Direct Energy, 2018a

4 ANTENNAS 69
4.2 Electromagnetic Radiation
4.2.1 Superposition
The physics of antenna operation is described by Maxwell's equations. Ampere's
law, one of Maxwell's equations, was introduced in Section 1.6.1.
−→ −→ −→ ∂ −→ D
∇× H = J + (4.1)
∂t
In Eq. 4.1, −→ H is the magnetic eld intensity in m A , −→ D is the displacement
ux density in m C , and −→ J is the current density in A 2 m . In the case of
2
a transmitting antenna, the current density in the antenna comes from a
known source, and the electromagnetic eld, described by −→ D and −→ H , can
be derived.
Using Maxwell's equations, we can algebraically derive the electromagnetic
eld only for very simple antennas. The simplest antenna is an in-
nitesimal dipole antenna, also known as a Hertzian dipole. References
[11] derives the electric eld intensity, −→ E in units m V , for an innitesimal
dipole antenna with length dl and sinusoidal current I 0 cos(ωt). The result
is given in spherical coordinates is
[
−→ sin(ωt−
E =
2π λ r )
2I 0·dl·cos θ
4πɛω
+ I 0·dl·sin θ
4πɛω
r 3 + 2π λ cos (ωt− 2π λ r )
r 2 ]
â r
[
sin(ωt− 2π λ r )
+ 2π λ cos (ωt− 2π λ r )
− ( 2π λ ) 2 sin(ωt− 2π λ r )
r 3 r 2 r
]
â θ .
(4.2)
In this expression, ω is frequency in rad
s , λ is the wavelength in meters,
ɛ is the permittivity of the material surrounding the antenna in m F , and
(r, θ, φ) are the coordinates of a point specied in spherical coordinates.
For complicated antennas, superposition is used to make the computation
feasible. To derive the electromagnetic radiation from a complicated
antenna, small straight antenna segments are considered [15, ch. 10]. The
electromagnetic radiation from each piece is found, and the principle of
superposition is the idea that the radiation fromthe entire antenna is the
sumof these pieces. The same idea applies to linear circuits. If a circuit has
a complicated input, the input can be broken up into simpler components.
Any voltage in the circuit can be found by nding the contribution due to
each of these components then summing.
4.2.2 Reciprocity
Reciprocity is the idea that the behavior of an antenna as a function of
angle is the same regardless of whether the antenna is being used to send