Practical_Antenna_Handbook_0071639586

94 P a r t I I : F u n d a m e n t a l s
The received E-field at a point far from a very short (relative to wavelength) pair of
rods oriented vertically and driven by a sine wave at frequency f 0 is described in spherical
coordinates by
60πI ⎧ ⎛ ⎞⎫
0h
r
E θ
= sin θsin ⎨2πf
⎜t
− ⎟⎬
(3.19)
0
λr
⎩ ⎝ c ⎠⎭
and the corresponding H-field is
H φ
I ⎧ ⎛ ⎞⎫
0h
r
= sin θsin ⎨2πf
⎜t
− ⎟⎬
(3.20)
0
2 λ r ⎩ ⎝ c ⎠⎭
These equations are not as complicated as they may seem; let’s break them down
piece by piece, looking at each term:
• 60 is the result of combining a bunch of physical constants into a single number.
We’ll say more about it shortly.
• I 0 is the amplitude of the drive current at the center of the two rods; it simply
tells us the obvious: Any field we measure at a distant point is going to be
directly proportional to the drive current at the source.
• h is the total height, or length, of the antenna formed by the two rods, expressed
as a fraction of the wavelength, l. Implicit in the approximations allowing us to
use this equation is the constraint that h is small with respect to l. A good value
might be l/20 or less.
• r is the distance from the center of the two rods to our distant point.
• q (pronounced “thay-ta”) is the angle from the axis of the antenna. The sinq
term tells us that the E-field drops to zero off the ends of the antenna and is
maximum broadside to the antenna, where sinq = 1. (See App. A for derivations
and meaning of sine and cosine functions.)
• sin{2pf 0 (t – r/c)} identifies the instantaneous amplitude of the E-field as a
function of the drive signal frequency and the distance of the receiving point
from the rods. Because this is an argument of a sine function, as it increases the
resulting sine function simply goes through multiple cycles from –1 to +1 and
back again.
• The subscript q for the E-field means the only component of the E-field that has
any amplitude far from the two rods is an E-field lying in the same direction as
the axis of the rods. Although the axis of the rods is typically described as lying
on the z axis of a cartesian coordinate system, the radiated wave is best analyzed
in spherical coordinates. If not, we run the risk of E q becoming replaced by
complicated equations for E x , E y , and E z . An E-field comprised of solely E q is
consistent with our earlier statement that the E- and H-fields are always at right
angles to the direction of wave propagation.
• Similarly, the subscript f for the H-field tells us that the only detectable component
of the magnetic field at a distant point is in the f direction as defined in