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Part III
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magnetization Br To bring the ux density to zero, we must further decrease the mmf (ie, produce a negative current), until the eld intensity reaches the value H0 (point on the curve) As the mmf is made more negative, the curve eventually reaches the point If the excitation current to the coil is now increased, the magnetization curve will follow the path = = = , eventually returning to the original point in the B-H plane, but via a different path The result of this process, by which an excess magnetomotive force is required to magnetize or demagnetize the material, is a net energy loss It is dif cult to evaluate this loss exactly; however, it can be shown that it is related to the area between the curves of Figure 1632 There are experimental techniques that enable the approximate measurement of these losses Figures 1633(a) (c) depict magnetization curves for three very common ferromagnetic materials: cast iron, cast steel, and sheet steel These curves will be useful in solving some of the homework problems
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Flux density (webers per square meter) Flux density (webers per square meter)
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Br H1 H0 '
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' H0 H1 A H t m
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Figure 1632 Hysteresis in magnetization curves
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Magnetization curve for cast iron
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Magnetization curve for cast iron
0 10,000 20,000 30,000 40,000 Field intensity (ampere turns per meter) (a) Flux density (webers per square meter) 150
1000 2000 3000 Field intensity (ampere-turns per meter) (b)
Magnetization curve for sheet steel
200 400 600 Field intensity (ampere-turns per meter) (c)
Figure 1633 (a) Magnetization curve for cast iron; (b) Magnetization curve for cast steel; (c) Magnetization curve for sheet steel
TRANSFORMERS
One of the more common magnetic structures in everyday applications is the transformer The ideal transformer was introduced in 7 as a device that can step an AC voltage up or down by a xed ratio, with a corresponding decrease or increase in current The structure of a simple magnetic transformer is shown in
16
Principles of Electromechanics
Figure 1634, which illustrates that a transformer is very similar to the magnetic circuits described earlier in this chapter Coil L1 represents the input side of the transformer, while coil L2 is the output coil; both coils are wound around the same magnetic structure, which we show here to be similar to the square doughnut of the earlier examples
i1 + v1 ~ + e1 L1 N2 turns N1 turns
i2 + + L2 e2 v2 Load
Figure 1634 Structure of a transformer
The ideal transformer operates on the basis of the same set of assumptions we made in earlier sections: the ux is con ned to the core, the ux links all turns of both coils, and the permeability of the core is in nite The last assumption is equivalent to stating that an arbitrarily small mmf is suf cient to establish a ux in the core In addition, we assume that the ideal transformer coils offer negligible resistance to current ow The operation of a transformer requires a time-varying current; if a timevarying voltage is applied to the primary side of the transformer, a corresponding current will ow in L1 ; this current acts as an mmf and causes a (time-varying) ux in the structure But the existence of a changing ux will induce an emf across the secondary coil! Without the need for a direct electrical connection, the transformer can couple a source voltage at the primary to the load; the coupling occurs by means of the magnetic eld acting on both coils Thus, a transformer operates by converting electric energy to magnetic, and then back to electric The following derivation illustrates this viewpoint in the ideal case (no loss of energy), and compares the result with the de nition of the ideal transformer in 7 If a time-varying voltage source is connected to the input side, then by virtue of Faraday s law, a corresponding time-varying ux d /dt is established in coil L1 : e1 = N1 d = v1 dt (1633)
But since the ux thus produced also links coil L2 , an emf is induced across the output coil as well: e2 = N2 d = v2 dt (1634)
This induced emf can be measured as the voltage v2 at the output terminals, and one can readily see that the ratio of the open-circuit output voltage to input terminal voltage is v2 N2 = v1 N1 (1635)
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