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Fig. 13-35 13.34 Referring to the two-port circuit of Fig. 13-36, nd the T-parameters as a function of ! and specify their values at ! 1, 103 , and 106 rad/s.
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A 1 10 9 !2 j10 9 !, B 10 3 1 j! , C 10 6 j!, and D 1. At ! 1 rad/s, A 1, B 10 3 1 j , C 10 6 j, and D 1. At ! 103 rad/s, A % 1, B % j, C 10 3 j, and D 1. At ! 106 rad/s, A % 103 , B % 103 j, C j, and D 1
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Fig. 13-36 13.35 A two-port network contains resistors, capacitors, and inductors only. With port #2 open [Fig. 13-37(a)], a unit step voltage v1 u t produces i1 e t u t mA and v2 1 e t u t (V). With port #2 shortcircuited [Fig. 13-37(b)], a unit step voltage v1 u t delivers a current i1 0:5 1 e 2t u t mA . Find i2 and the T-equivalent network. Ans: i2 0:5 1 e 2t u t [see Fig. 13-37(c)]
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Fig. 13-37 13.36 The two-port network N in Fig. 13-38 is speci ed by Z11 2, Z12 Z21 1, and Z22 4. Find I1 , I2 , and I3 . Ans: I1 24 A; I2 1:5 A; and I3 6:5 A
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14.1 MUTUAL INDUCTANCE The total magnetic ux linkage  in a linear inductor made of a coil is proportional to the current passing through it; that is,  Li (see Fig. 14-1). By Faraday s law, the voltage across the inductor is equal to the time derivative of the total in ux linkage; that is, v d di L dt dt
Fig. 14-1
The coe cient L, in H, is called the self-inductance of the coil. Two conductors from di erent circuits in close proximity to each other are magnetically coupled to a degree that depends upon the physical arrangement and the rates of change of the currents. This coupling is increased when one coil is wound over another. If, in addition, a soft-iron core provides a path for the magnetic ux, the coupling is maximized. (However, the presence of iron can introduce nonlinearity.) To nd the voltage-current relation at the terminals of the two coupled coils shown in Fig. 14-2, we observe that the total magnetic ux linkage in each coil is produced by currents i1 and i2 and the mutual linkage e ect between the two coils is symmetrical. 1 L1 i1 Mi2 2 Mi1 L2 i2 334
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CHAP. 14]
MUTUAL INDUCTANCE AND TRANSFORMERS
Fig. 14-2
where M is the mutual inductance (in H). The terminal voltages are time derivatives of the ux linkages. v1 t d1 di L1 1 M dt dt d2 di1 M L2 v2 t dt dt di2 dt di2 dt
The coupled coils constitute a special case of a two-port network discussed in 13. The terminal characteristics (2) may also be expressed in the frequency domain or in the s-domain as follows. Frequency Domain V1 j!L1 I1 j!MI2 V2 j!MI1 j!L2 I2 3 s-Domain V1 L1 sI1 MsI2 V2 MsI1 L2 sI2 4
The coupling coe cient M is discussed in Section 14.2. The frequency domain equations (3) deal with the sinusoidal steady state. The s-domain equations (4) assume exponential sources with complex frequency s.
EXAMPLE 14.1 Given L1 0:1 H, L2 0:5 H, and i1 t i2 t sin !t in the coupled coils of Fig. 14-2. v1 t and v2 t for (a) M 0:01 H, b M 0:2 H, and (c) M 0:2 H. a b v1 t 0:1 ! cos !t 0:01 ! cos !t 0:11 ! cos !t v2 t 0:01 ! cos !t 0:5 ! cos !t 0:51 ! cos !t V V Find
v1 t 0:1 ! cos !t 0:2 ! cos !t 0:3 ! cos !t V v2 t 0:2 ! cos !t 0:5 ! cos !t 0:7 ! cos !t V v1 t 0:1 ! cos !t 0:2 ! cos !t 0:1 ! cos !t v2 t 0:2 ! cos !t 0:5 ! cos !t 0:3 ! cos !t V V
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