Question

Prob.l. A rectangular magnetic core shown in the below Fig, has square cross-section of area 16 cm". An air-gap of 2 mm is cut across one of its limbs. Find the exciting current needed in the coil having 1000 turns wound on the core to create an airgap mu of 4 m Wb. The relative permeability of the core is 2000. N=1000 2 mm 25 cm - - 20 cm

Answer 1

Ans)

Given area of cross section $A_{c}=16\: \: cm^{2}=16*10^{-4}\: \: m^{2}$

length of airgap is $l_{g}=2\: \: mm=0.2\: \: cm=2*10^{-3}\: \: m$

N=1000 turns

airgap flux $\varphi _{g}=4\: \: mWb$

relative permeability $\mu _{r}=2000$

let the side is square cross section be 's'

$s^{2}=A_{c}=16\: \: cm^{2}$

$s=\sqrt{16}=4\: \: cm$

$s=4\: \: cm$ marked in figure shown

N 1000 s=4cm 2 mm Mean length of core 'l' -20 cm 25 cm

from the circuit mean length of the core is

$l=2*(20-\frac{s}{2}-\frac{s}{2})+(25-\frac{s}{2}-\frac{s}{2})+(25-\frac{s}{2}-\frac{s}{2}-l_{g})$

$l=2*(20-\frac{4}{2}-\frac{4}{2})+(25-\frac{4}{2}-\frac{4}{2})+(25-\frac{4}{2}-\frac{4}{2}-0.2)$

$l=2*(16)+(21)+(20.8)=73.8\: \: cm$

$l=73.8\: \: cm=73.8*10^{-2}\: m$

Now reluctance of the core is

$\Re _{c}=\frac{l}{A\mu_{o} \mu _{r}}=\frac{73.8*10^{-2}}{16*10^{-4}*4\pi *10^{-7}*2000}=183525.54\: \: AT/Wb$

$\Re _{c}=183525.54\: \: AT/Wb$

Reluctance of airgap is

$\Re _{g}=\frac{l_{g}}{A\mu_{o} \mu _{r}}=\frac{2*10^{-3}}{16*10^{-4}*4\pi *10^{-7}}=994718.39\: \: AT/Wb$

$\Re _{g}=994718.39\: \: AT/Wb$

Now equation for airgap flux is given as

$\varphi _{g}=\frac{NI}{\Re _{c}+\Re _{g}}$ where 'I is exciting current needed in the coil

$I=\frac{\varphi _{g}*(\Re _{c}+\Re _{g})}{N}=\frac{4m*(183525.54+994718.39)}{1000}$

$\mathbf{I=4.713\: \: A}$ is the current needed