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 139 feedpoint impedances substantially higher or lower and/or more reactive than the ideal l/2 dipole. So we can be reasonably certain that a matching network at the junction of the feedline and the antenna is likely to be needed. For most of us, the transmitters, transceivers, and amplifiers we are apt to buy or build are designed for maximum power transfer to a 50-Ω resistive load. In many cases we can simplify our task by choosing to use 52-Ω or 75-Ω transmission line from the transmitter to a matching network closer to the antenna. Most of the following techniques for obtaining a match assume just such a configuration. In general there are two broad categories of matching techniques: • Distributed matching systems (stubs, transmission line sections, etc.) • Lumped element tuners (antenna tuning units, or ATUs) The second category is covered in Chap. 24. R 2 Distributed Matching Networks Networks that depend on the propagation of the signal along a carefully chosen length of conductor for their matching capabilities are called distributed networks. The popular examples described in the following sections are typically employed either as a series element placed between the antenna and feedline or as a shunt element directly across the antenna feedpoint or a short distance along the feedline from it. Keep in mind that since the dimensions of most distributed matching networks are a function of the operating frequency (or wavelength), a specific implementation will work as advertised only over a limited frequency span centered on the design frequency. Most of the networks described here are most useful, therefore, as matching devices for single-band antennas and arrays. R 2 4R 1 Coaxial Cable Baluns Figure 4.11 describes a balun that transforms impedances at a 4:1 ratio, with R 2 = 4 × R 1 . The length of the balun (or U-shaped) section of coaxial cable is L = 492 F (4.41) Fv where L = measured physical length of U-shaped section of cable, in feet v F = velocity factor of coaxial cable (typically 0.66–0.85) F = operating frequency, in megahertz R 1 Figure 4.11 Coaxial balun transformer.
140 P a r t I I : F u n d a m e n t a l s In general, a balun can be used in either a step-up or a step-down configuration. For instance, this balun is frequently used as shown to transform the 300-Ω feedpoint impedance of a folded dipole down to 75 Ω for a coaxial transmission line. The radiation (or feedpoint) resistance of the dipole corresponds to R 2 in the figure, and R 1 represents the transmitter output impedance. However, the owner of a multielement Yagi or a short (< l/4) vertical monopole might employ the balun in a reverse configuration; now R 1 represents the low feedpoint impedance of the antenna, while R 2 corresponds to the Z 0 of a higher-impedance transmission line. Matching with Stubs The input impedance of a random length of lossless (or nearly so) transmission line is given by: ⎡ Z Z Z jZ l L + 0 tan( β ) ⎤ IN = 0 ⎢ ⎥ ⎣Z0 + jZL tan( βl) (4.42) ⎦ where Z IN = input impedance of line looking into end opposite load Z L = load impedance Z 0 = characteristic impedance of transmission line b = 2p/l l = length of transmission line, stated in terms of l The term (bl) in the argument of the tangent functions in Eq. (4.42) includes l in both the numerator and the denominator, so it does not matter whether we use the freespace wavelength or the wavelength in the transmission line, as long as the same one is used for both b and l. However, in some of the manipulations and outcomes that follow, l refers to the wavelength in the transmission line, so it is wise to always use that wavelength when working with stubs. Most of the time, Z IN will be complex—that is, having both a resistive (R) term and a reactive (jX) term—even for a purely resistive load, except when either of the following situations is true: • The input end of the line is one specific distance in the range 0 l l/4 from the load, or an integer multiple of l/2 beyond that distance. • The input end of the line is at one specific distance in the range l/4 l l/2 from the load, or an integer multiple of l/2 beyond that distance. Since there is an infinite number of possible impedances that Z IN can assume for an arbitrary line length, the odds of finding a purely resistive value of Z IN are extremely small—unless, of course, Z L = Z 0 . A very important characteristic of transmission lines comes into view if we examine Eq. (4.42) a little further. Suppose we are free to adjust the length (l) of the line so that the input end of the line is at a value of l that causes the tangent functions in Eq. (4.42) to be zero. Since that occurs when the argument (bl) of both tangent functions is zero or at phase angles that are integer multiples of π radians, l must be an integer multiple of l/2. In fact, when l = nl/2 (for integer values of n), Z IN = Z L , regardless of whether Z L is resistive, reactive, or complex.
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140 P a r t I I : F u n d a m e n t a l s<br />
In general, a balun can be used in either a step-up or a step-down configuration. For<br />
instance, this balun is frequently used as shown to transform the 300-Ω feedpoint impedance<br />
of a folded dipole down to 75 Ω for a coaxial transmission line. The radiation<br />
(or feedpoint) resistance of the dipole corresponds to R 2 in the figure, and R 1 represents<br />
the transmitter output impedance. However, the owner of a multielement Yagi or a<br />
short (< l/4) vertical monopole might employ the balun in a reverse configuration;<br />
now R 1 represents the low feedpoint impedance of the antenna, while R 2 corresponds to<br />
the Z 0 of a higher-impedance transmission line.<br />
Matching with Stubs<br />
The input impedance of a random length of lossless (or nearly so) transmission line is<br />
given by:<br />
⎡<br />
Z Z Z jZ l<br />
L<br />
+<br />
0<br />
tan( β ) ⎤<br />
IN<br />
=<br />
0 ⎢<br />
⎥<br />
⎣Z0<br />
+ jZL<br />
tan( βl)<br />
(4.42)<br />
⎦<br />
where Z IN = input impedance of line looking into end opposite load<br />
Z L = load impedance<br />
Z 0 = characteristic impedance of transmission line<br />
b = 2p/l<br />
l = length of transmission line, stated in terms of l<br />
The term (bl) in the argument of the tangent functions in Eq. (4.42) includes l in<br />
both the numerator and the denominator, so it does not matter whether we use the freespace<br />
wavelength or the wavelength in the transmission line, as long as the same one is<br />
used for both b and l. However, in some of the manipulations and outcomes that follow,<br />
l refers to the wavelength in the transmission line, so it is wise to always use that wavelength<br />
when working with stubs.<br />
Most of the time, Z IN will be complex—that is, having both a resistive (R) term and<br />
a reactive (jX) term—even for a purely resistive load, except when either of the following<br />
situations is true:<br />
• The input end of the line is one specific distance in the range 0 l l/4 from<br />
the load, or an integer multiple of l/2 beyond that distance.<br />
• The input end of the line is at one specific distance in the range l/4 l l/2<br />
from the load, or an integer multiple of l/2 beyond that distance.<br />
Since there is an infinite number of possible impedances that Z IN can assume for an arbitrary<br />
line length, the odds of finding a purely resistive value of Z IN are extremely<br />
small—unless, of course, Z L = Z 0 .<br />
A very important characteristic of transmission lines comes into view if we examine<br />
Eq. (4.42) a little further. Suppose we are free to adjust the length (l) of the line so that<br />
the input end of the line is at a value of l that causes the tangent functions in Eq. (4.42)<br />
to be zero. Since that occurs when the argument (bl) of both tangent functions is zero or<br />
at phase angles that are integer multiples of π radians, l must be an integer multiple of<br />
l/2. In fact, when l = nl/2 (for integer values of n), Z IN = Z L , regardless of whether Z L<br />
is resistive, reactive, or complex.