# Reluctance force

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The reluctance force or Maxwell's force arises due to the change in magnetic resistance , which is also referred to as reluctance . The reluctance force always works in such a way that the magnetic resistance decreases and the inductance increases and is attributable to the magnetostatics . ${\ displaystyle F _ {\ mathrm {R}}}$ This property is used in some types of electrical machines , for example switched reluctance machines , transverse flux machines , the synchronous reluctance motor or electromagnetic bearings .

A related force is the Lorentz force , which describes the effect of force on a moving electrical charge in an external electromagnetic field.

## Movable core

The reluctance force can be derived from the change in energy that results from an infinitesimal displacement of the movable piece to the side: ${\ displaystyle W}$ ${\ displaystyle dx}$ ${\ displaystyle F _ {\ mathrm {R}} = {\ frac {\ mathrm {d} W} {\ mathrm {d} x}}}$ ,
${\ displaystyle W = {\ frac {1} {2}} \ cdot I ^ {2} \ cdot L}$ ${\ displaystyle \ Rightarrow F _ {\ mathrm {R}} = {\ frac {1} {2}} \ cdot I ^ {2} \ cdot {\ frac {\ mathrm {d} L (x)} {\ mathrm {d} x}}}$ .

In it is

• ${\ displaystyle I}$ the electric current and
• ${\ displaystyle L}$ the inductance.

The inductance of a magnetic circuit with an air gap is given by ${\ displaystyle L}$ ${\ displaystyle L = {\ frac {N ^ {2}} {R_ {m, {\ text {core}}} + R_ {m, {\ text {air}}}}} \ approx {\ frac {N ^ {2}} {R_ {m, {\ text {air}}}}} = N ^ {2} \ cdot {\ frac {\ mu _ {0} \ cdot A} {l _ {\ text {air} }}}}$ With

• the number of coil turns${\ displaystyle N}$ • the magnetic resistance , the magnetic resistance of the core compared to that of the air gap being neglected for the approximation${\ displaystyle R_ {m}}$ • the magnetic field constant ${\ displaystyle \ mu _ {0}}$ • the face of the magnetic circuit at the air gap through which the field lines of the magnetic field pass${\ displaystyle A}$ • the sum of the size of both air gaps.${\ displaystyle l _ {\ text {air}}}$ The (idealized) area available for the magnetic circuit is given by

${\ displaystyle A = (x_ {0} - | x |) \ cdot y_ {0} \ = x_ {0} \ cdot y_ {0} - | x | \ cdot y_ {0}}$ ${\ displaystyle \ Rightarrow {\ frac {\ mathrm {d} A} {\ mathrm {d} | x |}} = \ left \ {{\ begin {matrix} -y_ {0}, \ quad {\ text { if}} | x |> 0 \\ 0, \ quad {\ text {if}} x = 0 \ end {matrix}} \ right.}$ The direction of the deflection is  irrelevant, hence the amount bars . The size denotes the depth. ${\ displaystyle x}$ ${\ displaystyle y_ {0}}$ Inserting supplies

${\ displaystyle {\ frac {\ mathrm {d} L} {\ mathrm {d} | x |}} = N ^ {2} \ cdot {\ frac {\ mu _ {0}} {l _ {\ text { Air}}}} \ cdot {\ frac {\ mathrm {d} A} {\ mathrm {d} | x |}} = - N ^ {2} \ cdot \ mu _ {0} \ cdot {\ frac { y_ {0}} {l _ {\ text {air}}}}}$ so that on the moving part of the deflected core a force

${\ displaystyle \ Rightarrow F _ {\ mathrm {R}} = - {\ frac {1} {2}} \ cdot (I \ cdot N) ^ {2} \ cdot \ mu _ {0} \ cdot {\ frac {y_ {0}} {l _ {\ text {Air}}}}}$ acts that pulls him towards the center. This is independent of the size of the deflection, except if the above derivation is no longer valid. This is the case when it gets too big. ${\ displaystyle {\ frac {\ mathrm {d} A} {\ mathrm {d} | x |}} = - y_ {0}}$ ${\ displaystyle | x |}$ ## Variable air gap

The same applies as above

${\ displaystyle F _ {\ mathrm {R}} = {\ frac {\ mathrm {d} W} {\ mathrm {d} l _ {\ text {air}}}} = {\ frac {1} {2}} \ cdot I ^ {2} \ cdot {\ frac {\ mathrm {d} L (l _ {\ text {air}})} {\ mathrm {d} l _ {\ text {air}}}}}$ .

The following applies approximately to the inductance

${\ displaystyle L \ approx {\ frac {N ^ {2}} {R_ {m, {\ text {Air}}}}} = N ^ {2} \ cdot A \ cdot \ mu _ {0} \ cdot {\ frac {1} {l _ {\ text {Air}}}}}$ .

With the power rule we get

${\ displaystyle {\ frac {\ mathrm {d} L} {\ mathrm {d} l _ {\ text {air}}}} = N ^ {2} \ cdot A \ cdot \ mu _ {0} \ cdot { \ frac {-1} {{l _ {\ text {Air}}} ^ {2}}}}$ .

Inserting it into the formula for gives the result: ${\ displaystyle F _ {\ mathrm {R}}}$ ${\ displaystyle F _ {\ mathrm {R}} = - {\ frac {1} {2}} \ cdot I ^ {2} \ cdot N ^ {2} \ cdot A \ cdot \ mu _ {0} \ cdot {\ frac {1} {{l _ {\ text {Air}}} ^ {2}}}}$ .

Since the inductance increases when the air gap is reduced, the reluctance force acts in this direction. The force decreases with the width of the air gap. The maximum of the reluctance force is reached when the air gap approaches zero. However, if the air gap is very small, the approximate formula for the inductance no longer applies, since the magnetic resistance of the core can then no longer be neglected.

## literature

• Hans-Dieter Stölting, Eberhard Kallenbach (ed.): Manual electrical small drives . 3. Edition. Hanser, ISBN 3-446-40019-2 , pp. 460 .