Magnetic circuit

Last updated
Magnetic field (green) induced by a current-carrying wire winding (red) in a magnetic circuit consisting of an iron core C forming a closed loop with two air gaps G in it. In an analogy to an electric circuit, the winding acts analogously to an electric battery, providing the magnetizing field
H
{\displaystyle H}
, the core pieces act like wires, and the gaps G act like resistors.
B - magnetic field in the core
BF - "fringing fields". In the gaps G the magnetic field lines "bulge" out, so the field strength is less than in the core: BF < B
BL - leakage flux; magnetic field lines which don't follow complete magnetic circuit
L - average length of the magnetic circuit. It is the sum of the length Lcore in the iron core pieces and the length Lgap in the air gaps G. Electromagnet with gap.svg
Magnetic field (green) induced by a current-carrying wire winding (red) in a magnetic circuit consisting of an iron core C forming a closed loop with two air gaps G in it. In an analogy to an electric circuit, the winding acts analogously to an electric battery, providing the magnetizing field , the core pieces act like wires, and the gaps G act like resistors.
B – magnetic field in the core
BF – "fringing fields". In the gaps G the magnetic field lines "bulge" out, so the field strength is less than in the core: BF < B
BLleakage flux; magnetic field lines which don't follow complete magnetic circuit
L – average length of the magnetic circuit. It is the sum of the length Lcore in the iron core pieces and the length Lgap in the air gaps G.

A magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The flux is usually generated by permanent magnets or electromagnets and confined to the path by magnetic cores consisting of ferromagnetic materials like iron, although there may be air gaps or other materials in the path. Magnetic circuits are employed to efficiently channel magnetic fields in many devices such as electric motors, generators, transformers, relays, lifting electromagnets, SQUIDs, galvanometers, and magnetic recording heads.

Contents

The relation between magnetic flux, magnetomotive force, and magnetic reluctance in an unsaturated magnetic circuit can be described by Hopkinson's law, which bears a superficial resemblance to Ohm's law in electrical circuits, resulting in a one-to-one correspondence between properties of a magnetic circuit and an analogous electric circuit. Using this concept the magnetic fields of complex devices such as transformers can be quickly solved using the methods and techniques developed for electrical circuits.

Some examples of magnetic circuits are:

Magnetomotive force (MMF)

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 around a closed loop is defined as:

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 ampere-turn (At), represented by a steady, direct electric current of one ampere flowing in a single-turn loop of electrically conducting material in a vacuum. The gilbert (Gb), established by the IEC in 1930, [1] is the CGS unit of magnetomotive force and is a slightly smaller unit than the ampere-turn. The unit is named after William Gilbert (1544–1603) English physician and natural philosopher.

[2]

The magnetomotive force can often be quickly calculated using Ampère's law. For example, the magnetomotive force of a long coil is:

where N is the number of turns and I is the current in the coil. In practice this equation is used for the MMF of real inductors with N being the winding number 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 flux through an element of area perpendicular to the direction of magnetic field is given by the product of the magnetic field and the area element. More generally, magnetic flux Φ is defined by a scalar product of the magnetic field and the area element vector. Quantitatively, the magnetic flux through a surface S is defined as the integral of the magnetic field over the area of the surface

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 SI unit of magnetic flux is the weber (in derived units: volt-seconds), and the unit of magnetic flux density (or "magnetic induction", B) is the weber per square meter, or tesla.

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.

Resistance–reluctance model

The resistance–reluctance model for magnetic circuits is a lumped-element model that makes electrical resistance analogous to magnetic reluctance.

Hopkinson's law

In electrical circuits, Ohm's law is an empirical relation between the EMF applied across an element and the current it generates through that element. It is written as:

where R is the electrical resistance of that material. There is a counterpart to Ohm's law used in magnetic circuits. This law is often called Hopkinson's law, after John Hopkinson, but was actually formulated earlier by Henry Augustus Rowland in 1873. [3] It states that [4] [5]

where is the magnetomotive force (MMF) across a magnetic element, is the magnetic flux through the magnetic element, and 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,

where is the magnetic resistance. This relationship is part of an electrical-magnetic analogy called the gyrator-capacitor model and is intended to overcome the drawbacks of the reluctance model. The gyrator-capacitor model is, in turn, part of a wider group of compatible analogies used to model systems across multiple energy domains.

Reluctance

Magnetic reluctance, or magnetic resistance, is analogous to resistance in an electrical circuit (although it does not dissipate magnetic energy). In likeness to the way an electric field causes an electric current to follow the path of least resistance, a magnetic field causes magnetic flux to follow the path of least magnetic reluctance. It is a scalar, extensive quantity, akin to electrical resistance.

The total reluctance is equal to the ratio of the MMF in a passive magnetic circuit and the magnetic flux in this circuit. In an AC field, the reluctance is the ratio of the amplitude values for a sinusoidal MMF and magnetic flux. (see phasors)

The definition can be expressed as:

where 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 Maxwell's equations, but the path of the loop depends on the reluctance of the surrounding materials. It is concentrated around the path of least reluctance. Air and vacuum have high reluctance, while easily magnetized materials such as soft iron 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 .

Its SI derived unit is the henry (the same as the unit of inductance, although the two concepts are distinct).

Permeability and conductivity

The reluctance of a magnetically uniform magnetic circuit element can be calculated as:

where

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 ]

Summary of analogy

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
MagneticElectric
NameSymbolUnitsNameSymbolUnits
Magnetomotive force (MMF) ampere-turn Electromotive force (EMF) volt
Magnetic field H ampere/meter Electric field E volt/meter = newton/coulomb
Magnetic flux weber Electric current I ampere
Hopkinson's law or Rowland's law ampere-turn Ohm's law
Reluctance 1/henry Electrical resistance R ohm
Permeance henry Electric conductance G = 1/R1/ohm = mho = siemens
Relation between B and HMicroscopic Ohm's law
Magnetic flux density BB tesla Current density J ampere/square meter
Permeability μ henry/meter Electrical conductivity σ siemens/meter

Limitations of the analogy

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:

Circuit laws

Magnetic circuit Magnetischer Kreis.svg
Magnetic circuit

Magnetic circuits obey other laws that are similar to electrical circuit laws. For example, the total reluctance of reluctances in series is:

This also follows from Ampère's law and is analogous to Kirchhoff's voltage law for adding resistances in series. Also, the sum of magnetic fluxes into any node is always zero:

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:

Magnetic circuits can be solved for the flux in each branch by application of the magnetic equivalent of Kirchhoff's voltage law (KVL) for pure source/resistance circuits. Specifically, whereas KVL states that the voltage excitation applied to a loop is equal to the sum of the voltage drops (resistance times current) around the loop, the magnetic analogue states that the magnetomotive force (achieved from ampere-turn excitation) is equal to the sum of MMF drops (product of flux and reluctance) across the rest of the loop. (If there are multiple loops, the current in each branch can be solved through a matrix equation—much as a matrix solution for mesh circuit branch currents is obtained in loop analysis—after which the individual branch currents are obtained by adding and/or subtracting the constituent loop currents as indicated by the adopted sign convention and loop orientations.) Per Ampère's law, the excitation is the product of the current and the number of complete loops made and is measured in ampere-turns. Stated more generally:

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.

Applications

Reluctance can also be applied to variable reluctance (magnetic) pickups.

See also

Related Research Articles

<span class="mw-page-title-main">Inductor</span> Passive two-terminal electrical component that stores energy in its magnetic field

An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a coil.

<span class="mw-page-title-main">Magnetic field</span> Spatial distribution of vectors allowing the calculation of the magnetic force on a test particle

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field. A permanent magnet's magnetic field pulls on ferromagnetic materials such as iron, and attracts or repels other magnets. In addition, a nonuniform magnetic field exerts minuscule forces on "nonmagnetic" materials by three other magnetic effects: paramagnetism, diamagnetism, and antiferromagnetism, although these forces are usually so small they can only be detected by laboratory equipment. Magnetic fields surround magnetized materials, electric currents, and electric fields varying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by a function assigning a vector to each point of space, called a vector field.

<span class="mw-page-title-main">Ohm's law</span> Law of electrical current and voltage

Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the three mathematical equations used to describe this relationship:

The oersted is the coherent derived unit of the auxiliary magnetic field H in the centimetre–gram–second system of units (CGS). It is equivalent to 1 dyne per maxwell.

<span class="mw-page-title-main">Electromagnetic induction</span> Production of voltage by a varying magnetic field

Electromagnetic or magnetic induction is the production of an electromotive force (emf) across an electrical conductor in a changing magnetic field.

<span class="mw-page-title-main">Electromotive force</span> Electrical action produced by a non-electrical source

In electromagnetism and electronics, electromotive force is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical transducers provide an emf by converting other forms of energy into electrical energy. Other electrical equipment also produce an emf, such as batteries, which convert chemical energy, and generators, which convert mechanical energy. This energy conversion is achieved by physical forces applying physical work on electric charges. However, electromotive force itself is not a physical force, and ISO/IEC standards have deprecated the term in favor of source voltage or source tension instead.

<span class="mw-page-title-main">Electromagnet</span> Magnet created with an electric current

An electromagnet is a type of magnet in which the magnetic field is produced by an electric current. Electromagnets usually consist of wire wound into a coil. A current through the wire creates a magnetic field which is concentrated in the hole in the center of the coil. The magnetic field disappears when the current is turned off. The wire turns are often wound around a magnetic core made from a ferromagnetic or ferrimagnetic material such as iron; the magnetic core concentrates the magnetic flux and makes a more powerful magnet.

<span class="mw-page-title-main">Inductance</span> Property of electrical conductors

Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The electric current produces a magnetic field around the conductor. The magnetic field strength depends on the magnitude of the electric current, and follows any changes in the magnitude of the current. From Faraday's law of induction, any change in magnetic field through a circuit induces an electromotive force (EMF) (voltage) in the conductors, a process known as electromagnetic induction. This induced voltage created by the changing current has the effect of opposing the change in current. This is stated by Lenz's law, and the voltage is called back EMF.

<span class="mw-page-title-main">Solenoid</span> Type of electromagnet formed by a coil of wire

A solenoid is a type of electromagnet formed by a helical coil of wire whose length is substantially greater than its diameter, which generates a controlled magnetic field. The coil can produce a uniform magnetic field in a volume of space when an electric current is passed through it.

<span class="mw-page-title-main">Ampère's circuital law</span> Concept in classical electromagnetism

In classical electromagnetism, Ampère's circuital law relates the circulation of a magnetic field around a closed loop to the electric current passing through the loop.

<span class="mw-page-title-main">Gyrator</span> Two-port non-reciprocal network element

A gyrator is a passive, linear, lossless, two-port electrical network element proposed in 1948 by Bernard D. H. Tellegen as a hypothetical fifth linear element after the resistor, capacitor, inductor and ideal transformer. Unlike the four conventional elements, the gyrator is non-reciprocal. Gyrators permit network realizations of two-(or-more)-port devices which cannot be realized with just the conventional four elements. In particular, gyrators make possible network realizations of isolators and circulators. Gyrators do not however change the range of one-port devices that can be realized. Although the gyrator was conceived as a fifth linear element, its adoption makes both the ideal transformer and either the capacitor or inductor redundant. Thus the number of necessary linear elements is in fact reduced to three. Circuits that function as gyrators can be built with transistors and op-amps using feedback.

<span class="mw-page-title-main">Magnetomotive force</span> Concept in physics

In physics, the magnetomotive force is a quantity appearing in the equation for the magnetic flux in a magnetic circuit, Hopkinson's law. It is the property of certain substances or phenomena that give rise to magnetic fields:

<span class="mw-page-title-main">Faraday's law of induction</span> Basic law of electromagnetism

Faraday's law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electric motors, generators and solenoids.

<span class="mw-page-title-main">Permeance</span>

Permeance, in general, is the degree to which a material admits a flow of matter or energy. Permeance is usually represented by a curly capital P: .

<span class="mw-page-title-main">Magnetostatics</span> Branch of physics about magnetism in systems with steady electric currents

Magnetostatics is the study of magnetic fields in systems where the currents are steady. It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast magnetic switching events that occur on time scales of nanoseconds or less. Magnetostatics is even a good approximation when the currents are not static – as long as the currents do not alternate rapidly. Magnetostatics is widely used in applications of micromagnetics such as models of magnetic storage devices as in computer memory.

<span class="mw-page-title-main">Magnetic reluctance</span> Resistance to magnetic flux

Magnetic reluctance, or magnetic resistance, is a concept used in the analysis of magnetic circuits. It is defined as the ratio of magnetomotive force (mmf) to magnetic flux. It represents the opposition to magnetic flux, and depends on the geometry and composition of an object.

<span class="mw-page-title-main">Magnetic complex reluctance</span> Measurement of a passive magnetic circuit

Magnetic complex reluctance is a measurement of a passive magnetic circuit dependent on sinusoidal magnetomotive force and sinusoidal magnetic flux, and this is determined by deriving the ratio of their complex effective amplitudes.[Ref. 1-3]

In electrical engineering the term flux linkage is used to define the interaction of a multi-turn inductor with the magnetic flux as described by the Faraday's law of induction. Since the contributions of all turns in the coil add up, in the over-simplified situation of the same flux passing through all the turns, the flux linkage is , where is the number of turns. The physical limitations of the coil and the configuration of the magnetic field make some flux to leak between the turns of the coil, forming the leakage flux and reducing the linkage. The flux linkage is measured in webers (Wb), like the flux itself.

<span class="mw-page-title-main">Gyrator–capacitor model</span> Model for magnetic circuits

The gyrator–capacitor model - sometimes also the capacitor-permeance model - is a lumped-element model for magnetic circuits, that can be used in place of the more common resistance–reluctance model. The model makes permeance elements analogous to electrical capacitance rather than electrical resistance. Windings are represented as gyrators, interfacing between the electrical circuit and the magnetic model.

An electropermanent magnet or EPM is a type of permanent magnet in which the external magnetic field can be switched on or off by a pulse of electric current in a wire winding around part of the magnet. The magnet consists of two sections, one of "hard" magnetic material and one of "soft" material. The direction of magnetization in the latter piece can be switched by a pulse of current in a wire winding about the former. When the magnetically soft and hard materials have opposing magnetizations, the magnet produces no net external field across its poles, while when their direction of magnetization is aligned the magnet produces an external magnetic field.

References

  1. "International Electrotechnical Commission".
  2. Matthew M. Radmanesh, The Gateway to Understanding: Electrons to Waves and Beyond, p. 539, AuthorHouse, 2005 ISBN   1418487406.
  3. Rowland H., Phil. Mag. (4), vol. 46, 1873, p. 140.
  4. "Magnetism (flash)".
  5. Tesche, Fredrick; Michel Ianoz; Torbjörn Karlsson (1997). EMC Analysis Methods and Computational Models. Wiley-IEEE. p. 513. ISBN   0-471-15573-X.