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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 Sales &
Manufacturing 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. This effect is more
closely examined in the section entitled "Permanent
Magnet Stability".
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.
Click
here to calculate Permeance Coefficients of Disc,
Rectangle,
Ring
(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.
The permeance coefficient at which BHmax occurs is given in
the material properties tables
.
We can compare the
varios 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.
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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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Ceramic
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
Click
here to calculate flux density of rectangular or cylindrical magnets in
various configurations (equations 4 through 9).
For magnet materials
with straight-line normal demagnetization curves such as Rare Earths and
Ceramics, it is possible to calculate with reasonable accuracy the flux
density at a distance X from the pole surface (where X>0) on the magnet's
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.
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x 10-6 
Equation
10
Equation
11