Practical_Antenna_Handbook_0071639586

120 P a r t I I : F u n d a m e n t a l s
Earlier we modeled the series impedance of a short length of transmission line as
R′ + jwL′, and the shunt admittance as G′ + jwC′. The propagation constant for a lossy
transmission line is defined as
γ = ( R' + jωL'
) ( G' + jωC'
)
(4.13)
The propagation constant can also be written in terms of the line attenuation constant a
and phase constant b:
γ = α + jβ (4.14)
Thus, when the line has loss, we can represent a sinusoidal (single-frequency) waveform
with the following equation:
−α
V( z, t) = V e z cos(2 πft −βz)
(4.15)
0
where V(z,t) = notation that tells us the observed amplitude of the wave is a function
of (i.e., depends upon) its location along the line (the z axis) and the
time of observation
V 0 = amplitude of the input waveform at the peak of its sine wave or cycle
e –az = decaying exponential term representing the effect of line loss, called
the attenuation factor; it is a function of wave frequency
2pft = portion of the cosine argument that mathematically accounts for the
sinusoidal or cyclic variation in the amplitude of the wave with the
passage of time
bz = portion of the cosine argument that mathematically accounts for the
sinusoidal spatial variation in the amplitude of the wave at any
instant; b is “shorthand” for 2p/l where l is the wavelength of the
wave in the transmission line
The loss term is exponential because the loss mechanism in a transmission line is a
linear function of distance along the line; that is, the peak amplitude of the wave a little
farther along the line is, say, 99 percent of the peak wave amplitude at this point on the
line. The peak amplitude of the wave at a second point farther along the line is then 99
percent of its amplitude at the previous point. And so on. In theory, the wave never
quite goes to zero, no matter how long the line may be. a is called the attenuation constant
of the transmission line, and it is a dimensionless number with units of nepers per
meter (Np/m). It is related to natural logarithms the same way bels and dB are related to
base-10 logarithms. (See App. A.)
Unfortunately, in the general case where we have no feeling for the relative importance
of the R′ and G′ terms, a and b are unbearably complicated and Z 0 is a complex
quantity—that is, it includes reactance terms. However, for the case of a line with just a
little bit of attenuation, we can “cheat” by allowing the argument of the cosine term to
be the same as for the lossless line and pretending the only significant effect of the loss
is the addition of the exponential term in Eq. (4.15).
In that case, a can be inferred from the attenuation specification for the line, and b
can be rewritten in terms of parameters we know. Specifically, we know the velocity of
propagation, v, in the transmission line is given by v = fl. Since b = 2p/l, we can write