lecture 4 -- electromagnetics and fdtd
TRANSCRIPT
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Lecture 4 Slide 1
EE 5303
Electromagnetic Analysis Using Finite‐Difference Time‐Domain
Lecture #4
Review of EM and Introduction to FDTD These notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited
Lecture Outline
Lecture 4 Slide 2
• Review– Maxwell’s equations– Physical boundary conditions
– The constitutive relations– Parameter relations– see Balanis Chapter 1
• Introduction to FDTD– Flow of Maxwell’s equations– Finite‐difference approximations
– The update equation– The FDTD algorithm…for now
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Lecture 4 Slide 3
Review of Maxwell’s Equations and
Electromagnetics
Maxwell’s Equations
4Lecture 4
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Lecture 4 Slide 5
Four Field Terms in Maxwell’s Equations
Electric field intensityInitial electric push
V mE
Displaced Charges
+ = D
2C m
Electric Field Quantities
Magnetic Field Quantities
Magnetic field intensityInitial magnetic push
A mH
Tilted Magnetic Dipoles
+ = B
2Wb m
H J D t
E B t
Gauss’s Law
Electric fields diverge from positive charges and converge on negative charges.
‐+
vD
yx zDD D
Dx y z
If there are no charges, electric fields must form loops.
6Lecture 4
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Gauss’s Law for Magnetism
Magnetic fields always form loops.
0B
yx zBB B
Bx y z
7Lecture 4
Consequence of Zero Divergence
The divergence theorems force the D and B fields to be perpendicular to the propagation direction of a plane wave.
k D
0
0jk r
D
de
d
no charges
0
0
jk d
k d
k
k
k B
0
0jk r
B
be
b
no charges
0
0
jk b
k b
k
k
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Ampere’s Law with Maxwell’s Correction
DH J
t
ˆ ˆ ˆy yx xz zx y z
H HH HH HH a a a
y z z x x y
Circulating magnetic fields induce currents and/or time varying electric fields.Currents and/or time varying electric fields induce circulating magnetic fields.
9Lecture 4
Faraday’s Law of Induction
BE
t
ˆ ˆ ˆy yx xz zx y z
E EE EE EE a a a
y z z x x y
Circulating electric fields induce time varying magnetic fields.Time varying magnetic fields induce circulating electric fields.
10Lecture 4
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Consequence of Curl Equations
The curl equations predict electromagnetic waves!!
Electric Field
Magnetic Field
11Lecture 4
Starting Point for Electromagnetic Analysis
Divergence Equations
0
v
B
D
DH J
t
BE
t
Curl Equations
Constitutive Relations
D t t E t
B t t H t
What produces fields
How fields interact with materials
means convolution
means tensor
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Lecture 4 Slide 13
Maxwell’s Equations in Cartesian Coordinates (1 of 4)Vector Terms
ˆ ˆ ˆ
ˆ ˆ ˆ
x x y y z z
x x y y z z
E E a E a E a
D D a D a D a
ˆ ˆ ˆ
ˆ ˆ ˆ
x x y y z z
x x y y z z
H H a H a H a
B B a B a B a
ˆ ˆ ˆx x y y z zJ J a J a J a
Divergence Equations
0
0yx z
D
DD D
x y z
0
0yx z
B
BB B
x y z
Lecture 4 Slide 14
Maxwell’s Equations in Cartesian Coordinates (2 of 4)Constitutive Relations
D E
ˆ ˆˆ ˆˆˆy y yx x xx x xy y xz zz z zx x zy y zx x yyx y z zy zy zzD D a Ea E ED a E E EE aa a E E
y yx x y
z zx x
x xx x xy y xz z
zy y zz z
y y yz zD E E
D E
D E E E
E
E E
B H x xx x xy y xz z
y yx x yy y yz z
z zx x zy y zz z
B H H H
B H H H
B H H H
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Lecture 4 Slide 15
Maxwell’s Equations in Cartesian Coordinates (3 of 4)Curl Equations
BE
t
ˆ ˆ ˆ ˆ ˆ ˆy yx xz zx y z x x y y z z
E EE EE Ea a a B a B a B a
y z z x x y t
ˆ ˆ ˆˆ ˆˆy xzx x
yx zy
y x zz zy
E E Ba a
E BEa
BE Ea a
y z tx tx ya
z t
y x
z
z
x
z
yx
yE BE
y z t
BE E
zE E B
x y
x t
t
Lecture 4 Slide 16
Maxwell’s Equations in Cartesian Coordinates (4 of 4)Curl Equations
DH J
t
y
yx zy
y x zz
xzx
DH HJ
z x tH H D
Jx y
H DHJ
y z
t
t
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆy yx xz zx y z x x y y z z x x y y z z
H HH HH Ha a a J a J a J a D a D a D a
y z z x x y t
ˆ ˆˆ ˆ ˆˆy x zy xzx xz z
zy y y z
yx
xDH H
a J az x t
H DHa J a
y z t
H H Da J a
x y t
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Lecture 4 Slide 17
Alternative Form of Maxwell’s Equations in Cartesian Coordinates (1 of 2)Alternate Curl Equations
EHt
ˆ
ˆ
ˆ
ˆ
ˆ ˆx zy
y
y xz
yx zzx zy zz
y yxz zx xx xy x
x
z x
zyx yy yz y
z
H EEH EH Ha
z x
E
H Ha
x y
EE Ea
t t t
E Ea
a az
t t
y t
t
t t
yx xz zyx yy y
y yx x zzx zy
y yxz zxx
zz
y
z
x xz
EH EH E
z x tH EH E E
x y t
H EEH E
y z
t
t
t
t t t
t
Lecture 4 Slide 18
Alternative Form of Maxwell’s Equations in Cartesian Coordinates (2 of 2)Alternate Curl Equations
HEt
ˆ
ˆˆ ˆ
ˆ
ˆ x y xz
y
y yxz zx xx xy xz
zy
yx zyx yy yz
x zzx zy zz
x
y
z
E EE HHE Ha aa
x y
E
y z t t t
Ea
z x
HH Ha
HH Ha
t t
t
t
t t
yx
y yx
x
y yx x z
z zxx x
zx zy zz
z zyx yy z
y z
y
x
E HE
HE HE H
z x t
H H
x y t t
E HHE
t
H
t
y t t t
t
z
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Lecture 4 Slide 19
Tensors
Tensors are a generalization of a scaling factor where the direction of a vector can be altered in addition to its magnitude.
V
aV
V a V
Scalar Relation
Tensor Relation
xx xy xz
yx yy yz
zx zy zz
x
y
z
a a a
a a a a
a a a
V
V V
V
The Constitutive Relations
20Lecture 4
Linear, isotropic and non‐dispersive materials:
Dispersive materials:
D t E t
Anisotropic materials:
D t t E t
D t E t
Nonlinear materials:
1 2 32 30 0 0e e eD t E t E t E t
We will use this in our FDTD codes
The point here is that you can wrap all of the complexities associated with modeling strange materials into this single equation. The implementation of Maxwell’s equations in FDTD will never change. This will make your code more modular and easier to modify. It may not be as efficient as it could be though.
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Anisotropic Materials
21Lecture 4
xx xy xz
yx yy yz
zx zy zz
D t E t
A generalized tensor for permittivity is written as
We see that E and D can be in different directions when the permittivity is anisotropic.
how much of contributes to ij j iE D
It greatly simplifies a finite‐difference method to consider only diagonal tensors. That is, all of the off‐diagonal terms will be set to zero.
0 0
0 0
0 0
xx x xx x
yy y yy y
zz z zz z
D t E t
D t E t D t E t
D t E t
Special Note: There are only three degrees of freedom for the tensor components. The nine elements cannot be chosen arbitrarily. It is always possible to choose a coordinate system that makes the tensor diagonal. The off diagonal terms only arise when the chosen coordinate system does not match the crystal axes of the anisotropic material. The simplification above restricts us to only be able to model anisotropic materials that align perfectly with our x, y, and z axes.
Simplifying Maxwell’s Equations
0
0
B
D
H D t
E B t
D t t E t
B t t H t
1. Assume no charges or current sources: 0v 0J
0
0
H t
E t
EH
t
HE
t
3. Sometimes the constitutive relations are substituted into Maxwell’s equations:
Note: It is helpful to retain μ and ε and not replace them with refractive index n.
22Lecture 4
D t E t
B t H t
2. Assume linear, isotropic, and non‐dispersive materials:Convolution becomes simple multiplication
0
0
B
D
H D t
E B t
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Physical Interpretation of E and D
E – Electric Field•A disturbance produced around charges or in the presence of a time‐varying magnetic field.•Think of E as a “push”•Units are volts per meter (V/m)
+‐
EV
d
V d
23Lecture 4
D – Electric Displacement Field• “D” stands for displacement• Includes E, but also accounts for displaced charges in a material (material polarization)• Equivalent to flux density• Think of “D” as displaced charge•Units are dipole moments per unit volume (C·m/m3), or just (C/m2)•We can make E look like an equivalent displaced charge through D=ε0E
Physical Interpretation of H and BH – Magnetic Field•A disturbance produced around currents or in the presence of a time‐varying electric field.•Think of H as a magnetic “push”•Units are amperes per meter (A/m)
B – Magnetic Displacement Field• Includes H, but also accounts for tilted magnetic dipoles in a material (magnetization)• Equivalent to flux density• Think of “B” as reoriented magnetic dipoles•Units are magnetic dipole moments per unit volume (W·m/m3), or just (W/m2)•We can make H look like an equivalent reoriented dipole through B=μ0H
C. A. Balanis, Advanced Engineering Electromagnetics, (Wiley, New York, 1989)
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Physical Boundary Conditions
Tangential components of E and Hare continuous across an interface.
1,TE 2,TE
1,TH 2,TH
1 1 and 2 2 and
Fields normal to the interface are discontinuous across an interface.
Note: Normal components of Dand B are continuous across the interface.
1 1,NE
1 1,NH
2 2,NE
2 2,NH
These are more complicated boundary conditions, physically and analytically.
1,Tk 2,TkTangential component of the wave vector is continuous across an interface.
25Lecture 4
Lecture 4 Slide 26
Parameter Relations
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The Dielectric Constant, r
0 r
Slide 27Lecture 4
120 8.854187817 10 F m
The dielectric constant of a material is its permittivity relative to the permittivity of free space.
1 r r is the relative permittivity or dielectric constant
The permittivity is a measure of how well a material stores electric energy. A circulating magnetic field induces an electric field at the center of the circulation in proportion to the permittivity.
EH
t
Table of Dielectric Constants
Slide 28Lecture 4
Constantine A. Balanis, Advanced Engineering Electromagnetics, Wiley, 1989.
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The Relative Permeability, r
0 r
Slide 29Lecture 4
60 1.256637061 10 H m
The relative permeability of a material is its permeability relative to the permeability of free space.
1 r r is the relative permeability
The permeability is a measure of how well a material stores magnetic energy. A circulating electric field induces a magnetic field at the center of the circulation in proportion to the permeability.
HE
t
Table of Permeabilities
Slide 30Lecture 4Constantine A. Balanis, Advanced Engineering Electromagnetics, Wiley, 1989.
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The Refractive Index
r rn
Slide 31Lecture 4
The permittivity and permeability appear in Maxwell’s equations so they are the most fundamental material properties. However, it is difficult to determine physical meaning from them in terms of how waves propagate (i.e. speed, loss, etc.). In this case, the refractive index is a more meaningful quantity.
Most materials exhibit a negligible magnetic response and the refractive index and dielectric constant are related through
2rn Hint: one of the most common mistakes made
in this course is using values of refractive index directly as permittivity.
Material Impedance
E H
Slide 32Lecture 4
The impedance of a material quantifies the relation between the electric and magnetic field of a wave travelling through that material. It is the most fundamental quantity that causes reflections and scattering.
The impedance can be written relative to the free space impedance as
00 0
0
376.73031346177 r
r
This shows that the electric field is around two to three orders of magnitude larger than the magnetic field.
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versus f
Lecture 4 Slide 33
is the angular frequency measured in radians per second. It relates more directly to phase and k. Think cos(t).
f is the ordinary frequency measured in cycles per second. It relates more directly to time. Think cos(2ft) and = 1/f.
2 f
Wavelength and Frequency
Lecture 4 Slide 34
The frequency f and free space wavelength 0 are related through
0 0c f Inside a material, the wave slows down according to the refractive index as follows.
0cvn
m0 s299792458 speed of light in vacuumc
The frequency is the most fundamental parameter because it is fixed. Inside a material, the wave slows down so the wavelength is reduced.
v f The free space wavelength 0 can be used interchangeably with frequency f. This is most common in optics.
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Sign Convention
Lecture 4 Slide 35
How do you define forward wave propagation? +z
Quantity ‐z +z
Wave Solution
Dielectric Function
Refractive Index
0
j t kzE E e
0j t kzE E e
j j
N n j N n j
Sign convention for this course
Summary of Parameter Relations
Slide 36Lecture 4
Permittivity
0
120 8.854187817 10 F m
r
Permeability
0
60 1.256637061 10 H m
r
Refractive Index Impedance
r rn 0
0 0 0 376.73031346177
r r
Wave Velocity
0
0 299792458 m s
cv
nc
Exact
Frequency and Wavelength
0 0
2 f
c f
00
Wave Number
2k
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Duality Between E‐D and H‐B
Electric Field Magnetic Field
E H
D B
P M
ε μ
37Lecture 4
Lecture 4 Slide 38
Introduction to Finite‐Difference Time‐Domain
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Lecture 4 Slide 39
Flow of Maxwell’s Equations
B tE t
t
A circulating E field induces a change in the B field at the center of circulation.
B t t H t
A B field induces an H field in proportion to the permeability.
D tH t
t
A circulating H field induces a change in the D field at the center of circulation.
D t t E t
A D field induces an E field in proportion to the permittivity.
Note: In reality, this all happens simultaneously. In FDTD, it follows this flow.
Lecture 4 Slide 40
Flow of Maxwell’s Equations Inside Linear, Isotropic and Non‐Dispersive Materials
H tE t
t
E tH t
t
A circulating E field induces a change in the H field at the center of circulation in proportion to the permeability.
A circulating H field induces a change in the E field at the center of circulation in proportion to the permittivity.
In materials that are linear, isotropic and non‐dispersive we have
t t
In this case, the flow of Maxwell’s equations reduces to
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Finite‐Difference Approximations
1.5 2 1df f f
dx x
1f2f
df
dx
x
second‐order accuratefirst‐order derivative
Slide 41Lecture 4
This derivative is defined to exist at the mid point between f1 and f2.
Stable Finite‐Difference Equations
Slide 42Lecture 4
0 Given 0 , , 2 ,f x
f x f f x f xx
0f x x f x
f xx
Exists at 2x x Exists at x
Your simulation will be unstable (i.e. explode).
0
2
f x x f x xf x
x
0
2
f x x f x f x x f x
x
Each term in a finite‐difference equation must exist at the same point in time and space.
Exists at 2x x Exists at 2x x
Exists at 2x x Exists at 2x x f(x) is only known at integer multiples of x. How do we calculate f(x+x/2)?
Example:
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Approximating the Time Derivative (1 of 3)
Slide 43Lecture 4
H tE t
t
E tH t
t
H t t H tE t
t
E t t E tH t
t
An intuitive first guess at approximating the time derivatives in Maxwell’s equations is:
This is an unstable formulation.
Exists at 2t t Exists at t
Exists at 2t t Exists at t
Approximating the Time Derivative (2 of 3)
Slide 44Lecture 4
H tE t
t
E tH t
t
2 2
H t H t t H t H t t
E tt
2
H t H t t E t t E t
t
We adjust the finite‐difference equations so that each term exists at the same point in time.
This works, but we will be doing more calculations than are necessary.
Is there a simpler approach?
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Approximating the Time Derivative (3 of 3)
Slide 45Lecture 4
H tE t
t
E tH t
t
2 2t t t t
t
H HE
t
2
t t t
t t
E EH
t
We stagger E and H in time so that E exists at integer time steps (0, t, 2t, …) and H exists at half time steps (t/2, t+t/2, 2t+t/2,…).
We will handle the spatial derivatives in × next lecture in a very similar manner.
Lecture 4 Slide 46
Derivation of the Update Equations
The “update equations” are the equations used inside the main FDTD loop to calculate the field values at the next time step.
They are derived by solving our finite‐difference equations for the fields at the future time values.
2 2t t t t
t
H HE
t
2
t t t
t t
E EH
t
2 2t t t t t
tH H E
2t t t t t
tE E H
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Lecture 4 Slide 47
Anatomy of the FDTD Update Equation
2t tt t t
tHE E
Field at the future time
step.
Field at the previous time
step.
Update coefficient
Curl of the “other” field at an intermediate time
step
To speed up simulation, we calculate these before iterating.
Lecture 4 Slide 48
The FDTD Algorithm…for now
Initialize Fields to Zero
2 2t t t t t
tH H E
0E H
Done?
Update H from E
Update E from H
2t t t t t
tE E H
2
0 0tt t
H E
2
2
3
2
32
2
52
2
53
2
tt t
tt t t
tt t t
tt t t
tt t t
tt t t
H E
E H
H E
E H
H E
E H
no
Finished!yes
Loop over time
t