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Hysteresis Loss
The hysteresis loss results from the BH characteristic following a different path for decreasing values of H than for increasing values of H. If H is carried through a complete cycle from +H_{max} to H_{max} and back to +H_{max}, the BH characteristic becomes a loop known as a hysteresis loop. Figure 337(a) shows a typical hysteresis loop.
Consider a unit volume of core material. Starting from point 1, Fig. 337(b), H is zero and is increased to H_{max}; the energy absorbed by the unit volume is
This amount of energy is represented by the area 124 in Fig. 337(b). If H is now decreased from H_{max} to 0, the path taken by the BH characteristic is from point 2 to point 3. The energy is represented by the shaded portion 234 in Fig. 337(c), which is
This energy input is negative since H is positive, but dB is negative, hence W_{2} represents the energy given up by a unit volume of the core. If H is now taken from zero to H_{max}, the path 35 is traced. The energy put into the core material is represented by the area 356 in Fig. 337(d). This area is positive and represents energy absorbed by the core, and
In Eq. 390, H is negative and dB is negative and the energy is therefore positive. If H is now increased from H_{max}, the core gives up an amount of energy represented by the shaded area 561 in Fig. 337(e), and
In Eq. 391, H is negative and dB is positive, hence the absorbed energy is negative. This means that the core gives up energy. If the sum of the four values of energy is taken with due regard for the signs, whether positive or negative, the total energy is represented by the area of the hysteresis loop in Fig. 337(a). Thus, for a core having a volume Yol, and a uniform flux density B throughout the entire volume, the energy loss is
where the line integral represents the area of the hysteresis loop. If H is in ampere turns per unit length and B is in webers per unit area and the volume is in the same system of units, Eq. 392 is valid for any system of dimensions. Thus, if H is in ampere turns per inch and B is in webers per square inch, then if the volume Vol is in cubic inches, W_{h} expresses the energy loss in joules for the entire volume. This is true also if the unit of length is the meter and the volume is in cubic meters. In the case of the flux undergoing a cyclic variation at a frequency f cps there are f hysteresis loops per second, so to speak, and the power is
or
In order to use the area of the loop in Eq. 393 it is necessary to take into account the scale to which the loop is plotted on the graph. Let p = the number of units of H per inch of graphthen W = pq(area of loop in square inches) The area of the loop may be determined from planimeter measurements or by counting squares. Then
The hysteresis loss, in a volume Vol in which the flux density is uniform and varies cyclically at a frequency of f cps, is expressed empirically as
The nature of the magnetic material determines the values of η and n. The exponent may vary from a value of 1.5 to 2.5 for different materials and is
usually not even constant for the same core. Steinmetz found in 1892 that the exponent n had a value of 1.6 for a range of flux densities from about 1,000 to 12,000 gausses for the magnetic materials then commonly in use. However, this exponent no longer applies to all present day materials. The hysteresis loss can, however, be computed from the area of the hysteresis loop for a given material if there are no reentrant loops. If, on the other hand, the hysteresis loop has reentrant loops, their areas must be added to that of the main loop. Reentrant loops result from a reduction in a positive value of H and then an increase before H_{max} is reached. This is shown in Fig. 338 where H, after having increased from 0 to a value H_{a}, is decreased to a value H_{b} and then increased to H_{max}. The locus then is from H = 0 at B_{r}, to H_{a} at B_{a}, back to H_{b} at B_{b}, back along the minor loop to Ha at Ba, then along the major loop from a to c at which we have B_{max} and H_{max}. A similar situation exists in this case in the bottom half of the loop.
Equation 396 is not valid for unsymmetrical loops such as occur when there is a dc component of flux in the presence of ac flux, a situation that exists in some choke coils and transformers in vacuum tube circuits. An unsymmetrical hysteresis loop is shown in Fig. 339.


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