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As you recall, any transmission line has a characteristic impedance, or Zo, that depends on the line construction. This property is sometimes used to make impedance transformers out of coaxial or parallel-wire line. Transmission-line transformers are always made from quarter-wave sections. From the previous chapter, remember the formula for the length of a quarter-wave section, that is,
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where Lft is the length of the section in feet, v is the velocity factor expressed as a fraction, and fo is the frequency of operation in megahertz. If the length Lm is in meters, then: Lm 75v/fo
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In the last chapter, you saw how a short circuit is changed into an open circuit, and vice versa, by a quarter-wave section of line. What happens to a pure resistive impedance at one end of such a line What will be seen at the opposite end Let a quarter-wave section of line, with characteristic impedance Zo, be terminated in a purely resistive impedance Rout. Then the input impedance is also a pure resistance Rin , and the following relationship holds: Zo2 RinRout
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This is illustrated in Fig. 18-11. This formula can be broken down to solve for Rin in terms of Rout , or vice versa: Rin and Rout Zo2/Rin. Zo2/Rout
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These relationships hold at the frequency, fo, for which the line is 1/4 wavelength long. Neglecting line losses, these relationships will also hold at the odd harmonics of fo, that is, at 3fo, 5fo, 7fo, and so on. At other frequencies, the line will not act as a transformer, but instead, will behave in complicated ways that are beyond the scope of this discussion. Quarter-wave transformers are most often used in antenna systems, especially at the higher frequencies, where their dimensions become practical.
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Problem 18-5
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An antenna has a purely resistive impedance of 100 . It is connected to a 1/4-wave section of 75- coaxial cable. What will be the impedance at the input end of the section Use the formula from above:
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Lft 246v/fo
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What about reactance 341
18-11 A quarter-wave matching section of transmission line. Abbreviations are discussed in the text.
Zo2/Rout 752/100 56
5625/100
Problem 18-6
An antenna is known to have a pure resistance of 600 . You want to match it to 50.0 pure resistance. What is the characteristic impedance needed for a quarter-wave matching section Use this formula: Zo2 Zo2 Zo2 RinRout 600 50 30,000 (30,000)1/2 173
The challenge is to find a line that has this particular Zo. Commercially manufactured lines come in standard Zo values, and a perfect match might not be obtainable. In that case, the closest obtainable Zo should be used. In this case, it would probably be 150 . If nothing is available anywhere near the characteristic impedance needed for a quarter-wave matching section, then a coil-type transformer will probably have to be used instead. A quarter-wave matching section should be made using unbalanced line if the load is unbalanced and balanced line if the load is balanced. The major disadvantage of quarter-wave sections is that they work only at specific frequencies. But this is often offset by the ease with which they are constructed, if radio equipment is to be used at only one frequency, or at odd-harmonic frequencies.
What about reactance
Things are simple when there is no reactance in an ac circuit using transformers. But often, especially in radio-frequency circuits, pure resistance doesn t occur naturally. It has to be obtained by using inductors and/or capacitors to cancel the reactance out.
342 Transformers and impedance matching Reactance makes a perfect match impossible, no matter what the turns ratio or Zo of the transformer. A small amount of reactance can be tolerated at lower radio frequencies (below about 30 MHz). A near-perfect match becomes more important at higher frequencies. The behavior of reactance, as it is coupled through transformer windings, is too complicated for a thorough analysis here. But if you re interested in delving into it, there are plenty of good engineering texts that deal with it in all its mathematical glory. Recall that inductive and capacitive reactances are opposite in effect, and that their magnitudes can vary. If a load presents a complex impedance R + jX, with X not equal to zero, it is always possible to cancel out the reactance X by adding an equal and opposite reactance ( X) in the circuit. This can be done by connecting an inductor or capacitor in series with the load. For radio communications over a wide band, adjustable impedance-matching and reactance-canceling networks can be placed between a transmitter and an antenna system. Such a device is called a transmatch and is popular among radio hams, who use frequencies ranging from 1.8 MHz to the microwave spectrum.
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