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 ...
[
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).
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.
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.
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
Coercivity in the SI
| Quantity name | coercivity |
|---|
| Quantity symbol | Hc |
|---|
| Unit name | amps per metre |
|---|
Unit
abbreviation | 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 |
|---|
Unit
abbreviation | m-1 |
|---|
| 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 |
|---|
Unit
abbreviation | m2 |
|---|
| 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 |
|---|
Unit
abbreviation | m |
|---|
| 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 = 
× (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 = 2×ln(R2/R1)/(1/R1-1/R2) m
Another common core type, the EE, is shown in Fig: EEE.
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)+((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 |
|---|
Unit
abbreviation | A |
|---|
| 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 |
|---|
Unit
abbreviation | 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 |
 = 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 -
= 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 |
|---|
Unit
abbreviation | Wb |
|---|
| 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.
= (1/N)
V.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 -
= 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 |
= 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 -
= Fm / Rm webers
Lastly, flux can also be derived by knowing the both the magnetic flux density and the area over which it applies:
= 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 |
|---|
Unit
abbreviation | T |
|---|
| Fundamental units | kg s-2
A-1 |
|---|
| Quantity | Unit | Formula |
|---|
| Magnetic flux density | webers per metre2 |
B = /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 = / 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 |
|---|
Unit
abbreviation | Wb-t |
|---|
| 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. 
In 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× or N×d/dt. For greater
accuracy you substitute λ or dλ/dt.
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Magnetization Curves
A Typical Magnetization Curve
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':
- Close to the origin a slow rise due to 'reversible growth'.
- A longer, fairly straight, stretch representing 'irreversible growth'.
- A slower rise representing 'rotation'.
- 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
Permeability in the SI
| Quantity name | permeability |
|---|
| Quantity symbol | µ |
|---|
| Unit name | henrys per metre |
|---|
Unit
abbreviation | 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 |
= 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 = 4
×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
Inductance in the SI
| Quantity name | Inductance |
|---|
| Quantity symbol | L |
|---|
| Unit name | henrys |
|---|
Unit
abbreviation | H |
|---|
| Fundamental units | kg m2 s-2
A-2 |
|---|
| Quantity | Unit | Formula |
|---|
| Inductance | webers per amp |
L = /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 × d/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 × /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
- Whether the coil has more than one layer of turns.
- The ratio of coil length to coil diameter.
- The shape of the cross section of a multi-layer winding.
- Whether the coil is wound on a circular, polygonal or rectangular
former.
- Whether the coil is open ended, or bent round into a toroid.
- Whether the cross section of the wire is round or rectangular, tubular or
solid.
- The permeability of the wire.
- The frequency of operation.
- The phase of the moon, direction of the wind etc..
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 |
|---|
Unit
abbreviation | nH |
|---|
| 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
Reluctance in the SI
| Quantity name | reluctance |
|---|
| Quantity symbol | Rm or ℜ |
|---|
| Unit name | ampere-turns per weber |
|---|
Unit
abbreviation | A-t Wb-1 |
|---|
| Fundamental units | A2 s2 kg-1
m-2 |
|---|
Reluctance is the ratio of mmf to flux -
Rm = Fm /
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
Remnance in the SI
| Quantity name | remnance |
|---|
| Quantity symbol | Br |
|---|
| Unit name | teslas |
|---|
Unit
abbreviation | T |
|---|
| 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.
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 = × (Rm_gap + Rm_core) ampere-turns
compare
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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E-mail:
R.Clarke@surrey.ac.uk
Last modified: 2001 May 24th.