Magnetic circuit

A magnetic circuit is made up of one or more closed loop paths containing a

.

The relation between

can be quickly solved using the methods and techniques developed for electrical circuits.

Some examples of magnetic circuits are:

• reluctance
  circuit)
• horseshoe magnet with no keeper (high-reluctance circuit)
• electric motor (variable-reluctance circuit)
• some types of pickup cartridge (variable-reluctance circuits)

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Similar to the way that electromotive force (EMF) drives a current of electrical charge in electrical circuits, magnetomotive force (MMF) 'drives' magnetic flux through magnetic circuits. The term 'magnetomotive force', though, is a misnomer since it is not a force nor is anything moving. It is perhaps better to call it simply MMF. In analogy to the definition of EMF, the magnetomotive force ${\displaystyle {\mathcal {F}}}$ around a closed loop is defined as:

${\displaystyle {\mathcal {F}}=\oint \mathbf {H} \cdot \mathrm {d} \mathbf {l} .}$

The MMF represents the potential that a hypothetical magnetic charge would gain by completing the loop. The magnetic flux that is driven is not a current of magnetic charge; it merely has the same relationship to MMF that electric current has to EMF. (See microscopic origins of reluctance below for a further description.)

The unit of magnetomotive force is the

(1544–1603) English physician and natural philosopher.

${\displaystyle {\begin{aligned}1\;{\text{Gb}}&={\frac {10}{4\pi }}\;{\text{At}}\\[2pt]&\approx 0.795775\;{\text{At}}\end{aligned}}}$^[2]

The magnetomotive force can often be quickly calculated using Ampère's law. For example, the magnetomotive force ${\displaystyle {\mathcal {F}}}$ of a long coil is:

${\displaystyle {\mathcal {F}}=NI}$

where N is the number of

of the inducting coil.

Magnetic flux

An applied MMF 'drives' magnetic flux through the magnetic components of the system. The magnetic flux through a magnetic component is proportional to the number of magnetic field lines that pass through the cross sectional area of that component. This is the net number, i.e. the number passing through in one direction, minus the number passing through in the other direction. The direction of the magnetic field vector B is by definition from the south to the north pole of a magnet inside the magnet; outside the field lines go from north to south.

The

of the magnetic field over the area of the surface

${\displaystyle \Phi _{m}=\iint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} .}$

For a magnetic component the area S used to calculate the magnetic flux Φ is usually chosen to be the cross-sectional area of the component.

The

.

Circuit models

The most common way of representing a magnetic circuit is the resistance–reluctance model, which draws an analogy between electrical and magnetic circuits. This model is good for systems that contain only magnetic components, but for modelling a system that contains both electrical and magnetic parts it has serious drawbacks. It does not properly model power and energy flow between the electrical and magnetic domains. This is because electrical resistance will dissipate energy whereas magnetic reluctance stores it and returns it later. An alternative model that correctly models energy flow is the gyrator–capacitor model.

The resistance–reluctance model for magnetic circuits is a

.

In electrical circuits, Ohm's law is an empirical relation between the EMF ${\displaystyle {\mathcal {E}}}$ applied across an element and the

${\displaystyle I}$ it generates through that element. It is written as: $${\displaystyle {\mathcal {E}}=IR.}$$ where R is the $${\displaystyle {\mathcal {F}}=\Phi {\mathcal {R}}.}$$ where ${\displaystyle {\mathcal {F}}}$ is the magnetomotive force (MMF) across a magnetic element, ${\displaystyle \Phi }$ is the magnetic flux through the magnetic element, and ${\displaystyle {\mathcal {R}}}$ is the magnetic reluctance of that element. (It will be shown later that this relationship is due to the empirical relationship between the H-field and the magnetic field B, B = μH, where μ is the permeability of the material). Like Ohm's law, Hopkinson's law can be interpreted either as an empirical equation that works for some materials, or it may serve as a definition of reluctance.

Hopkinson's law is not a correct analogy with Ohm's law in terms of modelling power and energy flow. In particular, there is no power dissipation associated with a magnetic reluctance in the same way as there is a dissipation in an electrical resistance. The magnetic resistance that is a true analogy of electrical resistance in this respect is defined as the ratio of magnetomotive force and the rate of change of magnetic flux. Here rate of change of magnetic flux is standing in for electric current and the Ohm's law analogy becomes, $${\displaystyle {\mathcal {F}}={\frac {d\Phi }{dt}}R_{\mathrm {m} },}$$ where ${\displaystyle R_{\mathrm {m} }}$ is the magnetic resistance. This relationship is part of an electrical-magnetic analogy called the

used to model systems across multiple energy domains.

Reluctance

Main article:

Reluctance

Magnetic reluctance, or magnetic resistance, is analogous to

, akin to electrical resistance.

The total reluctance is equal to the ratio of the MMF in a passive magnetic circuit and the

phasors

)

The definition can be expressed as: $${\displaystyle {\mathcal {R}}={\frac {\mathcal {F}}{\Phi }},}$$ where ${\displaystyle {\mathcal {R}}}$ is the reluctance in ampere-turns per weber (a unit that is equivalent to turns per henry).

Magnetic flux always forms a closed loop, as described by

have low reluctance. The concentration of flux in low-reluctance materials forms strong temporary poles and causes mechanical forces that tend to move the materials towards regions of higher flux so it is always an attractive force(pull).

The inverse of reluctance is called permeance. $${\displaystyle {\mathcal {P}}={\frac {1}{\mathcal {R}}}.}$$

Its

, although the two concepts are distinct).

Permeability and conductivity

The reluctance of a magnetically uniform magnetic circuit element can be calculated as: $${\displaystyle {\mathcal {R}}={\frac {l}{\mu A}}.}$$ where

• l is the length of the element,
• ${\displaystyle \mu =\mu _{r}\mu _{0}}$ is the permeability of the material (${\displaystyle \mu _{\mathrm {r} }}$ is the relative permeability of the material (dimensionless), and ${\displaystyle \mu _{0}}$ is the permeability of free space), and
• A is the cross-sectional area of the circuit.

This is similar to the equation for electrical resistance in materials, with permeability being analogous to conductivity; the reciprocal of the permeability is known as magnetic reluctivity and is analogous to resistivity. Longer, thinner geometries with low permeabilities lead to higher reluctance. Low reluctance, like low resistance in electric circuits, is generally preferred.^{[citation needed]}

The following table summarizes the mathematical analogy between electrical circuit theory and magnetic circuit theory. This is mathematical analogy and not a physical one. Objects in the same row have the same mathematical role; the physics of the two theories are very different. For example, current is the flow of electrical charge, while magnetic flux is not the flow of any quantity.

Analogy between 'magnetic circuits' and electrical circuits

Magnetic				Electric
Name	Symbol	Units		Name	Symbol	Units
Magnetomotive force (MMF)	${\displaystyle {\mathcal {F}}=\int \mathbf {H} \cdot \mathrm {d} \mathbf {l} }$	ampere-turn		Electromotive force (EMF)	${\displaystyle {\mathcal {E}}=\int \mathbf {E} \cdot \mathrm {d} \mathbf {l} }$	volt
Magnetic field	H	meter		Electric field	E	meter = newton/coulomb
Magnetic flux	${\displaystyle \Phi }$	weber		Electric current	I	ampere
Hopkinson's law or Rowland's law	${\displaystyle {\mathcal {F}}=\Phi {\mathcal {R}}_{m}}$	ampere-turn		Ohm's law	${\displaystyle {\mathcal {E}}=IR}$
Reluctance	${\displaystyle {\mathcal {R}}_{\mathrm {m} }}$	1/henry		Electrical resistance	R	ohm
Permeance	${\displaystyle {\mathcal {P}}={\frac {1}{{\mathcal {R}}_{\mathrm {m} }}}}$	henry		Electric conductance	G = 1/R	1/ mho = siemens
Relation between B and H	${\displaystyle \mathbf {B} =\mu \mathbf {H} }$			Microscopic Ohm's law	${\displaystyle \mathbf {J} =\sigma \mathbf {E} }$
Magnetic flux density B	B	tesla		Current density	J	square meter
Permeability	μ	meter		Electrical conductivity	σ	meter

The resistance–reluctance model has limitations. Electric and magnetic circuits are only superficially similar because of the similarity between Hopkinson's law and Ohm's law. Magnetic circuits have significant differences that need to be taken into account in their construction:

• Electric currents represent the flow of particles (electrons) and carry power, part or all of which is dissipated as heat in resistances. Magnetic fields don't represent a "flow" of anything, and no power is dissipated in reluctances.
• The current in typical electric circuits is confined to the circuit, with very little "leakage". In typical magnetic circuits not all of the magnetic field is confined to the magnetic circuit because magnetic permeability also exists outside materials (see
  leakage flux
  " in the space outside the magnetic cores, which must be taken into account but is often difficult to calculate.
• Most importantly, magnetic circuits are
  remanent magnetism
  is left in ferromagnetic materials, creating flux with no MMF.

Magnetic circuits obey other laws that are similar to electrical circuit laws. For example, the total reluctance ${\displaystyle {\mathcal {R}}_{\mathrm {T} }}$ of reluctances ${\displaystyle {\mathcal {R}}_{1},\ {\mathcal {R}}_{2},\ \ldots }$ in series is: $${\displaystyle {\mathcal {R}}_{\mathrm {T} }={\mathcal {R}}_{1}+{\mathcal {R}}_{2}+\dotsm }$$

This also follows from

for adding resistances in series. Also, the sum of magnetic fluxes ${\displaystyle \Phi _{1},\ \Phi _{2},\ \ldots }$ into any node is always zero: $${\displaystyle \Phi _{1}+\Phi _{2}+\dotsm =0.}$$

This follows from Gauss's law and is analogous to Kirchhoff's current law for analyzing electrical circuits.

Together, the three laws above form a complete system for analysing magnetic circuits, in a manner similar to electric circuits. Comparing the two types of circuits shows that:

• The equivalent to resistance R is the reluctance ${\displaystyle {\mathcal {R}}_{\mathrm {m} }}$
• The equivalent to current I is the magnetic flux Φ
• The equivalent to voltage V is the magnetomotive Force F

Magnetic circuits can be solved for the flux in each branch by application of the magnetic equivalent of

, the excitation is the product of the current and the number of complete loops made and is measured in ampere-turns. Stated more generally: $${\displaystyle F=NI=\oint \mathbf {H} \cdot \mathrm {d} \mathbf {l} .}$$

By Stokes's theorem, the closed line integral of H·dl around a contour is equal to the open surface integral of curl H·dA across the surface bounded by the closed contour. Since, from Maxwell's equations, curl H = J, the closed line integral of H·dl evaluates to the total current passing through the surface. This is equal to the excitation, NI, which also measures current passing through the surface, thereby verifying that the net current flow through a surface is zero ampere-turns in a closed system that conserves energy.

More complex magnetic systems, where the flux is not confined to a simple loop, must be analysed from first principles by using Maxwell's equations.

• Air gaps can be created in the cores of certain transformers to reduce the effects of saturation. This increases the reluctance of the magnetic circuit, and enables it to store more energy before core saturation. This effect is used in the flyback transformers of cathode-ray tube video displays and in some types of switch-mode power supply.
• Variation of reluctance is the principle behind the reluctance motor (or the variable reluctance generator) and the Alexanderson alternator.
• soft iron
  to minimize the stray magnetic field.

Reluctance can also be applied to variable reluctance (magnetic)

.

• Magnetic capacitance
• Magnetic complex reluctance
• Tokamak

External links

• Magnetic–Electric Analogs by Dennis L. Feucht, Innovatia Laboratories (PDF) Archived July 17, 2012, at the Wayback Machine
• Interactive Java Tutorial on Magnetic Shunts National High Magnetic Field Laboratory