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CHAPTER 10 Magnetic Coupling. Transformers
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Fig. 239
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Fig. 240
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Now, in order to increase the gain, we would naturally move the two coils closer to each other, thus increasing the value of k. However, as we continue to move the coils closer and closer together, a seemingly peculiar e ect takes place, as follows. At rst, as the value of k increases, the value of the gain at resonance also increases, as we would expect. This trend, however, continues only until we reach the condition of CRITICAL COUPLING, at which point the gain at resonance has reached its maximum possible value; this would be the condition for the example case of k kc 0:02 in Fig. 239. If, now, the value of k is increased BEYOND the value of kc , we nd that the gain at the normal center frequency decreases, while the gain versus frequency curve begins to show TWO SEPARATE RESONANT PEAKS, one below d 0 and one above d 0; an example of this condition is illustrated for the case of k 0:03 in Fig. 240. In this state the circuit is said to be overcoupled. We thus nd that, as the value of k increases (beyond the value of kc ), the gain at d 0 continues to decrease while the separation between the two high peaks continues to increase. Hence for large values of k the G versus d curve would become highly distorted, so that the circuit of Fig. 232 would become useless. Having said this, however, it should be noted that the use of a relatively SMALL AMOUNT of overcoupling (as in Fig. 240) can be used as a practical way to produce a relatively good type of band-pass circuit. In regard to the last statement, compare the curves for k 0:02 and k 0:03 in Figs. 239 and 240. Note that, compared with k 0:02, the somewhat overcoupled case of k 0:03 produces (for most practical purposes) a nearly at-topped gain curve from approximately d 1 to d 1. Also note that the gain falls o quite rapidly for frequencies beyond the region of the two peaks. Hence, by a proper choice of k, Fig. 232 can be made to serve as a reasonably good band-pass type of circuit. This makes the circuit of Fig. 232 especially useful as an ampli er of high-frequency modulated carrier waves. This is because such a circuit will pass, with almost uniform gain, both the carrier wave and the necessary side-bands either side of the carrier, while e ectively discriminating against possible nearby interfering signals. In closing, the explanation, in words, for the existence of a double-peaked gain curve can be summarized, very brie y, as follows. Taking Fig. 240 as an example, consider, rst, the condition of the circuit in the neighborhood of d 1. Since the frequency here is less than !0 , it follows that XC is greater than XL in both the primary and secondary circuits. Hence the reactance re ected into the primary coil is inductive in nature,* and in an amount su cient to increase the
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* See discussion note given with solution to problem 187.
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CHAPTER 10 Magnetic Coupling. Transformers
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inductive reactance of the primary coil to a point where it resonates with the primary capacitor at a frequency less than !0 . This causes increased voltage to appear across the primary load, and ultimately across the output capacitor on the secondary side at a frequency less than !0 . Now consider the circuit in the neighborhood of d 1. Since the frequency here is greater than !0 , it follows that XL is greater than XC in both the primary and secondary circuits. Hence the reactance re ected into the primary coil is capacitive in nature, and in an amount su cient to decrease the inductive reactance of the primary coil to a point where it resonates with the primary capacitor at a frequency greater than !0 . This again causes increased voltage to appear across the primary load, and ultimately across the output capacitor on the secondary side at a frequency greater than !0 . Lastly, consider the condition for d 0 in Fig. 240. Here the frequency is !0 , which is actually the basic resonant frequency of the circuit. This is because !0 is the only frequency at which the primary and secondary circuits are simultaneously resonant. We have discovered, however, that for larger values of k or M the gain at !0 can be less than the gain at the other two resonant frequencies above and below !0 . To see why this can be true recall, from eq. (387), that " " Zref !2 M 2 =Z2 " " where Z2 is the series impedance of the secondary circuit considered by itself. Since Z2 has "2 r at !0 ), it follows that a comparatively large value its minimum value at resonance (Z of resistance can be re ected into the primary coil at resonance, so that the primary current, for larger values of M, could be less at !0 than at the other two resonant frequencies. Thus the voltage induced into the secondary circuit at !0 could be less than at the other two resonant frequencies, causing a reduced value of secondary current, with the nal result of reduced voltage drop across the output capacitor. Problem 198 Suppose two coils are wound in the same sense on a cylindrical coil form, as in Fig. 241. Let it be given that an instrument for the measurement of inductance is available, and that the coils have been found to have inductances of L1 and L2 henrys, as shown.
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