Question

1-8. A core with three legs is shown in Figure P1-5. Its depth is 5 crn, and there are 100 turns on the leftmost leg. The relative permeability of the core can be assumed to be 2000 and constant. What flux exists in each of the three legs of the core? What is the flux density in each of the legs? 9 cm 9cm 25 cm 15 cm 25 cm 9 em 2 A 25 cm 100 tums 0.05 cm 9 em Core depth - 5 cm

Answer 1

Ans)

The core with mean lengths noted as well as areas is shown below

The core with mean lengths noted as well as areas is shown below Given N = 100 turns, i = 2 A, mu_0 = 4 pi * 10^-7 A mu_r = 2000 The reluctance's are R_1 = l_1/A_1 mu_0 mu_r = 34 * 10^-2/45 * 10^-4 * mu_0 * 2000 = 30062.6 A - t/Wb R_1 = 30062.6 A - t/Wb R_2 = l_2/A_2 mu_0 mu_r = 37 * 10^-2/45 * 10^-4 * mu_0 * 2000 = 32715.18 A - t/WB R_2 = 32715.18 A - t/WB R_3 = l_3/A_3 mu_0 mu_r = 33.95 * 10^-2/135 * 10^-4 * mu_0 * 2000 = 10006.13 A - t/Wb R_3 = 10006.13 A - t/Wb R_g = l_g/A_g mu_0 = 0.05 * 10^-2/135 * 10^-4 * mu_0 = 29473.14 A - t/Wb (air gap reluctance) R_g = 29473.14 A - t/Wb Now equivalent magnetic circuit is Now total reluctance seen by the mmf source is R_eq = R_1 + R_2 + R_2

Given $N=100 \: \: turns$ , $i=2 \: \: A,\: \: \mu_{0}=4 \pi *10^{-7} \:\:A\:\: \mu_{r}=2000$

The reluctance's are

$R_{1}=\frac{l_{1}}{A_{1}\mu_{0}\mu_{r}}=\frac{34*10^{-2}}{45*10^{-4}*\mu_{0}*2000}=30062.6\:\:A-t/Wb$

$R_{1}=30062.6\:\:A-t/Wb$

$R_{2}=\frac{l_{2}}{A_{2}\mu_{0}\mu_{r}}=\frac{37*10^{-2}}{45*10^{-4}*\mu_{0}*2000}=32715.18\:\:A-t/Wb$

$R_{2}=32715.18\:\:A-t/Wb$

$R_{3}=\frac{l_{3}}{A_{3}\mu_{0}\mu_{r}}=\frac{33.95*10^{-2}}{135*10^{-4}*\mu_{0}*2000}=10006.13\:\:A-t/Wb$

$R_{3}=10006.13\:\:A-t/Wb$

$R_{g}=\frac{l_{g}}{A_{g}\mu_{0}}=\frac{0.05*10^{-2}}{135*10^{-4}*\mu_{0}}=29473.14\:\:A-t/Wb$ (air gap reluctance)

$R_{g}=29473.14\:\:A-t/Wb$  

Now equivalent magnetic circuit is

Now total reluctance seen by the mmf source is

$R_{eq}=R_{1}+R_{2}+R_{2}+((R_{3}+R_{g})\parallel (R_{1}+R_{2}+R_{2}))$

$R_{eq}=R_{1}+2R_{2}+((R_{3}+R_{g})\parallel (R_{1}+2R_{2}))$

$R_{eq}=R_{1}+2R_{2}+(\frac{(R_{3}+R_{g})* (R_{1}+2R_{2})}{(R_{3}+R_{g})+ (R_{1}+2R_{2})})$

$R_{eq}=123424.58\:\: At-Wb$

Now flux in the left most leg is

$\varphi_{left}=\frac{NI}{R_{eq}}=\frac{200}{123424.58}=1.62 \:\: mWb$

$\mathbf{ \varphi_{left}=1.62 \:\: mWb}$

Now flux in the center leg is using flux division rule like current division in electric circuits

$\varphi_{center} =\varphi_{left}*\frac{R_{1}+2R_{2}}{R_{3}+R_{g}+R_{1}+2R_{2}}=1.146\:\: mWb$

$\mathbf{ \varphi_{center} =1.146\:\: mWb}$

Now flux in the right leg is

$\varphi_{right} =\varphi_{left}*\frac{R_{3}+R_{g}}{R_{3}+R_{g}+R_{1}+2R_{2}}=0.474\:\: mWb$

$\mathbf{ \varphi_{right} =0.474\:\: mWb}$

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Now flux densities in each of legs is

$B_{left}=\frac{ \varphi_{left}}{A_{1}}=0.36 \:\: T$

$\mathbf{B_{left}=0.36 \:\: T}$

$B_{center}=\frac{ \varphi_{center}}{A_{3}}=0.085 \:\: T$

$\mathbf{B_{center}=0.085 \:\: T}$

$B_{right}=\frac{ \varphi_{right}}{A_{1}}=0.105 \:\: T$

$\mathbf{B_{right}=0.105 \:\: T}$