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Fig. 241
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In the above, it s also possible to nd the value of the coe cient of coupling k by making two additional inductance measurements. The rst measurement is made with the coils connected so that their magnetic elds AID each other, and the second is made with the coils connected so that their magnetic elds OPPOSE each other; let us denote these results by Laid and Lopp . Find, now, the equation for k in terms of all of the above measurements. (Note: in doing this, it will be convenient to imagine that a generator of V volts is applied to each of the two di erent conditions.)
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CHAPTER 10 Magnetic Coupling. Transformers
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The Ideal Iron-Core Transformer
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In the following we ll be dealing, as usual, with rms sinusoidal voltages and currents. In order for a transformer to operate e ectively, the primary current must be able to induce an ADEQUATE MAGNITUDE OF VOLTAGE into the secondary coil. Thus, if " I1 is the primary current, then (see eq. (379)) the magnitude of voltage induced into the secondary coil is equal to V2 !MI1 or, by eq. (371), p V2 !k L1 L2 I1 415
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Fundamentally, the magnitude of V2 is proportional to the RATE OF CHANGE of primary current and, for a given situation, the rate of change of primary current increases as the frequency increases. (See discussion following Fig. 128 in section 8.1.) Thus, at the higher frequencies (let us say above the audio range), it s not di cult to obtain an adequate value of V2 . Let us, therefore, now consider the situation at the low frequencies such as, for example, at the power-line frequency of 60 Hz. Inspection of eq. (415) shows that, in order to produce an adequate value of V2 , we could take one or more of the following steps: 1. 2. 3. increase the primary current I1 ; increase the value of the coe cient of coupling k; increase the inductances of the primary and secondary coils.
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Possibility (1) must be avoided if at all possible, one reason being to prevent excessive power loss in the primary coil. Next, concerning possibility (2), we should strive to make the value of the coe cient of coupling, k, as close as possible to its maximum theoretical value of 1. This is done by winding the primary and secondary coils as close together as possible. Finally, and this is where the iron core comes into the picture, we must utilize possibility (3) and make the inductances of the primary and secondary coils as large as possible. From a practical standpoint, however, it would be di cult to get the large inductance values required for operation at low frequencies without using an iron core. This is because, in order to get the large number of ux linkages required for large L, we would have to use very large coils containing perhaps many hundreds of turns of wire, which would, in itself, introduce other problems and practical di culties. These di culties are avoided by winding the primary and secondary coils on a common, closed iron core, as illustrated in Fig. 242. The iron core of Fig. 242 is actually constructed of thin sheets or laminations of silicon steel bolted tightly together. (The laminated construction greatly reduces energy losses generated in the iron core by the rapidly changing magnetic ux.) In electrical diagrams, the presence of an iron core is indicated by drawing several vertical lines between the primary and secondary coils, as shown in Fig. 243. The use of the iron core with its high value of permeability allows the production of a large amount of ux with only a relatively few turns of wire. Hence, we can greatly reduce the number of turns and still have enough ux linkages to produce the large values of L1 and L2 required for operation at the low frequencies.
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