Week 3
Electromagnetics 2 (EM-2) 전자기학2
Maxwell’s Equations
And Retarded Potential
in time-varying fields
Maxwell’s Equations in a time-varying fields
Fields
Differential form of Maxwell EquationIntegral form of Maxwell Equation
E-field (전기장)
Divergence Flux divert from a charge
No curl Zero line integral thru closed loop
M-field (자기장)
No divergence No flux divert from a point within a closed surface Curl Line integral of H = Current
Gauss’s Law
Gauss’s Law
34
Ampere’s Law
Ampere’s Law
Faraday’s Law
Ampere-Maxwell’s Law
Other related equations
E-field Motional Force M-field (Lorentz’s law)
per unit volume
Vacuum vs. Dielectric
- - -
+ + +
Q
0Vacuum
V
Applied voltage- +
- - - - -
+ + + + +
Q
0+Q
pDielectric
V
Applied voltage- +
+ -
+ -
+ -
+ -
+ -
+ -
ε
0ε =
E D = ε
0E D
E P
E E
D
e r
r e
ε
ε χ ε
ε ε
ε ε ε
χ
=
∴
≡
≡
←
= +
=
0 0
0 0
, ) 1
(
ε
0ε ε =
rD
V
D
D <
Review on EM-1
Different kinds of dielectrics
- - - - -
+ + + + +
Q
0+Q
pDielectric
V
Applied voltage- +
+ -
+ -
+ -
+ -
+ -
+ -
E D
E E
D
r
r e
ε
ε ε
ε
ε ε ε
χ
=
∴
≡
←
= +
=
0
0
)
01 (
ε
0ε ε =
rReview on EM-1
Magnetic Field with a Wire Current
X
I
X
I
a a
b b
A Material
With Spinning electrons Vacuum = Material
Without Spinning electrons
H
B = µ
0B = µ
0( H + M )
M
Review on EM-1
Relative Permeability (µ µ µ µ
r)
and Magnetic Susceptability (X
m)
X
I
a
b
A Material
With Spinning electrons
)
0
( H M
B = µ +
M
m
m
or χ
χ H H M / M ≡ =
r m
m
µ χ
χ µ
≡
⇒ +
+
⇒ =
) 1
(
) 1
0
( H
B
Magnetic Susceptability
Relative Permeability
r
r
µ µ µ µ
µ
µ
0= ← ≡
0= H H
B
Review on EM-1
Example
2
1 N
N
B
B =
K H
H
t1−
t2=
(
Ht1 −Ht2)
= aN12 ×K = K×aN21Perfect Conductor
(E2=0, ∆∆∆∆V=0, J=0, H2=0)
Medium 1
Time-varying incident EM waves E-field: E
1(t), D
1(t)
M-field: B
1(t), H
1(t)
21
aN
Medium 2
≠ 0 K
≠ 0
ρ
S≠ 0
≠ 0
= 0
= 0
= 0
= 0
Transmission from antenna to antenna
EM waves
Applying AC signal (sinusoidal Voltage)
Receiving AC signal (sinusoidal Voltage)
Transmitting
Antenna Receiving
Antenna
Transmission EM waves
V , I V’ , I’
Station Radio
Distance: R Velocity: v
) / (
] [ ) / (
]
[ρv = f t −R v J = g t −R v v
Delay = R
Physics and Mathematical Formulation for signals
in Transmission Line
Transmission Line (TML) = Two wires
V
G~ Z
L+
-
+
-
V
OZ
Gf v
pλ =
f f T
p
π ω 2
1
=
=
-L 0
z
Reflected wave Incident wave
V
in+
-
Z
oΓ Γ Γ Γ
Z
inz
V(z,t)
V
I(z,t)
V
R(z,t)
Final Goal = Find V
O= V(0,t)
)
0
cos( t
V
V
G=
Gω
= V(-L,t)
Intermediate Goal = Find V(z,t)
Summary of TML Equations
( )
oi(
j z j z)
j tt j z
z
oi
e e e V e e e
V t
z
V ( , ) =
−γ+ Γ
+γ ω=
−(α+ β)+ Γ
+(α+ β) ω( )
j toi
O
V t V e
V = = + Γ
ω∴ ( 0 , ) 1
≠
= 0 0 α
α
: Lossless: Lossy
Attenuation constant:
Complex Form (more useful and easier for calculation)
(
j L j L)
j toi
in
V L t V e e e
V = ( − , ) =
(α+ β)+ Γ
−(α+ β) ω( )
j to
Out
V t V e
V = ( 0 , ) =
+1 + Γ
ω( + Γ ) = ( + Γ )
= 1
+1
,Out oi o
S
V V
V
Different notation:
Space-only form:
(
j L j L)
oi in
S V e e
V , = (α+ β) +Γ −(α+ β)
Space-only form:
Now, what we need to know in physical level
L = n ∆ ∆ ∆ ∆ z
Distributed lumped-element model Distributed line model
R, L, C, and G
per unit lengthCross-Relationships
= ?
α
= ?
β
= ? Γ
= ? Z
in= ? Z
O= ? Z
LR ( Ω /m) L ( Η /m) C (F/m) G (S/m)
Parameters Elements Model
= ?
γ
Cross-Relationships in equations (next Lectures)
Parameters
Elements
Model
O L
O L
Z Z
Z Z
+
= − Γ
+
= +
L jZ
L Z
L jZ
L Z Z
Z
L O
O L
O
in
β β
β β
sin cos
sin cos
C j G
L j R t
z I
t z V t
z I
t z Z V
R R I
I
O
ω
ω +
= +
≡ −
≡ ( , )
) , ( )
, (
) , (
given Z
L:
β α
ω ω
γ
= (R + j L)(G + j C) ≡ + jR (Ω/m), L (Η/m), C (F/m), G (S/m)
R = 0 and G = 0 in lossless propagation
C ZOlossless = L
LC j
LC j
C Lj
lossless j
ω ω ω β β ω
γ
= = = → =v β = ω
v LC1
=
