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Fig. 228
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Problem 191 " In Fig. 229, the generator produces V 50=08 volts at a frequency of 12,500 rad/sec. It is given that L1 0:0020 henry and L2 0:0045 henry, the resistance values being " in ohms. Find the magnitude of voltage Vo relative to ground. (Answer: 23.54 V)
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Fig. 229
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CHAPTER 10 Magnetic Coupling. Transformers
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Problem 192 Rework problem 191 with everything remaining the same EXCEPT let it be given that the secondary winding is now wound in the opposite sense from that assumed in Fig. 229. (Answer: 26.94 V) Problem 193 " In problem 191 show that Vo 23:54= 64:298. Problem 194 " In problem 192 show that Vo 26:94= 19:858.
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The T Equivalent of a Transformer
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In the analysis of transformer-coupled circuits it s often helpful to replace a transformer with its T-NETWORK EQUIVALENT. This can be done as follows, beginning with Fig. 230.
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Fig. 230
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In the gure, R1 resistance of the primary coil, R2 resistance of the secondary coil, L1 inductance of the primary coil, L2 inductance of the secondary coil, M mutual inductance between L1 and L2 . The conversion of a network into its T equivalent is done by using the procedure of section 9.2. Thus, for Fig. 230, we begin with the following relationships: " Z1O R1 jX1 (because, with terminals 2, 2 open, the secondary circuit has no e ect when we look into terminals 1, 1); " Z2O R2 jX2 (because, with terminals 1, 1 open, the primary circuit has no e ect when we look into terminals 2, 2) !2 M 2 " Z1S R1 jX1 R2 jX2 (here, looking into 1, 1 with 2, 2 shorted, we re making use of eq. (387)).
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CHAPTER 10 Magnetic Coupling. Transformers
Now, upon substituting the above values into eq. (282), you should nd that p " 390 * Z3 !2 M 2 j!M jXm Then, by eq. (283) " " " Z2 Z2O Z3 R2 jX2 jXm thus, " Z2 R2 j! L2 M and, by eq. (284) " " " Z1 Z1O Z3 R1 jX1 jXm thus, " Z1 R1 j! L1 M 392 Hence, by equations (390) through (392), we have that the T EQUIVALENT of the transformer of Fig. 230 is as shown in Fig. 231. 391
Fig. 231
Thus, while Figs. 230 and 231 are very di erent in appearance, they are equal in performance as far as alternating current (ac) is concerned, at a given frequency. Problem 195 Laboratory measurements on a certain transformer show that the primary winding has 14 ohms of resistance and 0.09 henry of inductance, while the secondary winding has 4 ohms of resistance and 0.04 henry of inductance. The coe cient of coupling is found to be 30%. Draw and label the T equivalent of the transformer, giving the correct values of resistance and inductance. Problem 196 Let a generator, producing 100 volts rms at a frequency of 500 radians/second, be connected to the primary terminals of the transformer in the preceding problem. The generator has negligible internal impedance. Now let a load of 2 ohms be connected to the secondary terminals. Making use of the T equivalent for the transformer found in the preceding problem, nd the magnitude of the voltage across the 2-ohm load. (Answer: 1.961 volts) Problem 197 Rework problem 196, this time using the basic coupled circuit formulas derived in section 10.3.
* Xm !M is called the mutual reactance, where M is the mutual inductance.
CHAPTER 10 Magnetic Coupling. Transformers
In closing, it should be noted that, in Fig. 231, the value of either (L1 M) or (L2 M) may come out to have a negative value. In such a case Fig. 231 is still, for purposes of analysis, a perfectly valid representation of the transformer, even though negative inductance does not physically exist. Thus, even though such an equivalent T could not be physically constructed, it would still be a perfectly valid representation of the actual transformer at a given frequency.
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