Design Guide
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.
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 Integrated
Magnetics has extensive experience in many types of
magnetic designs and is able to assist in the design
and execution of FEA models.
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.
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 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.
| Table
4.1 Flux Density vs. Material |
| Material
and Grade |
Residual
Flux Density, Br |
Flux
at distance of 0.050" from surface of magnet |
| Ceramic
1 |
2,200 |
629 |
| Ceramic
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 |
Rectangular Magnets
Equation
5(where
all angles are in radians)
For
a Magnet on a Steel Back plate
Equation
6 Substitute
2L for L in the above formulae.
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.
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
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.
Click here to calculate approximate
pull of a rectangular
or disc
magnet.
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 ceramic materials - and where the magnet length,
Lm, is kept within the bounds of normal,
standard magnet configurations.
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