An iron ring with a mean diameter of 25 cm is wound with 500 turns. Assuming the relative permeability of iron to be 400, find: . (iv) Find the value of coil current if the same flux density [as in part (i)] is to be maintained within the core if leakage factor is 1.2 (v) Find the value of the current required for maintaining the same flux density in the air gap.?

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Answer

Solution:

Given data:
Mean diameter of iron ring, d = 25 cm
Number of turns, N = 500
Relative permeability of iron, μr = 400
Leakage factor, δ = 1.2

(i) Find the flux density when a current of 5 A flows through the coil:
Current, I = 5 A
Magnetic field intensity, H = NI/l = (500 × 5)/π × 0.25 = 3183.1 A/m
Flux density, B = μ0μrH = 4π × 10^-7 × 400 × 3183.1 = 5.03 T

(ii) Find the flux linkage:
Flux linkage, Nϕ = NBA = (500 × π × 0.25^2 × 5.03) × 10^-4 = 0.989 Wb

(iii) Find the self-inductance of the coil:
Self-inductance, L = Nϕ/I = 0.989/5 = 0.198 H

(iv) Find the value of coil current if the same flux density [as in part (i)] is to be maintained within the core if leakage factor is 1.2:
Flux density in the core with leakage, B' = B/δ = 5.03/1.2 = 4.19 T
Magnetic field intensity with leakage, H' = B'/(μ0μr) = 2638.2 A/m
Current required, I' = H'l/N = (2638.2 × π × 0.25)/(500 × 5) = 0.52 A

(v) Find the value of the current required for maintaining the same flux density in the air gap:
Flux density in the air gap, B'' = B/μr = 5.03/400 = 0.0126 T
Magnetic field intensity in the air gap, H'' = B''/(μ0μr) = 7.92 A/m
Current required, I'' = H''l/N = (7.92 × π × 0.25)/(500 × 5) = 0.032 A

Explanation:
In an iron-core coil, the magnetic field is concentrated in the core due to its high permeability. But, some of the magnetic flux also leaks out of the core into the surrounding air or other nearby objects. The leakage factor is the ratio of the flux that leaks out of the core to the total flux produced by the coil. In this question, we have assumed a leakage factor of 1.2, which means that 20% of the flux produced by the coil is leaking out of the core.
To maintain the same flux density in the core with leakage, we need to increase the current in the coil because the effective magnetic field intensity in the core is reduced due to the leakage. On the other hand, to maintain the same flux density in the air gap, we need to reduce the current in the coil because the magnetic field intensity in the air gap is much lower than in the core due to the low permeability of air.

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