Formula sheet: Time Varying Electromagnetic Fields | Electromagnetics - Electronics and Communication Engineering (ECE) PDF Download

 Page 1


Formula Sheet for Time-Varying Electromagnetic
Fields (EMFT) – GATE
1. Basic Concepts
• Time-VaryingFields: Electricandmagnetic?eldsthatchangewithtime,leading
to electromagnetic wave propagation.
• Constants:
– Permittivity of free space: ?
0
= 8.854×10
-12
F/m.
– Permeability of free space: µ 0
= 4p×10
-7
H/m.
– Speed of light: c =
1
v
µ 0
?
0
˜ 3×10
8
m/s.
2. Maxwells Equations (Time-Varying Fields)
• Faradays Law (Di?erential Form):
?×E =-
?B
?t
• Amperes Law with Displacement Current:
?×H =J+
?D
?t
whereD = ?E,B = µ H.
• Gausss Law for Electric Field:
?·D = ?
• Gausss Law for Magnetic Field:
?·B = 0
• Integral Forms:
I
C
E·dl =-
d
dt
ZZ
S
B·dS
I
C
H·dl = I +
d
dt
ZZ
S
D·dS
3. Wave Equation
• Electric Field Wave Equation:
?
2
E-µ?
?
2
E
?t
2
= 0
1
Page 2


Formula Sheet for Time-Varying Electromagnetic
Fields (EMFT) – GATE
1. Basic Concepts
• Time-VaryingFields: Electricandmagnetic?eldsthatchangewithtime,leading
to electromagnetic wave propagation.
• Constants:
– Permittivity of free space: ?
0
= 8.854×10
-12
F/m.
– Permeability of free space: µ 0
= 4p×10
-7
H/m.
– Speed of light: c =
1
v
µ 0
?
0
˜ 3×10
8
m/s.
2. Maxwells Equations (Time-Varying Fields)
• Faradays Law (Di?erential Form):
?×E =-
?B
?t
• Amperes Law with Displacement Current:
?×H =J+
?D
?t
whereD = ?E,B = µ H.
• Gausss Law for Electric Field:
?·D = ?
• Gausss Law for Magnetic Field:
?·B = 0
• Integral Forms:
I
C
E·dl =-
d
dt
ZZ
S
B·dS
I
C
H·dl = I +
d
dt
ZZ
S
D·dS
3. Wave Equation
• Electric Field Wave Equation:
?
2
E-µ?
?
2
E
?t
2
= 0
1
• Magnetic Field Wave Equation:
?
2
H-µ?
?
2
H
?t
2
= 0
• Wave Speed in Medium:
v =
1
v
µ?
4. Electromagnetic Wave Propagation
• Plane Wave Solution:
E =E
0
e
j(?t-k·r)
, H =H
0
e
j(?t-k·r)
where ?: Angular frequency,k: Wave vector.
• Wave Number:
k =
?
v
= ?
v
µ?
• Wavelength:
? =
v
f
=
2p
k
• Phase Velocity:
v
p
=
?
k
5. Poynting Vector
• Power Flow:
S =E×H
(unit: W/mš).
• Average Poynting Vector (Time-Harmonic Fields):
S
avg
=
1
2
Re(E×H
*
)
6. Boundary Conditions
• Tangential Electric Field:
E
1t
=E
2t
• Tangential Magnetic Field:
H
1t
-H
2t
=J
s
(whereJ
s
: Surface current density).
2
Page 3


Formula Sheet for Time-Varying Electromagnetic
Fields (EMFT) – GATE
1. Basic Concepts
• Time-VaryingFields: Electricandmagnetic?eldsthatchangewithtime,leading
to electromagnetic wave propagation.
• Constants:
– Permittivity of free space: ?
0
= 8.854×10
-12
F/m.
– Permeability of free space: µ 0
= 4p×10
-7
H/m.
– Speed of light: c =
1
v
µ 0
?
0
˜ 3×10
8
m/s.
2. Maxwells Equations (Time-Varying Fields)
• Faradays Law (Di?erential Form):
?×E =-
?B
?t
• Amperes Law with Displacement Current:
?×H =J+
?D
?t
whereD = ?E,B = µ H.
• Gausss Law for Electric Field:
?·D = ?
• Gausss Law for Magnetic Field:
?·B = 0
• Integral Forms:
I
C
E·dl =-
d
dt
ZZ
S
B·dS
I
C
H·dl = I +
d
dt
ZZ
S
D·dS
3. Wave Equation
• Electric Field Wave Equation:
?
2
E-µ?
?
2
E
?t
2
= 0
1
• Magnetic Field Wave Equation:
?
2
H-µ?
?
2
H
?t
2
= 0
• Wave Speed in Medium:
v =
1
v
µ?
4. Electromagnetic Wave Propagation
• Plane Wave Solution:
E =E
0
e
j(?t-k·r)
, H =H
0
e
j(?t-k·r)
where ?: Angular frequency,k: Wave vector.
• Wave Number:
k =
?
v
= ?
v
µ?
• Wavelength:
? =
v
f
=
2p
k
• Phase Velocity:
v
p
=
?
k
5. Poynting Vector
• Power Flow:
S =E×H
(unit: W/mš).
• Average Poynting Vector (Time-Harmonic Fields):
S
avg
=
1
2
Re(E×H
*
)
6. Boundary Conditions
• Tangential Electric Field:
E
1t
=E
2t
• Tangential Magnetic Field:
H
1t
-H
2t
=J
s
(whereJ
s
: Surface current density).
2
• Normal Electric Displacement:
D
1n
-D
2n
= ?
s
(where ?
s
: Surface charge density).
• Normal Magnetic Flux Density:
B
1n
=B
2n
7. Electromagnetic Potentials
• Scalar Potential:
E =-?V -
?A
?t
• Vector Potential:
B =?×A
• Lorenz Gauge:
?·A+µ?
?V
?t
= 0
8. Wave Parameters in Lossy Media
• Complex Propagation Constant:
? = a+jß
where a: Attenuation constant, ß: Phase constant.
• Propagation Constant:
? = j?
v
µ?
r
1-j
s
??
where s: Conductivity.
• Intrinsic Impedance:
? =
r
µ ?
(free space: ?
0
˜ 377? )
9. Energy and Power
• Electric Energy Density:
w
e
=
1
2
D·E =
1
2
?E
2
• Magnetic Energy Density:
w
m
=
1
2
B·H =
1
2
B
2
µ • Power Conservation (Poynting Theorem):
I
S
S·dS =-
d
dt
ZZZ
V
 
1
2
?E
2
+
1
2
B
2
µ !
dV -
ZZZ
V
J·EdV
3
Page 4


Formula Sheet for Time-Varying Electromagnetic
Fields (EMFT) – GATE
1. Basic Concepts
• Time-VaryingFields: Electricandmagnetic?eldsthatchangewithtime,leading
to electromagnetic wave propagation.
• Constants:
– Permittivity of free space: ?
0
= 8.854×10
-12
F/m.
– Permeability of free space: µ 0
= 4p×10
-7
H/m.
– Speed of light: c =
1
v
µ 0
?
0
˜ 3×10
8
m/s.
2. Maxwells Equations (Time-Varying Fields)
• Faradays Law (Di?erential Form):
?×E =-
?B
?t
• Amperes Law with Displacement Current:
?×H =J+
?D
?t
whereD = ?E,B = µ H.
• Gausss Law for Electric Field:
?·D = ?
• Gausss Law for Magnetic Field:
?·B = 0
• Integral Forms:
I
C
E·dl =-
d
dt
ZZ
S
B·dS
I
C
H·dl = I +
d
dt
ZZ
S
D·dS
3. Wave Equation
• Electric Field Wave Equation:
?
2
E-µ?
?
2
E
?t
2
= 0
1
• Magnetic Field Wave Equation:
?
2
H-µ?
?
2
H
?t
2
= 0
• Wave Speed in Medium:
v =
1
v
µ?
4. Electromagnetic Wave Propagation
• Plane Wave Solution:
E =E
0
e
j(?t-k·r)
, H =H
0
e
j(?t-k·r)
where ?: Angular frequency,k: Wave vector.
• Wave Number:
k =
?
v
= ?
v
µ?
• Wavelength:
? =
v
f
=
2p
k
• Phase Velocity:
v
p
=
?
k
5. Poynting Vector
• Power Flow:
S =E×H
(unit: W/mš).
• Average Poynting Vector (Time-Harmonic Fields):
S
avg
=
1
2
Re(E×H
*
)
6. Boundary Conditions
• Tangential Electric Field:
E
1t
=E
2t
• Tangential Magnetic Field:
H
1t
-H
2t
=J
s
(whereJ
s
: Surface current density).
2
• Normal Electric Displacement:
D
1n
-D
2n
= ?
s
(where ?
s
: Surface charge density).
• Normal Magnetic Flux Density:
B
1n
=B
2n
7. Electromagnetic Potentials
• Scalar Potential:
E =-?V -
?A
?t
• Vector Potential:
B =?×A
• Lorenz Gauge:
?·A+µ?
?V
?t
= 0
8. Wave Parameters in Lossy Media
• Complex Propagation Constant:
? = a+jß
where a: Attenuation constant, ß: Phase constant.
• Propagation Constant:
? = j?
v
µ?
r
1-j
s
??
where s: Conductivity.
• Intrinsic Impedance:
? =
r
µ ?
(free space: ?
0
˜ 377? )
9. Energy and Power
• Electric Energy Density:
w
e
=
1
2
D·E =
1
2
?E
2
• Magnetic Energy Density:
w
m
=
1
2
B·H =
1
2
B
2
µ • Power Conservation (Poynting Theorem):
I
S
S·dS =-
d
dt
ZZZ
V
 
1
2
?E
2
+
1
2
B
2
µ !
dV -
ZZZ
V
J·EdV
3
10. Design Considerations
• Wave Propagation: Use plane wave solutions for free space or lossless media.
• Boundary Conditions: Apply at conductor-dielectric interfaces.
• Applications: Antennas, waveguides, RF circuits, electromagnetic compatibility.
• Lossy Media: Account for attenuation in conductive materials.
4