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CHAPTER 10 Magnetic Coupling. Transformers
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Fig. 242
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Fig. 243. Schematic symbol for Fig. 242.
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With these points in mind, the IDEAL IRON-CORE TRANSFORMER is de ned as a theoretical transformer having the following characteristics. ZERO ENERGY LOSSES, which means that the primary and secondary coils have zero resistance, and no energy loss in the iron core. (b) UNITY p p COEFFICIENT OF COUPLING, that is, k 1; hence, M k L1 L2 L1 L2 . (c) The inductive reactances X1 and X2 of the primary and secondary coils are in nitely great in value, but, for any given transformer, the RATIO of X1 to X2 is a constant nite number a; thus a X1 =X2 " Now let a nite load impedance of ZL R jX ohms be connected to the output terminals of an ideal transformer, as in Fig. 244. (a)
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Fig. 244
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Thus we have !L1 X1 inductive reactance of primary coil !L2 X2 inductive reactance of secondary coil p p p !M ! L1 L2 !L1 !L2 X1 X2 " Hence, by eq. (387) in section 10.3, the impedance Zin seen looking into terminals (1, 1) in Fig. 244 is equal to " Zin jX1
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2 X1 X2 aX2 jaX2 since X1 aX2 R j X2 X R j X2 X
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CHAPTER 10 Magnetic Coupling. Transformers
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Now, upon rationalizing and collecting like terms, you should nd that the above relationship becomes " # " # 2 2 X2 R2 X2 XX2 X 2 X2 " Zin aR ja 416 R2 X2 X 2 R2 X2 X 2 Now, in accordance with condition (C), we allow X2 to become in nitely great in value. Doing this, and holding everything else constant, notice that both the real and imaginary parts of eq. (416) take the form in nity over in nity, 1=1. Since in nitely great is not a speci c value, 1=1 is said to be an indeterminant form. This, however, does not mean that a de nite, speci c answer can never be found in such a case. If an answer does exist, it can sometimes be found by rst merely changing the form of the given expression and then allowing the variable to become in nitely great. This is true for the case of eq. (416) as follows. 2 First divide the numerators and denominators in both of the fractions by X2 . Doing this, and noting that, algebraically,
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2 X2 X 2 =X2 1 X=X2 2
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eq. (416) becomes
" Zin aR
2 R2 =X2 1 X=X2 2
# ja
R2 =X2 X X 2 =X2
2 R2 =X2 1 X=X2 2
Now let X2 become in nitely great, X2 ! 1. When this happens, note that 1=X2 approaches the value zero, that is, when X2 ! 1; then 1=X2 ! 0 and therefore (remembering that R and X, in Fig. 244, have only nite values) we see that, " for the IDEAL CASE of in nitely great X2 , Zin becomes equal to " " Zin a R jX aZL 417
Since a is a real number, eq. (417) shows that the ideal transformer is an impedancematching device that changes only the MAGNITUDE of the load impedance; that is, the " " phase angle of Zin is the same as the phase angle of ZL . While it s impossible, of course, to build a true ideal transformer, a well-designed ironcore transformer will come very close to being ideal at low frequencies. For example, a highquality audio transformer, designed for use in the ampli er of a high- delity sound system, may well have both an e ciency and a coe cient of coupling in excess of 99%. So, in many cases an actual transformer can be considered to be ideal, for practical engineering purposes. Let s continue now, rst with a brief mention of what is called the magnetizing current of an iron-core transformer. In doing this, we ll refer to Fig. 244 and assume that, for practical purposes, the transformer can be considered to be ideal. As we have seen, this means that the values of the reactances X1 and X2 must be very great. Since the primary and secondary coils are to have only relatively small numbers of turns of wire, this means that, to satisfy the requirement of large values of X1 and X2 , the iron core must possess a very high value of relative permeability.*
* Consider a coil of N turns. If Liron is the inductance of the coil with an iron core and Lair is the inductance with air as core, we ll de ne relative permeability of iron to air to be the ratio of Liron to Lair . This ratio can be in the order of 1000 to 1 for silicon steel. Thus, for a given number of N turns, the reactance of an iron-core coil could be 1000 times the reactance of the corresponding air-core coil.
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