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[CHAP. 14
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Fig. 14-12 Elimination of I2 and I among these equations results in V1 jXM =a R2 jX22 ZL  Zin R1 jX11 a2 I1 jXM =a R2 jX22 ZL If, instead, the mesh current equations for Fig. 14-11(b) are used to derive Zin , the result is Zin R1 jX1
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2 XM R2 jX2 ZL
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The reader may verify the equivalence of (14a) and (14b) see Problem 14.36.
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An ideal transformer is a hypothetical transformer in which there are no losses and the core has in nite permeability, resulting in perfect coupling with no leakage ux. In large power transformers the losses are so small relative to the power transferred that the relationships obtained from the ideal transformer can be very useful in engineering applications. Referring to Fig. 14-13, the lossless condition is expressed by
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1 V 2 I 2 2
(see Section 10.7).
But V1 E1 aE2 aV2
and so, a being real, V1 I2 a V2 I1 The input impedance is readily obtained from relations (15): Zin V1 aV2 V a2 2 a2 Z L I1 I2 =a I2 16 15
Fig. 14-13
CHAP. 14]
MUTUAL INDUCTANCE AND TRANSFORMERS
EXAMPLE 14.6 The ideal transformer may be considered as the limiting case of the linear transformer of Section 14.7. Thus, in (14a) set R1 R2 X11 X22 0 (no losses) and then let XM ! 1 (in nite core permeability), to obtain   jXM =a ZL Zin lim a2 a2 ZL XM !1 jXM =a ZL in agreement with (16)
Ampere-Turn Dot Rule Since a N1 =N2 in (15), N1 I1 N2 I2 that is, the ampere turns of the primary equal the ampere turns of the secondary. A rule can be formulated which extends this result to transformers having more than two windings. A positive sign is applied to an ampere-turn product if the current enters the winding by the dotted terminal; a negative sign is applied if the current leaves by the dotted terminal. The ampere-turn dot rule then states that the algebraic sum of the ampere-turns for a transformer is zero.
EXAMPLE 14.7 The three-winding transformer shown in Fig. 14-14 has turns N1 20, N2 N3 10. given that I2 10:0 53:138 A, I3 10:0 458 A. With the dots and current directions as shown on the diagram, Find I1
Fig. 14-14 N1 I1 N2 I2 N3 I3 0 from which 20I1 10 10:0 53:138 10 10:0 458 I1 6:54 j7:54 9:98 49:068 A
AUTOTRANSFORMER
An autotransformer is an electrically continuous winding, with one or more taps, on a magnetic core. One circuit is connected to the end terminals, while the other is connected to one end terminal and to a tap, part way along the winding. Referring to Fig. 14-15(a), the transformation ratio is V1 N1 N2 a 1 V2 N2 which exceeds by unity the transformation ratio of an ideal two-winding transformer having the same turns ratio. Current I1 through the upper or series part of the winding, of N1 turns, produces the ux 1 . By Lenz s law the natural current in the lower part of the winding produces an opposing ux
MUTUAL INDUCTANCE AND TRANSFORMERS
[CHAP. 14
Fig. 14-15
2 . Therefore, current In leaves the lower winding by the tap. The dots on the winding are as shown in Fig. 14-15(b). In an ideal autotransformer, as in an ideal transformer, the input and output complex powers must be equal.
1 2 V1 I1
1 V1 I 1 V2 I L ab 2 2 IL a 1 Iab
whence
That is, the currents also are in the transformation ratio. Since IL Iab Icb , the output complex power consists of two parts:
1 2 V2 IL
1 V2 I 1 V2 I 1 V2 I a 1 V2 I ab cb ab ab 2 2 2 2
The rst term on the right is attributed to conduction; the second to induction. Thus, there exist both conductive and magnetic coupling between source and load in an autotransformer.
REFLECTED IMPEDANCE
A load Z2 connected to the secondary port of a transformer, as shown in Fig. 14-16, contributes to its input impedance. This contribution is called re ected impedance. Using the terminal characteristics of the coupled coils and applying KVL around the secondary loop, we nd V1 L1 sI1 MsI2 0 MsI1 L2 sI2 Z2 I2 By eliminating I2 , we get
Fig. 14-16
CHAP. 14]
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