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18-10 At A, a balanced-to-unbalanced transformer. At B, an
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unbalanced-to-balanced transformer.
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296 Transformers and Impedance Matching
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Impedance Transfer Ratio In RF and AF systems, transformers are employed to match impedances. Thus, you will sometimes hear or read about an impedance step-up transformer or an impedance step-down transformer. The impedance transfer ratio of a transformer varies according to the square of the turns ratio, and also according to the square of the voltage-transfer ratio. If the primary (source) and secondary (load) impedances are purely resistive and are denoted Zpri and Zsec, then the following relations hold:
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Zpri/Zsec = (Tpri/Tsec )2 Zpri/Zsec = (Epri/Esec )2 The inverses of these formulas, in which the turns ratio or voltage-transfer ratio are expressed in terms of the impedance-transfer ratio, are: Tpri/Tsec = (Zpri/Zsec)1/2 Epri/Esec = (Zpri/Zsec)1/2
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Problem 18-3 Consider a situation in which a transformer is needed to match an input impedance of 50.0 , purely resistive, to an output impedance of 300 , also purely resistive. What is the required turns ratio Tpri/Tsec The required transformer will have a step-up impedance ratio of Zpri/Zsec = 50.0/300 = 1/6.00. From the preceding formulas:
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Tpri/Tsec = (Zpri/Zsec)1/2 = (1/6.00)1/2 = 0.166671/2 = 0.408 = 1/2.45
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Problem 18-4 Suppose a transformer has a primary-to-secondary turns ratio of 4.00:1. The load, connected to the transformer output, is a pure resistance of 37.5 . What is the impedance at the primary The impedance-transfer ratio is equal to the square of the turns ratio. Therefore:
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Zpri/Zsec = (Tpri/Tsec)2 = (4.00/1)2 = 4.00 2 = 16.0 We know that the secondary impedance, Zsec is 37.5 . Thus: Zpri = 16.0 Zsec = 16.0 37.5 = 600
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Radio-Frequency Transformers 297
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Radio-Frequency Transformers
In radio receivers and transmitters, transformers can be categorized generally by the method of construction used. Some have primary and secondary windings, just like utility and audio units. Others employ transmission-line sections. These are the two most common types of transformer found at radio frequencies.
Wire-Wound Types In wire-wound RF transformers, powdered-iron cores can be used up to quite high frequencies. Toroidal cores are common, because they are self-shielding (all of the magnetic flux is confined within the core material). The number of turns depends on the frequency, and also on the permeability of the core. In high-power applications, air-core coils are often preferred. Although air has low permeability, it has negligible hysteresis loss, and will not heat up or fracture as powdered-iron cores sometimes do. The disadvantage of air-core coils is that some of the magnetic flux extends outside of the coil. This affects the performance of the transformer when it must be placed in a cramped space, such as in a transmitter final-amplifier compartment. A major advantage of coil-type transformers, especially when they are wound on toroidal cores, is that they can be made to work over a wide band of frequencies, such as from 3.5 MHz to 30 MHz. These are called broadband transformers. Transmission-Line Types 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:
L ft = 246v/fo where L ft 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 specified in meters, then: Lm = 75v/fo Suppose that a quarter-wave section of line, with characteristic impedance Zo, is terminated in a purely resistive impedance R out. Then the impedance that appears at the input end of the line, Rin, is also a pure resistance, and the following relations hold: Zo2 = RinR out Zo = (RinR out)1/2 This is illustrated in Fig. 18-11. The first of the preceding formulas can be rearranged to solve for Rin in terms of Rout, or vice versa: Rin = Zo2/R out R out = Zo2/R in
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