lecture 7 the uniform plane wave - piazza
TRANSCRIPT
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 ๐๐.
We can express Eq. (8)๐๐ป๐ฆ
๐๐ง= โ๐0
๐๐ธ๐ฅ
๐๐ก(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,
๐ = ๐ธ๐ฅ๐๐ฅ ๐ธ๐ฅ = ๐ธ ๐ฅ, ๐ฆ, ๐ง cos ๐๐ก + ๐
Using Eulerโs Identity ๐๐๐๐ก = cos๐๐ก + ๐ sin๐๐ก
The field is ๐ธ๐ฅ = Re ๐ธ ๐ฅ, ๐ฆ, ๐ง ๐๐๐๐๐๐๐ก
The field in the phasor form is ๐ธ๐ฅ๐ = ๐ธ ๐ฅ, ๐ฆ, ๐ง ๐๐๐ ๐๐ = ๐ธ๐ฅ๐ ๐๐ฅ
For any sinusoidal time variation for field๐
๐๐ก๐ธ = ๐๐๐ธ and
๐
๐๐ก๐ป
= ๐๐๐ป
The Maxwellโs 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 Maxwellโs Equation the steady state wave equation can be
written as:
Since
๐๐, the free space wavenumber, ๐๐ = ๐ ๐๐๐๐
The wave equation for the ๐ฅ-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 EHโEโ-)Eโ(โ)Eโ(โ ooo-j
0Eโs s
2
s
2 EEโ ok
xs
2
xs
2 EEโ ok
xsoxsxsxs Ek
z
E
y
E
x
E 2
2
2
2
2
2
2
โ
โ
โ
โ
xsoxs Ek
dz
Ed 2
2
2
(31)
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Given E๐ , ๐๐ is most easily obtained from (24):
which is greatly simplified for a single ๐ธ๐ฅ๐ component varying only
with ๐ง,
Using (31) for๐ธ๐ฅ๐ = ๐ธ๐ฅ0๐โ๐๐0๐ง + ๐ธ๐ฅ0
โฒ ๐๐๐0๐ง, we have
In real instantaneous form, this becomes
where ๐ธ๐ฅ0 and ๐ธ๐ฅ0โฒ 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 ๐0 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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8
Thank you