Consider a Micrometals toroidal iron powder core with 52/μ75 material. The initial permeability versus magnetic field intensity curve is given in Figure 10.5.

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Accepted Answer

An inductor was constructed using a toroid with core length l[latex]_{c}[/latex] = 4.23 cm, core cross-sectional area [latex]A_{c}[/latex] = 0.179 [latex]cm^{2}[/latex] and with N = 72 turns. Calculate the inductance L and the effective inductance [latex]L_{\mathrm{eff}}[/latex] as a function of current for the range 1 to 4 A.

For H = 30 Oe, μ[latex]_{r}[/latex] is read from Figure 10.5:

i = [latex] \frac{Hl_{c}}{N}= \frac{(2387)(4.23 imes10^{-2})}{72} = 1.403[/latex] A

L(i) = [latex] \mu_{r}(i) imes \mu_{0} imes N^{2} imes \frac{A_{c}}{l_{c}} = (56.1)(4\pi imes10^{-7})(72)^{2}\left(\frac{0.179 imes 10^{-4}}{4.23 imes 10^{-2}} ight) = 154.7 \mu[/latex]H

L[latex]_{\mathrm{eff}}=L(i)+ i\frac{\Delta L}{\Delta i} = (154.7 imes10^{-6})+ (1.403)\frac{(154.7-176.7) imes10^{-6}}{(1.403-0.935)} = 88.7 \mu [/latex]H

The full set of calculations are summarized in Table 10.1, and L and L[latex]_{\mathrm{eff}}[/latex] are plotted in Figure 10.6.

Table 10.1 Calculations for inductance and effective inductance
H (Oe)	H (A/m)	[latex]\mu_{r}[/latex]	i (A)	L ([latex]\mu[/latex]H)	[latex]L_{\mathrm{eff}}(\mu H)[/latex]
1	80	75	0.047	206.8	206.8
1.4	111	75	0.065	206.8	206.8
2	159	74.9	0.094	206.5	205.6
3	239	74	0.14	204.0	196.6
5	398	73.5	0.234	202.6	199.2
7	557	72.5	0.327	199.9	190.2
10	796	71.1	0.468	196.0	183.1
14	1114	68.4	0.655	188.6	162.5
20	1592	64.1	0.935	176.7	137.2
30	2387	56.1	1.403	154.7	88.7
50	3979	44	2.338	121.3	37.9
70	5571	35	3.273	96.5	9.6
100	7958	27	4.675	74.4	0.9

Question 10.3: Consider a Micrometals toroidal iron powder core with 52/μ75...

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