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Unformatted text preview: 9 425 A t 163 A t/m 2.60 m c H l ⋅ = = = ⋅ F From the magnetization curve, 0.15 T B = and the total flux in the core is ( )( )( ) TOT 0.15 T 0.15 m 0.15 m 0.0033 Wb BA φ = = = The relative permeability of the core can be found from the reluctance as follows: A l r TOT TOT μ μ φ = = F R Solving for μ r yields ( )( ) ( ) ( ) ( )( ) TOT7 TOT 0.0033 Wb 2.6 m 714 425 A t 4 10 H/m 0.15 m 0.15 m r l A φ μ μ π = = = ⋅ × F The assumption that r μ = 1000 is not very good here. It is not very good in general. 113. A core with three legs is shown in Figure P110. Its depth is 8 cm, and there are 400 turns on the center leg. The remaining dimensions are shown in the figure. The core is composed of a steel having the magnetization curve shown in Figure 110 c . Answer the following questions about this core: (a) What current is required to produce a flux density of 0.5 T in the central leg of the core? (b) What current is required to produce a flux density of 1.0 T in the central leg of the core? Is it twice the current in part (a) ? (c) What are the reluctances of the central and right legs of the core under the conditions in part (a) ? (d) What are the reluctances of the central and right legs of the core under the conditions in part (b) ? (e) What conclusion can you make about reluctances in real magnetic cores? 10 S OLUTION The magnetization curve for this core is shown below: (a) A flux density of 0.5 T in the central core corresponds to a total flux of ( )( )( ) TOT 0.5 T 0.08 m 0.08 m 0.0032 Wb BA φ = = = By symmetry, the flux in each of the two outer legs must be 1 2 0.0016 Wb φ φ = = , and the flux density in the other legs must be ( )( ) 1 2 0.0016 Wb 0.25 T 0.08 m 0.08 m B B = = = The magnetizing intensity H required to produce a flux density of 0.25 T can be found from Figure 110 c . It is 50 A·t/m. Similarly, the magnetizing intensity H required to produce a flux density of 0.50 T is 70 A·t/m. Therefore, the total MMF needed is TOT center center outer outer H l H l = + F ( )( ) ( )( ) TOT 70 A t/m 0.24 m 50 A t/m 0.72 m 52.8 A t = ⋅ + ⋅ = ⋅ F and the required current is TOT 52.8 A t 0.13 A 400 t i N ⋅ = = = F (b) A flux density of 1.0 T in the central core corresponds to a total flux of ( )( )( ) TOT 1.0 T 0.08 m 0.08 m 0.0064 Wb BA φ = = = By symmetry, the flux in each of the two outer legs must be 1 2 0.0032 Wb φ φ = = , and the flux density in the other legs must be ( )( ) 1 2 0.0032 Wb 0.50 T 0.08 m 0.08 m B B = = = 11 The magnetizing intensity H required to produce a flux density of 0.50 T can be found from Figure 110 c . It is 70 A·t/m. Similarly, the magnetizing intensity H required to produce a flux density of 1.00 T is about 160 A·t/m. Therefore, the total MMF needed is TOT center center outer outer H I H I = + F ( )( ) ( )( ) TOT 160 A t/m 0.24 m 70 A t/m 0.72 m 88.8 A t = ⋅ + ⋅ = ⋅ F and the required current is TOT 88.8 A t 0.22 A 400 t i N φ ⋅ = = = This current is less not twice the current in part...
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 Spring '05
 Shaban
 Flux

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