In electromagnetism, the magnetic susceptibility (Latin: susceptibilis, “receptive”; denoted χ) is a measure of how much a material will become magnetized in an applied magnetic field. Mathematically, it is the ratio of magnetization M (magnetic moment per unit volume) to the applied magnetizing field intensity H. This allows a simple classification of most materials’ response to an applied magnetic field into two categories: an alignment with the magnetic field, χ > 0, called paramagnetism, or an alignment against the field, χ < 0, called diamagnetism.

Magnetic susceptibility indicates whether a material is attracted into or repelled out of a magnetic field. Paramagnetic materials align with the applied field and are attracted to regions of greater magnetic field. Diamagnetic materials are anti-aligned and are pushed away, toward regions of lower magnetic fields. On top of the applied field, the magnetization of the material adds its own magnetic field, causing the field lines to concentrate in paramagnetism, or be excluded in diamagnetism.^[1] Quantitative measures of the magnetic susceptibility also provide insights into the structure of materials, providing insight into bonding and energy levels. Furthermore, it is widely used in geology for paleomagnetic studies and structural geology.^[2]

The magnetizability of materials comes from the atomic-level magnetic properties of the particles of which they are made. Usually, this is dominated by the magnetic moments of electrons. Electrons are present in all materials, but without any external magnetic field, the magnetic moments of the electrons are usually either paired up or random so that the overall magnetism is zero (the exception to this usual case is ferromagnetism). The fundamental reasons why the magnetic moments of the electrons line up or do not are very complex and cannot be explained by classical physics (see Bohr–van Leeuwen theorem). However, a useful simplification is to measure the magnetic susceptibility of a material and apply the macroscopic form of Maxwell’s equations. This allows classical physics to make useful predictions while avoiding the underlying quantum mechanical details.

Definition

Volume susceptibility

Magnetic susceptibility is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field. A related term is magnetizability, the proportion between magnetic moment and magnetic flux density.^[3] A closely related parameter is the permeability, which expresses the total magnetization of material and volume.

The volume magnetic susceptibility, represented by the symbol χ_v (often simply χ, sometimes χ_m – magnetic, to distinguish from the electric susceptibility), is defined in the International System of Units — in other systems there may be additional constants — by the following relationship:^[4]

Here

• M is the magnetization of the material (the magnetic dipole moment per unit volume), measured in amperes per meter, and
• H is the magnetic field strength, also measured in amperes per meter.

χ_v is therefore a dimensionless quantity.

Using SI units, the magnetic induction B is related to H by the relationship

where μ₀ is the vacuum permeability (see table of physical constants), and (1 + χ_v) is the relative permeability of the material. Thus the volume magnetic susceptibility χ_v and the magnetic permeability μ are related by the following formula:

Sometimes^[5] an auxiliary quantity called intensity of magnetization I (also referred to as magnetic polarisation J) and measured in teslas, is defined as

This allows an alternative description of all magnetization phenomena in terms of the quantities I and B, as opposed to the commonly used M and H.

Mass susceptibility and molar susceptibility

There are two other measures of susceptibility, the mass magnetic susceptibility (χ_mass or χ_g, sometimes χ_m), measured in m³/kg (SI) and the molar magnetic susceptibility (χ_mol) measured in m³/mol that are defined below, where ρ is the density in kg/m³ and M is molar mass in kg/mol:

;

.

In CGS units

Note that the definitions above are according to SI conventions. However, many tables of magnetic susceptibility give cgs values (more specifically emu-cgs, short for electromagnetic units, or Gaussian-cgs; both are the same in this context). These units rely on a different definition of the permeability of free space:^[6]

The dimensionless cgs value of volume susceptibility is multiplied by 4π to give the dimensionless SI volume susceptibility value:^[6]

For example, the cgs volume magnetic susceptibility of water at 20 °C is 7.19×10⁻⁷, which is 9.04×10⁻⁶ using the SI convention.

In physics it is common to see cgs mass susceptibility given in cm³/g or emu/g·Oe⁻¹, so to convert to SI volume susceptibility we use the conversion ^[7]

where ρ^cgs is the density given in g/cm³, or

.

The molar susceptibility is measured cm³/mol or emu/mol·Oe⁻¹ in cgs and is converted by considering the molar mass.

Paramagnetism and diamagnetism

If χ is positive, a material can be paramagnetic. In this case, the magnetic field in the material is strengthened by the induced magnetization. Alternatively, if χ is negative, the material is diamagnetic. In this case, the magnetic field in the material is weakened by the induced magnetization. Generally, nonmagnetic materials are said to be para- or diamagnetic because they do not possess permanent magnetization without external magnetic field. Ferromagnetic, ferrimagnetic, or antiferromagnetic materials possess permanent magnetization even without external magnetic field and do not have a well defined zero-field susceptibility.

Experimental measurement

Volume magnetic susceptibility is measured by the force change felt upon a substance when a magnetic field gradient is applied.^[8] Early measurements are made using the Gouy balance where a sample is hung between the poles of an electromagnet. The change in weight when the electromagnet is turned on is proportional to the susceptibility. Today, high-end measurement systems use a superconductive magnet. An alternative is to measure the force change on a strong compact magnet upon insertion of the sample. This system, widely used today, is called the Evans balance.^[9] For liquid samples, the susceptibility can be measured from the dependence of the NMR frequency of the sample on its shape or orientation.^{[10][11][12][13][14]} Another method using NMR techniques measures the magnetic field distortion around a sample immersed in water inside an MR scanner. This method is highly accurate for diamagnetic materials with susceptibilities similar to water.^[15]

Tensor susceptibility

The magnetic susceptibility of most crystals is not a scalar quantity. Magnetic response M is dependent upon the orientation of the sample and can occur in directions other than that of the applied field H. In these cases, volume susceptibility is defined as a tensor

}

where i and j refer to the directions (e.g., x and y in Cartesian coordinates) of the applied field and magnetization, respectively. The tensor is thus rank 2 (second order), dimension (3,3) describing the component of magnetization in the ith direction from the external field applied in the jth direction.

Differential susceptibility

In ferromagnetic crystals, the relationship between M and H is not linear. To accommodate this, a more general definition of differential susceptibility is used

where χ^d
_ij is a tensor derived from partial derivatives of components of M with respect to components of H. When the coercivity of the material parallel to an applied field is the smaller of the two, the differential susceptibility is a function of the applied field and self interactions, such as the magnetic anisotropy. When the material is not saturated, the effect will be nonlinear and dependent upon the domain wall configuration of the material.

Several experimental techniques allow for the measurement of the electronic properties of a material. An important effect in metals under strong magnetic fields, is the oscillation of the differential susceptibility as function of 1/H. This behaviour is known as the de Haas–van Alphen effect and relates the period of the susceptibility with the Fermi surface of the material.

In the frequency domain

When the magnetic susceptibility is measured in response to an AC magnetic field (i.e. a magnetic field that varies sinusoidally), this is called AC susceptibility. AC susceptibility (and the closely related “AC permeability”) are complex number quantities, and various phenomena, such as resonance, can be seen in AC susceptibility that cannot in constant-field (DC) susceptibility. In particular, when an AC field is applied perpendicular to the detection direction (called the “transverse susceptibility” regardless of the frequency), the effect has a peak at the ferromagnetic resonance frequency of the material with a given static applied field. Currently, this effect is called the microwave permeability or network ferromagnetic resonance in the literature. These results are sensitive to the domain wall configuration of the material and eddy currents.

In terms of ferromagnetic resonance, the effect of an AC-field applied along the direction of the magnetization is called parallel pumping.

Examples

Magnetic susceptibility of some materials

Material	Temp.	Pressure	Molar susc., χmol		Mass susc., χmass		Volume susc., χv		Molarmass,M	Density, {\displaystyle \rho }
	(°C)	(atm)	SI (m3/mol)	CGS (cm3/mol)	SI (m3/kg)	CGS (cm3/g)	SI	CGS (emu)	(10−3kg/mol = g/mol)	(103kg/m3 = g/cm3)
Helium[16]	20	1	−2.38×10−11	−1.89×10−6	−5.93×10−9	−4.72×10−7	−9.85×10−10	−7.84×10−11	4.0026	1.66×10−4
Xenon[16]	20	1	−5.71×10−10	−4.54×10−5	−4.35×10−9	−3.46×10−7	−2.37×10−8	−1.89×10−9	131.29	5.46×10−3
Oxygen[16]	20	0.209	+4.3×10−8	+3.42×10−3	+1.34×10−6	+1.07×10−4	+3.73×10−7	+2.97×10−8	31.99	2.78×10−4
Nitrogen[16]	20	0.781	−1.56×10−10	−1.24×10−5	−5.56×10−9	−4.43×10−7	−5.06×10−9	−4.03×10−10	28.01	9.10×10−4
Air (NTP)[17]	20	1					+3.6×10−7	+2.9×10−8	28.97	1.29×10−3
Water[18]	20	1	−1.631×10−10	−1.298×10−5	−9.051×10−9	−7.203×10−7	−9.035×10−6	−7.190×10−7	18.015	0.9982
Paraffin oil, 220–260cSt[15]	22	1			−1.01×10−8	−8.0×10−7	−8.8×10−6	−7.0×10−7		0.878
PMMA[15]	22	1			−7.61×10−9	−6.06×10−7	−9.06×10−6	−7.21×10−7		1.190
PVC[15]	22	1			−7.80×10−9	−6.21×10−7	−1.071×10−5	−8.52×10−7		1.372
Fused silica glass[15]	22	1			−5.12×10−9	−4.07×10−7	−1.128×10−5	−8.98×10−7		2.20
Diamond[19]	r.t.	1	−7.4×10−11	−5.9×10−6	−6.2×10−9	−4.9×10−7	−2.2×10−5	−1.7×10−6	12.01	3.513
Graphite[20] χ∥ (to c-axis)	r.t.	1	−7.5×10−11	−6.0×10−6	−6.3×10−9	−5.0×10−7	−1.4×10−5	−1.1×10−6	12.01	2.267
Graphite[20] χ∥	r.t.	1	−3.2×10−9	−2.6×10−4	−2.7×10−7	−2.2×10−5	−6.1×10−4	−4.9×10−5	12.01	2.267
Graphite[20] χ∥	−173	1	−4.4×10−9	−3.5×10−4	−3.6×10−7	−2.9×10−5	−8.3×10−4	−6.6×10−5	12.01	2.267
Aluminium[21]		1	+2.2×10−10	+1.7×10−5	+7.9×10−9	+6.3×10−7	+2.2×10−5	+1.75×10−6	26.98	2.70
Silver[22]	961	1					−2.31×10−5	−1.84×10−6	107.87
Bismuth[23]	20	1	−3.55×10−9	−2.82×10−4	−1.70×10−8	−1.35×10−6	−1.66×10−4	−1.32×10−5	208.98	9.78
Copper[17]	20	1			−1.0785×10−9		−9.63×10−6	−7.66×10−7	63.546	8.92
Nickel[17]	20	1					600	48	58.69	8.9
Iron[17]	20	1					200000	15900	55.847	7.874

Sources of confusion in published data

The CRC Handbook of Chemistry and Physics has one of the only published magnetic susceptibility tables. Some of the data (e.g., for aluminium, bismuth, and diamond) is listed as cgs, which has caused confusion to some readers. “cgs” is an abbreviation of centimeters–grams–seconds; it represents the form of the units, but cgs does not specify units. Correct units of magnetic susceptibility in cgs is cm³/mol or cm³/g. Molar susceptibility and mass susceptibility are both listed in the CRC. Some table have listed magnetic susceptibility of diamagnets as positives. It is important to check the header of the table for the correct units and sign of magnetic susceptibility readings.

Application in the geosciences

Magnetism is a useful parameter to describe and analyze rocks. Additionally, the anisotropy of magnetic susceptibility (AMS) within a sample determines parameters as directions of paleocurrents, maturity of paleosol, flow direction of magma injection, tectonic strain, etc.^[2] It is a non-destructive tool, which quantifies the average alignment and orientation of magnetic particles within a sample.^[24]

There are two approaches to measuring paramagnetism that seem to be common. One is to use a balance to measure the slight attraction to a magnet – put sample in a balance, apply magnetic field, look for difference in weight of sample using a Gouy balance or use a torsion balance to observe the attraction in a horizontal plane which takes out the static weight of the sample.

The trouble with these two is the attraction due to paramagnetism is weak compared to the weight of the sample – these are lab bench instruments and the electromagnet consumes a lot of power. Although taking samples of soil is easy enough to bring back to the lab, one really shouldn’t be taking a hammer and chisel to ancient monuments to get a sample for a Gouy balance 😉

The other way of measuring volume magnetic susceptibility is to stick the sample into a coil and measure the inductance – with a different configuration  of the coil as a search coil it can be used to measure susceptibility at the rockface.

For a solenoid

L = µ₀µ_rN²A/l

L being the inductance,N the number of turns, A is the cross-sectional area and little l being the length – all of which are constants in any sensor

µ₀ being the permeability of free space, µ₀₌4π×10^-7 N A^-2

µ_r is the effective relative permeability of the sample such that

µ_r= χ_m+1 where

χ_m is the effective magnetic susceptibility of the sample

Bartington Instruments seem to be the go-to guys for magnetic susceptibility instrumentation. From them I learned that one shouldn’t use too high a frequency, which is a bear, My initial prototype ran at 52kHz which is ten to a hundred times too high.

HOW DO YOU MEASURE THE INDUCTANCE OF A COIL?

You resonate it with a capacitance and observe the frequency. It’s easy to measure frequency. The obvious way is that of time immemorial – I used a Colpitts oscillator of grid dip oscillator fame

and measured frequency with a frequency counter via a high impedance scope probe (10M/10pf) on the collector of the transistor. A respectably clean sine wave is to be seen

The first thing to note is this is not a big effect – 52,473 Hz unloaded as opposed to 52,441 with the sample. The sensor is a jam jar wound with enamelled wire

From the resonant frequency

f=1/(2π√(LC))

from which I infer the coil was about 0.56mH. Through a whole load of manipulating the formula for inductance

 

I can derive that if L1 is the inductance of the empty jar and L2 is the inductance of the jar with sample then the effective magnetic susceptibility of the sample χ_m is (L2-L1)/L1, which I then have to divide by 4π to convert from SI to CGS is about 97 µCGS¹ Phil Callahan would not approve…

All volcanic soil & rock is paramagnetic, (from 200 to 2,000 µCGS). According to Dr. Callahan’s research, a soil magnetic susceptibility reading of 0 – 100 µCGS would be poor; 100 – 300 µCGS good; 300 – 800 µCGS very good; & 800 -1200 µCGS above excellent. This force can be added to soil, where it has eroded away, by spreading ground-up paramagnetic rock (basalt, granite, etc.) into the soil.

Philip Callahan from the Pike Agri-Lab website

This generally squares with this chart I pinched from Bartington’s manual for the MS2 susceptibility system–

100 will take you to the granites and metamorphic rocks and 800 gets you about halfway to the left hand side.

There again, my sample takes up less than a quarter of the volume of the solenoid I guess, so if it filled the jar it would creep into very good category. That shows one area that needs thought – filling the sample vessel. A look at Bartington’s susceptibility product range of sensors is instructive – it shows me I want something like the MS2F point probe and maybe a search-coil-like MS2D surface scanning probe eventually. Bartington are good enough to tell me the frequency ranges  – the MS2F is 580Hz and the MS2D loop is 958Hz – presumably there are issues of getting the inductance high enough for the loop, hence going up in frequency. Or maybe the higher frequency suits the application better – Bartington seem to know their stuff.

ROOM FOR IMPROVEMENT

• there’s over 20V across the tank circuit, so EMC could be an issue[ref]It’s probably not that bad, the wavelength is in the order of 5km so most of the EMC will be due to the near field, it’s not going to propagate that much[/ref]. This will be even less if I go down in frequency
• The Q of the tank coil is damped by the circuit – I feel bad about the 470 ohm emitter resistor chucked across half the tapped capacitance which has an impedance of -92j at the operating frequency. I will lose sensitivity and pick up long-term drift because of that, though since this is a differential measurement the latter is not such a big deal.
• Form factor – I will usually be taking the sensor to the rock, not the other way round. There’s a sort of assumption that the rock will be bigger than the sensor.
• My frequency is way too high I am an order of magnitude off  – Bartington use 4.65kHz and 465 Hz in their MS2product, and indicate the lower frequency is more accurate.

The trouble with dropping frequency is that this is looking for a change at the 5 digit level – 1 part in 10,000. That’s two seconds to get a reading at 4.65 kHz (4650/10000) which is okay, but 20s at the lower frequency.

The obvious solution is to either use a frequency multiplier chain before the frequency measurement, or to put a PLL after it with a ÷N counter in the loop. And put a whole load of turns on the coil to get the inductance up

Magnetic properties of materials

An introduction for the designer of electrical wound components to the part played by materials within the magnetic field, and a summary of the related terminology.

• The scope of this page
• Magnetization curves
• Diamagnetic and paramagnetic materials
• Ferromagnetic materials
• Ferrimagnetic materials
• Saturation
• Materials classified
• Permeability
• Relative permeability
• Other variant forms of permeability and related quantities –
  • Initial permeability
  • Effective permeability
  • Permeability of a vacuum
  • Susceptibility
  • Mass susceptibility
• Terminology for intrinsic fields within materials –
  • Magnetic moment
  • Magnetization
  • Intensity of magnetization
  • Magnetic polarization

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 …
[ ↑ Producing wound components] [ Air coils] [ Power loss in wound components] [The force produced by a magnetic field] [ Faraday’s law] [A guide to unit systems]

The scope of this page

An entire sub-branch of physics is devoted to the study of the effects produced within various materials by the application of a magnetic field. These web pages make no attempt to cover the subject fully, and if you wish to explore it in greater depth then you should consult a text such as Jiles. What can be said here is that if you are restricted to just one parameter to describe this complexity then permeability is the one to choose. Most inductor calculations make use of it, or one of its multitudinous variants.

Emphasis goes on the aspects of practical importance in the design of wound components.

Magnetization Curves

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. Various methods are used to produce B-H curves, including one which you can easily replicate. Figure MPA shows how the B-H curve varies according to the type of material within the field.

The ‘curves’ here are all straight lines and have magnetic field strength as the horizontal axis and the magnetic flux density as the vertical axis. Negative values of H aren’t shown but the graphs are symmetrical about the vertical axis.

Fig. MPA a) is the curve in the absence of any material: a vacuum. The gradient of the curve is 4π.10^-7 which corresponds to the fundamental physical constant μ₀. More on this later. Of greater interest is to see how placing a specimen of some material in the field affects this gradient. 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.

Diamagnetic and paramagnetic materials

Imagine a hydrogen atom in which a nucleus with a single stationary and positively charged proton is orbited by a negatively charged electron. Can we view that electron in orbit as a sort of current loop? The answer is yes, and you might then think that hydrogen would have a strong magnetic moment. In fact ordinary hydrogen gas is only very weakly magnetic. Recall that each hydrogen atom is not isolated but is bonded to one other to form a molecule, giving the formula H₂ – because that has a lower chemical energy (for H by a whopping 218 kJ mol^-1) than two isolated atoms. It is not a coincidence that in these molecules the angular momentum of one electron is opposite in direction to that of its neighbour, leaving the molecule as a whole with little by way of magnetic moment. This behaviour is typical of many substances which are then said to lack a permanent magnetic moment.

When a molecule is subjected to a magnetic field those electrons in orbit planes at a right angle to the field will change their momentum (very slightly). This is predicted by Faraday’s Law which tells us that as the field is increased there will be a an induced E-field which the electrons (being charged particles) will experience as a force. This means that the individual magnetic moments no longer cancel completely and the molecule then acquires an induced magnetic moment. This behaviour, whereby the induced moment is opposite to the applied field, is present in all materials and is called diamagnetism. Hydrogen, ammonia, bismuth, copper, graphite and other diamagnetic substances, are repelled by a nearby magnet (although the effect is extremely feeble). Think of it as a manifestation of Lenz’s law. Diamagnetic materials are those whose atoms have only paired electrons.

In other molecules, however, such as oxygen, where there are unpaired electrons, the cancellation of magnetic moments belonging to the electrons is incomplete. An O₂ molecule has a net or permanent magnetic moment even in the absence of an externally applied field. If an external magnetic field is applied then the electron orbits are still altered in the same manner as the diamagnets but the permanent moment is usually a more powerful influence. The ‘poles’ of the molecule tend to line up parallel with the field and reinforce it. Such molecules, with permanent magnetic moments are called paramagnetic.

Although paramagnetic substances like oxygen, tin, aluminium and copper sulphate are attracted to a magnet the effect is almost as feeble as diamagnetism. The reason is that the permanent moments are continually knocked out of alignment with the field by thermal vibration, at room temperatures anyway (liquid oxygen at -183 °C can be pulled about by a strong magnet).

Particular materials where the magnetic moment of each atom can be made to favour one direction are said to be magnetizable. The extent to which this happens is called the magnetization. Fig. MPA b) above is the magnetization curve for diamagnetic materials. In diamagnetic substances the flux grows slightly more slowly with the field than it does in a vacuum. The decrease in gradient is greatly exaggerated in the figure – in practice the drop is usually less than one part in 6,000.

Fig. MPA c) is the curve for paramagnetic materials. Flux growth in this case is again linear (at moderate values of H) but slightly faster than in a vacuum. Again, the increase for most substances is very slight.

Although neither diamagnetic nor paramagnetic materials are technologically important (geophysical surveying is one exception), they are much studied by physicists, and the terminology of magnetics is enriched thereby. A short QuickTime movie (388 KB) demonstrates diamagnetism.

Ferromagnetic materials

The most important class of magnetic materials is the ferromagnets: iron, nickel, cobalt and manganese, or their compounds (and a few more exotic ones as well). The magnetization curve looks very different to that of a diamagnetic or paramagnetic material. We might note in passing that although pure manganese is not ferromagnetic the name of that element shares a common root with magnetism: the Greek mágnes lithos – “stone from Magnesia” (now Manisa in Turkey).

Figure MPB above shows a typical curve for iron. It’s important to realize that the magnetization curves for ferromagnetic materials are all strongly dependant upon purity, heat treatment and other factors. However, two features of this curve are immediately apparent: it really is curved rather than straight (as with non-ferromagnets) and also that the vertical scale is now in teslas (rather than milliteslas as with Figure MPA).

Figure MPB is a normal magnetization curve because it starts from an unmagnetized sample and shows how the flux density 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 paramagnetic behaviour and then μ₀ – the core can’t handle any more flux growth and has saturated.

At an atomic level ferromagnetism is explained by a tendency for neighbouring atomic magnetic moments to become locked in parallel with their neighbours. This is only possible at temperatures below the curie point, above which thermal disordering causes a sharp drop in permeability and degeneration into paramagnetism. Ferromagnetism is distinguished from paramagnetism by more than just permeability because it also has the important properties of remnance and coercivity.

Ferrimagnetic materials

Almost every item of electronic equipment produced today contains some ferrimagnetic material: loudspeakers, motors, deflection yokes, interference suppressors, antenna rods, proximity sensors, recording heads, transformers and inductors are frequently based on ferrites. The market is vast.

What properties make ferrimagnets so ubiquitous? They possess permeability to rival most ferromagnets but their eddy current losses are far lower because of the material’s greater electrical resistivity. Also it is practicable (if not straightforward) to fabricate different shapes by pressing or extruding – both low cost techniques.

What is the composition of ferrimagnetic materials? They are, in general, oxides of iron combined with one or more of the transition metals such as manganese, nickel or zinc, e.g. MnFe₂O₄. Permanent ferrimagnets often include barium. The raw material is turned into a powder which is then fired in a kiln or sintered to produce a dark gray, hard, brittle ceramic material having a cubic crystalline structure.

At an atomic level the magnetic properties depend upon interaction between the electrons associated with the metal ions. Neighbouring atomic magnetic moments become locked in anti-parallel with their neighbours (which contrasts with the ferromagnets). However, the magnetic moments in one direction are weaker than the moments in the opposite direction leading to an overall magnetic moment.

Saturation

Saturation is a limitation occurring in inductors having a ferromagnetic or ferrimagnetic core. Initially, as current is increased the flux increases in proportion to it (see figure MPB). At some point, however, further increases in current lead to progressively smaller increases in flux. Eventually, the core can make no further contribution to flux growth and any increase thereafter is limited to that provided by μ₀ – perhaps three orders of magnitude smaller. Iron saturates at about 1.6 T while ferrites will normally saturate between about 200 mT and 500 mT.

It is usually essential to avoid reaching saturation since it is accompanied by a drop in inductance. In many circuits the rate at which current in the coil increases is inversely proportional to inductance (I = V * T / L). Any drop in inductance therefore causes the current to rise faster, increasing the field strength and so the core is driven even further into saturation.

Core manufacturers normally specify the saturation flux density for the particular material used. You can also measure saturation using a simple circuit. There are two methods by which you can calculate flux if you know the number of turns and either –

1. The current, the length of the magnetic path and the B-H characteristics of the material.
2. The voltage waveform on a winding and the cross sectional area of the core – see Faraday’s Law.

Although saturation is mostly a risk in high power circuits it is still a possibility in ‘small signal’ applications having many turns on an ungapped core and a DC bias (such as the collector current of a transistor).

If you find that saturation is likely then you might –

• Run the inductor at a lower current
• Use a larger core
• Alter the number of turns
• Use a core with a lower permeability
• Use a core with an air gap

or some combination thereof – but you’ll need to re-calculate the design in any case.

Materials classification

Table MPJ categorizes (in a simplified fashion) each class assigned to a material according to its behaviour in a field.

Permeability

Although, as suggested above, magnetic permeability is related in physical terms most closely to electric permittivity, it is probably easier to think of permeability as representing ‘conductivity for magnetic flux‘; just as those materials with high electrical conductivity let electric current through easily so materials with high permeabilities allow magnetic flux through more easily than others. Materials with high permeabilities include iron and the other ferromagnetic materials. Most plastics, wood, non ferrous metals, air and other fluids have permeabilities very much lower: μ₀.

Just as electrical conductivity is defined as the ratio of the current density to the electric field strength, so the magnetic permeability, μ, of a particular material is defined as the ratio of flux density to magnetic field strength –

This information is most easily obtained from the magnetization curve. Figure MPC shows the permeability (in black) derived from the magnetization curve (in colour) using equation MPD. Note carefully that permeability so defined is not the same as the slope of a tangent to the B-H curve except at the peak (around 80 A m^-1 in this case). The latter is called differential permeability, μ′ = dB/dH.

In ferromagnetic materials the hysteresis phenomenon means that if the field strength is increasing then the flux density is less than when the field strength is decreasing. This means that the permeability must also be lower during ‘charge up’ than it is during ‘relaxation’, even for the same value of H. In the extreme case of a permanent magnet the permeability within it will be negative. There is an analogy here with electric cells, since they may be said to have ‘negative resistance’.

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. You can understand why by remembering that inductance is the ratio of flux to current. For a given core B is proportional to flux and H is proportional to the current so that inductance is also proportional to μ: the ratio of B to H.

Unlike electrical conductivity, permeability is often a highly non-linear quantity. Most coil design formulæ, however, pretend that μ is a linear quantity. If you were working at a peak value of H of 100 A m^-1, for example, then you might take an average value for μ of about 0.006 H m^-1. This is all very approximate, but you must accept inaccuracy if you insist on treating a non-linear quantity as though it was actually linear.

This form of permeability, where μ is written without a subscript, is known in SI parlance as absolute permeability. It is seldom quoted in engineering texts. Instead a variant is used called relative permeability described next.

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Relative permeability

Relative permeability is a very frequently used parameter. It is a variation upon ‘straight’ or absolute permeability, μ, but is more useful to you because it makes clearer how the presence of a particular material affects the relationship between flux density and field strength. The term ‘relative’ arises because this permeability is defined in relation to the permeability of a vacuum, μ₀

For example, if you use a material for which μ_r = 3 then you know that the flux density will be three times as great as it would be if we just applied the same field strength to a vacuum. This is simply a more user friendly way of saying that μ = 3.77×10^-6 H m^-1. Note that because μ_r is a dimensionless ratio that there are no units associated with it.

Many authors simply say “permeability” and leave you to infer that they mean relative permeability. In the CGS system of units these are one and the same thing really. If a figure greater than 1.0 is quoted then you can be almost certain it is μ_r.

Note that, unlike μ₀, μ_r 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 sheet is often made by cold rolling to orient the grains along the laminations (rather than allowing them to lie randomly) giving increased μ. We call such material anisotropic.

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 permeability of your core then build a simple circuit to measure it.

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Variant forms of permeability and related quantities

Initial permeability

Initial permeability describes the relative permeability of a material at low values of B (below 0.1T). The maximum value for μ in a material is frequently a factor of between 2 and 5 or more above its initial value.

Low flux has the advantage that every ferrite can be measured at that density without risk of saturation. This consistency means that comparison between different ferrites is easy. Also, if you measure the inductance with a normal component bridge then you are doing so with respect to the initial permeability.

* Although initial permeability is usually relative to μ₀, you may see μ_i as an absolute permeability.

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Effective permeability

Effective permeability is seen in some data sheets for cores which have air gaps. Coil calculations are easier because you can simply ignore the gap by pretending that you are using a material whose permeability is lower than the material you actually have.

* Effective permeability is usually relative to μ₀.

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Permeability of a vacuum in the SI

The permeability of a vacuum has a finite value – about 1.257×10^-6   H m^-1 – and in the SI system (unlike the cgs system) is denoted by the symbol μ₀. Note that this value is constant with field strength and temperature. Contrast this with the situation in ferromagnetic materials where μ is strongly dependant upon both. Also, for practical purposes, most non-ferromagnetic substances (such as wood, plastic, glass, bone, copper aluminium, air and water) have a permeability almost equal to μ₀; that is, their relative permeability is 1.0.

In Fig. MPZ you see, in cross section, two long, straight conductors spaced one metre apart in a vacuum. Both carry one ampere. The field strength due to the current in conductor A at a distance of one metre may be found, using Ampère’s Law –

where I is the current in conductor A and d is the path length around the circular field line. We know, from the definition of the ampere, that the force on conductor C is 2×10^-7 newtons per metre of its length. However, flux density, B, is also defined in terms of the force F, in newtons, exerted on a conductor of unit length and carrying unit current –

Since we now know both B and H at a distance of 1 metre from A we calculate the permeability of a vacuum as –

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Susceptibility

Although susceptibility is seldom directly important to the designer of wound components it is used in most textbooks which explain the theory of magnetism. When you work with non-ferromagnetic substances the permeability is so close to μ₀ that characterizing them by μ is inconvenient. Instead use the magnetic susceptibility, χ – via the permeability

In paramagnetic and diamagnetic materials the susceptibility is given by

Susceptibilities of some other substances are given in table MPS where the paramagnetic substances have positive susceptibilities and the diamagnetic substances have negative susceptibilities. The susceptibility of a vacuum is then zero.

Susceptibility is a strong contender for the title of ‘most confusing quantity in all science’. There are five reasons for this:

1. The counterpart to permeability in electrostatics has a distinct name: permittivity. Unfortunately, the counterpart to susceptibility in electrostatics has the same name. However, the electrostatic susceptibility should be given the symbol χ_e. Susceptance has nothing to do with susceptibility.
2. There are variant forms of susceptibility, the main two of which are listed below. Authors do not always explicitly state which variant is being used and, worse still, there is incomplete agreement about the names and symbols of each variant. The symbol χ_m is somewhat overloaded: magnetic susceptibility, mass susceptibility, or molar susceptibility? Take your pick.
3. Most reference books, and many instruments, still present susceptibility figures in CGS units. Often, the units are not made explicit and you are left to deduce them from the context or the values themselves. The procedures for converting to SI are not obvious.
4. Some quite prestigious publications incorrectly abbreviate the units to ‘per gram’ or ‘per kilogram’. 🙁
5. Measurement of susceptibility is notoriously difficult. The slightest whiff of contamination by iron in the sample will send the experimental results off into the twilight. Published figures frequently show differences of 5%; and 50% is not rare.

The international symbol for susceptibility of the ordinary (‘volumetric’) kind is simply χ without any subscript. ISO suggests χ_m to distinguish magnetic susceptibility from electric susceptibility but this may risk confusion with mass or molar susceptibility. Some writers have used χ_v to indicate volumetric susceptibility. Although electromagnetism is already up to its ears in subscript soup, this seems a good solution.

where ρ is the density of the substance in kg m^-3 and W_a is the molar mass in kg mol^-1.

To appreciate the difference for each variant think of it as being a separate way to get the total magnetic moment for a magnetic field strength of one amp per metre. With bulk susceptibility you start from a known volume, with mass susceptibility you start from a known mass and from molar susceptibility you start from a known number of moles. Depending upon your application one form will be more convenient than another. Physicists like molar susceptibility because their calculations derive from atomic properties. Geologists like mass susceptibility because they know the weight of their sample.

 The definition of susceptibility given here accords with the Sommerfeld SI variant. In the Kennelly variant χ has a different definition.

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Mass susceptibility

Magnetic mass susceptibility is simply

where χ_v is the ordinary (‘volumetric’) susceptibility and ρ is the density of the material in kg per cubic metre. Unfortunately some tables of mass susceptibility, even in prestigious publications, abbreviate the units to ‘susceptibility per gram’ or ‘susceptibility per kilogram’ which is, at best, a source of confusion. Take care to distinguish χ_ρ from χ_M or molar susceptibility; that is a different quantity.

So, if you know the mass of your material sample you need only multiply by χ_ρ to find its magnetic moment when the field strength is one amp per metre.

 This definition of susceptibility accords with the Sommerfeld SI variant. In the Kennelly variant χ has a different definition.
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Terminology for intrinsic fields within materials –

Magnetic moment

The concept of magnetic moment is the starting point when discussing the behaviour of magnetic materials within a field. If you place a bar magnet in a field then it will experience a torque or moment tending to align its axis in the direction of the field. A compass needle behaves the same. This torque increases with the strength of the poles and their distance apart. So the value of magnetic moment tells you, in effect, ‘how big a magnet’ you have.

It is also well known that a current carrying loop in a field also experiences a torque (electric motors rely on this effect). Here the torque, τ, increases with the current, i, and the area of the loop, A. θ is the angle made between the axis of the loop normal to its plane and the field direction.

The unit of τ is the newton metre. This puzzling quantity appears to have the dimensions of force times distance … which is energy. Hmmm.

The quantity i × A is defined as the magnetic moment, m. This gives

Particular materials where the magnetic moment of each atom can be made to favour one direction are said to be magnetizable. The extent to which this happens is called the magnetization. Magnetic moment is a vector quantity which has both direction and magnitude. This is important because although the atoms in most materials may have magnetic moments they are not easily brought into alignment in one direction, so the moments cancel each other, leading to weak magnetization.

The Earth has a magnetic moment of 8×10²² A m².   A single electron has a magnetic moment due to its orbit around the nucleus which is a multiple of 9.27×10^-24 A m² (known as the Bohr magneton, μ_B).

We have, then, two ways of looking at the basis of magnetism: one is the idea of a pair of opposing poles and the other is current circulation. Each viewpoint has some advantages over the other; and this gave science a hard time deciding which to prefer. The reason this is worth mentioning is that different definitions arose for several quantities in magnetics. Both models, however, function more as convenient mathematical abstractions rather than literal descriptions of the physical origins of magnetism.

 The definition given above accords with the Sommerfeld variant of the SI system of units. In the Kennelly variant m is in weber metres.

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Magnetization

Magnetic fields are caused by the movement of charge, normally electrons. This movement may take place in a wire carrying current. The wire then develops a surrounding magnetic field which is given the symbol, H.

In a bar magnet you may not think that there need be any current but the magnetic field here is also due to moving charge: the electrons circling around the nuclei of the iron atoms or simply spinning about their own axis. Atoms like this are said to possess a magnetic moment. The average field strength due to these moments at any particular point is called magnetization and given the symbol M.

In most materials the moments are oriented almost at random – which leads to weak magnetization and ‘non-magnetic’ properties. In iron the moments readily align themselves along an applied field so inducing a large value of M and the familiar characteristics in the presence of a field.

Magnetization is defined –

where m is the total vector sum for the magnetic moments of all the atoms in a given volume V (in m³) of the material. We can then say –

This equation is of theoretical importance because it highlights a closeness between H and M. The H field is related to ‘free currents’: for example those flowing from a battery along a piece of wire. M, on the other hand, is related to the ‘bound’ (‘Ampèrian’) currents of electron orbitals within magnetized materials.

In practice, with ferromagnetic materials, M tends to be a very complex function of H – including values of H in the past. As a designer of wound components you therefore pretend instead that B = μH … and hope for the best!

Magnetization occurs not just in materials having permanent magnetic moments but also in any magnetizable material in a field which can induce a magnetic moment in its constituent atoms. In the special case of paramagnetic and diamagnetic materials this magnetization is given by

 The definition given above accords with the Sommerfeld variant of the SI system of units. The Kennelly variant M is in tesla.

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Intensity of magnetization

Intensity of magnetization functions in the Kennelly variant of the SI as an alternative to the Sommerfeld variant for magnetization, M.

We can then say –

Note the units of I: teslas, not amps per metre as in the Sommerfeld magnetization. So don’t confuse intensity of magnetization with magnetization.

Magnetic polarization

Magnetic polarization is a synonym for intensity of magnetization in the Kennelly variant of the SI