1.0 Introduction
Magnets
are an important part of our daily lives, serving as essential
components in everything from electric motors, loudspeakers,
computers, compact disc players, microwave ovens and the family car,
to instrumentation, production equipment, and research. Their
contribution is often overlooked because they are built into devices
and are usually out of sight.
Magnets
function as transducers, transforming energy from one form to
another, without any permanent loss of their own energy. General
categories of permanent magnet functions are:
Mechanical to mechanical - such as attraction
and repulsion.
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Mechanical to electrical - such as generators
and microphones.
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Electrical to mechanical - such as motors,
loudspeakers, charged particle deflection.
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Mechanical to heat - such as eddy current and
hysteresis torque devices.
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Special effects - such as magneto resistance,
Hall effect devices, and magnetic resonance. |
The following sections will provide a brief
insight into the design and application of permanent magnets. The
Design Engineering team at Magnet Applications Group will be happy
to assist you further in your applications.
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2.0 Modern Magnet Materials
There
are four classes of modern commercialized magnets, each based on
their material composition. Within each class is a family of grades
with their own magnetic properties. These general classes
are:
• Neodymium Iron Boron
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• Samarium
Cobalt
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• Ferrite
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• Alnico
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NdFeB
and SmCo are collectively known as Rare Earth magnets because they
are both composed of materials from the Rare Earth group of
elements. Neodymium Iron Boron (general composition
Nd2Fe14B, often abbreviated to NdFeB) is the
most recent commercial addition to the family of modern magnet
materials. At room temperatures, NdFeB magnets exhibit the highest
properties of all magnet materials. Samarium Cobalt is manufactured
in two compositions: Sm1Co5 and
Sm2Co17 - often referred to as the SmCo 1:5 or
SmCo 2:17 types. 2:17 types, with higher Hci values,
offer greater inherent stability than the 1:5 types. Ferrite, also
known as Ferrite, magnets (general composition
BaFe2O3 or SrFe2O3) have
been commercialized since the 1950s and continue to be extensively
used today due to their low cost. A special form of ferrite magnet
is "Flexible" material, made by bonding ferrite powder in a flexible
binder. Alnico magnets (general composition Al-Ni-Co) were
commercialized in the 1930s and are still extensively used
today.
These
materials span a range of properties that accommodate a wide variety
of application requirements. The following is intended to give a
broad but practical overview of factors that must be considered in
selecting the proper material, grade, shape, and size of magnet for
a specific application. The chart below shows typical values of the
key characteristics for selected grades of various materials for
comparison. These values will be discussed in detail in the
following sections.
Table 2.1 Magnet Material
Comparisons |
Material |
Grade |
Br |
Hc |
Hci |
BHmax |
Tmax (Deg C)* |
NdFeB |
39H |
12,800 |
12,300 |
21,000 |
40 |
150 |
SmCo |
26 |
10,500 |
9,200 |
10,000 |
26 |
300 |
NdFeB |
B10N |
6,800 |
5,780 |
10,300 |
10 |
150 |
Alnico |
5 |
12,500 |
640 |
640 |
5.5 |
540 |
Ferrite |
8 |
3,900 |
3,200 |
3,250 |
3.5 |
300 |
Flexible |
1 |
1,600 |
1,370 |
1,380 |
0.6 |
100 |
| * Tmax (maximum
practical operating temperature) is for
reference only. The maximum practical operating
temperature of any magnet is dependent on the circuit
the magnet is operating
in. |
|
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3.0 Units of Measure
Three
systems of units of measure are common: the cgs (centimeter, gram,
second), SI (meter, kilogram, second), and English (inch, pound,
second) systems. This catalog uses the cgs system for magnetic
units, unless otherwise specified.
| Table 3.1 Units of
Measure Systems |
| Unit |
Symbol |
cgs System |
SI System |
English System |
| Flux |
ø |
maxwell |
weber |
Maxwell |
| Flux Density |
B |
gauss |
tesla |
lines/in2 |
| Magnetomotive Force |
F |
gilbert |
ampere turn |
ampere turn |
| Magnetizing Force |
H |
oersted |
ampere turns/m |
ampere turns/in |
| Length |
L |
cm |
m |
in |
| Permeability of a vacuum |
µv |
1 |
0.4![]() x
10-6 |
3.192 |
| Table
3.2 Conversion Factors |
| Multiply |
By |
To obtain |
| inches |
2.54 |
centimeters |
lines/in2 |
0.155 |
Gauss |
| lines/in2 |
1.55 x 10-5 |
Tesla |
| Gauss |
6.45 |
lines/in2 |
| Gauss |
0-4 |
Tesla |
| Gilberts |
0.79577 |
ampere turns |
| Oersteds |
9.577 |
ampere turns /m |
| ampere turns |
0.4 |
Gilberts |
| ampere turns/in |
0.495 |
Oersteds |
| ampere turns/in |
39.37 |
ampere
turns/m |
|
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4.0 Design Considerations
Basic problems
of permanent magnet design revolve around estimating the
distribution of magnetic flux in a magnetic circuit, which may
include permanent magnets, air gaps, high permeability conduction
elements, and electrical currents. Exact solutions of magnetic
fields require complex analysis of many factors, although
approximate solutions are possible based on certain simplifying
assumptions. Obtaining an optimum magnet design often involves
experience and tradeoffs.
4.1 Finite Element
Analysis
Finite Element
Analysis (FEA) modeling programs are used to analyze magnetic
problems in order to arrive at more exact solutions, which can then
be tested and fine tuned against a prototype of the magnet
structure. Using FEA models flux densities, torques, and forces may
be calculated. Results can be output in various forms, including
plots of vector magnetic potentials, flux density maps, and flux
path plots. The Design Engineering team at Magnet Applications has
extensive experience in many types of magnetic designs and is able
to assist in the design and execution of FEA models.
4.2 The B-H
Curve
The basis of
magnet design is the B-H curve, or hysteresis loop, which
characterizes each magnet material. This curve describes the cycling
of a magnet in a closed circuit as it is brought to saturation,
demagnetized, saturated in the opposite direction, and then
demagnetized again under the influence of an external magnetic
field.
The second
quadrant of the B-H curve, commonly referred to as the
"Demagnetization Curve", describes the conditions under which
permanent magnets are used in practice. A permanent magnet will have
a unique, static operating point if air-gap dimensions are fixed and
if any adjacent fields are held constant. Otherwise, the operating
point will move about the demagnetization curve, the manner of which
must be accounted for in the design of the device.
The three most
important characteristics of the B-H curve are the points at which
it intersects the B and H axes (at Br - the residual
induction - and HC - the coercive force - respectively),
and the point at which the product of B and H are at a maximum
(BHmax - the maximum energy product). Br represents the maximum flux the magnet is able to produce
under closed circuit conditions. In actual useful operation
permanent magnets can only approach this point. HC represents the point at which the magnet becomes demagnetized under
the influence of an externally applied magnetic field.
BHmax represents the point at which the product of B and
H, and the energy density of the magnetic field into the air gap
surrounding the magnet, is at a maximum. The higher this product,
the smaller need be the volume of the magnet. Designs should also
account for the variation of the B-H curve with temperature.
When plotting
a B-H curve, the value of B is obtained by measuring the total flux
in the magnet (ø)and then dividing this by the magnet pole area (A)
to obtain the flux density (B=ø/A). The total flux is composed of
the flux produced in the magnet by the magnetizing field (H), and
the intrinsic ability of the magnet material to produce more flux
due to the orientation of the domains. The flux density of the
magnet is therefore composed of two components, one equal to the
applied H, and the other created by the intrinsic ability of
ferromagnetic materials to produce flux. The intrinsic flux density
is given the symbol Bi where total flux B = H +
BI, or, BI = B - H. In normal operating
conditions, no external magnetizing field is present, and the magnet
operates in the second quadrant, where H has a negative value.
Although strictly negative, H is usually referred to as a positive
number, and therefore, in normal practice, BI = B + H. It
is possible to plot an intrinsic as well as a normal B-H curve. The
point at which the intrinsic curve crosses the H axis is the
intrinsic coercive force, and is given the symbol Hci.
High Hci values are an indicator of inherent stability of
the magnet material. The normal curve can be derived from the
intrinsic curve and vice versa. In practice, if a magnet is operated
in a static manner with no external fields present, the normal curve
is sufficient for design purposes. When external fields are present,
the normal and intrinsic curves are used to determine the changes in
the intrinsic properties of the material.
4.3 Magnet
Calculations
In the absence
of any coil excitation, the magnet length and pole area may be
determined by the following equations:
Equation 1
and
Equation 2
where
Bm = the flux density at the operating point,
Hm = the magnetizing force at the operating point,
Ag,
= the air-gap area,
Lg = the air-gap length,
Bg = the gap flux density,
Am = the magnet pole area,
and
Lm = the magnet length.
Combining the
two equations, the permeance coefficient Pc may be
determined as follows:
Equation 3
Strictly, 
where µ is
the permeability of the medium, and k is a factor which takes
account of leakage and reluctance that are functions of the geometry
and composition of the magnetic circuit.
(The intrinsic
permeance coefficient Pci = B i/H. Since the
normal permeance coefficient PC = B/H, and B = H + B i, PC = (H + B i)/H or PC = 1 + B i /H. Even though the
value of H in the second quadrant is actually negative, H is
conventionally referred to as a positive number. Taking account of
this convention, PC = 1 - B i /H, or B i /H = Pci = PC + 1. In other words, the
intrinsic permeance coefficient is equal to the normal permeance
coefficient plus 1. This is a useful relationship when working on
magnet systems that involve the presence of external
fields.)
The permeance
coefficient is a useful first order relationship, helpful in
pointing towards the appropriate magnet material, and to the
approximate dimensions of the magnet. The objective of good magnet
design is usually to minimize the required volume of magnet material
by operating the magnet at BHmax.
We can compare
the various magnet materials for general characteristics using
equation 3 above.
Consider that
a particular field is required in a given air-gap, so that the
parameters Bg, Hg (air-gap magnetizing force),
Ag, and Lg are known.
| • |
Alnico 5 has the ability to provide very high levels of
flux density BM, which is often desirable in high
performance electromechanical devices. This is accompanied,
however, by a low coercivity Hm, and so some
considerable magnet length will be required.
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| • |
Alnico 8 operates at a higher magnetizing force,
Hm, needing a smaller length Lm, but
will yield a lower BM, and would therefore require
a larger magnet area Am.
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| • |
Rare Earth materials offer reasonable to high values of
flux density at very high values of magnetizing force.
Consequently, very short magnet lengths are needed, and the
required volume of this material will be small.
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| • |
Ferrite operates at relatively low flux densities, and
will therefore need a correspondingly greater pole face area,
Am.
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The permeance
coefficient method using the demagnetization curves allows for
initial selection of magnet material, based upon the space available
in the device, this determining allowable magnet
dimensions.
4.3.1 Calculation Of Flux
Density On A Magnet's Central Line
For magnet
materials with straight-line normal demagnetization curves such as
Rare Earths and ferrites, it is possible to calculate with
reasonable accuracy the flux density at a distance X from the pole
surface (where X>0) on the magnetic centerline under a variety of
conditions.
a.
Cylindrical Magnets
Equation 4
Table 4.1
shows flux density calculations for a magnet 0.500" in diameter by
0.250" long at a distance of 0.050" from the pole surface, for
various materials. Note that you may use any unit of measure for
dimensions; since the equation is a ratio of dimensions, the result
is the same using any unit system. The resultant flux density is in
units of gauss.
| Table 4.1 Flux Density
vs. Material |
| Material and
Grade |
Residual Flux Density,
Br |
Flux at distance of 0.050" from
surface of magnet |
| Ferrite 1 |
2,200 |
629 |
| Ferrite 5 |
3,950 |
1,130 |
| SmCo 18 |
8,600 |
2,460 |
| SmCo 26 |
10,500 |
3,004 |
| NdFeB 35 |
12,300 |
3,518 |
| NdFeB 42H |
13,300 |
3,804 |
|
b.
Rectangular Magnets
Equation
5
(where all angles are in radians)
c.
For a Magnet on a Steel Back plate
Equation 6 Substitute 2L
for L in the above formulae.
d.
For Identical Magnets Facing Each Other in Attracting
Positions
Equation 7 The value of
Bx at the gap center is double the value of Bx in case 3. At a point P, Bp is the sum of
B(x-p) and B(XP), where (X+P) and (XP)
substitute for X in case 3.
e.
For Identical, Yoked Magnets Facing Each Other in Attracting
Positions
Equation 8 Substitute 2L
for L in case 4, and adopt the same procedure to calculate BP
4.3.2 Force Calculations
The
attractive force exerted by a magnet to a ferromagnetic material may
be calculated by:
Equation 9
where F is the force in pounds, B is the flux density in
Kilogauss, and A is the pole area in square inches. Calculating B is
a complicated task if it is to be done in a rigorous manner.
However, it is possible to approximate the holding force of
certain magnets in contact with a piece of steel using the
relationship:
Equation 10
where
Br is the residual flux density of the material, A is the
pole area in square inches, and Lm is the magnetic
length.
This formula is only intended to give an order of
magnitude for the holding force that is available from a magnet with
one pole in direct contact with a flat, machined, steel surface. The
formula can only be used with straight-line demagnetization curve
materials - i.e. for rare earth and ferrite materials - and where
the magnet length, Lm, is kept within the bounds of
normal, standard magnet configurations.
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