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CHAPTER 10 Magnetic Coupling. Transformers
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components. In doing this, we ll use rms values of voltage and current instead of peak values, and thus, letting " Vpri rms vector value of induced voltage appearing in PRIMARY coil, and " Vsec rms vector value of induced voltage appearing in SECONDARY coil, in the primary coil, " " " Vpri !L1 I1 908 !M I2 908 in the secondary coil, " " " Vsec !M I1 908 !L2 I2 908 " " where notation of the form X I 908 is used to denote that a vector voltage drop X I " by 908. However, since we re representing vector quantities by leads the current vector I complex numbers, all we need to do, to rotate a vector through 908, is to multiply the vector by j (as discussed in connection with eq. (196) in Chap. 8). Thus the preceding two equations become " " " Vpri j!L1 I1 j!M I2 " " " Vsec j!M I1 j!L2 I2 378 379
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in which !L1 and !L2 are the inductive reactances of the primary and secondary coils, each considered separately (as if they were not part of the transformer), and where the quantity !M is measured in ohms and called the mutual reactance. Equations (378) and (379) are expressions connecting the four inductive voltage drops present in a transformer. We ll make use of these equations in section 10.3, to derive the basic equations for the sinusoidal steady-state analysis of coupled circuits. Before going on to the next section, however, there are some more points we should mention regarding the polarity dots associated with a transformer. First, it should be noted that, in some simple problems involving only a single transformer, in which we need to nd only the magnitude of the secondary current, it will not be necessary to be concerned about polarity dots at all. In such a case we can simply assume that the primary and secondary currents ow into like-marked terminals. However, in more complicated cases, in which, for example, we have two or more transformers having a common secondary current, then we must take the dot-marks into account. This can be a problem, as it may not be possible to actually see the senses in which the primary and secondary coils of a transformer are wound (because the windings may be covered with a strong tape, used both to insulate the windings and to keep them rmly in place). If the manufacturer has not dot-marked such a transformer, this can be done by experimental means, by the user, in several ways. One such way is explained with the aid of Figs. 220 and 221, as follows, in which V is a constant value of ac voltage and I is the reading of an ac ammeter.
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Fig. 220
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Fig. 221
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CHAPTER 10 Magnetic Coupling. Transformers
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Note that in the test setup the primary and secondary coils are connected in series in both gures, and hence, in each gure, the primary current is equal to the secondary current. Now, in the gures, let the arrows denote the direction of current at a particular moment. Notice that the direction of the primary current is the same in both gures, but the secondary current ows in opposite directions. This means that in one of the gures the mutual magnetic e ects of the primary and secondary currents will be ADDITIVE, while in the other gure they will be SUBTRACTIVE. Since the total reactance seen by the generator is greater in the additive case than in the subtractive case, it follows that the ammeter will read a smaller value of current in the additive case than in the subtractive case. With these points in mind, the actual test procedure is as follows. First, to one of the primary leads attach a tag having a dot, then attach tags to the secondary leads, one labeled a and the other b, so that we have the condition shown in the gures. Now apply the ac voltage V rst to the setup of Fig. 220, and then to that of Fig. 221. Upon doing this, suppose it is found that I1 is smaller than I2 ; this would mean that current owing into the b terminal has the same e ect, magnetically, as current owing into the dot-marked terminal, and thus terminal b should be marked with a dot. Or, if the opposite is found to be true (I2 less than I1 ), then terminal a would be marked with a dot.
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