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Machines (EE352)
جامعة طرابلس
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Example 1 Figure E1 represents the magnetic circuit of a primitive relay. The coil has 500 turns and the mean core path is lc 360 mm. When the air gap lengths are 1 mm each, a flux density of 0 tesla is required to actuate the relay. The core is cast steel. (a) Find the current in the coil. (b) Compute the values of permeability and relative permeability of the core. (c) If the air gap is zero, find the current in the coil for the same flux density (0 T) in the core. Core Mean path N FIGURE E1 N 500 turns, 1c 36 cm. Example 1 Figure E1 represents the magnetic circuit of a primitive relay. The coil has 500 turns and the mean core path is lc 360 mm. When the air gap lengths are 1 mm each, a flux density of 0 tesla is required to actuate the relay. The core is cast steel. (a) Find the current in the coil. (b) Compute the values of permeability and relative permeability of the core. (c) If the air gap is zero, find the current in the coil for the same flux density (0 T) in the core. Core Mean path N FIGURE E1 N 500 turns, 1c 36 cm. 1 1 Silicon sheet steel 1 1 Cast steel 1 0 0 0 0 0 Cast iron 0 0 0 0 0 200 400 600 800 Field intensity, H 1 1 Silicon sheet steel 1 1 Cast steel 1 0 0 0 0 0 Cast iron 0 0 0 0 0 200 400 600 800 Field intensity, H (b) Permeability of core: He Bc 510 0 1 x (c) Relative permeability of core: 10 1250 Hdc 0x0 184 At 500 184 0 A Note that if the air gap is not present, a much smaller current is required to establish the same flux density in the magnetic circuit. (b) Permeability of core: He Bc 510 0 1 x (c) Relative permeability of core: 10 1250 Hdc 0x0 184 At 500 184 0 A Note that if the air gap is not present, a much smaller current is required to establish the same flux density in the magnetic circuit. Example 2 A ferromagnetic core is shown. Three sides of this core are of uniform width, while the fourth side is somewhat thinner. The depth of the core is 10cm, and the other dimensions are shown in the figure. There is a 200 turn coil wrapped around the left side of the core. Assuming relative permeability H, of 2500, how much flux will be produced a 1A input current? Solution: 3 sides of the core have the same cross sectional area, while the 4th side has a different area. Thus the core can be divided into 2 regions: (1) the single thinner side (2) the other 3 sides taken together Example 2 A ferromagnetic core is shown. Three sides of this core are of uniform width, while the fourth side is somewhat thinner. The depth of the core is 10cm, and the other dimensions are shown in the figure. There is a 200 turn coil wrapped around the left side of the core. Assuming relative permeability H, of 2500, how much flux will be produced a 1A input current? Solution: 3 sides of the core have the same cross sectional area, while the 4th side has a different area. Thus the core can be divided into 2 regions: (1) the single thinner side (2) the other 3 sides taken together The magnetic circuit corresponding to this core: R 1 F Ni) R2 (b) The magnetic circuit corresponding to this core: R 1 F Ni) R2 (b) R1 uA, l1 l1 H,H,A, 0 m (2500)(4 X m²) 14,300 A . R2 uA2 12 H,HOA2 12 1 m (2500)(4 X m² 27,600 A R1 uA, l1 l1 H,H,A, 0 m (2500)(4 X m²) 14,300 A . R2 uA2 12 H,HOA2 12 1 m (2500)(4 X m² 27,600 A Example 2 Figure shows a ferromagnetic core whose mean path length is 40cm. There is a small gap of 0 in the structure of the otherwise whole core. The csa of the core is 12cm², the relative permeability of the core is and the coil of wire on the core has 400 turns. Assume that fringing in the air gap increases the effective csa of the gap Given this information, find 1 total reluctance of the flux path (iron plus air gap) 2 current required to produce a flux density of 0 in the air gap. Example 2 Figure shows a ferromagnetic core whose mean path length is 40cm. There is a small gap of 0 in the structure of the otherwise whole core. The csa of the core is 12cm², the relative permeability of the core is and the coil of wire on the core has 400 turns. Assume that fringing in the air gap increases the effective csa of the gap Given this information, find 1 total reluctance of the flux path (iron plus air gap) 2 current required to produce a flux density of 0 in the air gap. N 400 turns B 0 cm A 12 cm2 to 40 cm (a) N 400 turns B 0 cm A 12 cm2 to 40 cm (a) R lc 0 m X m² 66,300 A . la Ra HOA m (4TT X m ² A . R lc 0 m X m² 66,300 A . la Ra HOA m (4TT X m ² A . Therefore, the total reluctance of the flux path is Req 66,300 A . A . 382,300 A . BAR i N (0 T)(0 m²)(382,300 A . 400 turns 0 A Therefore, the total reluctance of the flux path is Req 66,300 A . A . 382,300 A . BAR i N (0 T)(0 m²)(382,300 A . 400 turns 0 A N If 200 A 12 cm2 turns lr 5 cm 1g 0 cm lc 50 cm N If 200 A 12 cm2 turns lr 5 cm 1g 0 cm lc 50 cm Solution: To determine the flux density in the air gap, it is necessary to first calculate the mmf applied to the core and the total reluctance of the flux path. With this information, the total flux in the core can be found. Finally, knowing the csa of the air gaps enables the flux density to be calculated. The magnetic ckt corresponding to this machine is shown below. Rs Stator reluctance Air gap 1 reluctance Rar F Ni) Rotor reluctance R, Air gap 2 reluctance Raz Solution: To determine the flux density in the air gap, it is necessary to first calculate the mmf applied to the core and the total reluctance of the flux path. With this information, the total flux in the core can be found. Finally, knowing the csa of the air gaps enables the flux density to be calculated. The magnetic ckt corresponding to this machine is shown below. Rs Stator reluctance Air gap 1 reluctance Rar F Ni) Rotor reluctance R, Air gap 2 reluctance Raz Ra ,, la m (1)(4 X m² A . R101 Rs Ra R, 16,600 A . A . The net magnetomotive force applied to the core is F Ni (200 turns)(1 A) 200 A . turns Ra ,, la m (1)(4 X m² A . R101 Rs Ra R, 16,600 A . A . The net magnetomotive force applied to the core is F Ni (200 turns)(1 A) 200 A . turns Therefore, the total flux in the core is F 200 A . turns A . Wb Finally, the magnetic flux density in the air gap is Wb B A 0 cm2 0 T Therefore, the total flux in the core is F 200 A . turns A . Wb Finally, the magnetic flux density in the air gap is Wb B A 0 cm2 0 T
