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[CHAP. 14
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while that of the active loop is R1 i1 L1 di1 di M 2 v1 dt dt
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Writing the above equations in the s-domain with the initial conditions i1 0 i2 0 0 and eliminating I1 s , we nd H s  response I s Ms 2 excitation V1 s L1 L2 M 2 s2 R1 L2 R2 L1 s R1 R2
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and from the poles of H s we have the natural frequencies of i2 .
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The sign on a voltage of mutual inductance can be determined if the winding sense is shown on the circuit diagram, as in Figs. 14-4 and 14-5. To simplify the problem of obtaining the correct sign, the coils are marked with dots at the terminals which are instantaneously of the same polarity. To assign the dots to a pair of coupled coils, select a current direction in one coil and place a dot at the terminal where this current enters the winding. Determine the corresponding ux by application of the right-hand rule [see Fig. 14-7(a)]. The ux of the other winding, according to Lenz s law, opposes the rst ux. Use the right-hand rule to nd the natural current direction corresponding to this second ux [see Fig. 14-7(b)]. Now place a dot at the terminal of the second winding where the natural current leaves the winding. This terminal is positive simultaneously with the terminal of the rst coil where the initial current entered. With the instantaneous polarity of the coupled coils given by the dots, the pictorial representation of the core with its winding sense is no longer needed, and the coupled coils may be illustrated as in Fig. 14-7(c). The following dot rule may now be used: (1) (2) when the assumed currents both enter or both leave a pair of coupled coils by the dotted terminals, the signs on the M-terms will be the same as the signs on the L-terms; but if one current enters by a dotted terminal while the other leaves by a dotted terminal, the signs on the M-terms will be opposite to the signs on the L-terms.
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Fig. 14-7 EXAMPLE 14.3 The current directions chosen in Fig. 14-8(a) are such that the signs on the M-terms are opposite to the signs on the L-terms and the dots indicate the terminals with the same instantaneous polarity. Compare this to the conductively coupled circuit of Fig. 14-8(b), in which the two mesh currents pass through the common element in opposite directions, and in which the polarity markings are the same as the dots in the magnetically coupled circuit. The similarity becomes more apparent when we allow the shading to suggest two black boxes.
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ENERGY IN A PAIR OF COUPLED COILS The energy stored in a pair
The energy stored in a single inductor L carrying current i is 0.5Li2 J. of coupled coils is given by
CHAP. 14]
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Fig. 14-8
1 1 2 2 W L1 i1 L2 i2 Mi1 i2 2 2
where L1 and L2 are the inductances of the two coils and M is their mutual inductance. The term Mi1 i2 in (9) represents the energy due to the e ect of the mutual inductance. The sign of this term is (a) positive if both currents i1 and i2 enter either at the dotted or undotted terminals, or (b) negative if one of the currents enters at the dotted terminal and the other enters the undotted end.
EXAMPLE 14.4 In a pair of coils, with L1 0:1 H and L2 0:2 H, at a certain moment, i1 4 A and p i2 10 A.p Find the total energy in the coils if the coupling coe cient M is (a) 0.1 H, (b) 2=10 H, (c) 0:1 H, and (d) 2=10 H. From (9), a b c d W 0:5 0:1 42 0:5 0:2 102 0:1 10 4 14:8 J W 16:46 J W 6:8 J W 5:14 J p 2=10 and perfect
The maximum and minimum energies occur in conjunction with perfect positive coupling M p negative coupling M 2=10 .
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