Practical_Antenna_Handbook_0071639586

722 A p p e n d i x A : U s e f u l M a t h
so
h = 2a 2 = a 2 = 1.414a
(A.4.10)
The ability to calculate lengths and angles in triangles and squares is useful for such
diverse antenna tasks as laying out arrays of vertical antennas and obtaining their required
phasing line lengths, establishing the locations of guy anchors for a guyed tower,
determining the minimum support height needed for a delta loop, quad, or invertedvee
antenna, and estimating the length of strut support cables for a long-boom Yagi, to
name but a few.
A.5 Exponentials
As discussed in Chap. 3, if a capacitor has a net surplus of positive charges on one of its
plates and a net surplus of negative charges on its other plate, a voltage will exist between
the two plates indefinitely. If a resistor is connected across the two plates, however,
current will flow in the resistor as long as there is a voltage across it. After an
infinite interval of time, the voltage across the capacitor will be zero and there will be
equal numbers of positive and negative charges on both plates. As current flows in the
resistor, it does so according to Ohm’s law:
V
= IR
(A.5.1)
Because the initial flow of current results in a smaller surplus of charges on the
plates of the capacitor for the next short interval, the voltage across the capacitor is
slightly less at the end of the first interval. In fact, at the end of any short interval, the
voltage across (and, hence, the current through) the resistor is slightly less than at the
end of the immediately preceding interval, since R is constant throughout the discharge
process. As a result, the current or voltage in any given interval is a percentage (less
than 100 percent) of the current or voltage in the preceding interval. When this occurs,
the resulting current or voltage waveform is known as an exponential waveform, and it
has the general shape of the voltage shown in Fig. A.5.1A.
The speed of the discharge (and, hence, the scale of the x axis, which represents time
(t) is determined by the capacitance C and the resistance R according to Eq. (A.5.2):
−α t
V ( t) = V e
(A.5.2)
C
0
where V C (t) = voltage across capacitor as a function of time
V 0 = initial voltage across capacitor at time t = 0
e = Euler’s number, 2.7618
a = constant determined by circuit configuration and component values
t = time, in seconds
For this simple circuit,
1
α = ⎛ ⎝ ⎜ RC
⎞
⎟
⎠
(A.5.3)