Magnetic Circuit Given the current i = 5 A, and μr = 2000, find the mmf, the total reluctance, Φ, B, and H for the magnetic circuit in Figure 12.11.

Question

Magnetic Circuit

Given the current i = 5 A, and [latex]\mu_r[/latex] = 2000, find the mmf, the total reluctance, Φ, B, and H for the magnetic circuit in Figure 12.11.

Accepted Answer

Using Equation (12.11a), the mmf corresponds to:

[latex]\Im=N \cdot i=5 \cdot 700=3500 A \cdot t[/latex]

To find the total reluctance, consider the mean length, which is calculated based on Figure 12.11. The mean path lengths are:

[latex]l_1=4 cm, ext { and } l_2=5 cm[/latex]

The cross section of the area is:

A = 0.01 . 0.01 = 0.0001 m²

Thus, the reluctances for the core segments are:

[latex]\begin{aligned}&\Re_1=\frac{l_1}{\mu_0 \mu_{r} A}=\frac{0.04}{4 \pi imes 10^{-7} imes 2000 imes 0.0001}=159.15 imes 10^3 A \cdot t / Wb \&\Re_2=\frac{l_2}{\mu_0 \mu_{\mathrm{r}} A}=\frac{0.05}{4 \pi imes 10^{-7} imes 2000 imes 0.0001}=198.94 imes 10^3 A \cdot t / Wb\end{aligned}[/latex]

In addition, the total reluctance of all segments is:

[latex]\Re_{ ext {total }}=2 \Re_1+2 \Re_2=716.18 imes 10^3 A \cdot t / Wb[/latex]

Using Equation (12.16b), the magnetic flux corresponds to:

[latex]\Im=\Re \cdot \phi[/latex] (12.16b)

[latex]\varphi=\frac{\Im}{\Re_{ ext {total }}}=\frac{3500}{716.18 imes 10^3}=4.887 mWb[/latex]

Using Equations (12.13) and (12.14), the magnetic flux density and field intensity correspond to:

[latex]H=\frac{B}{\mu}=\frac{\phi}{A\mu}[/latex] (12.14)

[latex]B=\frac{\phi}{A}=\frac{4.887 imes 10^{-3}}{0.0001}=48.87 Wb / m^2[/latex]

And:

[latex]H=\frac{B}{\mu}=\frac{48.87}{4 \pi imes 10^{-7} imes 2000}=19.44 imes 10^3 A \cdot t / m[/latex]

Question 12.8: Magnetic Circuit Given the current i = 5 A, and μr = 2000, f...

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