,"

Electromagnetics, vol. 16, no. 3, July/August, 1996.

S. D. Gedney, "A Uniaxial PML Absorbing Media for FDTD Simulations," IEEE Transactions on Antennas and Propagation, vol. 44, pp. , December 1996.

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EE699 S. Gedney

Absorbing Boundary Conditions 7-35

UNIAXIAL PML - THEORY

Incident Plane Wave

Uniaxial Medium

Reflected Wave

x

z

• Assume a time-harmonic arbitrarily polarized plane wave, incident on a material half space described as a uniaxial anisotropic medium:

i ix zj x j zinc

oH H e β β− −=

• The interface between the two media is the z = 0 plane.

• The fields excited within the uniaxial medium are plane wave in nature and satisfy Maxwell's equations. In the plane wave space the curl equations are expressed as:

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EE699 S. Gedney

Absorbing Boundary Conditions 7-36

,a ao r o rE H H Eβ ωµ µ µ β ωε ε ε× = × = −

where: ˆ ˆa i a

x zx zβ β β= + , and:

0 00 00 0

aa

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

, 0 0

0 00 0

cc

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

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EE699 S. Gedney

Absorbing Boundary Conditions 7-37

UNIAXIAL PML - THEORY (cont'd) • Derive the wave equation in the Uniaxial Medium equation:

1 2 0a a H k Hβ ε β µ−× × + = where: 2 2

o r o rk ω µ µ ε ε= . • Expressing the cross products as matrix operators, the wave equation can be expressed

more suitably in matrix form as: 2

2 2

2

2 1 1

2 1 1

1 2 1

0

0 0 0

0

a i az x z x

a iz x y

i a izx z x

k c a a Hk c a b H

Ha k d a

β β β

β β

β β β

− −

− −

− −

⎡ ⎤− ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥− − =⎢ ⎥⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎢ ⎥⎣ ⎦

• The dispersion relationship for the uniaxial medium is derived from the determinant of the matrix operator:

( )( )2 2 2 22 1 1 2 1 1 0a i a iz x z xk c a b k a c dβ β β β− − − −− − − − =

• Conveniently, these solutions can be decoupled into forward and backward TEy and TMy modes, which satisfy the dispersion relationships:

( )2 2 2 22 1 1 2 1 1y y0 for TE , & 0 for (TM )a i a i

z x z xk c a b k a c dβ β β β− − − −− − = − − =

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EE699 S. Gedney

Absorbing Boundary Conditions 7-38

UNIAXIAL PML - THEORY (cont'd)

• Initially, assume a TEy incident wave impinging on the interface:

( )2

1

2 21

ˆ (1 )

ˆ ˆ(1 ) (1 )

i iix zz

ii i ii ixz x zz z

j x j zj zo

j x j zj z j zo

H yH e e

E x e z e H e

β ββ

ββ β ββ βωε ωε

− −

− −

= + Γ

= − Γ − + Γ

( )11

2

2

ˆ

ˆ ˆ

i ax z

ia i axz x z

j x j zo

ba j x j zo

H yH e

E x z H e

β β

ββ β βωε ωε

τ

τ−−

− −

− −

=

= −

• Enforcing field continuity at the z = 0 interface: 1

1 1

2, 1i a iz z zi a i az z z z

aa a

β β βτβ β β β

− −

−Γ = = + Γ =

+ +

• The underlying objective is to determine if there exists a choice of constitutive parameters for which Γ = 0 for all angles of incidence.

• A sufficient condition is if 1i az z aβ β −= .

• Given the TEy dispersion relationship: 2 22 1a i

z xk ca abβ β−= − or 2 22 1 1 1i i

z xk ca b aβ β− − −= − • Finally, if c = a and b = a-1, then:

2 22 !i iz xkβ β= −

• Repeating for the TMy polarization, reflectionless conditions holds if c = a and d = c-1.

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EE699 S. Gedney

Absorbing Boundary Conditions 7-39

UNIAXIAL ABSORBING MEDIA • In conclusion, if:

1

0 00 00 0

aa

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

, 1

0 00 00 0

aa

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

a plane wave of arbitrary polarization and angle of incidence will be purely transmitted into the uniaxial medium.

• For FDTD applications, the perfectly matched uniaxial medium is extremely useful if it is highly lossy such that any wave entrant upon the medium will quickly attenuate while no physical reflections will be encountered due to the interface.

• Terminating the uniaxial slab with a hard boundary such as a perfect electrical conductor (PEC), small reflections will be encountered due to the finite depth. However, if the medium is highly lossy, these reflections can be made to be extremely small.

• For a lossy uniaxial medium, one obvious choice for the constitutive parameters is a = 1 z

ojσωε+ . This leads to the relative permittivity and permeability tensors:

1 0 0

0 1 0

0 0 1/(1 )

z

o

z

o

z

o

j

j

j

σωε

σωε

σωε

ε µ

⎡ ⎤+⎢ ⎥⎢ ⎥= + =⎢ ⎥

+⎢ ⎥⎣ ⎦

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EE699 S. Gedney

Absorbing Boundary Conditions 7-40

UNIAXIAL ABSORBING MEDIA (cont'd)

• The dispersion relationship is then expressed as: 2 22 2/(1 )z

o

a iz xjk σ

ωεβ β= + +

which leads to: (1 )z

o

a iz zj σ

ωεβ β= ± − • This is identical to that derived for Berenger's PML. • Given a TEy incident wave, the field intensities in the uniaxial medium are given by:

2 ˆi ix z zj x j z z

oH yH e eβ β α− − −=

(1 )

2 ˆ ˆi zi i ix j oz x z z

o r o r

j x j z zoE x z H e e

σωεββ β β α

ωε ε ωε ε

+ − − −⎛ ⎞= −⎜ ⎟⎝ ⎠

where: cosz

o

i iz z z o r

σωεα β σ η ε θ= =

• Which, again, is identical to that derived using Berenger's split field PML. In fact, the uniaxial PML can be derived from the split field formulation through a normalization of the normal field components†.

• The principal advantage of the Uniaxial PML is that it is posed in a Maxwellian form. This lends to further physical insight to its properties, and has lead to the generalization of the PML to more generalized media, such as lossy, dispersive, and anisotropic media†.

† S. D. Gedney, "An Anisotropic PML Absorbing Media for FDTD Simulation of Fields in Lossy Dispersive Media," Electromagnetics, vol. 16, no. 3, July/August, 1996.

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EE699 S. Gedney

Absorbing Boundary Conditions 7-41

Corner Regions

σ x

σ x

σz σz = 0

σ x = 0

σz

FDTD Space

PML d

PEC Wall

• In General, the FDTD lattice must be terminated by PML on all 6 sides. In the corner

regions there are multiple interface boundaries. • The constitutive relations in the corner regions are simply derived by matching two

uniaxial media. This leads to the general expressions for Maxwell's curl equations: ,o r oH j E E j Hωε ε ε ωµ µ∇ × = ∇ × = −

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EE699 S. Gedney

Absorbing Boundary Conditions 7-42

where:

0 0

0 0

0 0

y z

x

x z

y

x y

z

s ss

s ss

s ss

ε µ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

, (1 ), (1 ), (1 )yx z

o o ox y zj j js s sσσ σωε ωε ωε= + = + = +

• sx, sy, and sz are associated with the x, y, and z-normal planes, respectively. Outside of

these regions, the respective σi = 0.

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EE699 S. Gedney

Absorbing Boundary Conditions 7-43

IMPLEMENTATION IN DISCRETE SPACE • In the uniaxial medium, Ampère's law is expressed as:

0 0

0 0

0 0

y z

xz yy z x

x zx z o r yz x

yy x zx y

x y

z

s ssH H E

s sH H j Es

H H Es ss

∂ ∂∂ ∂

∂ ∂∂ ∂∂ ∂∂ ∂

ωε ε

⎡ ⎤⎢ ⎥⎢ ⎥−⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥− = ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

• Let's examine the update expression for Ez.

• Initially, define the constitutive relationship:

xz o r z

z

sD Es

ε ε=

where from Ampere's law, Dz satisfies:

y

oy x y z z zx yH H j s D j D Dσ∂ ∂∂ ∂ εω ω− = = +

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EE699 S. Gedney

Absorbing Boundary Conditions 7-44

• Dz is then updated using the standard explicit FDTD formulation:

, , 1/ 2 , , 1/ 2

1 1 1 1 1 1 1 1, , , , , , , ,2 2 2 2 2 2 2 2

1/ 2 1/ 22 22 2i j k i j k

i j k i j k i j k i j k

o yn n oz z

o y o y

n n n ny y x x

t tD Dt t

H H x H H y

ε σ εε σ ε σ+ +

+ + − + + + − +

+ −− ∆ ∆= + ⋅

+ ∆ + ∆

⎧ ⎫⎛ ⎞ ⎛ ⎞− ∆ − − ∆⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎩ ⎭

• Next, Ez must be determined from Dz. Recall,

, or

(1 ) (1 ) , or

.

xz

o o

xz

o o

z z o r x z

z o r zj j

z z o r z o r z

s D s E

D E

j D D j E E

σσωε ωε

σσε ε

ε ε

ε ε

ω ωε ε ε ε

=

+ = +

+ = +

• Transforming into the time-domain: jω t

∂∂→ .

xz

o oz z o r z o r zD D E Et t

σσε ε

∂ ∂ ε ε ε ε∂ ∂

+ = +

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EE699 S. Gedney

Absorbing Boundary Conditions 7-45

• Approximating the time derivatives using a central difference approximation and averaging z

o zDσε and y

o zEσε in time:

1 12 2

1 12 2

22

2 1 (1 ) (1 )2 2 2

n no xz z

o x

n nz zz z

o x r o o

tE Et

t tD Dt

ε σε σ

σ σε σ ε ε ε

+ −

+ −

− ∆= +

+ ∆

⎛ ⎞∆ ∆+ − −⎜ ⎟+ ∆ ⎝ ⎠

• Two-Step Update Process:

- Maintains Second-Order Accuracy. - Requires Storage of Dz and Ez. - Stable within the Courant Limit.

• Similarly Hz is updated using a two-step process.

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EE699 S. Gedney

Absorbing Boundary Conditions 7-46

SAMPLE CODE FOR UPML-FDTD UPDATES: do 10 k = 2,nz-1 + Exclude k = 1 and k = nz since PEC bndaries are assumed do 10 j = 2,ny-1 + Exclude j = 1 and j = ny since PEC bndries are assumed do 10 i = 1,nx-1 dxs = dex(i,j,k) dex(i,j,k)=aexz(k)*dex(i,j,k)+bexz(k)*(hz(i+1,j+1,k)-1 hz(i+1,j,k)-hy(i+1,j,k)+hy(i+1,j,k-1))* 2 epsc(iepx(i,j,k)) ex(i,j,k)=aezy(j)*ex(i,j,k)+dexy(j)*(eexx(i)*dex(i,j,k) 1 +fexx(i)*dxs)

10 continue do 20 k = 2,nz-1 do 20 j = 1,ny-1 do 20 i = 2,nx-1 dys = dey(i,j,k) dey(i,j,k)=aeyx(i)*dey(i,j,k)+beyx(i)*(hx(i,j+1,k)- 1 hx(i,j+1,k-1)-hz(i+1,j+1,k)+hz(i,j+1,k))* 2 epsc(iepy(i,j,k)) ey(i,j,k)=aexz(k)*ey(i,j,k)+deyz(k)*(eeyy(j)*dey(i,j,k) 1 +feyy(j)*dys)

20 continue do 30 k = 1,nz-1 do 30 j = 2,ny-1 do 30 i = 2,nx-1 dzs = dez(i,j,k) dez(i,j,k)=aezy(j)*dez(i,j,k)+bezy(j)*(hy(i+1,j,k)- 1 hy(i,j,k)-hx(i,j+1,k)+hx(i,j,k))* 2 epsc(iepz(i,j,k)) ez(i,j,k)=aeyx(i)*ez(i,j,k)+dezx(i)*(eezz(k)* 1 dez(i,j,k)+fezz(k)*dzs)

30 continue

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EE699 S. Gedney

Absorbing Boundary Conditions 7-47

Actual Performance of PML • Example: 50 Ω Microstrip Line printed on a 10 mil Alumina Substrate (εr = 9.8)

Terminating into a PML Interface. Side Walls and Ceiling also Terminating by PML walls.

10 cells10 cells

10 cells5

cells5

cells

• End Wall Reflection Error:

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Absorbing Boundary Conditions 7-48

-160

-140

-120

-100

-80

-60

-40

-20

0 10 20 30 40 50

σmax

= 10

σmax

= 20

σmax

= 30

σmax

= 40

σmax

= 50

σmax

= 60

σmax

= 70

σmax

= 80

σmax

= 90R

efle

ctio

n E

rror

(dB

)

f (GHz)

Reflection error due to a 10 cell thick uniaxial PML termination of the 50 Ω microstrip line (m = 4) for various values of σmax. (10 mil strip width).

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EE699 S. Gedney

Absorbing Boundary Conditions 7-49

• Comparison of the reflection error at f = 0 computed using the split field PML method compared with the theoretical reflection error due to a 10 cell thick PML termination of the microstrip line for m = 4 and m = 2, and due to a 5 and 10 cell thick PML termination of the microstrip line (∆z = 0.12 mm, and εreff = 6.62).

-200

-150

-100

-50

0

0 20 40 60 80 100

Γ2(PML) (10-cell)

R2(0) (10-cell)

Γ4(PML) (10 cell)

R4(0) (10 cell)

Γ4(PML) (5-cell)

R4(0) (5-cell)

Ref

lect

ion

Err

or (d

B)

σmax

• Comparison of the reflection error due to a 10 cell thick PML termination of a strip line

with σmax = 90, m = 4 encountered using the uniaxial PML method and Berenger's PML method.

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EE699 S. Gedney

Absorbing Boundary Conditions 7-50

-110

-105

-100

-95

-90

-85

-80

0 5 10 15 20 25 30 35 40

uniaxial PML

Berenger's PML

Ref

lect

ion

Err

or (d

B)

f (GHz)

Choosing the PML Material Parameters • Due to the discontinuity in material profile at the PML interface, a large reflection can

occur due to discretization error. It was proposed by Berenger, to spatially scale the conductivity profile to minimize this error, i.e., ( )xσ .

• the reflection due to a purely propagating wave impinging at angle θ is then expressed as:

( ) 02 cos ( )

dr x dx

R eηε θ σ

θ− ∫=

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EE699 S. Gedney

Absorbing Boundary Conditions 7-51

• Polynomial Scaling:

( ) max2 cos /( 1)max ,( ) r d m

m

R exxd

ηε σ θθσ σ − +⎛ ⎞=⎜ ⎟

⎝ ⎠=

• Where d is the thickness of the PML slab.

• Geometric Scaling:

( ) ( )1 2 ( 1)cos / ln,( ) Nx o

xx g g

og R ex ησ θσ θσ ∆ − ∆ −==

• Specifically, the conductivity scales from σo at the PML interface to gNσo at the PEC boundary, where d = N∆x is the thickness of the PML slab.

• The parameters can be determined for a given error estimate. For example, given m and

d for polynomial scaling, it is desired to achieve a reflection error of R(0), then maxσ is:

maxln( (0))( 1)

2 r

R md

σηε

+= −

• For geometric scaling, one would predetermine g, d, and R(0). Then, ln( (0))ln( )

2 ( 1)o Nr

R gx g

σηε

= −∆ −

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EE699 S. Gedney

Absorbing Boundary Conditions 7-52

• The design of the optimal PML results in a delicate balance of the theoretical reflection error ( )R θ and discretization error. If maxσ is small, the predominant reflection of the finite PML will be due to PEC backed wall.

• In practice one would rather choose maxσ to be as large as possible to minimize ( )R θ . Unfortunately, if maxσ is too large, then discretization error due to the FDTD approximation will dominate. Subsequently, there is an optimal choice for ( )R θ which balances reflection from the PEC wall and discretization error.

• Through extensive experimental study, it has been found that for a broad range of applications an optimal choice for a 10 cell thick polynomially scaled PML is 16(0)R e−≈ . For a 5 cell thick PML, 8(0)R e−≈ is optimal. This leads to an optimal choice for maxσ :

opt( 1)

150 r

mx

σπ ε

+=

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EE699 S. Gedney

Absorbing Boundary Conditions 7-53

More Examples of PML • TE-polarized wave excited by a vertically directed electric current source in a 2D region.

The working volume is 40 mm x 40mm, and is surrounded by PML layers of thickness d. The source is located at the center of the region. The fields are probed at two observation points A and B.

40 mm

40 m

m

d

d

PML Region

A

B

J

2 mm

2 m

m2

mm

• Relative error at points A and B over 1,000 time iterations for 5 and 10 cell thick PMLs

terminating a 40 x 40 cell lattice excited by a small dipole with σmax = σopt and κ = 1.

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EE699 S. Gedney

Absorbing Boundary Conditions 7-54

10 -8

10 -7

10 -6

10 -5

0.0001

0.001

0.01

0 500 1000 1500 2000 2500

5 cells (A)5 cells (B)10 cells (A)10 cells (B)

Rel

ativ

e E

rror

t (ps) • Maximum relative error due to a 5 and 10 cell thick PML termination of a 40 x 40 cell

lattice excited by a small dipole over 1,000 time iterations versus σmax/σopt. Polynomial spatial scaling with m = 4.

10 -5

0.0001

0.001

0.01

0.1

1

0 0.5 1 1.5 2 2.5 3

5 cells (A)5 cells (B)10 cells (A)10 cells (B)

Rel

ativ

e E

rror

σmax

/σopt

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EE699 S. Gedney

Absorbing Boundary Conditions 7-55

• Maximum relative error due to a 15 thick PML termination of a 40 x 40 cell lattice excited by a small dipole over 1,000 time iterations versus σmax/σopt . Polynomial spatial scaling with m = 4.

10 -6

10 -5

0.0001

0.001

0.01

0.1

0 0.5 1 1.5 2 2.5 3

15 cells (A)15 cells (B)

Rel

ativ

e E

rror

σmax

/σopt

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EE699 S. Gedney

Absorbing Boundary Conditions 7-56

• Contour plot of the maximum relative error at points A and B, respectively, due to a polynomial scaled 10 cell thick PML termination of a 40 x 40 cell lattice excited by a small dipole over 1,000 time iterations versus σmax/σopt and m.

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EE699 S. Gedney

Absorbing Boundary Conditions 7-57

• Maximum relative error at points A and B, respectively, due to a geometrically scaled 10

cell thick PML termination of a 40 x 40 cell lattice excited by a small dipole over 1,000 time iterations versus g and ln(R(0)).

ln( ) ln( )

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EE699 S. Gedney

Some Advanced PML Concepts 8-1

Alternate Forms of the PML Metric Tensor o Berenger Originally proposed:

1 xx

o

sjσωε

= +

o We found that for a plane wave incident on the boundary with normal dependence ixj xe β− ,

the propagation constant in the PML will be 1

cosi xx ii i

o x rx x xj x

j xj s x j xe e e eσ

βωε σ η ε θβ β

⎛ ⎞− +⎜ ⎟ −− −⎝ ⎠= =

o Now consider a wave impinging on the boundary that has a complex wave number (e.g.,

lossy medium, guided mode below cutoff, or the evanescent part of a wave expansion): ( )i ii x xx

j xxe e α βγ − +− = . o If the wave is incident on the PML boundary, it can be shown that the wave transmitted in

the PML is still perfectly matched with propagate along the normal direction as:

( ) ( ) 1cos

i i x i xi i x x x ii i ix x x o x rx x x o xj x j xj s x j xs x x j xe e e e e e e

σ σα β αα β ωε σ η ε θγ α ωε β⎛ ⎞

− + +⎜ ⎟− + −− − −⎝ ⎠= = = o What if the wave was purely evanescent ( 0i

xβ = )? The transmitted wave is: i xxi

x oj x

xe eσ

αα ωε−

which has no additional attenuation! o Propose instead:

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EE699 S. Gedney

Some Advanced PML Concepts 8-2

xx x

o

sjσκωε

= +

where 1xκ ≥ . o Then, the transmitted wave behaves as:

( ) ( ) cosi i x i xi i x x x x ii i ix x x o x rx x x x o x x

j x j xj s x j xs x x j xe e e e e e eσ σα β κ αα β ωε σ η ε θγ α κ ωε κ β

⎛ ⎞− + +⎜ ⎟− + −− − −⎝ ⎠= = =

Now, the attenuation is amplified by xκ . Also note that the wave speed is effectively reduced. Once again, xκ cannot be increased indiscriminately. In fact, if xκ becomes too large, it can actually degrade the reflection coefficient of a propagating wave impinging on the boundary, since it increases the time interaction with the PML medium.

o In practice, xκ must be scaled to reduce discretization error. A recommended scaling is between 1 (at the front boundary) and max

xκ at the PEC wall. For a PML of depth d with boundary interface x = 0, set:

( ) max1 ( 1)m

xxxd

κ κ ⎛ ⎞= + − ⎜ ⎟⎝ ⎠

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EE699 S. Gedney

Some Advanced PML Concepts 8-3

Example – Rectangular Waveguide o Consider a lossless rectangular waveguide with a height of 0.5 cm and a width of 1.0 cm.

For these dimensions, the cutoff frequency of the TE10 mode is 15 GHz. o The FDTD lattice has the uniform dimensions ∆x = ∆y = ∆z = 0.25 mm. o The TE10 mode is excited in a rectangular waveguide. The time-dependent source was a

modulated Gaussian pulse, with center frequency of 20 GHz and half bandwidth of 30 GHz. The field was measured 20 cells from the source (0.5 cm) and three cells from the PML interface. The PML medium was 10 cells thick.

o Reflection error (relative) as σmax is varied between 5 and 50 and κmax = 1.

-120

-100

-80

-60

-40

-20

0

0 10 20 30 40 50

σmax = 5

σmax = 10

σmax = 20

σmax = 30

σmax = 40

σmax = 50

Ref

lect

ion

Erro

r (dB

)

f (Ghz)

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EE699 S. Gedney

Some Advanced PML Concepts 8-4

o Above cutoff, the mode is well absorbed. Below cutoff, only attenuation is due to the waveguide mode.

o Now increase κ

-40

-35

-30

-25

-20

-15

-10

-5

0

0 5 10 15 20

κmax

= 1

κmax = 4

κmax

= 6κ

max = 8

κmax

= 10

Ref

lect

ion

Erro

r (dB

)

f (GHz) -30

-25

-20

-15

-10

-5

0

0 5 10 15

Analytical ( σ = 0, κ = 1)σ

max = 5, κ = 1

σmax

= 40, κ = 1

σmax

= 40, κmax

= 10

Ref

lect

ed P

ower

(dB

)

f (Ghz) max 40σ =

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EE699 S. Gedney

Some Advanced PML Concepts 8-5

PML for Lossy (Conducting) Media o Consider the application of PML in a Lossy medium:

0 0

0 0

0 0

y z

xz yy z x

x zx z o r yz x

o yy x zx y

x y

z

s ssH H E

s sH H j Ej s

H H Es ss

∂ ∂∂ ∂

∂ ∂∂ ∂∂ ∂∂ ∂

σωε εωε

⎡ ⎤⎢ ⎥⎢ ⎥−⎡ ⎤ ⎡ ⎤⎢ ⎥⎛ ⎞⎢ ⎥ ⎢ ⎥− = + ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎢ ⎥⎝ ⎠⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

o Previously, we introduced the constitutive relation for the lossless medium: x

z o r zz

sD Es

ε ε= . For a lossy medium, this would be: (1 ) xz o r z

o z

sD Ej sσε εωε

= + . This

would lead to a second-order auxiliary equation. o Rather, use the constitutive relations:

xz z

z

sP Es

= , z y zQ s P=

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EE699 S. Gedney

Some Advanced PML Concepts 8-6

o Given ii i

o

sjσκωε

= + , we can rewrite these as:

( ) ( )xzz z x z z z x z o z z z o x x z

o o

s P s E P E j P j Ej j

σσκ κ ωε κ σ ωε κ σωε ωε

⎛ ⎞ ⎛ ⎞= ⇒ + = + ⇒ + = +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

( )yz y z o z o y y z

o

Q P j Q j Pjσ

κ ωε ωε κ σωε

⎛ ⎞= + ⇒ = +⎜ ⎟⎝ ⎠

o In the time domain, these are expressed as: z z

o z z z o x x zP EP Et t

ε κ σ ε κ σ∂ ∂+ = +

∂ ∂

z zo o y y z

Q P Pt t

ε ε κ σ∂ ∂= +

∂ ∂

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EE699 S. Gedney

Some Advanced PML Concepts 8-7

o Update procedure (z-projection):

, , 1/ 2 , , 1/ 2

1 1 1 1 1 1 1 1, , , , , , , ,2 2 2 2 2 2 2 2

1/ 2 1/ 22 22 2i j k i j k

i j k i j k i j k i j k

n no oz z

o y o

n n n ny y x x

t tQ Qt t

H H x H H y

ε σ εε σ ε σ+ +

+ + − + + + − +

+ −− ∆ ∆= + ⋅

+ ∆ + ∆

⎧ ⎫⎛ ⎞ ⎛ ⎞− ∆ − − ∆⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎩ ⎭

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

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

1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2

1/ 2 1/ 2 1/ 2

22 22 2

i j k i j k i j k i j k i j k i j k

i j k i j k i j k

n n n n n nz z z z z z

o y y o

o y yn n noz z z

o y y o y y

P P P P Q Q

t tt

P P Q Qt t

ε κ σ ε

ε κ σ εε κ σ ε κ σ

+ + + + + +

+ + +

+ − + − + −

+ − +

− + −+ =

∆ ∆− ∆

⇒ = + −+ ∆ + ∆ ( ), , 1/ 2

1/ 2i j k

nz +

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

, , 1/ 2 , , 1

1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2

1/ 2

2 222

i j k i j k i j k i j k i j k i j k i j k i j k

i j k i j k

n n n n n n n nz z z z z z z z

o z z o x x

n o x xz z

o x x

P P P P E E E E

t ttE Et

ε κ σ ε κ σ

ε κ σε κ σ

+ + + + + + + +

+ +

+ − + − + − + −

+

− + − ++ = +

∆ ∆− ∆

⇒ =+ ∆

( ) ( )( )/ 2 , , 1/ 2 , , 1/ 2

1/ 2 1/ 2 1/ 21 2 22 i j k i j k

n n no z z z o z z z

o x x

t P t Pt

ε κ σ ε κ σε κ σ + +

− + −+ + ∆ − − ∆+ ∆

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EE699 S. Gedney

Some Advanced PML Concepts 8-8

o Repeated for the rest of the E and H-field projections. o Note that the conductivity requires an additional auxiliary variable.

o Three-step update process for each field projection

o Nevertheless, medium is perfectly matched. The κ terms aide in the absorption of the

attenuating and evanescent waves.

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EE699 S. Gedney

Some Advanced PML Concepts 8-9

EXAMPLE – Dipole over a Lossy Half Space

16 mm

16 m

m

d

d

PML Region

A

B

J

0.8 mm0.

8 m

m

ε r +σ

jωε o

ε o

Geometry for the FDTD simulation of a TEz polarized wave excited by a vertical

electric current source above a lossy half space. The working volume was 40 cells by 40 cells (∆x = ∆y = 0.4 mm), and the PML layers are 10 cells thick (d = 4 mm). (εr = 10, σ = 0.3 S/m). (Lossy medium extends thru PML to PEC boundaries)

( )2( , , ) 2

t totwo

y o ow

t tJ x y t et

−−−= − , with tw = 2.65258 x 10-11 s & to = 4 tw

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EE699 S. Gedney

Some Advanced PML Concepts 8-10

Reflection Error Point A ( 1κ = ) Point B ( 1κ = )

Point A ( 3m = ) Point B ( 3m = )

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EE699 S. Gedney

Some Advanced PML Concepts 8-11

o Vertical electric field at points A and B due to the electric current dipole radiating above

the lossy half space computed with the PML termination as compared with a reference solution computed on an extremely large grid. 10 cell PML, σmax = 1.1 σopt, m = 3.5, κmax = 7.

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0 200 400 600 800 1000

Pt. A - ReferencePt. AE y (V

/m)

t (ps)

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0 200 400 600 800 1000

Pt. B - ReferencePt. B

E y (V/m

)t (ps)

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EE699 S. Gedney

Some Advanced PML Concepts 8-12

Complex-Frequency Shifted PML Tensor

o The PML still suffers from o Large reflections for very slowly varying wave (low frequency, large evanescence,

highly-oblique angles of incidence) o The majority of the reflection error comes from the front boundary interface

c.f., [J. P. Berenger, "Evanescent waves in PML's: Origin of the numerical reflection in wave-structure interaction problems," IEEE Transactions on Antennas and Propagation, vol. 47, pp. 1497-1503, 1999.]

o This is partly related to the pole in the metric tensor at the origin ( 0ω = ). o We can circumvent this by introducing a metric tensor coefficient of the form:

xx x

x o

sj

σκα ωε

= ++

o This is referred to as the Complex-Frequency-Shifted (CFS) PML tensor coefficient. First proposed by M. Kuzuoglu and R. Mittra, in "Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers," IEEE Microwave and Guided Wave Letters, vol. 6, pp. 447-449, 1996, for frequency dependent FEM. First proposed for FDTD, with implementation, by Gedney in Chapter 7 of Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, Allen Taflove, Ed., Artech House, Boston, 1998.

o Consider a wave that is incident on the PML with a complex wave-number:

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EE699 S. Gedney

Some Advanced PML Concepts 8-13

( ) ( ) cosi i x i ix xi i x x x xi i ix x x x ox x x x x o x x x oj x x j xj s x js x x j j x je e e e e e e

σ σ σα β κ α ω µε θα β α ωεγ κ α α ωε κ β α ωε⎛ ⎞

− + + − −⎜ ⎟− + +− − + − +⎝ ⎠= = = o At very low frequency ( 0ω → ):

ixxi

x x xx

xe eσ

ακ α α

−−≈

o Which is purely attenuating. In fact, now both xκ and xσ attenuate the wave.

o At higher frequencies, where o xωε α>> , the CFS PML behaves as the original PML.

o The consequence of this is that we can better attenuate low-frequency interactions with the PML. A rigorous study of this was presented by Berenger in: “Numerical reflection from FDTD-PMLs: A comparison of the split PML with the unsplit and CFSPML,” IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, vol. 50 (3): pp. 258-265 MAR 2002

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EE699 S. Gedney

Some Advanced PML Concepts 8-14

o Caveat: For a purely propagating wave, the CFS PML tensor can provide little to no

attenuation. Consider a TEM mode with propagation j xe β− , where β ω µε= .

o In the CFS PML, the wave propagate as: x

x x oj x

j x je eσ

βκ β α ωε

−− + .

o In the limit 0ω → , this becomes: x

x xj x

j xe eσ

βκ β α

−− , which has no attenuation at all!

o To circumvent this, we need to scale xα . It is recommended to scale xα to be maximum at the interface, and minimum at the back wall.

o Two different polynomial scalings:

( ) max 1m

x xxxd

α α⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

, ( ) max 1m

x xxxd

α α ⎛ ⎞= −⎜ ⎟⎝ ⎠

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EE699 S. Gedney

Some Advanced PML Concepts 8-15

Implementing the CFS PML Method 1: Auxiliary Equation Approach

o Given the CFS tensor coefficients:

, ,yx zx x y y z z

x o y o z o

s s sj j j

σσ σκ κ κα ωε α ωε α ωε

= + = + = ++ + +

o Lets work with the x-projection of Ampere’s Law:

yzx

HHj sEy z

ωε∂∂

= −∂ ∂

, where 0 0

0 00 0

y z x

z x y

x y z

s s ss s s s

s s s

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

o Make the substitution: 1 , ,z y z

x x x z x x y xx

P E P s P P s Ps

= = =

o Then, from y zx y xP s P=

( ) ( )yy z y zx y x y o x y y o y x

y o

P P j P j Pj

σκ α ωε κ α ωε σ

α ωε⎛ ⎞

⎡ ⎤= + ⇒ + = + +⎜ ⎟ ⎣ ⎦⎜ ⎟+⎝ ⎠

o Transforming this to the time domain:

( )y y z zy x o x y y y x t o xP P P P

t t∂ ∂α ε κ α σ κ ε∂ ∂

+ = + +

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EE699 S. Gedney

Some Advanced PML Concepts 8-16

o Therefore, to write down the CFS equations, we can use the following procedure:

1. yy zx

HHj Py z

ωε∂∂

= −∂ ∂

2. ( ) z z y yy y y x y o x y x o xP P P P

t t∂ ∂κ α σ κ ε α ε∂ ∂

+ + = +

3. ( ) z zz z z x z o x z x o xP P P P

t t∂ ∂κ α σ κ ε α ε∂ ∂

+ + = +

4. ( )x x o x x x x x x o xE E P Pt t∂ ∂α ε κ α σ κ ε∂ ∂

+ = + +

o We can discretize this in a second-order accurate manner as before. o Problem: We’ve increased to 4 unknowns per field projection! In 3D, this increases the

number of base field vectors from 6 to 24! o A more efficient procedure was introduced in: [Roden JA, Gedney SD, “Convolution PML

(CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media”, Microwave and Optical Technology Letters, vol. 27, no. 6, pp. 334-339, Dec. 5, 2000]

o Note that the CPML procedure improves efficiency of the algorithm, but exactly the same reflection properties.

Method 1: Convolutional PML - CPML

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EE699 S. Gedney

Some Advanced PML Concepts 8-17

o If we combine the split field projections of Berenger, we can express Maxwell’s equations in the PML region in a “stretched coordinate form”. Thus,

1 1 1 1,

1 1 1 1,

1 1 1 1,

y yz zx x

y z y z

x xz zy y

z z z x

y yx xz z

x y x y

H EH Ej E j Hs y s z s y s z

H EH Ej E j Hs z s x s z s x

H EH Ej E j Hs x s y s x s y

ωε ωµ

ωε ωµ

ωε ωµ

∂ ∂∂ ∂= − − = −

∂ ∂ ∂ ∂

∂ ∂∂ ∂= − − = −

∂ ∂ ∂ ∂

∂ ∂∂ ∂= − − = −

∂ ∂ ∂ ∂

o Note, this is derived by first dividing both sides by ks , then adding split fields. o We can then express this in the time domain as:

ε ∂ ∂ ∂= ∗ − ∗

∂ ∂ ∂( ) ( )x y z z yE s t H s t H

t y z

o Where ( )ys t is the inverse Fourier transform of 1/ ys , and similarly for ( )zs t

o Given that , ( , , or )ii i

i o

s i x y zj

σκα ωε

= + =+

, we can show that

( ) ( ) ( )2 ( ) ( )

i i

o i ot

ii i

i o i i

t ts t e u t t

σ αε κ εδ δσ ζ

κ ε κ κ

⎛ ⎞− +⎜ ⎟⎝ ⎠= − = +

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EE699 S. Gedney

Some Advanced PML Concepts 8-18

( ) ( ) ( )2 ( ) ( )

i i

o i ot

ii i

i o i i

t ts t e u t t

σ αε κ εδ δσ ζ

κ ε κ κ

⎛ ⎞− +⎜ ⎟⎝ ⎠= − = +

o Now, we can rewrite the x-projection of Ampere’s law as:

ε ζ ζκ κ

∂ ∂ ∂ ∂ ∂= − + ∗ − ∗

∂ ∂ ∂ ∂ ∂1 1 ( ) ( )x z y y z z yy z

E H H t H t Ht y z y z

o At this point, we need to compute the convolutions efficiently. o Define the “Discrete Impulse Response” for ζ ( )i t as:

( ) ( ) ( )1 1

0 2( )i

t t o i o

it t

i t

i o

m mi

im mo i

m

i

Z m d e d

a e

σ α τε κ ε

σα

κ ε

σζ τ τ τε κ

⎛ ⎞− +⎜ ⎟+ ∆ + ∆⎝ ⎠

∆ ∆

⎛ ⎞ ∆− +⎜ ⎟⎝ ⎠

= = −

=

∫ ∫

where ( )2

1.0i t

ii oi

ii i i i

a eσ

ακ εσ

σ κ κ α

⎛ ⎞∆− +⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟= −⎜ ⎟+ ⎝ ⎠

o Then, the discrete convolution of ζ ( )i t with a discrete function f(t) sampled at n t∆ is:

( ) ( ) ( )1

0( )

i

N

i om

f t t Z m f n mζ−

=

∗ = −∑

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EE699 S. Gedney

Some Advanced PML Concepts 8-19

o Inserting this into Ampere’s law, and discretizing in time leads to: o

( ) ( )

1 1 1 1 1 1 1 1 1 1, , , , , , , , , , , ,2 2 2 2 2 2 2 2 2 2

1 1 1 1 1 1, , , , , ,2 2 2 2 2 2

1 1 1 11 2 2 2 2

0

1 1 12 2 2

1 1

0 0

i j k i j k i j k i j k i j k i j k

i j k i j k i j k i

y z

n n n nn nx x z z y y

rt y y z z

n m n m n mN Nz z y y

o om my

E E H H H H

H H H HZ m Z m

ε εκ κ

+ + + + + − + + + −

+ + + − + + +

+ + + ++

− + − + − +− −

= =

− − −= −

∆ ∆ ∆

− −+ −

∆∑ ∑1 1, ,2 2

12

j k

n m

z

− +

o This is still too expensive, since it requires a full summation at every time step!

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EE699 S. Gedney

Some Advanced PML Concepts 8-20

o Fortunately, given the properties of ( )ioZ m , this can be performed in a simplified manner

using the “Recursive Convolution Method” [R. J. Luebbers and F. Hunsberger, "FDTD for Nth-Order Dispersive Media," IEEE Transactions on Antennas and Propagation, vol. 40, pp. 1297-1301, 1992.]. Thus, the update can be re-written as:

1 1 1 1 1 1 1 1 1 1, , , , , , , , , , , ,2 2 2 2 2 2 2 2 2 2

1 1, , , ,2 2

1 1 1 11 2 2 2 21 12 2

0i j k i j k i j k i j k i j k i j k

e exy xzi j k i j k

n n n nn nx x z z y y n n

rt y y z z

H H H HE Eε ε ψ ψ

κ κ+ + + + + − + + + −

+ +

+ + + ++

+ +− −−

= − + −∆ ∆ ∆

where

1 1 1 1 11 , , , , , ,, , 2 2 2 2 22

1 1 1 12 2 2 2

xy xyi j k i j k i j ki j k

n n n ny y ye e z zb a H Hψ ψ

+ + + + −+

+ − + +⎛ ⎞= + − ∆⎜ ⎟

⎝ ⎠

1 1 1 11 1 , , , ,, , , , 2 2 2 22 2

1 1 1 12 2 2 2

xz xz i j k i j ki j k i j k

n n n nz z ze e y yb a H Hψ ψ

+ + + −+ +

+ − + +⎛ ⎞= + − ∆⎜ ⎟

⎝ ⎠

( )σ ακ ε

⎛ ⎞∆− +⎜ ⎟⎝ ⎠= =, , , or i ti

i oib e i x y z

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EE699 S. Gedney

Some Advanced PML Concepts 8-21

o Advantages of the CPML formulation: 1. Reduced the number of auxiliary variables from 3 to 2 2. Lossy medium is similarly treated. Updates do not require additional terms!:

1 1 1 1 1 1 1 1 1 1 1 1, , , , , , , , , , , , , , , ,2 2 2 2 2 2 2 2 2 2 2 2

1 1, , , ,2 2

1 1 1 11 1 2 2 2 21 12 2

0 2i j k i j k i j k i j k i j k i j k i j k i j k

e exy xzi j k i j k

n n n nn n n nx x x x z z y y n n

rt y y z z

H H H HE E E Eε ε σ ψ ψ

κ κ+ + + + + + + − + + + −

+ +

+ + + ++ +

+ +− −− −

+ = − + −∆ ∆ ∆

3. No more expensive than standard PML terminating a lossy medium 4. The convolutional terms ψ are simply added in as soft source terms – standard

FDTD update loops are not corrupted. 5. Very easy to implement in a standard FDTD code with inhomogeneous and lossy

medium.

o In practice, , ,i i iα κ σ are scaled along the i-direction. The same rule applies to CPML as PML. You sample , ,i i iα κ σ at the edge-centers for E-field updates, and you sample

, ,i i iα κ σ at the face-centers for H-field updates. o The update coefficients can them be pre-calculated and stored as one-dimensional

vectors

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EE699 S. Gedney

Some Advanced PML Concepts 8-22

EXAMPLE – Scattering by a PEC Plate

a = 25 mm, b = 100 mm, PML is 10 cells thick, and 3 cells from plate in all dimensions

Maximum relative reflection error over 2,000 time steps for 10 cell thick PML

Time (seconds)0.0 2.0e-9 4.0e-9 6.0e-9 8.0e-9 1.0e-8 1.2e-8

Ref

lect

ion

Erro

r (dB

)

-150

-140

-130

-120

-110

-100

-90

-80

-70

-60

-50

-40

PML, α=0.0

CPML, α=0.05

o A significant improvement is the late time reflection error.

CPML

,ε σ

sJ

2 c⋅

b ca

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EE699 S. Gedney

Some Advanced PML Concepts 8-23

0α = 0.05α =

σmax/σopt

0.5 1.0 1.5

κ max

5

10

15

20

25

-42

-42

-42

-42

-42

-42

-42

-42

-36

-36

-36

-36

-36

-36

-36

-30

-30

-48

-30

-30

-30

-30

-24

-24

-24

-24

-18

-18

-18

-18

-12

-12-6

σmax/σopt

0.5 1.0 1.5

κ max

5

10

15

20

25

-48

-48

-48

-48

-48

-54

-54

-54

-54

-60

-60-60

-60

-60

-60

-66

-66

-66

-66

-66

-48-54-54

-42

-42

-42

-42

-36

-36

-36-30

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EE699 S. Gedney

Some Advanced PML Concepts 8-24

Maximum relative reflection error over 2,000 time steps for 10 cell PML PML 3 cells from plate & 0.05α = PML 17 cells from plate & 0α =

σmax/σopt

0.5 1.0 1.5

κ max

5

10

15

20

25

-48

-48

-48

-48

-48

-54

-54

-54

-54

-60

-60-60

-60

-60

-60

-66

-66

-66

-66

-66

-48-54-54

-42

-42

-42

-42

-36

-36

-36-30

σmax/σopt

0.5 1.0 1.5

κ max

5

10

15

20

25

-54

-54

-54

-54

-54

-54

-54-48

-48

-48

-48

-48

-60

-60

-60

-60

-60

-60

-60

-42

-42

-42

-42

-66

-36

-36

-36

-36

-30

-30

-30

-24

(note PML boundary is equidistant from plate on all sides) Lattice 1: × ×126 51 26 Lattice 2: × ×154 79 54 Memory Savings Factor: 2× × ×154 79 54 /3× × ×126 51 26 = 2.6

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EE699 S. Gedney

Some Advanced PML Concepts 8-25

Late Time-Stability of PML

o Consider a PML termination of a single boundary with no overlapping corner regions (a bounded wave guide would be an example).

o For simplicity, assume a lossless, homogeneous region, with a z-normal: 0 0

0 00 0 1/

z yy z z x

x z o z yz x

y x z zx y

H H s EH H j s EH H s E

∂ ∂∂ ∂

∂ ∂∂ ∂∂ ∂∂ ∂

ωε

−⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥− =⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥− ⎣ ⎦ ⎣ ⎦⎣ ⎦

, 0 0

0 00 0 1/

z yy z z x

x z o z yz x

y x z zx y

E E s HE E j s HE E s H

∂ ∂∂ ∂

∂ ∂∂ ∂∂ ∂∂ ∂

ωµ

−⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥− = −⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥− ⎣ ⎦ ⎣ ⎦⎣ ⎦

o The PML updates for the transverse fields appear in the form of a lossy medium. The

normal fields, as a Debye medium. We write these in a two step update process: &y x o z z z zx yH H j P E s P∂ ∂

∂ ∂ ωε− = = o In the time-domain:

&z z z zy x o zx y

o

P E PH H Pt t t

∂ ∂∂ ∂

σεε

∂ ∂ ∂− = = +

∂ ∂ ∂

o Comparing these two equations, we recognize that zPt

∂∂

is proportional to the transverse

derivatives of the transverse magnetic field. We have a dual situation from Faradays law

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EE699 S. Gedney

Some Advanced PML Concepts 8-26

o In the very late time, as the fields reach steady state, the transverse derivatives of the transverse fields will tend to zero.

o As a consequence 0 zy x ox y

PH Ht

∂ ∂∂ ∂ ε ∂

− = =∂

o Therefore, the auxiliary equation reduces: z zz

o

E Pt

σε

∂=

∂.

o Where, since 0zPt

∂=

∂, then zP is a constant in time. Thus, the solution to the ODE

z zz

o

E Pt

σε

∂=

∂ is simply:

0

zz z z

o

E P t Eσε

= + , which is a linear function in time! Similarly,

for the dual field 0

zz z z

o

H Q t Hσε

= + . Therefore, in the event that the transverse gradient

of the fields is zero, the normal fields in the PML can grow linearly in time!

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EE699 S. Gedney

Some Advanced PML Concepts 8-27

o This has been the topic of some publications:

S. Abarbanel, D. Gottlieb, and J. S. Hesthaven, “Long time behavior of the PML equations in computational electromagnetics,” Journal of Scientific Computing, vol. 17, no. 1-4, pp. 405-422, 2002. E. Becache, P. Petropoulos, & S. D. Gedney, “On the long-time behavior of unsplit perfectly matched layers,” IEEE Transactions on Antennas and Propagation, to appear. S. J. Yakura, D. Dietz, and A. D. Greenwood, “A dynamic stability analysis of the PML method,” Interaction Note Series Note 582, Kirtland Air Force Base, Albuquerque, NM, November 1, 2002

o Consequences of the late time instability: 1. The late time instability does not occur until after the fields in the working volume are

~ zero, which is beyond steady state for a radiation/scattering problem 2. The linear growth of the fields only occurs in the PML, when using the UPML

formulation. ∴ it does not corrupt the fields in the working volume. For the split-field PML, the linear growth can affect the fields in the working volume.

3. If the PML boundaries are overlapping, this problem does not occur. . 4. Late time instability does not occur for CFS PML. Why? The late time-stability is

related to the weakly causal nature of the constitutive parameters.

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EE699 S. Gedney

Some Advanced PML Concepts 8-28

Examples from Bécache, Petropoulos & Gedney

TMz polarization. 1x y mm∆ = ∆ = , 122.35 10 st −∆ = × . ( )( )2

2

2 sin(2 )o

w

t tt

y cw

tJ t e f tt

π− −

= − ,

with 103.183 10 swt−= × , 4o wt t= , and 93 10 Hzcf = × . Standard PML with m = 4,

max 10.61 S/mσ = , max 1κ = . We are monitoring the fields inside the PML in the very late time (~100,000 time steps).

60 mm

60 mm

PML 10 mm

x

z

y

( )yJ tPEC

PML

( )o , oε µ

-6 10-6

-4 10-6

-2 10-6

0

2 10-6

4 10-6

6 10-6

0 5 10-8 1 10-7 1.5 10-7 2 10-7

Hz(A/m)Hx(A/m)Ey(V/m)

t(sec) Normal field shows linear growth in the late time as predicted!

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EE699 S. Gedney

Some Advanced PML Concepts 8-29

-4 10-9

-3 10-9

-2 10-9

-1 10-9

0

1 10-9

2 10-9

3 10-9

4 10-9

0 5 10-8 1 10-7 1.5 10-7 2 10-7

Hz (A/m)Hx (A/m)Ey (V/m)

t(sec) Repeat the same experiment with CFS PML, with 0.08α = . The field remains constant – no linear growth. Here we have:

( )z z zz z z z z z z o z z z z z

z o

H QH s Q Q H Qj t t

σκ α ε κ α σ σα ωε

⎛ ⎞ ∂ ∂= = + ⇒ + = + +⎜ ⎟+ ∂ ∂⎝ ⎠

In the late time, if 0zQt

∂=

∂, then z

z z o z zHH Qt

α ε σ∂+ =

∂, which has solution: z

z zz

H Qσα

= .

Therefore, zH is a non-zero constant in the late time. Nevertheless, it is quite small.

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-1

FDTD ANALYSIS OF PRINTED CIRCUITS • Microwave Circuits: * Communications Systems * Microstrip, Stripline, Co-Planar Wave Guide * Passive and Active Circuit Devices * Printed Antennas

Wilkinson power divider printed on a 15 mil TMM substrate

Port 1

36.1 mil

19.5 mil 100 Ω

Port 3

Port 2

10.5 milgap

• Printed Circuit Boards or Packages: * EMC Modeling (Radiation/Coupling) * Signal Integrity on Electrical Interconnect networks with Passive and active Loads

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-2

SOURCE MODELING • Choice of the source will be problem dependent. • Introduced a number of sources Section 6 of the notes. Most commonly used sources

for this type of modeling are:

* Voltage Source (Soft Source) * Thévenin Source * Current Sheet

• Analysis of Circuits with linear loads will typically be source independent. • Typically choose a broad band pulse (Gaussian) providing a broad-band analysis of the

device with a single simulation.

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-3

PARAMETERIZATION OF CIRCUITS • Fundamental Measured Quantities:

* Line Voltage and Current • Uniform Transmission Line Analysis

* Characteristic Line Impedance * Propagation Constant (axial) and effective dielectric

• Network Analysis (General N-port)

* Input Impedance * Scattering Parameters * Impedance/Admittance Matrix * Radiated Fields

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-4

FUNDAMENTAL MEASURED QUANTITIES • Line Voltage

* Quasi-Static Approximation

V12 = −

r E ⋅ d

r l

2

1

* Example: Microstrip Circuit

y = 0

y = h

w

, ,

1

10i j k

h ny

yjy

V E d E y==

=− ⋅ =− ∆∑∫

where, ny1 = h / ∆y

* Valid if dimensions of the cross-section of the device are electrically small. * Most accurate if performed at the physical center of the microstrip

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-5

• Line Current * Quasi-Static Approximation

I =

r J ⋅ dr s

S∫ ≈

r H ⋅ d

r l

C∫

* Example: Microstrip Circuit

y = 0

y = h

w

C

* Approximations:

• Discrete Tangential H-fields are 1/2 cell removed from the conductor surface. Therefore, there is a small amount of displacement current present in the line integral as well.

* Evaluation:

, 1, 1 , 1 1, 1 1 1, 1, 1 2, 1, 1

2

1i j k i j k i j k i j k

i

x x y yi iC

H d x H H y H H− −

=

⎡ ⎤ ⎡ ⎤⋅ ≈ ∆ − +∆ −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∑∫

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-6

, 1, 1 , 1 1, 1 1 1, 1, 1 2, 1, 1

2

1i j k i j k i j k i j k

i

x x y yi iC

H d x H H y H H− −

=

⎡ ⎤ ⎡ ⎤⋅ ≈ ∆ − +∆ −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∑∫

Hxi1, j1, k1

Hxi , j1−1,k 1Hy i1

−1,j

1,k1

x

y

C

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-7

LINE IMPEDANCE • The Generalized Line Impedance can simply be Calcuated as

Z(ω ) =V ω( )I(ω)

where, V(ω) and I(ω) are the line voltage and current at a fixed point on the transmission line, and are found by Fourier transforming the time-dependent voltage or currents.

• A difficulty encountered with this formulation is that the transverse magnetic fields and the transverse electric fields are separated by one-half of a cell.

• The voltage and current are further separated by one-half of a time-step. • This leads to error in the line impedance. Namely, in the discrete space

˜ Z (ω ) = V (ω )e−α (z+∆z/ 2)e− jβ (z+∆z/2)e jω(t +∆t /2)

I(ω )e−αze− jβ ze jωt

= V (ω)I(ω)

e−α∆z /2e− jβ∆z/ 2e jω∆t/2

where α and β are the attenuation and propagation constants • An alternate scheme of computing the line impedance proposed by Fang [IEEE MGWL,

vol. 5, pp. 1-6, 1995] helps to overcome this phase shift error.

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-8

ACCURATE EVALUATION OF THE LINE IMPEDANCE

I1 I2

V

Top View

• Compute I1 and I2 on either side of the voltage probe • Compute Characteristic Impedance as:

˜ Z (ω ) =V(ω)

I1(ω )I2 (ω)

• In an ideal sense: ˜ Z (ω ) ≈

V(ω )e−α (z+∆z/ 2)e− jβ (z+∆z/ 2)e jω(t +∆t /2)

I(ω )e−αze− jβze jωt ⋅ I(ω)e−α (z+∆z )e− jβ(z+∆z)e jω (t+∆t)=

V(ω )I(ω )

• Thus, the phase Errors Cancel Out, greatly improving the accuracy • For a uniform line matched at both ends, ˜ Z (ω ) represents the characteristic line

impedance • For a uniform line terminated into some load such as a lumped load, antenna, or N-port

network, ˜ Z (ω ) represents the input impedance of that port.

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-9

EXAMPLE OF COMPUTING THE CHARACTERISTIC LINE IMPEDANCE VIA THE FDTD METHOD

• Grounded Co-Planar WaveGuide (GCPWG) • Cross section of the uniform GCPWG: w = 10 mils, s = 5 mils, and wf = 40 mils, and the

50-mil spaced vias were placed at the centers of the outer conductors

ε = 9.8r25 mils

wf s

w

I1 =

r H ⋅ d

r l ∫

I2 =r H ⋅ d

r l ∫

V =

r E ⋅ d

r l ∫

I fin =

r H ⋅ d

r l ∫

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-10

• Cross section of the uniform GCPWG: w = 10 mils, s = 5 mils, and wf = 40 mils, and the

50-mil spaced vias were placed at the centers of the outer conductors

ε = 9.8r25 mils

wf s

w

44

45

46

47

48

49

50

51

0 10 20 30 40 50

SQRT(I1 I

2)

I1

|Z|

(ohm

s)

f(GHz)

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-11

• Comparison of the Line Impedance of the GCPWG loaded with vias, and without vias.

45

46

47

48

49

50

51

0 10 20 30 40 50

No ViaWith Via

Line

Impe

danc

e (o

hms)

f(GHz)

49.1 (0.22 o)

49.43 (0.200 o)

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-12

PROPAGATION CONSTANT AND EFFECTIVE DIELECTRIC • The propagation constant of the fundamental mode can be estimated by simply probing

the line voltage or current on a uniform line that is matched at both the source and load ends

• Assuming: ˜ V (ω , z) = Voe

−(α z+ jβz)z • Probe the voltage at two points z1 and z2 (where z2 > z1): • Then:

˜ V (ω , z2 )˜ V (ω , z1)

≅Voe−(αz + jβz )z2

Voe−(α z + jβz)z1= e−(αz + jβz )(z2− z1)

∴ α z = −Re ln˜ V (ω ,z2 )˜ V (ω ,z1)

⎝ ⎜

⎠ ⎟

⎧ ⎨ ⎩

⎫ ⎬ ⎭

/ z2 − z1( ), βz = −∠˜ V (ω, z2 )˜ V (ω, z1)

⎝ ⎜

⎠ ⎟ / z2 − z1( )

• Then, assuming a quasi-static or TEM approximation,

( )

2

2

z o o eff

zeff

o o

β ω µ ε ε

βεω µ ε

∴ ≈

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-13

SCATTERING PARAMETERS • A convenient characterization of a microwave circuit device is through its scattering

parameters, or the S-parameters • Scattering Parameters are the elements of the [S]-matrix

N-port

V1+

V1−

V2−

V2+ V N

+

V N−

• Given an N-port Network, let Vi

+ be the input or incident voltage, and Vi- be the reflected

voltage on the i-th port.

V1−

V2−

M

V N−

⎢ ⎢ ⎢ ⎢ ⎢

⎥ ⎥ ⎥ ⎥ ⎥

=

S11 S12 L S1NS21 S22 S2 NM M

SN1 L SNN

⎢ ⎢ ⎢

⎥ ⎥ ⎥

V1+

V2+

M

VN+

⎢ ⎢ ⎢ ⎢ ⎢

⎥ ⎥ ⎥ ⎥ ⎥

• Each term of the S-matrix can be determined as

Sij =Vi

V j+

Vk+ =0 for k ≠ j

• Note that these are frequency dependent values, and it is assumed that all ports have the same characteristic line impedance

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-14

SCATTERING PARAMETERS (CONT'D)

Sij =Vi

V j+

Vk+ =0 for k ≠ j

• To compute Si,j, excite the j-th port, and terminate all ports with a matched load (including the j-th port to avoid multiple reflections).

• Matched condition is most often simulated by terminating the uniform line representing

each port into an absorbing boundary wall. This represents a true matched condition, rather than an approximate matched condition which can be achieved using a lumped load.

• Problem: On the j-th port, we can only measure the total line voltage, which is a

superposition of the incident and reflected voltage waves. (A soft source is used.) • To circumvent this, perform an independent simulation of a uniform line for the incident

field. To this end, the separation between the source and the port, and the source and the ABC wall must remain unchanged. Otherwise, there will be a phase shift between the incident and total line voltages, leading to errors.

• Then

Sij =Vi

V j+

Vk+ =0 for k ≠ j and i≠ j

, Sjj =V j

t − V j+

V j+

Vk+ =0 for k≠ j

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-15

EXAMPLE - 3 dB WILKINSON POWER DIVIDER 100 Ω

36 mils

19.74 mils

194.

4 m

ils

36 mils

15 mils ε r = 3.25

Side View

Top View

Port 1

Port 2Port 3

7.2 mils

• Simulation Procedure:

1. Excite Port 1 with an ideal voltage source with a Gaussian pulse profile, match all three ports into ABC walls, and probe the line voltage at all three ports

2. Excite Port 2 with an ideal voltage source, and probe the line voltages at all three ports which are again matched into ABC walls.

3. Solve an Auxiliary problem consisting of a uniform microstrip line representing ports 1 & 2. Excite ports with the same pulse, and measure the voltage at port (V1+ or V2+)

4. Fourier transform the line voltages, and compute the frequency dependent S-parameters S11, S21, and S31, S22, and S32. By symmetry S12 = S21 = S13, etc.

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-16

TIME-DEPENDENT PORT VOLTAGES WITH PORT 1 EXCITED BY GAUSSIAN SOURCE

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-17

• Magnitude of the S-parameters of the Wilkinson power divider illustrated in Fig. 7

modeled using the FDTD algorithm.

-40

-35

-30

-25

-20

-15

-10

-5

0

0 5 10 15 20 25 30 35 40

|S11

||S

21|

|S31

||S

32|

|S33

|

dB

f (GHz)

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-18

• Compared to HPADs Simulation of the Wilkinson Power Divider:

-40

-35

-30

-25

-20

-15

-10

-5

0

0 5 10 15 20 25 30 35 40

|S11

||S

21|

|S31

||S

32|

|S33

|

dB

f (GHz)

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-19

8-Way Power Divider

-35

-30

-25

-20

-15

-10

-5

0

-20

-15

-10

-5

0 5 10 15 20 25

FDTD

PGY

FDTD

PGY

|S11

| (d

B) |S

21 | (dB)

f (GHz)

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-20

EXAMPLE - 1 PORT PATCH ANTENNA

-20

-15

-10

-5

0

5

0 5 10 15 20

|S11

|

f (GHz)

2.46 mm

7.90 mm

12.45 mm

16.00 mm

• |S11| of a microstrip fed patch antenna (superimposed) printed on a 31.25 mil Duroid

subsrate(εr = 2.2) computed via the FDTD method. The FDTD lattice is terminated by a 10-cell thick uniaxial PML layer which is placed: 3 cells from the edge of the patch and 5 cells above the patch, and 10 cells from the edges of the patch.

• ∆x = 0.389 mm, ∆y = 0.4 mm, ∆z = 0.1588 mm, and ∆t = 0.441 ps. • How can you compute the input impedance from S11?

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-21

RADIATION FROM PRINTED STRUCTURES

• Previously we detailed the near-field to far-field transformation for radiating structures in a

homogeneous space • Cannot directly apply to structures printed on infinite substrates since a closed surface will

require an infinite planar surface, or a surface in an inhomogeneous space. • From the time-varying fields, equivalent surface currents on the surfaces of conductors, and

displacement currents within dielectrics are easily extracted. The far field radiation can then be computed from these currents.

• As previously seen, the far fields can be computed in either the time-domain or the

frequency domain. However, for infinite substrates the time delay of the radiated fields is dependent on the observation angle due to multiple reflections in the substrate, it has been elected to first Fourier transform the time-varying currents and compute the far fields in the frequency domain.

• The FDTD method can be used to model printed antennas within multilayered stratified

media. If the planar slabs are infinite in extent, then the radiated fields are computed using the surface currents (on the surface of the strips) and the layered Green's function.

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-22

RADIATION FROM ANTENNAS ON INFINITE SUBSTRATES • Given a printed patch antenna on a single substrate that is infinitely planar. The

conductors are assumed to be infinitessimally thin and lossless. The equivalent surface current density is computed as

r J (x, y, z,t) = ˆ n × (

r H +(x, y, z+ ,t) −

r H − (x, y, z− ,t))

PEC

• The currents are Fourier Transformed using a DFT or FFT • The far electric-fields are then computed via the integrals

Eφ =jωµo

4πre− jkor F(θ) sinφJx − G(θ )cosφJy( )e jko ′ r cosψ dA

S∫

Eθ = −jωµo

4πre− jkor G(θ )cosφJx + F(θ )sinφJy( )e jko ′ r cos ψ dA

S∫

where 1

1

2 tan( )cos( ) ,tan( )cos ( )

z

z

hFh j

β θθβ θ θ

=− Θ

1

1

2 tan( )cos( ) ,tan( ) cos

( )

z

rz

hGh j

β θθ εβ θθ

=−

Θ

and βz1 = koΘ(θ), and, Θ(θ) = εr − sin2 θ .

and θ and φ are the zenith and azimuthal angles, (origin at center of the antenna) • These integrals can be derived from a far-field approximation of the exact Green's

function using the method of steepest decent [W. Chew, Waves in Inhomogeneous Media, Van Nostrand Reihnold, 1989], or using Reciprocity [Jackson and Alexopoulos, IEEE Trans. on A.P., Sept. 85, pp. 976-987]

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-23

EFFICIENT COMPUTATION • If the far fields are needed over a very broad band, the time-varying currents can be

Fourier transformed using a fast Fourier-transform (FFT) algorithm. This, however, will require an extremely large data base since the complete time-history for each current element will need to be stored.

• An alternative, is to choose a finite number of discrete frequency points for which the far-

field will be evaluated, and perform a discrete Fourier-transform (DFT) throughout the time-updates. Since this involves updating an integral, only one complex number per frequency and per current element need be stored. This is easily stored in core memory.

• Simulation Procedure:

1. Excite structure with an independent source using standard FDTD lattice terminated with absorbing boundary conditions (open regions), or natural boundary conditions (e.g., ground plane).

2. Compute the instantaneous current densities on the conductor surfaces from the local tangential H-fields..

3. Assuming a recursive expression for the DFT, update the DFT at selected discrete frequencies.

4. After steady state conditions are reached, stop the time-domain simulation, and compute the far fields at the selected angles.

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-24

EFFICIENT COMPUTATION OF RADIATED FIELDS (CONT'D) • For the previous method, assume that there are Mxx My surface currents to store, times

Mf frequencies. • For the radiation pattern, there will be Mφ azimuthal angles, and Mθ zenith angles to

compute. If a full three-dimensional pattern is desired, typically, one would choose Mθ = 180 (0.5o increments), and Mφ = 720, or a total of ~130 k points. Often, only an E and H plane cut are desired, and then Mφ = 2, so only 360 points may be necessary.

• In either case, if Mxx My >> Mθx Mφ, then a more efficient scheme which would reduce storage would be the following: • Excite structure with an independent source. • Compute the instantaneous current densities on the conductor surfaces from the local

tangential H-fields.. • Again assume a recursive expression for the DFT. However, move the summation

over the surface inward, and the DFT summation outward. • Recursively update the fields for each angle θj and φk:

Eφi(θ j ,φ k ) = Eφi

(θ j ,φk ) + jωiµo

4πre− jkor F(θ j )sin φk Jx

S∑

− G(θ j )cosφk Jye jko ′ r cosψ j,k ∆x∆ye jωit∆t

and similarly for Eθ. • After steady state conditions are reached, stop the time-domain simulation. The fields

are already stored in local memory.

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-25

SINGLE PATCH ANTENNA

2.334 mm

1.945 mm

12.448 mm

16 mm

.795mm ε r = 2.2

-60

-50

-40

-30

-20

-10

0

Eθ( φ =0)

20 lo

g10

|E(

θ)|

0

30

60

90

-30

-60

-50

-40

-30

-20

-10

0

Eθ( φ =90)

20 lo

g10

|E(

θ)|

0

30

60

90

-30

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-26

4 Element Patch Array Antenna

• Center Fed Probe Source (50 Ω internal impedance) • 4 Element Array: Ez just beneath the substrate/conductor surface at time step 1000 (75

ps)

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-27

-40-35-30-25-20-15-10

-50

0

30

60

90

120

210

240

270

300

330

4 Element Patch Array Antennaf = 34 GHz, φ = 0o PlaneE

φ ( φ = 0)

Eθ ( φ = 0)

20 lo

g10

(|E(

θ)|)

Far field radiation pattern of the four element patch array antenna at 34 GHz – φ = 0o. Note that the electric fields are normalized to the maximum value.

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EE699 - FDTD S. Gedney

FDTD Modeling of Microwave Circuit Devices 9-28

-50

-40

-30

-20

-10

0

0

30

60

90

120

210

240

270

300

330

4 Element Patch Array Antennaf = 34 GHz, φ = 90 o PlaneE

φ ( φ = 90)

Eθ ( φ = 90)

20 lo

g10

(|E(

θ)|)

Far field radiation pattern of the four element patch array antenna at 34 GHz – φ = 90o. Note that the electric fields are normalized to the maximum value.

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-1

NON-UNIFORM FDTD METHODS • The Previous Lectures have dealt explicitly with FD-TD algorithms based on orthogonal,

regular Cartesian lattices. • Leads to a second-order accurate solution in both space and time. This error, however, is

only associated with the differential operators. • Additional error can be encountered through the boundary conditions imposed on the

discrete fields - specifically when boundaries do not align with the uniform grid. • This has lead to a number of studies of FD-TD based algorithms which rely on boundary

fitted lattice grids. * Non-Uniform Grid FDTD. * Non-Orthogonal Structured Grid Methods. * Non-Orthogonal Unstructured Grid Methods. * Finite Volume Time-Domain Methods (Unstructured Grids). • These Techniques offer near second-order accuracy, but offer the advantage of conformal

grids.

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-2

NON-UNIFORM ORTHOGONAL GRID FDTD METHODS • Non-Uniform Gridding Can be Achieved Via a Number of Techniques. * Subgridding Methods - Local Refinement of Mesh Based on an Integer Factor:

∆x

∆y

∆x/3

∆y/3

The field can be interpolated in a second-order accurate manner at the interface of the two grids. Time subsampling is used for the refined region. If the grid is subsampled at ∆x/m and ∆y/m, then ∆ts = ∆t/m.1,2

* Non-Uniform Gridding - Irregular Grid Spacing to conform to problem boundaries: - Method is numerically stable. - Method is Supraconvergent: Locally First Order, Global Second-Order Accuracy3. 1 D. T. Prescott and N. V. Shuley, " A method for incorporating different sized cells into the finite-difference time-domain analysis

technique", IEEE Microwave and Guided Wave Letters, Vol 2, No. 11, Nov. 1992. 2 S. S. Zivanovic, K. S. Yee, and K. K. Mei, "A subgridding method for the time-domain finite-difference method to solve

Maxwell's equations," IEEE Trans Microwave Theory Tech., Vol. 39,pp. 471-479, Mar. 1991. 3 P. Monk and E. Suli, "A convergence analysis of Yee's scheme on non-uniform grids," SIAM Journal on Numerical Analysis, vol.

31, pp. 393-412, April 1994.

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-3

IRREGULAR ORTHOGONAL GRID FDTD METHODS

xi , y j, zk

∆y j

∆xi

xi + ∆xi / 2

• Vertices of Irregular Grid Defined as 1-Dim. Arrays:

xi ; i = 1, Nx, yj ; j = 1, Ny, zk ; k = 1, Nz. • The edge lengths between vertices is also defined as:

1 1 1 ; 1, 1, ; 1, 1, ; 1, 1.i i i x j j j y k k k zx x x i N y y y j N z z z k N+ + +∆ = − = − ∆ = − = − ∆ = − = −

• The cell and edge centers are Defined by the Coordinates: xi+ 1

2= xi + ∆xi / 2, yj + 1

2= yj + ∆yj / 2, zk + 1

2= zk + ∆zk / 2

• Introduce a set of dual edge lengths representing the distances between the edge centers:

1 1

1

( ) / 2; 2, , ( ) / 2; 2, ,

( ) / 2; 2,

x yi i i x j j j y

zk k k z

h x x i N h y y j N

h z z k N− −

= ∆ + ∆ = = ∆ + ∆ =

= ∆ + ∆ =

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-4

∆ xi

∆ yj

S

Ex i + 12 , j,k

Exi + 12 , j+1,k

E yi,

j+1 2,

k

Ey i+

1,j+

1 2,k

Hz i+ 12, j+ 1

2 ,k

h xi

∆ xi−1 / 2 ∆ xi / 2

′ S

h yj

∆y j

−1/2

∆y j

/ 2

Hx i, j+ 12,k+ 1

2

Hx i, j− 12 ,k+ 1

2

Hy

i +1 2

,j, k

+1 2

Hy i−

1 2,

j, k+

1 2

Ez i, j,k+ 12

(a) (b)

Lattice faces bound by lattice edges defining surfaces of integration bound by closed

contours. (a) Lattice cell face bound by grid edges and a dual lattice edge passing through its center. (b) Dual lattice face bound by dual edges.

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-5

• The electric and magnetic fields, in the discrete non-uniform lattice space, are expressed as:

Exi+1

2 , j,k

n ≈ Ex (xi+ 12yj ,zk ,n∆t), Hx

i, j+12 ,k+ 1

2

n+ 12 ≈ Hx (xi , yj + 1

2, zk+ 1

2,(n + 1

2 )∆t),

Eyi, j +1

2,k

n≈ Ey(xi , yj+ 1

2, zk ,n∆t), Hy

i+ 12, j,k+1

2

n+ 12 ≈ Hy(xi+1

2, y j ,zk+ 1

2,(n + 1

2)∆t),

Ezi, j,k+1

2

n ≈ Ez (xi , yj ,zk + 12, n∆t), Hz

i+12, j +1

2,k

n+ 12 ≈ Hz (xi+ 1

2, yj + 1

2, zk ,(n + 1

2 )∆t).

• The non-uniform FD-TD algorithm is based on a discretization of Maxwell's equations expressed in their integral form, specifically, Faraday's law and Ampère's law:

E ⋅ d

′ C ∫ = −

∂∂t

B ⋅ d s

′ S ∫∫ −

M ⋅d s

′ S ∫∫ ,

H ⋅ d

C∫ =

∂∂t

D ⋅d s

S∫∫ + σ

E ⋅ d s

S∫∫ +

J ⋅ d s

S∫∫ .

• The surface integrals are performed over a primary and secondary lattice cell face, respectively, and the contour integral is performed over the edges bounding the faces.

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-6

• Evaluating the time derivatives using central difference approximations leads to the

Discrete approximation:

Exi+1

2, j+1,k

n ∆xi − Exi+1

2, j,k

n ∆xi − Eyi+1, j+1

2 ,k

n∆y j + Ey

i, j+12 ,k

n∆yj

⎣ ⎢

⎦ ⎥ =

− µ i+ 12, j+ 1

2,k

Hzi+1

2 ,j +12,k

n+ 12 − Hz

i+12, j +1

2,k

n− 12

∆t+ Mz

i+12, j+1

2 ,k

n+ 12

⎢ ⎢ ⎢

⎥ ⎥ ⎥ ∆xi∆yj

Hxi, j+1

2 ,k+12

n+ 12 hxi

− Hxi, j−1

2,k+12

n+ 12 hxi

− Hyi+1

2, j,k+12

n+ 12 hyj

+ Hyi −1

2, j,k+ 12

n+ 12 hyj

⎣ ⎢

⎦ ⎥ =

ε i, j,k+ 12

Ezi, j,k+1

2

n+1 − Ezi, j,k+1

2

n

∆t+

σi, j,k + 12

2

Ezi, j,k+1

2

n+1 + Ezi,j,k+1

2

n

∆t+ Jz

i, j,k+12

n+ 12

⎢ ⎢ ⎢

⎥ ⎥ ⎥ hxi

hyj

• ε i, j,k+ 1

2, σ i, j,k+ 1

2, and µi+ 1

2, j+ 12,k are averaged permittivity, conductivity, and permeability,

respectively, about the grid edges as done for the Yee-Algorithm.

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-7

• This finally leads to an explicit update scheme:

Hzi +1

2, j+12 ,k

n+ 12 = Hz

i+ 12, j+1

2,k

n− 12 − ∆t

µi+1

2, j+ 12,k

Exi +1

2, j+1,k

n − Exi+ 1

2, j,k

n⎛

⎝ ⎜

⎠ ⎟ ∆y j −

⎣ ⎢

Eyi +1, j +1

2,k

n− Ey

i, j +12,k

n⎛

⎝ ⎜

⎠ ⎟ ∆xi + Mz

i+12, j +1

2,k

n+ 12

⎦ ⎥ ⎥

Ezi,j,k+1

2

n+1 =2ε i, j,k+ 1

2− ∆tσi, j,k + 1

2

2ε i, j,k+ 12

+ ∆tσi, j,k + 12

⎝ ⎜ ⎜

⎠ ⎟ ⎟ Ez

i, j,k+12

n + 2∆t2ε i, j,k + 1

2+ ∆tσ i, j,k +1

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟ ⋅

Hxi,j +1

2,k+12

n+ 12 − Hx

i,j −12,k+1

2

n+ 12

⎝ ⎜

⎠ ⎟ hyj

− Hyi+ 1

2, j,k+12

n+ 12 − Hy

i−12, j,k+1

2

n+ 12

⎝ ⎜

⎠ ⎟ hxi

− Jzi,j,k+1

2

n+ 12

⎣ ⎢ ⎢

⎦ ⎥ ⎥

• Recall that the time-space stability derived using the Von-Neumann analysis was based on the Local discrete field characteristics. This is a result of the explicit nature of the algorithm. Therefore, the stability relationship is governed by the minimum grid spacing:

∆t < 1 co 1 ∆ximin( )2+1 ∆yjmin( )2

+1 ∆zkmin( )2

where ∆ximin

, ∆yjmin, and ∆zkmin

are the edge lengths along the x, y, and z directions, respectively, of the smallest grid cell in the non-uniform grid.

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-8

EFFICIENT IMPLEMENTATION • Normalize the Fields by their respective edge lengths, i.e.,

˜ e xi+1

2 ,j +1,k

n= ∆xi Ex

i+ 12,j +1,k

n , ˜ h xi, j +1

2,k +12

n+ 12 = hxi

Hxi, j+ 1

2,k+ 12

n+ 12

• Then:

˜ h zi+1

2, j+ 12,k

n+ 12 = ˜ h z

i+12, j +1

2,k

n− 12 −

hzk∆t

∆xi∆yjµi+ 12, j+ 1

2,k

˜ e xi+ 1

2,j +1,k

n − ˜ e xi+ 1

2, j,k

n⎛

⎝ ⎜

⎠ ⎟ −

⎣ ⎢

˜ e yi +1, j +1

2,k

n− ˜ e y

i, j+12,k

n⎛

⎝ ⎜

⎠ ⎟ + ˜ M z

i+12, j+ 1

2,k

n+ 12

⎦ ⎥ ⎥

˜ e zi,j,k+1

2

n+1 =2ε i, j,k+ 1

2− ∆tσi, j,k + 1

2

2ε i, j,k+ 12

+ ∆tσ i, j,k + 12

⎝ ⎜ ⎜

⎠ ⎟ ⎟ e z

i, j,k+12

n + ∆zk

hxihyj

2∆t2ε i, j,k+ 1

2+ ∆tσi, j,k + 1

2

⎝ ⎜ ⎜

⎠ ⎟ ⎟ ⋅

˜ h xi, j+1

2,k+12

n+ 12 − ˜ h x

i, j−12 ,k+1

2

n+ 12

⎝ ⎜

⎠ ⎟ − ˜ h y

i +12, j,k+1

2

n+ 12 − ˜ h y

i−12 ,j,k+1

2

n+ 12

⎝ ⎜

⎠ ⎟ − ˜ J z

i,j,k+12

n+ 12

⎣ ⎢ ⎢

⎦ ⎥ ⎥

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-9

• Or:

˜ h zi+1

2, j+ 1

2,k

n+ 12 = ˜ h z

i+12

, j +12

,k

n− 12 − Bi+1

2 , j +12 ,k ˜ e x

i+ 12

, j+1,k

n − ˜ e xi +1

2, j,k

n⎛

⎝ ⎜

⎠ ⎟ −

⎣ ⎢

˜ e yi +1, j +1

2,k

n− ˜ e y

i, j+12,k

n⎛

⎝ ⎜

⎠ ⎟ + ˜ M z

i+12, j+ 1

2,k

n+ 12

⎦ ⎥ ⎥

˜ e zi, j,k+1

2

n+1 = Di, j,k+ 12

˜ e zi, j,k+1

2

n + Ci, j,k + 12

˜ h xi, j+1

2,k+12

n+ 12 − ˜ h x

i, j−12 ,k+1

2

n+ 12

⎝ ⎜

⎠ ⎟ − ˜ h y

i +12, j,k+1

2

n+ 12 − ˜ h y

i−12 , j,k+1

2

n+ 12

⎝ ⎜

⎠ ⎟ − ˜ J z

i, j,k+12

n+ 12

⎣ ⎢ ⎢

⎦ ⎥ ⎥

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-10

EFFICIENT IMPLEMENTATION (CONT'D) • High-Level Language Program Structure: * Generate Irregular Grid to Best Fit Boundaries of Problem Geometry. * Identify Source and Probe Locations. * Compute Edge Lengths and Edge Centers and Store in 1D Arrays. * Compute the Averaged Material Parameters. * Compute the Coefficient Arrays. * Initialize Coefficient Arrays for ABC (Note that Higdon ABC or PML is preferable,

and uniform grid spacing near the exterior boundary along the normal axis should be maintained).

* Perform the Explicit Time-Dependent Solution of the Discrete Fields. • Note that the recursive update algorithm has the same computational complexity as the

regular grid FDTD method with three-dimensional material parameters! • Furthermore, with the use of an irregular grid, the overall grid size can be greatly

reduced, while the time step can be about the same as the regular FDTD method, or even larger!

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-11

EXAMPLE 1: PEC CAVITY • Rectangular cavity with dimensions 0.5 m x 1.0 m x 0.25 m bound by perfect electrical

conducting (PEC) walls. • A random grid spacing for xi, yj, and zk, is assumed, such that:

xi = (i −1)∆x + 12 ℜ∆x ; i = 1, Nx

y j = ( j −1)∆y + 12 ℜ∆y; j =1, Ny

zk = (k −1)∆z + 12 ℜ∆z ; k = 1, Nz

where ℜ is a random number, and − 12 ≤ ℜ ≤ 1

2 (note that the boundaries remain fixed). • The cavity is excited with a time varying electric dipole placed off a center axis in the

cavity. The electric current is z-directed and has a Gaussian time variation:

J (t ) = ˆ z e−( t− to )2

T2

where T = 0.15 ns, and to = 0.45 ns.

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-12

0

50

100

150

200

0 1 10 8 2 10 8 3 10 8 4 10 8 5 10 8 6 10 8

Four

ier S

pect

rum

f (Hz)

TE110

TE120

TE130

TE210

Amplitude of the Fourier transform of the time-varying electric field probed in the resonant cavity for a 21 x 41 x 11 non-uniform lattice with random grid spacing.

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-13

0.001

0.01

0.1

0.01 0.1

Error (non-uniform FD-TD)h 2 error

Erro

r (TE

110 m

ode)

h (m)

Error convergence of the resonant frequency of the TE110 mode computed using

the non-uniform FD-TD algorithm.

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-14

EXAMPLE 2: LOW PASS FILTER • Microstrip Low Pass Filter printed on 0.794 mm Duroid substrate(εr = 2.2). • The total lattice had a dimension of 64 x 76 x 16 (compared to 110 x 80 x 16 using

traditional FDTD method). • A global grid size of ∆x = 0.64, ∆y = 0.635, ∆z = 0.265 was used. A time-step of 0.441

ps was used, resulting in 4000 time steps for this simulation. The grid was also locally refined to model the fields more accurately near the microstrip edges.

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EE699 S. Gedney

Non-Uniform FDTD Methods 10-15

-40

-30

-20

-10

0

10

0 5 10 15 20

Uniform Grid

Non_Uniform|S

11|

(dB

)

f (GHz)

-150

-100

-50

0

50

100

150

0 5 10 15 20

Uniform GridNon-Uniform

Phas

e (S

11)

(deg

rees

)

f (GHz) (a) (b)

-50

-40

-30

-20

-10

0

0 5 10 15 20

Uniform GridNon-Uniform

|S21

| (d

B)

f (GHz)

-150

-100

-50

0

50

100

150

0 5 10 15 20

Uniform Grid

Non-Uniform

Phas

e (S

21)

(deg

rees

)

f (GHz) (c) (d)

S-parameters for the low-pass filter computed using the traditional FD-TD algorithm and the non-uniform grid FD-TD algorithm. (a) |S11|, (b) ∠S11, (c) |S21|, (d) ∠S21.

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EE699 S. Gedney

Subcell FDTD Models: 11-1

Subcell Modeling Techniques • The Previous Lectures have dealt explicitly with FD-TD algorithms based on orthogonal

Cartesian lattices. • A challenge is presented when modeling structures with fine geometric detail that is very

small relative to a wavelength, and to the global problem size. • Global cell size and time step can be bound by the smallest geometric detail • Need sufficient representation to accurately model boundary and singular field

behavior near discontinuities • Subcell modelling techniques attempt to accurately model the fields near small

geometrical discontinuities without the reduction of the global grid. Namely, the model is based on a local approximation of the fields within the cell which the boundary lies. Hence, the methods are referred to as "subcell" models.

• We will discuss subcell models for: • Thin wire models • Thin slots • Thin material sheets • Contour Patch model for skew boundaries

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EE699 S. Gedney

Subcell FDTD Models: 11-2

Thin Wire Subcell Models • Many applications require the accurately modeling of field interactions with thin metallic

wires • Dipole/Monopole Antennas, Wire bonds for PCBs and microwave circuits, Vias or

through holes • If the radius of the wire (a) << λ, the following thin wire model can be employed:

a

, , 1i j kxE +

, ,i j kxE

, ,i j kzE1, ,i j kzE +

, ,i j kyH

• B.C.:

, ,0

i j kzE = on the surface of the wire. Realizing this, we will:

1. We will deform the cell to be x a∆ − wide 2. Model the 1/r singular behavior of Ex and Hy in the edge normal to the wire axis 3. Derive an explicit update expression based on Faraday's law expressed in its integral

form.

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EE699 S. Gedney

Subcell FDTD Models: 11-3

a

, , 1i j kxE +

, ,i j kxE

, ,i j kzE1, ,i j kzE +

, ,i j kyH

0x = • Let:

, , , ,

1'2i j k i j ky yxH Hx

∆= , & , , , ,

1'2i j k i j kx xxE Ex

∆=

• Then, from Faraday's Law: H ds E dt

µ ∂− ⋅ = ⋅ ⇒∂∫∫ ∫

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EE699 S. Gedney

Subcell FDTD Models: 11-4

1 12 2

, , , ,

, , 1 , , 1, ,

ln2

ln2

i j k i j k

i j k i j k i j k

n ny y

n n nx x z

z x xH Ht ax xE E E z

a

µ

+ +

+ − ⎛ ⎞⎡ ⎤ ∆ ∆ ∆ ⎟⎜− − =⎢ ⎥ ⎟⎜ ⎟⎜⎝ ⎠⎢ ⎥ ∆⎣ ⎦⎛ ⎞∆ ∆⎡ ⎤ ⎟⎜− − ∆⎢ ⎥ ⎟⎜ ⎟⎜⎝ ⎠⎣ ⎦

• This leads to the explicit update expression:

1 1 , , 1 , , 1, ,2 2, , , ,

ln2

i j k i j k i j ki j k i j k

n n nx x zn n

y y

t E E tEH H x xz

aµ µ

+ ++ −⎡ ⎤∆ − ∆⎢ ⎥⎣ ⎦= − + ⎛ ⎞∆ ∆∆ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

• Note that the update for , ,i j kxE is unchanged by the scaling.

• The application of the thin wire model then simply requires a modification of the coefficients of the update of the four magnetic fields surrounding the wire. This is a preprocessing step. Further, the axial electric field is simply zeroed out

• As a result, this is a preprocessing step, and the fundamental FDTD code is unchanged.

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EE699 S. Gedney

Subcell FDTD Models: 11-5

Example: Input Impedance of a Thin Wire Dipole

The input impedance of a thin-wire dipole is antenna is computed via the FDTD method. The dipole is excited by a voltage gap source. The results below are compared to a Method of Moment calculation when:

/11, /21, /41, /81,x y z∆ =∆ =∆ = Watanabe and Taki, "An improved FDTD Model for the Feeding

Gap of a Thin-Wire Antenna," Microwave and Guided Wave Letters, Vol 8, pp. 152-154, April 1998.

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EE699 S. Gedney

Subcell FDTD Models: 11-6

This model assumes a delta gap source, where:

( ) ( )z oE gap V zδ=− Thus, in the line integral:

, , 1 , , 1, ,ln

2i j k i j k i j kn n n nx x z

E d

x xE E E z Va+ +

⋅ =

⎛ ⎞∆ ∆⎡ ⎤ ⎟⎜− − ∆ −⎢ ⎥ ⎟⎜ ⎟⎜⎝ ⎠⎣ ⎦

1 1 , , 1 , , 1, ,2 2, , , ,

/

ln2

i j k i j k i j ki j k i j k

n n nx x zn n

y y

t E E tE V zH H x xz

aµ µ

+ ++ −⎡ ⎤∆ − ∆ + ∆⎢ ⎥⎣ ⎦= − + ⎛ ⎞∆ ∆∆ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

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EE699 S. Gedney

Subcell FDTD Models: 11-7

Modeling of Thin Slots • Next, we consider another class of problems of interest, which involve the

electromagnetic interaction with a thin slot. Some examples of these applications include: • EMC: EM penetration through thin slots and seams in metallic enclosures • Microwave circuits and antennas: EM coupling through slots in a ground plane, or EM

radiation from slots in a waveguide wall or ground plane • We will focus on problems where the slot is narrow and thick relative to the grid size.

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EE699 S. Gedney

Subcell FDTD Models: 11-8

C1

C2

• Consider the grids to the right,

with C1 and C2 C1:

1 12 2

, , , ,

, , 1 , ,

1, , , ,

i j k i j k

i j k i j k

i j k i j k

n ny y

n nx x

n nz z

H Hx z

tE x E g

E z E z+

+

+ −−∆ ∆ =

∆⎡ ⎤∆ −⎢ ⎥⎢ ⎥− ∆ + ∆⎢ ⎥⎢ ⎥⎣ ⎦

C2:

1 12 2

, , , ,

, , 1 , ,

i j k i j k

i j k i j k

n ny y

n nx x

H Hg z

t

E g E g+

+ −−∆ =

∆⎡ ⎤−⎢ ⎥⎣ ⎦

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EE699 S. Gedney

Subcell FDTD Models: 11-9

• This leads to the explicit updates: C1:

1 1 , , 1 , , 1, , , ,2 2, , , ,

i j k i j k i j k i j ki j k i j k

n n n nx x z zn n o

y y

gE E E ExH H

t z xµ + ++ −

⎡ ⎤− −⎢ ⎥∆⎢ ⎥= − −⎢ ⎥∆ ∆ ∆⎢ ⎥⎣ ⎦

C2:

1 1 , , 1 , ,2 2, , , ,

i j k i j ki j k i j k

n nx xn n o

y y

E EH H

t zµ ++ −

⎡ ⎤−⎢ ⎥= − ⎢ ⎥∆ ∆⎢ ⎥⎣ ⎦

• Note that again, the field updates resemble those of the standard FDTD update scheme.

The one difference is that , ,i j knxE is weighted by g

x∆. However,

, ,i j knxE is properly

weighted (by its edge length), then again, the main FDTD update loops are unchanged. • It is interesting to observe that the update about C2 is independent of g. The physical

width of the slot is actually realized by the slot aperture fields at either end. Only the tangential field is modeled within the slot thereafter.

• For this approximation to be valid, g must be << λ , such that the axial field can be neglected.

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EE699 S. Gedney

Subcell FDTD Models: 11-10

Example: Electromagnetic Interaction with a Thin, Narrow Slot

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EE699 S. Gedney

Subcell FDTD Models: 11-11

Modeling of Thin Material Sheets • Assume a thin sheet of lossy dielectric material ( , , )ots tsε σ µ passing through the FDTD

lattice that is normal to the x-axis. The sheet has a thickness d < x∆ , and d λ<< . • The boundary conditions at the surface of the discontinuity is that the tangential electric

and magnetic fields are continuous. The normal electric flux density is also continuous - hence, the normal electric field is discontinuous.

• The normal electric field is separated into two contributions, ,x inE and ,x outE - namely, the normal field "in" and "out" of the dielectric sheet. This allows for the jump discontinuity in xE .

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Subcell FDTD Models: 11-12

• Field Updates: 1 1 1 12 2 2 2

, , , 1, , , , , 1

, , , ,

1, ,

i j k i j k i j k i j k

i j k i j k

n n n nz z y yn n

x out x outo z

H H H HtE Eyε

− −

+ + + ++

⎡ ⎤− −⎢ ⎥∆ ⎢ ⎥= + −⎢ ⎥∆ ∆⎢ ⎥⎢ ⎥⎣ ⎦

1 1 1 12 2 2 2

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

1, ,

12

1 12 2

i j k i j k i j k i j ki j k i j k

ts n n n nz z y yn nts ts

x in x ints ts o zts ts

t tH H H HtE Et t y

σε ε

σ σ εε ε

− −+ + + +

+

⎛ ⎞ ⎛ ⎞∆ ∆ ⎡ ⎤⎟ ⎟⎜ ⎜− ⎟ ⎟⎜ ⎜ − −⎢ ⎥⎟ ⎟∆⎜ ⎜⎟ ⎟⎜ ⎜ ⎢ ⎥= + −⎟ ⎟⎜ ⎜⎟ ⎟∆ ∆ ⎢ ⎥⎜ ⎜⎟ ⎟ ∆ ∆⎜ ⎜⎟ ⎟+ + ⎢ ⎥⎟ ⎟⎜ ⎜⎟ ⎟ ⎣ ⎦⎜ ⎜⎟ ⎟⎝ ⎠ ⎝ ⎠

[Note that the continuity of Hy and Hz is enforced.] • Tangential Electric Field Updates:

1 1 1 12 2 2 2

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

11

2

1 12 2

i j k i j k i j k i j ki j k i j k

av n n n nx x z zn nav av

y yav av

av av

t tH H H H

E Et t z x

σε ε

σ σε ε

− −+ + + +

+

⎛ ⎞ ⎛ ⎞∆ ∆ ⎡ ⎤⎟ ⎟⎜ ⎜− ⎟ ⎟⎜ ⎜ − −⎢ ⎥⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜ ⎢ ⎥= + −⎟ ⎟⎜ ⎜⎟ ⎟∆ ∆ ⎢ ⎥⎜ ⎜⎟ ⎟ ∆ ∆⎜ ⎜⎟ ⎟+ + ⎢ ⎥⎟ ⎟⎜ ⎜⎟ ⎟ ⎣ ⎦⎜ ⎜⎟ ⎟⎝ ⎠ ⎝ ⎠

1 1 1 12 2 2 2

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

11

2

1 12 2

i j k i j k i j k i j ki j k i j k

av n n n ny y x xn nav av

z zav av

av av

t tH H H H

E Et t x y

σε ε

σ σε ε

− −+ + + +

+

⎛ ⎞ ⎛ ⎞∆ ∆ ⎡ ⎤⎟ ⎟⎜ ⎜− ⎟ ⎟⎜ ⎜ − −⎢ ⎥⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜ ⎢ ⎥= + −⎟ ⎟⎜ ⎜⎟ ⎟∆ ∆ ⎢ ⎥⎜ ⎜⎟ ⎟ ∆ ∆⎜ ⎜⎟ ⎟+ + ⎢ ⎥⎟ ⎟⎜ ⎜⎟ ⎟ ⎣ ⎦⎜ ⎜⎟ ⎟⎝ ⎠ ⎝ ⎠

where,

1 ,av o avts tsd d dx x x

ε ε ε σ σ⎛ ⎞⎟⎜= − + =⎟⎟⎜⎝ ⎠∆ ∆ ∆

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Subcell FDTD Models: 11-13

Tangential Magnetic Field Updates:

( )( ) ( )( )

1 1 , , 1 , , , , 1 , ,2 2, , , ,

1, , , ,

, , , ,i j k i j k i j k i j k

i j k i j k

i j k i j k

n n n nx out x out x in x inn n

y y n no z z

x d E E d E EtH Hx z z E Eµ

+ +

+

+ −⎡ ⎤∆ − − + −⎢ ⎥⎛ ⎞∆ ⎟⎜ ⎢ ⎥= − ⎟⎜ ⎟⎟⎜ ⎢ ⎥∆ ∆⎝ ⎠ −∆ −⎢ ⎥⎣ ⎦

( ) ( )( )( )

1 1 1, , , , , 1, , ,2 2, , , ,

, 1, , ,

, ,

, ,

i j k i j k i j k i j k

i j k i j k

i j k i j k

n n n ny y x out x outn n

z z n no x in x in

y E E x d E EtH Hx y d E Eµ

+ +

+

+ −⎡ ⎤∆ − − ∆ − −⎢ ⎥⎛ ⎞∆ ⎟⎜ ⎢ ⎥= − ⎟⎜ ⎟⎟⎜ ⎢ ⎥∆ ∆⎝ ⎠ − −⎢ ⎥⎣ ⎦

• Observations: • Can still preserve the fundamental FDTD update loops • ,

nx inE requires a separate update loop and separate storage

• updates involving ,nx inE can be added in separately after the main update loops of H

• Tangential E field updates are modified only through the material coefficients. • Accurately represents the affect of the thin material sheet providing d << λ

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Subcell FDTD Models: 11-14

Example: Resistive Sheet Loaded Parallel Plate Waveguide†

†Maloney and Smith, "The Efficient Modeling of Thin Material Sheets in the FDTD Method," IEEE Transactions on Antennas and Propagation, Vol. 40, pp. 323-330, March 1992.

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Subcell FDTD Models: 11-15

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Subcell FDTD Models: 11-16

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EE699 S. Gedney

Subcell FDTD Models: 11-17

Contour Path Modeling of Irregular Surfaces

• Since God did not make our world rectangular, the FDTD method is forced to approximate irregularly shaped surfaces with staircase approximations. • Staircasing leads to boundary error which can supercede dispersive errors • To reduce boundary error sufficiently, grid lattice may need to be dramatically refined • This leads to additional memory and FLOP's to compute an accurate solution

• This can be alleviated using several techniques

1. Non-Orthogonal FDTD Methods 2. Contour-Patch FDTD Methods • Deformed rectangular lattices

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Subcell FDTD Models: 11-18

Contour Path FDTD (CPFDTD) Method • Original CPFDTD method was introduced by Jurgens, et al [1]. • Method successfully demonstrated to model irregularly shaped objects • Suffers from late time instabilities

• Origin of late time instabilities was shown in [2] to be due to the non-reciprocal nature of the CPFDTD method. A stable and accurate scheme was introduced [3,4]. • Scheme is cumbersome to implement, and intrusive of the base FDTD algorithm

• Stable and simple CPFDTD scheme was recently introduced by Dey and Mittra [5]. • Scheme is stable, and the most accurate of all the CPFDTD methods introduced [6] • This scheme will be the topic of our discussion

[1] T. Jurgens and A. Taflove, "Three-dimensional contour FDTD modeling of scattering from single and multiple bodies," IEEE Trans. Antennas and Prop., vol. 41, pp. 1803-1708, Dec. 1993

[2] I. J. Craddock, C. J. Railton, and J. P. BcGeehan, "Derivation and application of a passive equivalent circuit for the FDTD algorithm," IEEE Microwave and Guided Wave Letters, vol. 6, pp. 40-42, Jan. 1996.

[3] C. J. Railton and I. J. Craddock, "Analysis of general 3-D PEC structures using an improved CP-FDTD algorithm," Electronics Letters, vol. 31, pp. 1553-1554, Sept. 1995.

[4] C. J. Railton and I. J. Craddock, "Stabilized CP-FDTD algorithm for the analysis of arbitrary 3D PEC structures," Proc. Inst. Elect. Eng., vol. 143, pt. H, pp. 367-372, Oct. 1996.

[5] S. Dey and R. Mittra, "A locally conformal FDTD algorithm for modeling 3D PEC objects," IEEE Microwave and Guided Wave Letters, vol. 7, pp. 273-275, Sept. 1997.

[6] C. J. Railton and J. Schneider, "An analytical and numerical analysis of several locally conformal FDTD schemes," IEEE Trans. On Microwave Theory and Techniques, vol. 47, pp. 56-66, Jan. 1999.

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Subcell FDTD Models: 11-19

Contour Path FDTD (CPFDTD) Method (cont'd) • Consider the cross-section of a quadrant of a spherical surface positioned within an

FDTD lattice • Near the surface of the boundary, the cells are distorted. • The electric fields are located on the edges of the cells (as indicated by the arrows) • The magnetic fields are assumed to be positioned at the cell centers - even if the cell

center happens to lie outside of the PEC boundary (as indicated by the filled circles) • Faraday's law is performed in its integral form over the deformed cell faces • Ampere's law is represented in differential form, assuming undeformed cells

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EE699 S. Gedney

Subcell FDTD Models: 11-20

Field Updates • From Faraday's Law:

1 12 2

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

i j k i j k i j k i j k i j k i j k i j k i j k i j k i j kn n n n n nz z y y y y x x x x

i j k

tH H E l E l E l E lAµ + + + +

+ − ∆ ⎡ ⎤= − − − +⎢ ⎥⎣ ⎦

where , ,i j kA is the face area

• From Ampere's Law:

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

1 i j k i j k i j k i j ki j k i j k

n n n nz z y yn n

x x

H H H HtE Ey zε

− −+⎡ ⎤− −∆ ⎢ ⎥= + −⎢ ⎥∆ ∆⎢ ⎥⎣ ⎦

• Observations: • Scheme can be incorporated directly into the FDTD algorithm. Only the update

coefficients need to be modified - a preprocessing step • Scheme is stable, since method is reciprocal • Dey and Mittra report:

1. If the area of the smallest distorted cell is > 1.5 % of the undistorted cell area, the time step should be reduced by 50 % of the Courant Limit

2. If the area of the smallest distorted cell is > 2.5 % of the undistorted cell area, the time step should be reduced by 30 % of the Courant Limit

3. If the ratio between the maximum length of the side of a cell and its area is < 15, the time step should be reduced by 50 % of the Courant Limit

4. If the ratio between the maximum length of the side of a cell and its area is < 10, the time step should be reduced by 30 % of the Courant Limit

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Subcell FDTD Models: 11-21

Examples of the CPFDTD Method Resonant Frequency of a Cylindrical Resonant Frequency of a Spherical Cavity [5]: Cavity [5]:

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Subcell FDTD Models: 11-22

Resonant Frequency of a Tilted Cylindrical Cavity [6]:

• How would you modify the CPFDTD method to open PEC boundaries rather than closed?

0.3 m

0.19 m

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Subcell FDTD Models: 11-23

Conformal FDTD Model of Dielectric Surfaces • Previous discussion was limited to non-penetrable conductor surfaces • Not applicable to penetrable materials such as dielectrics • Applications: • Dielectric Resonators - microwave circuit applications, Photonic Bandgap (PBG)

structures, antenna radomes or sheaths • Conformal FDTD method will be based on both deforming the cell and introducing an

effective tensor representation [1,2] • Effective permittivity tensor is necessary due to the different properties of the normal

and tangential electric field [0] Nadobny J, Sullivan D, Wlodarczyk W, et al. ,”A 3-D tensor FDTD-formulation for treatment of sloped interfaces in electrically inhomogeneous media,” IEEE T. Antennas and Propagation, vol. 51 (8), pp. 1760-1770, August 2003 [1] J.-Y. Lee and N.-H. Myung, "Locally tensor conformal FDTD method for arbitrary dielectric surfaces," Microwave and Optical Technology Letters, vol. 23, No. 4, pp. 245-249, Nov. 20, 1999. [2] R. D. Meade, et al., "Accurate theoretical analysis of photonic band-ap materials,"

Phys. Rev. B, vol. 48, pp. 8434-8437, 1993. [3] D. E. Aspnes, " Local-field effects and effective-medium theory: a microscopic

perspective," American Journal of Physics, vol. 50, pg. 704-709, 1982.

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Subcell FDTD Models: 11-24

• The behavior of the electric field at the surface of a dielectric interface will depend on the

field polarization • We've seen that we can average the permittivity in a cell • (a) Tangential (b) Normal

1 2tan 1 2

A AA A

ε ε ε= + 1 2

n 1 2

1 1 1A AA Aε ε ε

= +

• Skew (c) • The skew boundary is a combination of the tangential and normal contributions • Based on effective media theory [1-3]:

( )( )

2 2tan tan

2 2tan tan

tan

00

0 0

n x y n x y

n x y n y xeff

n n n nn n n n

ε ε ε εε ε ε ε ε

ε

⎡ ⎤+ −⎢ ⎥⎢ ⎥= − +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

where, the unit vector normal to the surface is: ˆ ˆ ˆx yn n x n y= + and tanε and nε are defined above

• This can be generalized to a three-dimensional surface

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Subcell FDTD Models: 11-25

Consider the limiting case when 1, 0x yn n= =

tantan

0 00 00 0

n

eff

εε ε

ε

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

• Similarly along other axes • The field update is performed in the following order:

1. Update the electric flux densities 1, ,ni j kD + using the direct FDTD updates based on

Ampere's law 2. Compute the electric fields from 1

, ,ni j kD + using the constitutive relationship

3. Update the magnetic fields based on Faraday's law

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Subcell FDTD Models: 11-26

• Constitutive Relationship 1 1 1

, , , ,n ni j k i j kE Dε+ − +=

where ( )

( )

2 2tantan

2 21 tan tan

tan

0

010 0

n x yn y x

n x y n x yeff

n nn n

n n n n

ε εε ε

ε ε ε εε

ε

⎡ ⎤−+⎢ ⎥−⎢ ⎥∆ ∆⎢ ⎥− +⎢ ⎥= −⎢ ⎥∆ ∆⎢ ⎥

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

and, where

( ) ( )24 4 2 2 2 2tan tan tan2n x y y x n x y nn n n n n nε ε ε ε ε ε∆ = + + = + =

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Subcell FDTD Models: 11-27

Constitutive Relationships:

, , , , , ,

, , , , , ,

, , , ,

221 1 1

tan tan22

1 1 1

tan tan

1 1

tan

1 1

1 1

1

i j k i j k i j k

i j k i j k i j k

i j k i j k

yn n nxx x x y y

n n

yn n nxy x y x y

n n

n nz z

nnE D n n D

nnE n n D D

E D

ε ε ε ε

ε ε ε ε

ε

+ + +

+ + +

+ +

⎛ ⎞ ⎛ ⎞⎟⎜ ⎟⎜⎟ ⎟⎜= + + −⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜⎟⎜ ⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎟ ⎜= − + +⎜ ⎟⎟ ⎜⎜ ⎟⎟⎜ ⎟⎜⎝ ⎠ ⎝ ⎠

=

where, and , , , ,

1 1i j k i j kn nx yD D+ + are averaged as:

1 =4, , , , , 1, 1, , 1, 1,

1 1 1 1 1i j k i j k i j k i j k i j kn n n n nx x x x xD D D D D

+ − − ++ + + + +⎡ ⎤+ + +⎢ ⎥⎣ ⎦

1 =4, , , , 1, , , 1, 1, 1,

1 1 1 1 1i j k i j k i j k i j k i j kn n n n ny y y y yD D D D D

+ − + −+ + + + +⎡ ⎤+ + +⎢ ⎥⎣ ⎦

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Subcell FDTD Models: 11-28

Example 1: Rotated Planar Interface [1]

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Subcell FDTD Models: 11-29

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Subcell FDTD Models: 11-30

Example 2: Dielectric Cylinder [1]

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Subcell FDTD Models: 11-31

• Observations • Method appears to offer excellent accuracy as compared to a staircase approximation

with a coarse grid • Late time stability? It is uncertain given the averaging terms. Needs to be validated. • Method could be extended to 3-D

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EE699 S. Gedney

Subcell FDTD Models: 11-32

Assume a three-dimensional surface with unit normal: cos sin sin sin cosˆ ˆ ˆ ˆn x y zφ θ φ θ θ= + +

Define the rotational tensor:

cos sin 0sin 0 cos0 1 0 sin cos 0

0 0 1cos 0 sinT

φ φθ θφ φ

θ θ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

and the permittivity tensor:

tantan

0 00 00 0

n

eff

εε ε

ε

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

Then applying rotation of coordinates, the constitutive relation can be defined as:

effTE TDε = Finally, define:

( ) ( ) ( )( ) ( ) ( )( ) ( )

1

2 2 2 2 2 2tan tan tan

2 2 2 2 2 2tan tan tan

2tan tan

cos sin sin sin cos cos sin sin cos sin coscos sin sin sin sin cos sin cos sin sin coscos sin cos sin sin cos cos

eff eff

n n n

n n n

n n

T Tε ε

φ θε φ θ θ ε φ φ θ ε ε φ θ θ ε εφ φ θ ε ε φ θε φ θ θ ε φ θ θ ε εφ θ θ ε ε φ θ θ ε ε

−= =

+ + − −− + + −− − 2

tansinnθε θε

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥+⎢ ⎥⎣ ⎦

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Subcell FDTD Models: 11-33

This can be more conveniently expressed in terms of the unit normal vector:

( ) ( )( ) ( )

( ) ( ) ( )

2 2 2tan tan tan

2 2 2tan tan tan

2 2 2tan tan tan

( )( )

x n y z x y n x z n

x y n y n x z y z neff

x z n y z n z n x y

n n n n n n nn n n n n n nn n n n n n n

ε ε ε ε ε εε ε ε ε ε ε ε

ε ε ε ε ε ε

⎡ ⎤+ + − −⎢ ⎥⎢ ⎥= − + + −⎢ ⎥⎢ ⎥− − + +⎢ ⎥⎣ ⎦

This leads to a full, three-dimensional tensor relationship.

effE Dε =