lecture 1a -- maxwell's equations · maxwell’s equations scaling properties of maxwell’s...
TRANSCRIPT
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Advanced Computation:
Computational Electromagnetics
Maxwell’s Equations
Outline
• Maxwell’s equations
• Physical Boundary conditions
• Preparing Maxwell’s equations for CEM
• Scaling properties of Maxwell’s equations
Slide 2
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Slide 3
Maxwell’s Equations
Born
Died
June 13,1831Edinburgh, Scotland
November 5, 1879Cambridge, England
James Clerk Maxwell
Sign Conventions for Waves
Slide 4
To describe a wave propagating the positive z direction, we have two choices:
, cosE z t A t kz
, cosE z t A t kz
Most common in engineering
Most common science and physics
Both are correct, but we must choose a convention and be consistent with it. For time‐harmonic signals, this becomes
expE z A jkz
expE z A jkz
Negative sign convention
Positive sign convention
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Time‐Harmonic Maxwell’s Equations
Slide 5
Time‐Domain
0
v
BD E
t
DB H J
t
Frequency‐Domain(e+jkz convention)
0vD E j B
B H J j D
Frequency‐Domain(e-jkz convention)
0vD E j B
B H J j D
Summary of Sign Conventions
Slide 6
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Lorentz Force Law
Slide 7
F qE qv B
Magnetic ForceElectric Force
One additional equation is needed to completely describe classical electromagnetism...
Gauss’s Law
Slide 8
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
Slide 9
Magnetic fields always form loops.
0B
yx zBB B
Bx y z
Consequence of Zero Divergence
Slide 10
The divergence theorems force the 𝐷 and 𝐵 fields to be perpendicular to the propagation
direction 𝑘 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
Slide 11
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.
H
DJ
t
Faraday’s Law of Induction
Slide 12
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.
E
B t
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Consequence of Curl Equations
Slide 13
The curl equations predict electromagnetic waves!!
Electric Field
Magnetic Field
The Constitutive Relations
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Electric Response
ED
Magnetic Response
HB
• Electric field intensity (V/m)• Initial electric “push.”• Induced electric field.• Electric energy in vacuum.
• Permittivity (F/m)• Measure of how well a material stores electric energy.
• Electric flux density (C/m2)• Pretends as if all electric energy is displaced charge.
• Includes electric energy in vacuum and matter.
• Magnetic field intensity (A/m)• Initial magnetic “push.”• Induced magnetic field.• Magnetic energy in vacuum.
• Permeability (H/m)• Measure of how well a material stores magnetic energy.
• Magnetic flux density (Wb/m2)• Pretends as if all magnetic energy is tilted magnetic dipoles.
• Includes magnetic energy in vacuum and matter.
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Material Classifications
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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 almost exclusively
A key point is that you can wrap all of the complexities associated with modeling strange materials into this single equation. This will make your code more modular and easier to modify. It may not be as efficient as it could be though.
All Together Now…
Slide 16
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 17
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 18
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 19
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
yzy xy zz xxBE E
z t
E E BE BE
y z t x xt y
Maxwell’s Equations in Cartesian Coordinates (4 of 4)
Slide 20
Curl Equations
DH J
t
y xzx
zyx zy
y xz
DH HJ
z x tz
H H DJ
x
H D
y
HJ
y 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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Alternative Form of Maxwell’s Equations in Cartesian Coordinates (1 of 2)
Slide 21
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 22
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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Slide 23
Physical Boundary Conditions
Physical Boundary Conditions
Slide 24
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Slide 25
Preparing Maxwell’s Equationsfor CEM
Simplifying Maxwell’s Equations
Slide 26
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: 0, 0v J
0
0
H
E
H j E
E j H
3. Substitute constitutive relations into Maxwell’s equations:
Note: It is useful to retain μ and ε and not replace them with refractive index n.
0
0
B
D
H j D
E j B
D E
B H
2. Transform Maxwell’s equations to frequency‐domain:Convolution becomes simple multiplication
Note: We have chosen to proceed with the negative sign convention.
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Isotropic Materials
Slide 27
xx xy xz xx xy xz
yx yy yz yx yy yz
zx zy zz zx zy zz
For anisotropic materials, the permittivity and permeability terms are tensor quantities.
For isotropic materials, the tensors reduce to a single scalar quantity.
0 0 0 0
0 0 0 0
0 0 0 0
Maxwell’s equations can then be written as
r
r
0
0
H
E
0 r
0 r
H j E
E j H
0 and 0 dropped from these equations because they are constants and do not vary spatially.
Expand Maxwell’s Equations
Slide 28
Divergence Equations
r
rr r
0
0yx z
H
HH H
x y z
0 r
0 r
0 r
0 r
yzx
x zy
y xz
E j H
EEj H
y z
E Ej H
z xE E
j Hx y
Curl Equations
r
rr r
0
0yx z
E
EE E
x y z
0 r
0 r
0 r
0 r
yzx
x zy
y xz
H j E
HHj E
y z
H Hj E
z xH H
j Ex y
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Normalize the Magnetic Field
Slide 29
0 rE j H
0 rH j E
Standard form of “Maxwell’s Curl Equations”
Normalized Magnetic Field
377E
nH
0
0
H j H
Normalized Maxwell’s Equations
0 rE k H
0 rH k E
0 0 0k Note:
Equalizes E and H amplitudes•Eliminates j•No sign inconsistency• Just have k0
Starting Point for Most CEM
Slide 30
0
0
0
yzxx x
x zyy y
y xzz z
HHk E
y z
H Hk E
z x
H Hk E
x y
0
0
0
yzxx x
x zyy y
y xzz z
EEk H
y z
E Ek H
z xE E
k Hx y
0
0
negative sign convnetion
positive sign convnetion
j HH
j H
We arrive at the following set of equations that are the same regardless of the sign convention used.
The manner in which the magnetic field is normalized does depend on the sign convention chosen.
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Slide 31
Scaling Properties ofMaxwell’s Equations
Scaling Properties of Maxwell’s Equations
Slide 32
There is no fundamental length scale in Maxwell’s equations.
Devices may be scaled to operate at different frequencies just by scaling the mechanical dimensions or material properties in proportion to the change in frequency.
This assumes it is physically possible to scale systems in this manner. In practice, building larger or smaller features may not be practical. Further, the properties of the materials may be different at the new operating frequency.
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Scaling Dimensions
Slide 33
We start with the wave equation and write the parameters dependence on position explicitly.
20 0 r
r
1E r r E r
r
Next, we scale the dimensions by a factor a.
20 0 r
r
1a a E r a r a E r a
r a
The scale factors multiplying the operators are moved to multiply the frequency term.
2
0 0 rr
1E r r E r
r a
1 stretch dimensions
1 compress dimensions
a
a
The effect of scaling the dimensions is just a shift in frequency.
rr
a
Visualization of Size Scaling
Slide 34
fc = 500 MHz
a = 1.0
fc = 1000 MHz
a = 0.5
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Scaling and
Slide 35
We apply separate scaling factors to and .
20 0 r
r
1E a E
a
The scale factors are moved to multiply the frequency term.
2
0 0 rr
1E a a E
The effect of scaling the material properties is just a factor in frequency.
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