Calculation of Magnetic Circuits without Air Gaps

Magnetic circuits are constructed in a variety of shapes. In a-c equipment and in devices in which rapid response is required the magnetic core is usually built up of laminations having coated surfaces to provide interlaminar resistance. If the magnetic structure in a-c magnets were solid, large objectionable currents, which lead to excessive losses and heating, would be induced in the iron. In such d-c magnets where rapid response is required the currents induced in the magnetic cores would, in accordance with Lenz's Law, oppose changes in the magnetic flux and thus retard the building up of the flux. A few typical shapes of laminations used in motors, generators, relays, and transformers are shown in Fig. 3-14. Some of these shapes are somewhat complex and the magnetic calculations involved are far from simple since the flux density is not uniform throughout the structure. Thus, some parts of the structure may be highly saturated while others carry only moderate flux densities. In addition there is fringing of flux at the sides of air gaps as well as flux leakage across air spaces. Figure 3-15 shows small magnetic cores built by Arnold Engineering. Several thousand of the smallest of these may be used in a single computer or data-processing machine. Such cores are also applicable to high-frequency magnetic amplifiers where high gain is needed.

Example 3-1:	Consider the simple core shown in Fig. 3-16 comprised of U.S.S. Transformer 72, 29 (0.0140") Gage Steel. Determine the current necessary to produce a flux of 25,000 maxwells (lines) in the core.
Solution:	Figure 3-15. Collection of magnetic cores. (Courtesy of The Arnold Engineering Co.) Figure 3-16. Laminated electromagnet From Fig. 3-12 H = 124 amp turns per in., magnetic field intensity F = NI = HI = 124 x 9.0 = 1,116 amp turns, mmf

The shape of the core shown in Fig. 3-17 is that used in the core-type transformer. It is also used in certain kinds of reactors or chokes. In the

Figure 3-17. Three-legged core and winding

shell-type transformer and in some reactors the shape of the core is as shown in Fig. 3-17, Reactors are used in circuits as current-limiting devices and to smooth out ripples in direct current.

Example 3-2:

Figure 3-17 shows a three-legged core comprised of U.S.S. Transformer 72, 29 Gage Steel. The winding has 100 turns and carries a current of 0.64 amp. Determine the flux in the center leg and in each outer leg. Assume a stacking factor of 0.95 for the core.

Solution:

Mean length of flux path as indicated by the broken lines in Fig. 3-17 is 16 in., and the magnetizing force or magnetic field intensity is From Fig. 3-12 the flux density corresponding to a magnetizing force of 4.0 amp turns per in. is 60,000 lines per sq in. Hence B = 60,000 Gross area of center leg of core = 1.5 x 2 = 3.00 sq in. Then if the stacking factor is 0.95 Net area of center leg of core = 3.00 x 0,95 = 2.85 sq in. Since the outer legs are one-half as wide as the center leg Net area of each outer leg = 1.50 x 0.95 = 1.425 sq in. Flux in center leg = BA = 60,000 x 2.85 = 171,000 lines Flux in each outer leg = 60,000 x 1.425 = 85,500 lines