lecture 7 the uniform plane wave - piazza
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Electromagnetic
Field Theory
Ahmed Farghal, Ph.D.Electrical Engineering, Sohag University
Lecture 7
The Uniform Plane Wave
Nov. 13, 2019
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2
Taking the partial derivative of any field quantity w. r. t. time is
equivalent to multiplying the corresponding phasor by ππ.
We can express Eq. (8)ππ»π¦
ππ§= βπ0
ππΈπ₯
ππ‘(using sinusoidal fields) as
where, in a manner consistent with (19):
On substituting the fields in (21) into (20), the latter equation
simplifies to
Phasor Form
(20)
(22)
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Phasor Form
3
For sinusoidal or cosinusoidal time variation,
π = πΈπ₯ππ₯ πΈπ₯ = πΈ π₯, π¦, π§ cos ππ‘ + π
Using Eulerβs Identity ππππ‘ = cosππ‘ + π sinππ‘
The field is πΈπ₯ = Re πΈ π₯, π¦, π§ πππππππ‘
The field in the phasor form is πΈπ₯π = πΈ π₯, π¦, π§ πππ ππ = πΈπ₯π ππ₯
For any sinusoidal time variation for fieldπ
ππ‘πΈ = πππΈ and
π
ππ‘π»
= πππ»
The Maxwellβs equation in phasor form
Eqs. (23) through (26) may be used to obtain the sinusoidal
steady-state vector form of the wave equation in free space.
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Wave Equation
4
Using Maxwellβs Equation the steady state wave equation can be
written as:
Since
ππ, the free space wavenumber, ππ = π ππππ
The wave equation for the π₯-components is
Using the definition of Del operator
The field components does not change with x or y so the wave
equation is lead to ordinary differential equation
the solution of which
Vector Helmholtz Wave Equation
s
2
ss
2
ss EHβEβ-)Eβ(β)Eβ(β ooo-j
0Eβs s
2
s
2 EEβ ok
xs
2
xs
2 EEβ ok
xsoxsxsxs Ek
z
E
y
E
x
E 2
2
2
2
2
2
2
β
β
β
β
xsoxs Ek
dz
Ed 2
2
2
(31)
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Given Eπ , ππ is most easily obtained from (24):
which is greatly simplified for a single πΈπ₯π component varying only
with π§,
Using (31) forπΈπ₯π = πΈπ₯0πβππ0π§ + πΈπ₯0
β² πππ0π§, we have
In real instantaneous form, this becomes
where πΈπ₯0 and πΈπ₯0β² are assumed real.
WAVE PROPAGATION IN FREE SPACE
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In general, we find from (32) that the electric and magnetic field
amplitudes of the forward-propagating wave in free space are
related through
We also find the backward-propagating wave amplitudes are related
through
where the intrinsic impedance of free space is defined as
The dimension of π0 in ohms is immediately evident from its
definition as the ratio of E (in units of V/m) to H (in units of A/m).
WAVE PROPAGATION IN FREE SPACE
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The fields
7
The uniform plane cannot exists because it extends to infinity in
two dimensions at least and represents an infinite a mount of
energy.
The far field of the antenna can be represent by a uniform plane
wave
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Thank you
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