Magnetic Circuit

Problem Statement

Problem 2.16 from Electric Machinery and Transformers, 3rd ed:

A magnetic circuit is given in Figure P2.16. What must be the current in the 1600-turn coil to set up a flux density of 0.1 T in the air-gap? All dimensions are in centimeters. Assume that magnetic flux density varies as $B=[1.5H/(750+H)]$.<ref>Guru and Huseyin, Electric Machinery and Transformers, 3rd ed. (New York: Oxford University Press, 2001), 129.</ref>

Solution

First, we note that the problem statement is incomplete. Assume that the core has a relative permeability of 500. Hence, for all magnetic sections excluding the air gap,

$\mu =\mu _{r}\mu _{0}=(500)(4\pi \times 10^{-7})$

Also, as recommended in the text, we will neglect fringing.

The lengths and areas of each of the sections to be evaluated are given in the following table.

Table 1: Lengths and Areas for the pertinent secions of the magnetic circuit.

Section	fg	def	ghc	dc	dabc
Length $l$ (m)	0.01	0.555	0.555	0.48	1.4
Area $A$ (m2)	0.0032	0.0032	0.0032	0.0096	8.0e-4

We must now work backward from the air-gap, since the value of the flux-density is given there. All units are understood to be standard units.

Air Gap:

${\mathcal {R}}={\frac {l_{fg}}{\mu A_{f}g}}=4973.6$

$\Phi _{fg}=B_{fg}A_{fg}=3.2\times 10^{-4}$

${\mathcal {F}}={\mathcal {R}}_{fg}\Phi _{fg}=1.5915$ Right Arms:

$\Phi _{def}=\Phi _{ghc}=\Phi _{fg}=3.2\times 10^{-4}$

${\mathcal {R}}_{def}={\mathcal {R}}_{ghc}={\frac {l_{def}}{\mu A_{def}}}=276030$

${\mathcal {F}}_{def}={\mathcal {F}}_{ghc}={\mathcal {R}}_{def}\Phi _{def}=88.331$

Center Column

${\mathcal {F}}_{dc}={\mathcal {F}}_{def}+{\mathcal {F}}_{fg}+{\mathcal {F}}_{ghc}=178.25$

Calculations were performed using the following Magnetic Circuit Matlab Script.

References

<references />