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Time-Harmonic Maxwell’s Equations
- We studied the Maxwell’s equations in differential forms and
integral forms for general time varying fields
- However, many practical systems and sources are time-
harmonic (Fields have variations in sinusoidal or cosinusoidal
forms)
- Such time variations can be represented as e
j w t
- In this case, we can simplify matters by using Maxwell’s
equations in the frequency-domain
- Maxwell’s equations in the frequency-domain are relationships
between the phasor representations of the fields.
Ratio of conduction Current Density J
c
to Displacement
Current Density J
d
t c d
J J J
J E
c
j E
t
D
J
d
w
J E j E
t
w
w
tan
d
c
J
J
So as frequency increases displacement current J
d
becomes equally important w
Maxwell’s Equations in Differential Form for Time-
Harmonic Fields
Source Free (𝜌 = 0 ), (
ഥ
𝐽 = 0 ) and loss less ( =0)
Eqns.(1) and (2) are coupled Equations
Taking Curl on ( 1 ) and ( 2 ) we get
......( 2 )
......( 1 )
H j E
E j H
w
w
......( 4 )
......( 3 )
H j E
E j H
w
w
In phasor notation:
Unbound, Homogeneous, Isotropic
No source No charge and No current
Maxwell’s Equations in Differential Form for Time-
Harmonic Fields
Similarly
(. ) ( )......( 5 )
......( 3 )
2
E E j j E
E j H
w w
w
......( 6 )
2 2
E w E
(. ) ( )......( 7 )
......( 4 )
2
H H j j H
H j E
w w
w
......( 8 )
2 2
H w H
0
0
Charge Free
Charge Free
Maxwell’s Equations in Differential Form for Time-
Harmonic Fields: The Wave Equations
Solution to wave equations:
E E z x
ˆ ( )
Expanding incartesiancoordinates
2
2
2
2
2
2
2
2
2 2
2 2
x x
x x
E E
x y z
E E
E E
w
w
w
E field is oriented in x direction and varying along z
e
j w t
is implicit
x
x
x
E
E
x y
E
w
2
2
2
2
2
2
2
dz
d
Since isa functionof (z)only, 0
Maxwell’s Equations in Differential Form for Time-
Harmonic Fields: The Wave Equations
2 2
w
Where
Propagation constant square
w w
w
j
2
2 2
Propagation constant
j
Attenuation constant
Phase constant
Maxwell’s Equations in Differential Form for Time-
Harmonic Fields: The Wave Equations
Solution to wave equations:
x
x
E
E
w
2
2
2
dz
d
w 0
dz
d
2
2
2
x
x
E
E
j z
x
j z
x x
E z E e E e
Traveling wave in (-)ve z direction Traveling wave in (+)ve z direction
Simplest solution that can exist in a unbound media is an E field which is constant in a plane containing the
field vector and have variation perpendicular to the vector
Solution
Maxwell’s Equations in Differential Form for Time-
Harmonic Fields: The Wave Equations
Taking curl of E , we get
j H
E z
z
x y z
x
w
( ) 0 0
0 0
ˆ ˆ ˆ
E e E e y j H
z
j z
x
j z
x
w
ˆ
E j w H
Since E , has no variation in x, and y directions, we get
y
j z
x
j z
x
x y z
j z
x
j z
x
j E e j E e j H
E e E e y j H x H y H z
z
w
w
ˆ ˆ ˆ ˆ
Maxwell’s Equations in Differential Form for Time-
Harmonic Fields: The Wave Equations
Assignment
Find similar solution for a ŷ oriented E field
- So E and H are always perpendicular to each other and propagate along a direction perpendicular to them
TEM wave (Transverse ElectroMagnetic wave) in an unbound media ( may not free space )
- The ratio of the magnitude of E and H field proves the intrinsic impedance
- Moreover once and are known we can calculate
- If we know E we an calculate H
Maxwell’s Equations in Differential Form for Time-
Harmonic Fields
Intrinsic impedance of the medium
0
0
0
If medium is in free space
Free space impedance
H m
F m
4 10 /
10 /
36
1
7
0
9
0
?
0
Free space
impedance
Wave propagation in Unbound Media with Finite
Conductivity
D E E
r
0
Now Electric Displacement vector:
J E
c
c
J
So, we have
- Conduction current density –
- Displacement current density -
t
D
For Time Harmonic Fields:
H E j E
t
D
H E
t
D
H J
r
c
w
0
Wave propagation in Unbound Media with Finite
Conductivity
No conductivity ( J
c
= 0 )
..........( 9 )
0
0
0
0
0
H j j E
E
j
H j
H E j E
r
r
r
w
w
w
w
w
j E ......( 10 )
t
D
H w
..........( 9 )
0
0
H j j E
r
w
w
Compare Equation ( 9 ) with
Dielectric constant or relative permittivity of the medium
Because of Finite Conductivity, the dielectric constant is now a complex quantity
Wave propagation in Unbound Media with Finite
Conductivity
Complex Dielectric Constant or Complex Permittivity
Any medium which has finite conductivity will behave like a conductor if we go to the lower end of the
spectrum
At higher end of the frequency spectrum where w is very large then for any finite value of conductivity, the
medium will behave like a dielectric
If conduction current >> displacement current : medium is a conductor w
If conduction current << displacement current : medium is a dielectric
w
So, high conductor value does not always justify that the medium is a conductor, it depends on the value of
frequency w
r
c
r
w
w w
0
0
when
Transition angular
frequency
c
c
w w
w w
Dielectric
Conductor
Wave propagation in Unbound Media with Finite
Conductivity
So, sea water below 225 KHz will behave like a conductor and above 225 KHz will behave like a dielectric
Copper : = 5.8 x 10
7
Siemens/m (assume
r
= 1 )
Find f
c
????
Sea water : = 1 x 10
Siemens/m (
r
= 81 )
Find f
c
????
