where: $E$ - stored energy (J), $B$ - flux density in the air gap (T), $V$ - volume of the air gap (m3), $μ_0$ - permeability of free space (H/m).

The operating principle of flyback transformers used in switched-mode power supplies is more akin to energy-storing inductors than to classical transformers. For this reason should be more correctly referred to as coupled inductor. 33) All energy must be first stored in the magnetic field in the first part of the cycle, and only passed to the secondary winding in the second part of the cycle. The energy storing capability is usually achieved by means of an air gap (or more precisely: a gap filled with non-magnetic material).

Flux fringing is caused by the fact that the reluctance of the concentrated air gap is much greater than that of the core. The flux tries to spread as wide as possible in order to minimise the drop of magnetomotive force across the air gap. As a result of flux fringing the total reluctance of the circuit is somewhat lower. This has several major effects.

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The effects depend on the magnetic path length of the core and the material permeability, and are more pronounced for physically smaller cores with high permeability. Even small contamination on the mating surfaces (dust, adhesive, etc.) can have significant impact on the measured values of permeability, inductance or the AL factor. Before a precise measurement is attempted, it is recommended to clean the mating phases and/or rub the two halves together in an elliptical motion.7)

Air gap is frequently used as means of tuning, or adjusting inductance to the required value. The variation can be done just once, for instance with a fixed shim introduced between parts of the core to create an air gap. But in some cases the variation must be carried out more frequently, or even cyclically. An important phenomena which necessitates tuning of an LC circuit, is to either achieve the resonance point (e.g. signal transmission through electromagnetic waves) or to be as far as possible from it (e.g. for filtering reasons).

Due to increased reluctance of an air gap the flux spreads into the surrounding medium causing the flux fringing effect. It is generally an unwanted phenomenon which usually increases proximity and eddy current loss in conductors located in the vicinity of the air gap.

In rotating machines the air gap is usually unwanted yet unavoidable due to the necessity of physical movement required between the stator and the rotor. The smallest practical air gap for industrial machines is around 0.2 mm.8) But with larger machines there are greater dimensional tolerances and the gap is sized accordingly to the diameter of the rotor to be at the order of 0.1% of the machine diameter.

The second effect is additional copper loss due to the fact that fringing flux “bulges away” from the air gap. Usually most of the core window is occupied by windings and if they are exposed to fast-changing fringing flux (e.g. in flyback transformers) this causes additional eddy current losses in the windings.53)

Rogowski coil is wound on a non-magnetic former - it is therefore an air-cored transformer, so that its whole magnetic circuit consist of an air gap. Because the core is non-magnetic then magnetic saturation does not occur and very high alternating currents (hundreds of kA) can be measured with high accuracy.

If operation with high currents is required then the air gap might be very large, so that the magnetic circuit is quite “open”. For instance, a common design for electronic chokes is to place a winding on a magnetic rod. The magnetic field lines must close through the surrounding air (outside of the winding), so the length of the air gap is comparable with the length of the rod.16)

An air gap is a practically unavoidable part of any magnetic circuit in which there is a relative movement between different parts (e.g. in motors, generators, relays, etc).

A magnetic circuit behaves like a “conductor” so that the magnetic field can be efficiently guided along desired path. If a high-permeability material is used then very little energy will be stored in the magnetic core. However, an air gap introduces a discontinuity and due to its low permeability stores significant amount of magnetic energy, as compared to the same volume of magnetic core. Using analogy to an electrical circuit - a good conductor such as thick copper wire (low resistance) does not dissipate or store much energy, but a resistor or other load (high impedance) will dissipate or store large amount of energy.

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For a simple magnetic circuit with a single air gap (see the first image at the top), for which the core is made out of high-permeability material such that $μ_{material} >> μ_0$, with the air gap itself and the flux density in the air gap being uniform, and if the flux fringing can be neglected, it can be derived that the stored energy is: 49)

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A common performance expected from an electromagnet is to generate magnetic field within a given volume of an air gap. This could be done for a number of tasks, for instance:

Any air gap in the magnetic core increases leakage inductance and stores additional energy, which needs to be cyclically transferred or dissipated. All these factors impact efficiency of energy transformation.

In some cases the currents are so high that it is very difficult or cost prohibitive to design the inductor with a magnetic core. In such case a so-called “air core” is used, where the windings are supported by a non-magnetic structure, and the whole magnetic circuit is effectively one big air gap.

The distribution of air gap can be also extended even further. There are magnetic materials, which are made from small particles (mostly based on powder iron, sendust or moly permalloy powder) bound together in such a way as to contain certain percentage of non-magnetic volume in them. The resultant effective permeability is much lower, but the air gap is uniformly distributed throughout the whole material.18) The fringing effect and leakage flux is greatly reduced, which is especially important for high-frequency applications. Such cores are usually more expensive (either in initial production or further coil winding) than alternative technologies with a single gap.

The B-H loop of a magnetic circuit is affected by the presence of an air gap. Permeability of non-magnetic material is low and therefore it requires greater values of $H$ to obtain the same value of $B$ as compared with magnetically soft materials. With the introduction of an air gap the B-H loop of a magnetic circuit gets "sheared" (slanted), hence the value of its slope proportional to the effective permeability is reduced. The amount of “shearing” is proportional to the length of the air gap - the larger the air gap the lower the slope. For air core coil (no magnetic material present) the B-H characteristics become by definition the same as for the non-magnetic material encircled by the winding (e.g. air).

Multiple approaches are used in current sensors. The importance and influence of the gap depends on the given technology.35) 36)37)

The air gap can take different form, shape and size depending on the type of magnetic circuit and its shape. In some circuits it might be actually an integral part ensuring correct performance of the device, but in other cases it should be as small as possible. Such requirements will be determined by the operating principle, performance, size, efficiency, and many other technological factors.

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Another common approach is the “closed loop sensing”. This technique also uses a magnetic core, and an air gap with a sensor.

The air gap is therefore “distributed” by introducing several smaller air gaps, whose total length is comparable to one large air gap. This reduces the fringing effect considerably.

Completely air-cored inductors can also be employed in radio frequencies, where values of inductance are low, and linearity and stability is more important than other factors.21)

In energy-storing inductors the inductance is related to the reluctance of the air gap. The fringing lowers the overall reluctance, so that the resulting inductance is somewhat higher. This needs to be taken into account so that the inductance value is appropriate for a given design. There are various empirical equations suggested in literature for calculating the correction of this effect.

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Induction heating employs a coil (or set of coils) which generates alternating magnetic field, usually at kHz frequency. The coil does not need to directly touch the heated object, because the electromagnetic field induces enough power loss (e.g. through eddy currents) in the object that it is possible even to achieve melting temperatures of metals.

This energy storing property is utilised for instance in energy storing inductors and flyback transformers, in which air gap in a pivotal design parameter. On the one hand, the air gap is used for storing the actual energy, but on the other it changes operating characteristics of the B-H curve and allows driving the inductor at higher currents hence higher magnetic field strength thus extending the range before magnetic saturation occurs.

In the powder cores the transition into magnetic saturation (when overexcited) is less sharp, which is also an important feature reducing severity of faults in electronic circuits using such solutions. However, such cores are usually more expensive and exhibit different magnetic loss (e.g. higher than soft ferrites), so magnetic design must take these factors into account.

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where: $F$ - factor by which the inductance is increased (unitless), $l_{gap}$ - length of the air gap (m), $A$ - cross-section area of the core (m2), $l_{window}$ - length of the inside (in the window) of the core leg in which the gap is present (m).

Split cores made out of ferrite are widely used in electronic transformers because they greatly simplify assembly, so that low-cost devices are possible. However, by definition, a split core comprises some small air gap, even if the mating faces are very carefully polished or lapped. Typical air gaps for such ferrite cores are as follows:6)

The cylindrical shapes of rotating machines can be manufactured with tighter tolerances than it is the case for instance for linear motors. As a consequence the linear machines must operate with larger air gaps, which might impact on their efficiency.12)

In classical transformers air gap is usually avoided. The role of transformer is to deliver the energy from the primary winding to the secondary winding instantaneously, without the need for energy storage.

Magnetic cores of high-frequency chokes can be made from powder cores. High-permeability magnetic particles are compressed into a core, but a certain amount of non-magnetic volume is left on purpose. This causes a very good distribution of the resulting air gap over the whole volume of such core. As a result the fringing flux greatly reduced (as compared to a core with a concentrated gap), yet a large amount of energy can be stored.

In relays the air gap is usually an integral part, as it facilitates the movement between the fixed parts (e.g. winding and magnetic core) and the active armature, which mechanically drives the main electrical contacts to be connected or disconnected.

In order to reduce the flux fringing effect and the losses associated with it, in some inductors the gap is distributed into many smaller gaps.17)

Air gaps are an integral part of gapped inductors. The gap reduces effective permeability of a given magnetic circuit and allows storing much greater energy before saturation is reached. Increasing the gap reduces the inductance, so the winding must have more turns to compensate accordingly.14)

Depending on application, air gap may be filled with a non-magnetic material such as gas, water, vacuum, plastic, wood etc. and not necessarily just with air.3)4)

If a transformer core is made of laminations, then they are cut and assembled in a way as to stagger the gaps, to minimise any localised effects.29) For large power transformers the laminations are cut in a specific way to accommodate such overlapping. Similarly for smaller E-I or U-I cores the assembly can be carried out such that the gaps are staggered every other lamination.

However, all such equations are only approximate, and usually work only under the assumption that the length of the air gap is much smaller than any of the dimensions of the core.

However, special designs (for instance used in protection) might include air gap to lower the effective permeability and widen the operating range of primary current (linear CTs, transactors), but also to reduce the remanence (anti-remanence CTs), tune capacitive voltage transformers, etc. 32)

Air gap, also airgap1) or air-gap2) - is a non-magnetic part of a magnetic circuit. It is usually connected magnetically in series with the rest of the circuit, so that a substantial part of the magnetic flux (or magnetic field) flows through the gap.

However, there are also special motors with oversized air gap required for non-magnetic reasons (electrical isolation, access to other parts, etc.)11)

LC resonance and tuning is very important for many electromagnetic circuits, ranging from millimetre-sized low-power high-frequency signal transmission19), to high-power devices which work with parts of electricity grid (several kilometres long), as is the case for instance for Peterson coil.20)

However, the air gap also increases leakage inductance, which generates an unwanted back EMF at the instant of switch off, and this voltage can be high enough to damage the switching transistor if not designed correctly. So the construction of a flyback transformer must take into account a compromise between the required energy storage and any voltage spikes. Rapidly changing currents and high-frequency magnetic field “leaking” from the air gaps might cause problems with electromagnetic compatibility (EMC). Low-energy supplies might not need any EMC measures, higher energy might require filters in form of bead ferrites, but at for the highest energies a conductive (flux band) or magnetic shields can be used. 34)

However, there is a compensating winding on the core and the sensor is used as a zero detector - for a condition where the magnetic field from the primary current is compensated by the current in the winding located on the core. The value of the compensating current is proportional and hence a measure of the primary current. Closed-loop technique offers better linearity and accuracy than the open loop, but it is more costly to implement.

In the simplest form a plunger is being pulled into the space inside the coil. However, as with relays, there are multiple different designs and approaches, including those with a rotating armature, whose construction becomes therefore close to electric motors.25)

Solenoids and linear actuators work similarly to relays. The magnetic circuit comprises yoke or core and a movable armature, usually called plunger.

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In high-power energy storing inductors the air gaps can be quite large. This would cause for the inductance to differ by unacceptable amount and also the losses would be too large.

The air gap lowers the total inductance of the primary winding and causes an increase in apparent power 27) through the increase of magnetising current.28)

A common approach used by sensor manufacturers is the “open loop sensing” principle. a magnetic core as means of concentrating the magnetic flux, which is forced to flow through the air gap. A sensor of flux density $B$ (or magnetic field strength $H$) is placed in the gap. Since the gap non-magnetic (and hence there is a linear response) the measured value of $B$ or $H$ can be related to the value of the primary current.

Another example is when the area of the air gap is scaled according to its length. For instance if the magnetic core cross-section is a rectangle then Kazimierczuk proposes to use the following calculation: 51)

Due to low permeability of the non-magnetic core the sensitivity does not allow to measure very low currents, and coils with “high sensitivity” can typically measure currents of around 300 mA even if amplification and filtering is employed. 39) 40) For this reason for low currents Rogowski coils cannot compete with ordinary current transformers.

The air gap also provides electrical insulation, so that the coil is not short-circuited by the heated object or flux concentrator. 43)

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In rotating machines it may lead to increased magnetic loss on the surface of the rotor due to increased space harmonics. Thin laminations are thus required to reduce the effect of eddy currents in the parts of the motor which are exposed to AC flux.10)

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For accurate measurements the Rogowski coil must fully enclose the current to be measured. But the coil must also open to enable the coil to be closed around a conductor. Therefore, there is a gap between the meeting ends, and this introduces a measurement error. There are various ways of dealing with this gap, for instance overwinding of the coil ends or positioning them in a specific way. 38)

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Current and voltage transformers of the common design follow the same principles as regular transformers - the air gap is to be minimised as it limits the measurement accuracy of both amplitude and phase.

In relays, the length of the gap is a compromise between the required mechanical force, mechanical movement, and the available excitation. There are numerous constructions which allow optimisation of these different factors, according to requirement of specific application22) 23) including such that the movable armature is magnetised.24)

The third effect is that the fringing flux enters the core perpendicularly to the normal flow of magnetic field. In soft ferrites this is not a problem. But in laminated cores this flux does not travel along the laminations, but enters them perpendicularly to their surface, resulting in a large value of normal component, inducing elevated eddy currents and thus additional iron loss. A distributed air gap is employed in order to reduce this effect (see next section).

Because of the level of current the coils are commonly water cooled from inside. Apart from the heated object, the coils often do not have any magnetic core. Popularity of the so-called flux concentrators 41) keeps growing, but applying them to already existing designs can change inductance of the coil, which might require re-tuning of the system for resonating power supplies. 42)

Conversely, if no air gap is present then the slope becomes as steep as possible, and the B-H loop will represent the closest approximation of the characteristic of the magnetic material (for a given shape of the magnetic circuit). For this reason international standards defining magnetic measurements specify procedures to ensure that the influence of non-magnetic parts of a magnetic circuit remain within negligible levels.

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This can be achieved for instance by careful polishing or lapping of the flat faces, in order to reduce the surface roughness and the amount of space between the magnetic surfaces. 5)

Trains using magnetic levitation technology are propelled by using the principle of linear motors. The air gap between the train suspension and the tracks can be as large as 13 mm for electromagnetic suspension (Germany) or 102 mm for electrodynamic suspension (Japan).13)

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