review of electromagnetics and introduction to fdtd...11/2/2020 1 computational science:...
TRANSCRIPT
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Computational Science:
Introduction to Finite‐Difference Time‐Domain
Review of Electromagnetics and Introduction to FDTD
Lecture Outline
• 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
Slide 2
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Slide 3
Review of Maxwell’s Equations and
Electromagnetics
Maxwell’s Equations
4
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Four Field Terms in Maxwell’s Equations
Slide 5
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
6
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.
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Gauss’s Law for Magnetism
7
Magnetic fields always form loops.
0B
yx zBB B
Bx y z
Consequence of Zero Divergence
8
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
9
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.
Faraday’s Law of Induction
10
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.
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How Waves Propagate
Slide 11
Start with an oscillating
electric field.
How Waves Propagate
Slide 12
This induces a circulating
magnetic field.
H j E
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How Waves Propagate
Slide 13
Now let’s examine the magnetic field
on axis.
How Waves Propagate
Slide 14
This induces a circulating
electric field.E j H
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How Waves Propagate
Slide 15
Now let’s examine the electric field on axis.
How Waves Propagate
Slide 16
This induces a circulating
magnetic field.
H j E
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How Waves Propagate
Slide 17
…and so on…
Starting Point for Electromagnetic Analysis
18
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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Maxwell’s Equations in Cartesian Coordinates (1 of 4)
Slide 19
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
Maxwell’s Equations in Cartesian Coordinates (2 of 4)
Slide 20
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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Maxwell’s Equations in Cartesian Coordinates (3 of 4)
Slide 21
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
yx zBE E
z x t
y xzE BE
y z t
y x zE E B
x y t
Maxwell’s Equations in Cartesian Coordinates (4 of 4)
Slide 22
Curl Equations
DH J
t
yx zy
DH HJ
z x 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
y xzx
H DHJ
y z t
y x z
z
H H DJ
x y t
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Alternative Form of Maxwell’s Equations in Cartesian Coordinates (1 of 2)
Slide 23
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
Alternative Form of Maxwell’s Equations in Cartesian Coordinates (2 of 2)
Slide 24
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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Tensors
Slide 25
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
26
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 e e e1D t E t E t E t
This will be used throughout this course.
The point is that all of the complexities associated with modeling strange materials are incorporated into this single equation. The implementation of Maxwell’s equations in FDTD will never change as different materials are introduced. Keeping this as a separate step will make the FDTD code more modular and easier to modify. It may not be as efficient as it could be though.
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Anisotropic Materials
27
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 chosencoordinate 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
28
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.
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
29
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
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 (Cꞏm/m3), or just (C/m2)
•We can make 𝐸 look like an equivalent displaced
charge through 𝐷 𝜀 𝐸.
Physical Interpretation of 𝐻 and 𝐵
30
𝐻 – 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 (Wꞏm/m3), or just (W/m2)
•We can make 𝐻 look like an equivalent reoriented dipole through 𝐵 𝜇 𝐻.
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Physical Boundary Conditions
31
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.
Slide 32
Parameter Relations
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The Dielectric Constant, r
Slide 33
0 r 12
0 8.854187817 10 F m
The dielectric constant 𝜀 of a material is its permittivity relative to the permittivity of free space 𝜀 .
r1 r is the relative permittivity or dielectric constant
The permittivity 𝜀 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 34Constantine A. Balanis, Advanced Engineering Electromagnetics, Wiley, 1989.
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The Relative Permeability, r
Slide 35
0 r 6
0 1.256637061 10 H m
The relative permeability 𝜇 of a material is its permeability relative to the permeability of free space 𝜇 .
r1 r is the relative permeability
The permeability 𝜇 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 36Constantine A. Balanis, Advanced Engineering Electromagnetics, Wiley, 1989.
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The Refractive Index
Slide 37
r rn
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
Slide 38
E H
The impedance 𝜂 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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𝜔 versus 𝑓
Slide 39
𝜔 is the angular frequency measured in radians per second. It relates more directly to phase and 𝑘. Think cos 𝜔𝑡 .
𝑓 is the ordinary frequency measured in cycles per second. It relates most directly to time 𝑡. Think cos 2 𝜋𝑓𝑡 and 𝜏 1/𝑓.
2 f
Wavelength and Frequency
Slide 40
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 is often used interchangeably with frequency f. This is most common in optics.
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Sign Convention
Slide 41
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 42
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
43
Electric Field Magnetic Field
E H
D B
P M
ε μ
Slide 44
Introduction to Finite‐Difference Time‐Domain
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Flow of Maxwell’s Equations
Slide 45
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.
Flow of Maxwell’s Equations Inside Linear, Isotropic and Non‐Dispersive Materials
Slide 46
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
Slide 47
1.5 2 1df f f
dx x
1f2f
df
dx
x
second‐order accuratefirst‐order derivative
This derivative is defined to exist at the mid point between f1 and f2.
Stable Finite‐Difference Equations
Slide 48
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 49
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 50
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 51
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
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.
Derivation of the Update Equations
Slide 52
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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Anatomy of the FDTD Update Equation
Slide 53
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.
The FDTD Algorithm…for now
Slide 54
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