Practical_Antenna_Handbook_0071639586

C h a p t e r 4 : T r a n s m i s s i o n L i n e s a n d I m p e d a n c e M a t c h i n g 115
In other words, each short section of any transmission line can be represented by a
(relatively) simple RLC network. If we then use network analysis and calculus to examine
the current and voltage relationships at the input of a long transmission line made up
of an infinite number of infinitesimally short sections daisy-chained together, we find
the transmission line exhibits an impedance at its input terminals that is independent of
the length of the line. This characteristic impedance Z 0 , also sometimes called the line’s
surge impedance, is a function of the four parameters previously defined, which in turn
are a function of the physical geometry of the line, including conductor size, shape, and
spacing, and the dielectric constant of the insulating material between the conductors.
Z
0
=
R'
G'
+
+
jwL'
jwC'
(4.1)
where Z 0 = characteristic impedance of line, in ohms
R′ = total series resistance per unit length of two conductors, in ohms
G′ = shunt conductance between two conductors per unit length, in mhos
L′ = total series inductance per unit length of two conductors, in henrys
C′ = shunt capacitance per unit length between conductors, in farads
j = imaginary number -1
w = angular frequency in radians per second (w = 2πf )
In the general case, Z 0 is complex; that is, it has both a resistive component and a
reactance component. Most real transmission lines fall in this category. The existence of
a reactance component leads to attenuation and delay across each unit length that is
often frequency dependent. When that is the case, signals applied to one end of the line
are said to suffer dispersion or frequency dispersion as they propagate along the line. Thus,
a complex waveform consisting of multiple-frequency components will become distorted
by the time it reaches the far end of the line, since the different frequencies will
have experienced differing amounts of attenuation over their common path.
For certain values of R′, G′, L′, and C′, the transmission is lossless and Z 0 is purely
resistive or so nearly so that we can work with it as if it were. Three such cases are:
• R′ = G′ = 0. In an ideal cable, the series resistance is zero and the shunt resistance
is infinite, so Eq. (4.1) reduces to the following simplified form for a lossless cable:
Z = 0
L'
C'
(4.2)
• wL′ >> R′ and wC′ >> G′. Although not zero, the series resistance and shunt
conductance are negligible with respect to the series inductance and shunt
capacitance, respectively, at the frequencies of operation. Note, however, that R′
is a function of frequency because the effective resistance of a wire at RF is
modified by a frequency-dependent phenomenon called skin effect. It is not
appropriate, for instance, to blindly use the dc or ohmic value of R at RF.
• R′/L′ = G′/C′ and both R′ and G′ are small, but not necessarily negligible. If this
condition is met, the line is distortionless or dispersionless even though it may not
be lossless.