Editing Electromagnet (section)

=== Magnetic circuit ===
[[Image:Electromagnet with gap.svg|thumb|upright=1.7|Figure 1. Magnetic field (<span style="color:green;">green</span>) of a typical electromagnet, with the iron core ''C'' forming a closed loop with two air gaps ''G'' in it.<br /> 
''B'' – magnetic field in the core<br />
''B<sub>F</sub>'' – fringing fields; in the gaps ''G'', the magnetic field lines bulge out, so the field strength is less than in the core: ''B<sub>F</sub>''&nbsp;<&nbsp;''B''<br />
''B<sub>L</sub>'' – [[leakage flux]]; magnetic field lines which do not follow complete magnetic circuit<br />
''L'' – average length of the magnetic circuit (used in {{EquationNote|3|Eq. 3}}). It is the sum of the length ''L<sub>core</sub>'' in the iron core pieces and the length ''L<sub>gap</sub>'' in the air gaps ''G''.<br />
Both the leakage flux and the fringing fields get larger as the gaps are increased, reducing the force exerted by the magnet. ]]

In many practical applications of electromagnets, such as motors, generators, transformers, lifting magnets, and loudspeakers, the iron core is in the form of a loop or [[magnetic circuit]], possibly broken by a few narrow air gaps. Iron presents much less "resistance" ([[reluctance]]) to the magnetic field than air, so a stronger field can be obtained if most of the magnetic field's path is within the core.<ref name="Merzouki" /> Since the magnetic field lines are closed loops, the core is usually made in the form of a loop.

Since most of the magnetic field is confined within the outlines of the core loop, this allows a simplification of the mathematical analysis.<ref name="Merzouki" /> A common simplifying assumption satisfied by many electromagnets, which will be used in this section, is that the magnetic field strength ''<math>B</math>'' is constant around the magnetic circuit (within the core and air gaps) and zero outside it. Most of the magnetic field will be concentrated in the core material (''C'') (see Fig. 1). Within the core, the magnetic field (''B'') will be approximately uniform across any cross-section; if the core also has roughly constant area throughout its length, the field in the core will be constant.<ref name="Merzouki" />

At any air gaps (''G'') between core sections, the magnetic field lines are no longer confined by the core. Here, they bulge out beyond the core geometry over the length of the gap, reducing the field strength in the gap.<ref name="Merzouki" /> The "bulges" (''B<sub>F</sub>'') are called fringing fields.<ref name="Merzouki" /> However, as long as the length of the gap is smaller than the cross-section dimensions of the core, the field in the gap will be approximately the same as in the core.

In addition, some of the magnetic field lines (''B<sub>L</sub>'') will take "short cuts" and not pass through the entire core circuit, and thus will not contribute to the force exerted by the magnet. This also includes field lines that encircle the wire windings but do not enter the core. This is called [[leakage flux]].

The equations in this section are valid for electromagnets for which:
# the magnetic circuit is a single loop of core material, possibly broken by a few air gaps;
# the core has roughly the same cross-sectional area throughout its length;
# any air gaps between sections of core material are not large compared with the cross-sectional dimensions of the core;
# there is negligible leakage flux.

==== Magnetic field in magnetic circuit ====
The magnetic field created by an electromagnet is proportional to both the number of turns of wire <math>N</math> and the current <math>I</math>; their product, <math>NI</math>, is called [[magnetomotive force]]. For an electromagnet with a single [[magnetic circuit]], Ampere's Law reduces to:<ref name="Merzouki" /><ref>{{cite book
  | last = Feynman
  | first = Richard P.
  | title = Lectures on Physics, Vol. 2
  | publisher = Addison-Wesley
  | year = 1963
  | location = New York
  | pages = 36–9 to 36–11, eq. 36–26
  | isbn =  978-8185015842
  | url = https://www.feynmanlectures.caltech.edu/II_36.html#Ch36-S5
}}</ref><ref name="Fitzgerald">{{cite book
  | last1 = Fitzgerald
  | first1 = A.
  | last2 = Kingsley |first2=Charles |last3=Kusko |first3=Alexander
  | title = Electric Machinery, 3rd Ed
  | publisher = McGraw-Hill
  | year = 1971
  | location = USA
  | pages = 3–5}}</ref>

:<math>NI = H_{\mathrm{core}} L_{\mathrm{core}} + H_{\mathrm{gap}} L_{\mathrm{gap}}</math>
{{NumBlk|:|<math>NI = B \left(\frac{L_{\mathrm{core}}}{\mu} + \frac{L_{\mathrm{gap}}}{\mu_0} \right) </math>|{{EquationRef|3}}}}

This is a [[nonlinear equation]], because the permeability of the core <math>\mu</math> varies with <math>B</math>. For an exact solution, <math>\mu(B)</math> must be obtained from the core material [[Hysteresis loop|hysteresis curve]].<ref name="Merzouki" /> If <math>B</math> is unknown, the equation must be solved by [[Numerical analysis|numerical methods]].  

However, if the magnetomotive force is well above saturation (so the core material is in saturation), the magnetic field will be approximately the material's saturation value <math>B_{sat}</math>, and will not vary much with changes in <math>NI</math>. For a closed magnetic circuit (no air gap), most core materials saturate at a magnetomotive force of roughly 800&nbsp;ampere-turns per meter of flux path.

For most core materials, the relative permeability <math>\mu_r \approx 2000 \text{–} 6000\,</math>.<ref name="Fitzgerald" /> So in ({{EquationNote|3|Eq. 3}}), the second term dominates. Therefore, in magnetic circuits with an air gap, <math>B</math> depends strongly on the length of the air gap, and the length of the flux path in the core does not matter much. Given an air gap of 1mm, a magnetomotive force of about 796&nbsp;ampere-turns is required to produce a magnetic field of 1&nbsp;T.

==== Closed magnetic circuit ====
[[File: Lifting electromagnet cross section.png|thumb|Cross section of a lifting electromagnet, showing the cylindrical construction. The windings (''C'') are flat copper strips to withstand the [[Lorentz force]] of the magnetic field. The core is formed by the thick iron housing (''D'') that wraps around the windings.]]

For a closed magnetic circuit (no air gap), such as would be found in an electromagnet lifting a piece of iron bridged across its poles, equation ({{EquationNote|3|Eq. 3}}) becomes:

{{NumBlk|:|<math>B = \frac{NI\mu}{L} </math>|{{EquationRef|4}}}}

Substituting into ({{EquationNote|2|Eq. 2}}), the force is:

{{NumBlk|:|<math>F = \frac{\mu^2 N^2 I^2 A}{2\mu_0 L^2} </math>|{{EquationRef|5}}}}

To maximize the force, a core with a short flux path ''<math>L</math>'' and a wide cross-sectional area ''<math>A</math>'' is preferred (this also applies to magnets with an air gap). To achieve this, in applications like lifting magnets and [[loudspeaker]]s, a flat cylindrical design is often used. The winding is wrapped around a short wide cylindrical core that forms one pole, and a thick metal housing that wraps around the outside of the windings forms the other part of the magnetic circuit, bringing the magnetic field to the front to form the other pole.