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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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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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