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(c) 8.87 m (d) 2.84 m 19. What is the fourth harmonic of 800 kHz (a) 200 kHz (b) 400 kHz (c) 3.20 MHz (d) 4.00 MHz 20. Suppose you want to build a 1 2-wave dipole antenna designed to have a fundamental resonant frequency of 3.60 MHz. How long should you make it, as measured from end to end in feet (a) 130 ft (b) 1680 ft (c) 39.7 ft (d) 515 ft
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CHAPTER
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Transformers and Impedance Matching
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TRANSFORMERS ARE USED TO OBTAIN THE OPTIMUM VOLTAGE FOR THE OPERATION OF A CIRCUIT OR
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system. Transformers can also match impedances between a circuit and a load, or between two different circuits. Transformers can be used to provide dc isolation between electronic circuits while letting ac pass. Another application is to mate balanced and unbalanced circuits, feed systems, and loads.
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Principle of the Transformer
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When two wires are near each other and one of them carries a fluctuating current, a fluctuating current is induced in the other wire. This effect is known as electromagnetic induction. All ac transformers work according to the principle of electromagnetic induction. If the first wire carries sine-wave ac of a certain frequency, then the induced current is sine-wave ac of the same frequency in the second wire. The closer the two wires are to each other, the greater is the induced current, for a given current in the first wire. If the wires are wound into coils and placed along a common axis (Fig. 18-1), the induced current will be greater than if the wires are straight and parallel. Even more coupling, or efficiency of induced-current transfer, is obtained if the two coils are wound one atop the other.
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Primary and Secondary The two windings, along with the core on which they are wound, constitute a transformer. The first coil is called the primary winding, and the second coil is known as the secondary winding. These are often spoken of simply as the primary and the secondary. The induced current in the secondary creates a voltage between its end terminals. In a step-down transformer, the secondary voltage is less than the primary voltage. In a step-up transformer, the secondary voltage is greater than the primary voltage. The primary voltage is abbreviated Epri, and the secondary voltage is abbreviated Esec. Unless otherwise stated, effective (rms) voltages are always specified. The windings of a transformer have inductance, because they are coils. The required inductances of the primary and secondary depend on the frequency of operation, and also on the resistive part of the impedance in the circuit. As the frequency increases, the needed inductance decreases. At high resistive impedances, more inductance is generally needed than at low resistive impedances.
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Copyright 2006, 2002, 1997, 1993 by The McGraw-Hill Companies, Inc. Click here for terms of use.
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Principle of the Transformer 287
18-1 Magnetic lines of flux
between two aligned coils of wire when one of the coils carries fluctuating or alternating current.
Turns Ratio The primary-to-secondary turns ratio in a transformer is the ratio of the number of turns in the primary, Tpri, to the number of turns in the secondary, Tsec. This ratio is written Tpri:Tsec or Tpri/Tsec. In a transformer with excellent primary-to-secondary coupling, the following relationship always holds:
Epri/Esec = Tpri/Tsec That is, the primary-to-secondary voltage ratio is always equal to the primary-to-secondary turns ratio (Fig. 18-2).
Problem 18-1 Suppose a transformer has a primary-to-secondary turns ratio of exactly 9:1. The ac voltage at the primary is 117 V rms. Is this a step-up transformer or a step-down transformer What is the voltage across the secondary This is a step-down transformer. Simply plug in the numbers in the preceding equation and solve for Esec, as follows:
Epri/Esec = Tpri/Tsec 117/Esec = 9.00 1/Esec = 9.00/117 Esec = 117/9.00 = 13.0 V rms
18-2 The primary voltage (Epri) and
secondary voltage (Esec) in a transformer depend on the number of turns in the primary winding (Tpri) versus the number of turns in the secondary winding (Tsec).
288 Transformers and Impedance Matching
Problem 18-2 Consider a transformer with a primary-to-secondary turns ratio of exactly 1:9. The voltage at the primary is 121.4 V rms. Is this a step-up transformer or a step-down transformer What is the voltage at the secondary This is a step-up transformer. Plug in numbers and solve for Esec, as follows:
121.4/Esec = 1/9.000 Esec/121.4 = 9.000 Esec = 9.000 121.4 = 1093 V rms Sometimes the secondary-to-primary turns ratio is given, rather than the primary-to-secondary turns ratio. This is written Tsec/Tpri. In a step-down unit, Tsec/Tpri is less than 1. In a step-up unit, Tsec/Tpri is greater than 1. When you hear someone say that such-and-such a transformer has a certain turns ratio, say 10:1, be sure of which ratio is meant, Tpri/Tsec or Tsec/Tpri! If you get it wrong, you ll have the secondary voltage wrong by a factor of the square of the turns ratio.
Ferromagnetic Cores If a ferromagnetic substance such as laminated iron or powdered iron is placed within the pair of coils, the extent of coupling is increased far above that possible with an air core. But this improvement in coupling is obtained at a price. Some energy is invariably lost as heat in the core. Also, ferromagnetic cores limit the maximum frequency at which a transformer will work well. The schematic symbol for an air-core transformer consists of two inductor symbols back-toback (Fig. 18-3A). If a laminated iron core is used, two parallel lines are added to the schematic symbol (Fig. 18-3B). If the core is made of powdered iron, the two parallel lines are broken or dashed (Fig. 18-3C). In transformers for 60-Hz utility ac, and also for low audio-frequency (AF) use, sheets of an alloy called silicon steel, glued together in layers, are often employed as transformer cores. The silicon steel is sometimes called transformer iron. The reason layering is used, rather than making the core from a single mass of metal, is that the magnetic fields from the coils cause currents to flow in a solid core. These eddy currents go in circles, heating up the core and wasting energy that would otherwise be transferred from the primary to the secondary. Eddy currents are choked off by breaking up the core into layers, so that currents cannot flow very well in circles. A rather esoteric form of loss, called hysteresis loss, occurs in all ferromagnetic transformer cores, but especially laminated iron. Hysteresis is the tendency for a core material to be sluggish in accept-
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