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DC current for ring core with 2 mm air gap

Short answer

Using mean diameter 4.5 cm, iron length 0.1394 m, area 7.854×10⁻⁵ m², flux density B = 0.3 mWb/area = 3.82 T. MMF for air gap 6079 AT, iron 212 AT, total 6291 AT. Required direct current I = 6291/1000 = 6.29 A.

A magnetic core consists of circular ring with outer diameter 5.5 cm and inner diameter 3.5 cm. The relative permeability of the iron is 2000. A radial airgap of 2 mm is cut in this core. Calculate the direct current that will be required in a coil of 1000 turns uniformly distributed around the core to produce a magnetic flux of 0.3 mWb in the airgap. Assume the magnetic leakage is negligible.

Assumption: the ring has a circular cross-section whose diameter equals the radial width, (5.5−3.5)/2=1(5.5 - 3.5)/2 = 1 cm.

Dimensions

Dmean=5.5+3.52=4.5 cmlmean=πDmean=π×0.045=0.1414 mli=0.1414−0.002=0.1394 mA=π(0.01)24=7.854×10−5 m2\begin{aligned} D_{mean} &= \frac{5.5 + 3.5}{2} = 4.5\ \text{cm} \\ l_{mean} &= \pi D_{mean} = \pi\times0.045 = 0.1414\ \text{m} \\ l_i &= 0.1414 - 0.002 = 0.1394\ \text{m} \\ A &= \frac{\pi(0.01)^2}{4} = 7.854\times10^{-5}\ \text{m}^2 \end{aligned}

Flux density

B=ϕA=0.3×10−37.854×10−5=3.82 TB = \frac{\phi}{A} = \frac{0.3\times10^{-3}}{7.854\times10^{-5}} = 3.82\ \text{T}

Ampere-turns

ATg=Bμ0lg=3.824π×10−7×0.002=3.0396×106×0.002=6079 ATATi=Bμ0μrli=3.0396×1062000×0.1394=1519.8×0.1394=211.8 ATNI=6079+211.8=6291 AT\begin{aligned} AT_g &= \frac{B}{\mu_0} l_g = \frac{3.82}{4\pi\times10^{-7}}\times0.002 = 3.0396\times10^{6}\times0.002 = 6079\ \text{AT} \\ AT_i &= \frac{B}{\mu_0\mu_r} l_i = \frac{3.0396\times10^{6}}{2000}\times0.1394 = 1519.8\times0.1394 = 211.8\ \text{AT} \\ NI &= 6079 + 211.8 = 6291\ \text{AT} \end{aligned}

Current

I=62911000=6.29 AI = \frac{6291}{1000} = 6.29\ \text{A}

Answer: I ≈ 6.29 A (dc). (B = 3.82 T is beyond the saturation of real iron; the answer assumes μr stays 2000. If a square 1 cm × 1 cm section is assumed instead, A = 1 cm², B = 3 T and I ≈ 4.94 A.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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