how to generate barcode in ssrs report COUPLING COEFFICIENT in Software

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COUPLING COEFFICIENT
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A coil containing N turns with magnetic ux  linking each turn has total magnetic ux linkage  N. By Faraday s law, the induced emf (voltage) in the coil is e d=dt N d=dt . A negative sign is frequently included in this equation to signal that the voltage polarity is established according to Lenz s law. By de nition of self-inductance this voltage is also given by L di=dt ; hence, L di d N dt dt or L N d di 5a
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MUTUAL INDUCTANCE AND TRANSFORMERS
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[CHAP. 14
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The unit of  being the weber, where 1 Wb 1 V s, it follows from the above relation that 1 H 1 Wb=A. Throughout this book it has been assumed that  and i are proportional to each other, making L N  constant i 5b
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In Fig. 14-3, the total ux 1 resulting from current i1 through the turns N1 consists of leakage ux, 11 , and coupling or linking ux, 12 . The induced emf in the coupled coil is given by N2 d12 =dt). This same voltage can be written using the mutual inductance M: e M di1 d12 N2 dt dt or M N2 d12 di1 6
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Fig. 14-3
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Also, as the coupling is bilateral, M N1 d21 di2 7
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The coupling coe cient, k, is de ned as the ratio of linking ux to total ux: k 12 21 1 2
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where 0 k 1. Taking the product of (6) and (7) and assuming that k depends only on the geometry of the system,          d12 d21 d k1 d k2 d1 d2 M 2 N2 N1 N2 N1 k2 N1 N2 k2 L1 L2 di1 di2 di1 di2 di1 di2 p p or XM k X1 X2 (8) from which M k L1 L2 p Note that (8) implies that M L1 L2 , a bound that may be independently derived by an energy argument. If all of the ux links the coils without any leakage ux, then k 1. On the other extreme, the coil axes may be oriented such that no ux from one can induce a voltage in the other, which results in k 0. The term close coupling is used to describe the case where most of the ux links the coils, either by way of a magnetic core to contain the ux or by interleaving the turns of the coils directly over one another. Coils placed side-by-side without a core are loosely coupled and have correspondingly low values of k.
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14.3 ANALYSIS OF COUPLED COILS Polarities in Close Coupling In Fig. 14-4, two coils are shown on a common core which channels the magnetic ux . This arrangement results in close coupling, which was mentioned in Section 14.2. To determine the proper signs on the voltages of mutual inductance, apply the right-hand rule to each coil: If the ngers wrap
CHAP. 14]
MUTUAL INDUCTANCE AND TRANSFORMERS
Fig. 14-4
around in the direction of the assumed current, the thumb points in the direction of the ux. Resulting positive directions for 1 and 2 are shown on the gure. If uxes 1 and 2 aid one another, then the signs on the voltages of mutual inductance are the same as the signs on the voltages of self-inductance. Thus, the plus sign would be written in all four equations (2) and (3). In Fig. 14-4, 1 and 2 oppose each other; consequently, the equations (2) and (3) would be written with the minus sign.
Natural Current Further understanding of coupled coils is achieved from consideration of a passive second loop as shown in Fig. 14-5. Source v1 drives a current i1 , with a corresponding ux 1 as shown. Now Lenz s law implies that the polarity of the induced voltage in the second circuit is such that if the circuit is completed, a current will pass through the second coil in such a direction as to create a ux opposing the main ux established by i1 . That is, when the switch is closed in Fig. 14-5, ux 2 will have the direction shown. The right-hand rule, with the thumb pointing in the direction of 2 , provides the direction of the natural current i2 . The induced voltage is the driving voltage for the second circuit, as suggested in Fig. 14-6; this voltage is present whether or not the circuit is closed. When the switch is closed, current i2 is established, with a positive direction as shown.
Fig. 14-5
EXAMPLE 14.2 Suppose the switch in the passive loop to be closed at an instant t 0 when i1 0. the sequence of the passive loop is (see Fig. 14-6). di2 di M 1 0 dt dt
For t > 0,
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