Magnetism: quantities, units and relationships

If you occasionally need to design a wound component, but do not deal with the science of magnetic fields on a daily basis, then you may become confused about what the many terms used in the data sheet for the core represent, how they are related and how you can use them to produce a practical inductor. (All right, I became confused. If you're not then congratulations!)

About fonts: if the character in brackets here [ × ] does not look like a multiplication sign then try setting your browser to use the Unicode character set (view:character set menu on Netscape 4). Also this character [ Φ ] is the Greek letter 'phi' in modern browsers.

See also ...
[Up sign Producing wound components] [ Air coils] [ Power loss in wound components] [The force produced by a magnetic field] [ Faraday's law]

Index to magnetic units

This set of web pages uses the system of units known as the SI (Système International).

Magnetic quantities in the SI
Quantity Quantity
coercivity Hc
effective area Ae
effective permeability µe
induced voltage E
inductance factor Al
magnetic flux greek letter phi
magnetomotive force Fm
permeability of vacuumµ0
reluctance Rm
Magnetic quantities in the SI
Quantity Quantity
core factor Σl/A
effective length le
flux linkage λ
inductance L
magnetic field strength H
magnetic flux density B
permeability µ
relative permeability µr
remnance Br

Magnetics in the CGS system

Other publications, particularly from the US, employ the cgs (centimetre, gram, second) system of units. If you have a quantity expressed in cgs units then multiply by k to find the equivalent in SI units.

Magnetics in the CGS system
Quantity CGS Unit k SI Unit
magnetomotive force gilberts 0.4π ampere-turns
magnetic field strength oersteds 79.6 amps per metre
magnetic flux density gauss 10-4 tesla
magnetic flux maxwells 10-8 weber

Converting formulae between the two systems requires care. You may also see flux being specified in 'lines' - this is synonymous with maxwells.

An Example Toroid Core

As a concrete example for the calculations throughout this page we consider the 'recommended' toroid, or ring core, used in this School. Manufacturers use toroids to derive material characteristics because there is no gap, even a residual one. Such tests are done using fully wound cores rather than just the two turns here; but, providing the permeability is high then the error will be small.

Toroid with two turns

Data for approved toroid
Parameter Symbol Value
Effective magnetic path length le 27.6×10-3 m
Effective core area Ae 19.4×10-6 m2
Relative permeability µr 2490
Inductance factor Al 2200 nH
saturation flux density Bsat 360 mT

Let's take a worked example to find the inductance for the winding shown with just two turns (N=2).

Σl/A = le / Ae = 27.6×10-3 / 19.4×10-6 = 1420   m-1

µ = µ0 × µr = 1.257×10-6 × 2490 = 3.13×10-3   Hm-1

Rm = (Σl/A) / µ = 1420 / 3.13×10-3 = 4.55×105   A-t Wb-1

Al = 109 / Rm = 109 / 4.55×105 = 2200   nH per turn2

L = Al × N2 = 2200 × 10-9 × 22 = 8.8 μH

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Coercivity in the SI
Quantity name coercivity
Quantity symbol Hc
Unit name amps per metre
A m-1
Fundamental units A m-1

Coercivity is the field strength which must be applied in order to reduce (or coerce) a remnant flux to zero. Materials with high coercivity (such as those used for permanant magnets) are sometimes called hard. Conversely, materials with low coercivity (such as those used for transformers) are called soft. See the section on hysteresis for more details.

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Core Factor

Core Factor in the SI
Quantity name core factor or
geometric core constant
Quantity symbol Σl/A
Unit name per metre
Fundamental units m-1

The idea of core factor is, apart from adding to the jargon :-( , to encapsulate in one figure the contribution to core reluctance made by the size and shape of the core. It is usually quoted in the data sheet but it is calculated as -

Σl/A = le / Ae   m-1
So for our example toroid we find -
Σl/A = 27.6×10-3 / 19.4×10-6 = 1420   m-1
If the core factor is specified in millimetres-1 then multiply by 1000 before using it in the formula for reluctance.

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Effective Area

Effective Area in the SI
Quantity name effective Area
Quantity symbol Ae
Unit name square metres
Fundamental units m2

The 'effective area' of a core represents the cross sectional area of one of its limbs. Usually this corresponds closely to the physical dimensions of the core but because flux may not be distributed completely evenly the manufacturer will specify a value for Ae which reflects this.

In the example toroid the area could be determined approximately as -

Ae = 6.3 × ((12.7 - 6.3) / 2) = 20.2   mm2
However, because the flux concentrates where the path length is shorter it is better to use the value stated by the manufacturer - 19.4 mm2. For the simple toroidal shape Ae is calculated as
Ae = h×ln2(R2/R1) / (1/R1-1/R2)   m2
This assumes square edges to the toroid; real ones are often rounded.

Important: effective area is usually quoted in millimetres squared. Many formulae in data books implicitly assume that a numerical value in mm2 be used. Other books, and these notes, assume metres squared.

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Effective Length

Effective Length in the SI
Quantity name effective length
Quantity symbol le
Unit name metres
Fundamental units m

The 'effective length' of a core is a measure of the distance which flux lines travel in making a complete circuit of it. Usually this corresponds closely to the physical dimensions of the core but because flux has a tendency to concentrate on the inside corners of the path the manufacturer will specify a value for le which reflects this.

In the toroid example the path length could be determined approximately as -

le = greek letter pi × (12.7 + 6.3) / 2 = 29.8 mm
However, because the flux concentrates where the path length is shorter it is better to use the value stated by the manufacturer - 27.6 mm. For a simple toroidal shape le is calculated as
le = 2pi×ln(R2/R1)/(1/R1-1/R2)   m
Another common core type, the EE, is shown in Fig: EEE. Path length in an EE core The red line represents the shortest path which a flux line could take to go round the core. The green line is the longest. Shown in blue is a path whose length is that of the short path plus four sectors whose radius is sufficient to take the path mid-way down the limbs.
le = 2(3.8+1.2)+pi((2.63-1.2)/2)   mm
This is all a bit approximate; but bear in mind that since manufacturing tolerances on permeability are often 25% there isn't much point in being more exact.

Important: effective length is usually quoted in millimetres. Many formulae in data books implicitly assume that a numerical value in mm be used. Other books, and these notes, assume metres.

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Magnetomotive Force

Magnetomotive Force in the SI (see also cgs)
Quantity name magnetomotive force
Quantity symbol Fm or ℑ
Unit name amperes
Fundamental units A

Quantity Unit Formula
Magnetomotive force amperes Fm = H × le
Electromotive force volts E = D (Electric field strength) × l (distance)
Duality with the Electric World

MMF can be thought of as the magnetic equivalent of electromotive force. Calculate MMF as the product of the current flowing in a coil and the number of turns it has-

Fm = I × N   ampere turns
The units of MMF are often stated as ampere turns (A-t) because of this. In the example toroid core-
Fm = 0.25 × 2 = 0.5   ampere turns
Don't confuse MMF with magnetic field strength. As an analogy think of the plates of a capacitor with a certain electromotive force (EMF) between them. How high the electric field strength is will depend on the distance between the plates. Similarly, the magnetic field strength in a transformer core depends not just on the MMF but also on the distance that the flux must travel round it.

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Magnetic Field Strength

Magnetic Field Strength in the SI (see also cgs)
Quantity name magnetic field strength
Quantity symbol H
Unit name amperes per metre
A m-1
Fundamental units A m-1

Quantity Unit Formula
Magnetic field strength amperes per metre H = Fm/le
Electric field strength volts per metre epsilon = E/d
Duality with the Electric World

Whenever current flows it is always accompanied by a magnetic field. The strength, or intensity, of this field is exactly proportional to the amount of current but inversely proportional to the distance from the conductor.

Magnetic field strength is analogous to electric field strength. Where an electric field is set up between two plates separated by a distance d and having a potential difference, E, between them the electric field is given by -

epsilon = E / d   V m-1
Similarly, magnetic field strength is -
H = Fm / le   A m-1
Where le is the length of the field line.

In the example the field strength is then -

H = 0.5 / 27.6×10-3 = 18.1   A m-1

Do not confuse magnetic field strength with flux density, B. This is closely related to field strength but depends also on the material within the field.

H = B/µ
Flux also emerges from a permanent magnet even when there are no wires about to impose a field.

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Magnetic Flux

Magnetic Flux in the SI (see also cgs)
Quantity name magnetic flux
Quantity symbol Φ
Unit name webers
Fundamental units kg m2 s-2 A-1

We talk of magnetism in terms of lines of force or flow or flux. Although the Latin fluxus, means 'flow' the English word is older and unrelated. Flux, then, is a measure of the number of these lines - the total amount of magnetism.

You can calculate flux as the time integral of the voltage on a winding divided by the number of turns, N.

greek letter phi = (1/N)Time integralV.dt   webers
This is one form of Faraday's law. If a constant voltage is applied for a time T then this boils down to -
greek letter phi = V × T / N   Wb
How much simpler can the maths get? Because of this relationship flux is sometimes specified as volt seconds.

Quantity Unit Formula
Magnetic flux volt seconds greek letter phi = V × T
Electric charge amp seconds (= coulombs) Q = I × T
Duality with the Electric World

Although as shown above flux corresponds in physical terms most closely to electric charge, you may find it easiest to envisage flux flowing round a core in the way that current flows round a circuit. When a given voltage is applied across a component with a known resistance then a specific current will flow. Similarly, application of a given magnetomotive force across a ferromagnetic component with a known reluctance results in a specific amount of magnetic flux -

greek letter phi = Fm / Rm   webers
Lastly, flux can also be derived by knowing the both the magnetic flux density and the area over which it applies:
greek letter phi = Ae×B   webers

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Magnetic Flux Density

Magnetic Magnetic Flux Density in the SI (see also cgs)
Quantity name Magnetic flux density
Quantity symbol B
Unit name teslas
Fundamental units kg s-2 A-1

Quantity Unit Formula
Magnetic flux density webers per metre2 B = greek letter phi/Area
Electric flux density coulombs per metre2 D = C/Area
Duality with the Electric World

Flux density is simply the total flux divided by the cross sectional area of the part through which it flows -

B = greek letter phi / Ae   teslas
Thus 1 weber per square metre = 1 tesla.
Flux density is related to field strength via the permeability
B = µ × H   teslas
So for the example core -
B = 3.13×10-3 × 18.1 = 0.0567   teslas

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Flux Linkage

Flux Linkage in the SI
Quantity name flux linkage
Quantity symbol λ
Unit name weber-turns
Fundamental units kg m2 s-2 A-1

In an ideal inductor the flux generated by one of its turns would be encircle all the other other turns. Real coils come close to this ideal when the cross sectional dimensions of the winding are small compared with its diameter, or if a high permeability core guides the flux right the way round. Example of imperfect
flux linkageIn longer air-core coils the situation is likely to be nearer to that shown in Fig.TFK: Here we see that the flux density decreases towards the ends of the coil as some flux takes a 'short cut' bypassing the outer turns. Let's assume that the current into the coil is 5 amperes and that each flux line represents 7 mWb.

The central three turns all 'link' four lines of flux: 28 mWb.
The two outer turns link just two lines of flux: 14 mWb.

We can calculate the total 'flux linkage' for the coil as:

λ = 3×28 + 2×14 = 112   mWb-t
The usefulness of this result is that it enables us to calculate the total self inductance of the coil, L:
L = λ/I = 112/5 = 22.4   mH
In general, where an ideal coil is assumed, you see expressions involving N×greek letter phi or N×dgreek letter phi/dt. For greater accuracy you substitute λ or dλ/dt.

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Magnetization Curves

A Typical Magnetization Curve

BH graph Any discussion of the magnetic properties of a material is likely to include the type of graph known as a magnetization or B-H curve. Manufacturers of a particular grade of ferrite material usually provide this curve because the shape reveals how the core material in any component made from it will respond to changes in applied field. The curves have magnetic field strength as the horizontal axis and the magnetic flux density as the vertical axis. The curve tells you the permeability at any point by the ratio of absolute value of flux density to that of field strength:
µ = B/H   Hm-1
This is not the same as the slope of a tangent to the curve, unless it is 'incremental permeability' which is specifically being refered to.

The figure above is an initial magnetisation curve because it starts from an unmagnetised sample and shows how the flux increases as the field strength is increased. You can identify four distinct regions in most such curves. These can be explained in terms of changes to the material's magnetic 'domains':

  1. Close to the origin a slow rise due to 'reversible growth'.
  2. A longer, fairly straight, stretch representing 'irreversible growth'.
  3. A slower rise representing 'rotation'.
  4. An almost flat region corresponding to µ0 - the core can't handle any more flux growth and has saturated.
The curve shown is approximately correct for the example core at 25 deg C. and tells us that the flux density must be kept below about 350mT to avoid saturation.

There are two other common types of magnetization curve: the hysteresis loop and the normal magnetization curve. A circuit you can use to plot magnetization is described here.

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Permeability in the SI
Quantity name permeability
Quantity symbol µ
Unit name henrys per metre
H m-1
Fundamental units kg m s-2 A-2

Quantity Unit Formula
Permeability henrys per metre µ = L/d
Permittivity farads per metre epsilon = C/d
Duality with the Electric World

Permeability is defined as the ratio of flux density to field strength.

µ = B / H   Hm-1
Permeability is determined by the type of material within the magnetic field. Think of permeability as a sort of 'resistance to magnetic flux'; just as those materials with high conductivity let electric current through easily so different materials allow flux through more easily than others. The analogy is not that good because for most materials (or no material at all, i.e. a vacuum - 'free space') the permeability is non-zero and is called µ0 -
µ0 = 4pi ×10-7 = 1.257×10-6   henrys per metre.
However, for iron, nickel, cobalt and manganese, or their compounds, the permeability may be significantly greater. These materials are classed as ferromagnetic. Using a ferromagnetic core will result in a higher value for µ. Iron has about 3.5 times the permeability of nickel.

It is quite unusual to see a straight or absolute permeability figure in data sheets. It's more common for permeability to be expressed as the ratio over µ0. The factor by which permeability increases above µ0 is called the relative permeability, µr.

µ = µ0 × µr   Hm-1

So for the toroid example permeability is then:

µ = 1.257×10-6 × 2490 = 3.13×10-3   Hm-1
Many authors simply say "permeability" and leave you to infer that they mean relative permeability. If a figure greater than 1.0 is quoted then you can be almost certain it is µr. If you use a core with a high value of permeability then fewer turns will be required to produce a coil with a given value of inductance.

Material µr Application
Ferrite U 60 8 UHF chokes
Ferrite M33 750 Resonant circuit RM cores
Nickel 2000 -
Ferrite N41 3000 Power circuits
Ferrite T38 10000 Broadband transformers
Silicon GO steel 40000 Dynamos, mains transformers
supermalloy 1000000 Recording heads
Approximate maximum relative permeabilties

Note that, unlike µ0, µr is not constant and changes with flux density. Also, if the temperature is increased from, say, 20 to 80 centigrade then a typical ferrite can suffer a 25% drop in permeability. This is a big problem in high-Q tuned circuits.

Another factor, with steel cores especially, is the microstructure, in particular grain orientation. Silicon steel is often made with the grains oriented along the laminations (rather than alowing them to lie randomly) giving increased µ. We call such material anisotropic.

Other types of permeability may be seen in data sheets. A common one is effective permeability, µe. This is often quoted when a core has an air gap which causes an apparent reduction in µ. Another is initial permeability, µi which is the relative permeability measured at low values of B (below 0.1T). The maximum value for µ in a material is frequently a factor 5 or more above its initial value.

Before you pull any value of µ from a data sheet ask yourself if it is appropriate for your material under the actual conditions under which you use it. Finally, if you do not know the permeabilty of your core then build a simple circuit to measure it.

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Inductance in the SI
Quantity name Inductance
Quantity symbol L
Unit name henrys
Fundamental units kg m2 s-2 A-2

Quantity Unit Formula
Inductance webers per amp L = greek letter phi/I
Capacitance coulombs per volt C = Q/V
Duality with the Electric World

Inductance is the rate of change of flux with current -

L = N × dgreek letter phi/dI   henrys.
Where I is the current flowing in the winding. If the material permeability is constant then the relation between flux and current is linear so:
L = N × greek letter phi/I   henrys.
In practice, where a high permeability core is used, inductance is usually determined knowing the number of turns in a coil, N, and the Al value specified by the manufacturer for the core -
L = Al × N2    nanohenrys
Inductance for the toroid example is then:
L = 2200 × 10-9 × 22 = 8.8 μH
If there is no ferromagnetic core so µr is 1.0 (the coil is 'air cored') then a variety of formulae are available to estimate the inductance. The correct one to use depends upon Most of these variants are described in early editions of Terman or successor publications. There are too many formulae to reproduce here. You can find them all in Grover.

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Inductance Factor

Inductance Factor in the SI
Quantity name inductance factor
Quantity symbol Al
Unit name nanohenrys
Fundamental units kg m2 s-2 A-2

Al is usually called the inductance factor. If you know the inductance factor then you can multiply by the square of the number of turns to find the inductance. In our example core Al = 2200, so the inductance is -

L = 2200 × 10-9 × 22 = 8800nH = 8.8 µ H
The core manufacturer may directly specify an Al value, but frequently you must derive it via the reluctance, Rm. The advantage of this is that only one set of data need be provided to cover a range of cores having identical dimensions but fabricated using materials having different permeabilities.
Al = 109 / Rm   nH per turn2
So, for our example toroid core -
Al = 109 / 4.55×105 = 2200   nH per turn2
If you have no data on the core at all then wind ten turns of wire onto it and measure the inductance (in henrys) using an inductance meter. The Al value will be 107 times this reading.

Al values are, like permeability, a non-linear function of flux. The quoted values are usually measured at low (<0.1mT) flux.

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Reluctance in the SI
Quantity name reluctance
Quantity symbol Rm or ℜ
Unit name ampere-turns per weber
A-t Wb-1
Fundamental units A2 s2 kg-1 m-2

Reluctance is the ratio of mmf to flux -

Rm = Fm / greek letter phi   ampere-turns per weber
Reluctance in a magnetic circuit corresponds to resistance in an electric circuit. It is proportional to the core factor, Σl/A, but inversely proportional to permeability -
Rm = (Σl/A) / µ   A-t Wb-1
Take care to use the absolute rather than the relative permeability here. So for the toroid example reluctance is then:
Rm = 1420 / 3.13×10-3 = 4.55×105   A-t Wb-1
Although it can be a useful concept when analysing series or parallel combinations of magnetic components reluctance is, like permeability, non-linear and must be used carefully.

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Remnance in the SI
Quantity name remnance
Quantity symbol Br
Unit name teslas
Fundamental units kg s-2 A-1

Remnance (or remanance) is the flux density which remains in a magnetic material when any externally applied field is removed (H = 0). For materials used in permanant magnets you usually need a high value of remnance. For transformers you need low remnance. See the section on hysteresis for more details.

Remnance is also what makes possible all magnetic recording technologies; including the hard disk drive on which this text was stored until you loaded it into your browser.
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Relationships between magnetic quantities

Flux, field strength, permeability, reluctance ..... it's easy to go into jargon overload. Snelling lists over 360 different symbol uses connected with ferromagnetics. There isn't even agreement about what to call some properties (I say remnance, you say remanance, he says retentivity). You will cope better if you can form a mental picture of the party that these names throw when they get together inside your transformer.

Analogy with electric quantities:

You may find it easier to obtain an intuitive grasp of the relationships between magnetic quantities by thinking in terms of 'magnetic circuits' with flux flowing round a core in a fashion analogous to current flowing round an electric circuit.

magnetomotive force electromotive force (voltage)
magnetic field strength electric field strength
permeability conductivity
magnetic flux current
magnetic flux density current density
reluctance resistance
Electric analogues

For example, if you have a transformer with a gapped core then imagine that the core and the gap form a series magnetic circuit with the same flux flowing through both reluctance components in an analogous fashion to a series electric circuit in which the same current flows through two resistors -

Fm = greek letter phi × (Rm_gap + Rm_core)   ampere-turns
V = I × (R1 + R2)   volts
There's an entire family of formulae which take similar forms in both the electric and magnetic worlds. Kraus lists most of them.

All analogies break down when pushed too far. This one falls rather quickly if you realise that curent flowing through a resistor dissipates energy while flux flowing through a reluctance does not. In fact you can ask whether flux is a real physical effect at all (in the way that electron flow is).

Sequence of operation

In transformer design you would normally like to deal in terms of the voltages on the windings. However, the key to understanding what happens in a transformer (or other wound component) is to realise that what the transformer really cares about is the current in the windings; and that everything follows on from that. If you can follow this five step sequence then building a mental image of a magnetic component becomes simpler. Remember, you put in a current and get back an induced voltage. In fact, if you can treat the permeability as being linear, then the constants N, le, µ and Ae can be lumped together into one constant for the winding which is called (surprise!) Inductance, L -
L = µ × Ae × N2 / le     henrys
I give the fundamental units for all the quantities in this equation; enabling thrill-seekers to make a dimensional analysis verifying that it is consistent. Right, so then our five step relationship between current and EMF boils down to:
E = L × dI/dt   volts

You may be about to complain that you know the EMF on your winding but don't know the current in it. The answer is that the process then works in reverse - the current will build up until the induced voltage is sufficient to oppose the applied voltage. You can find out more by looking at Faraday's law.

How do you take into account the presence of the secondary windings in a transformer? One way is to take the first four steps of the sequence above and apply them separately to each winding (whether primary or secondary). The arithmetic sum over all windings gives total core flux. From the time rate of change of flux you then have the induced voltage in each winding (since you also know the number of turns for each). There are less tedious methods of analysing transformer operation which you would probably do better using. But they are another story.

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Last modified: 2001 May 24th.