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The permeance in magnetic circuits
Journal of Magnetism and Magnetic Materials 19 (1980) 323-325 © North-Holland Publishing Company
THE PERMEANCE IN MAGNETIC CIRCUITS Stig BERGLUND Telefonaktiebolaget L.M. Ericsson, S-126 25 Stockholm, Sweden
The field distribution of a magnetic circuit, and the forces acting on a movable armature, is always possible to solve when all permeances and their derivatives are known. An airgap can be seen as an ideal permeance surrounded by fringe and edge permeances, and accurate formulas have been found by conformal mapping. A suitable computer program is indicated.
1. Method
bridge into a cell with known character. Enger recommends a flux potential diagram in logarithmic scale as in good aid to perform the solution of a network. When an implicit (exact) solution is impossible, a preliminary flux distribution can be assumed and the necessary adjustments are then rather easily found from the diagram - a graphical solution. Another way is to perform such adjustments numerically, by recalculating the flux distribution of the circuit several times using a computer, until a nearly exact solution is found, satisfying the conditions o f the whole network. This has now been successfully done and is used for optimization in our miniature relay construction. This is an example o f a circuit type where the influence of even small permeances must be considered.
A magnetic device presents a three-dimensional field problem of intricate nature. But it is shown by Enger [1], that the problem can be solved with sufficient accuracy by a systematic approach. He has put together a lot of experience so that his paper can be used as a manual for calculating: 1) The distribution of magnetic flux and magnetomotances in magnetic apparatus when permeances in all parts are known; 2) Mechancial forces in an air gap, when the permeance function is known; 3) The permeance in various forms of air gaps as the sum of an ideal permeance coupled parallel to fringe and edge permeances. The principle used is to split up the entire field of a magnetic device into flux tubes and these in turn into flux cells, limited by flux surfaces and equipotential surfaces. Thus the whole three-dimensional field can be transformed into a compound net i.e. a net composed of cells coupled in parallel and in series. Cells may be passive or active (containing magnetomotance). The relationship between flux and magnetic potential in each cell is given by the permeance, and is linear in air cells and usually non-linear in ironfilled cells. The number of necessary cells will not be too large. This makes a solution possible, as all compound nets are hierarchic structures of branches and may be solved by starting at the branches of lowest degree in the hierarchy. If the field contains bridges i.e. if cross connections exist between parallel branches, methods are given for the evaluation of such a
2. Permeances The magnetic flux in an airgap may be seen as built up by three parts: an ideal flux, the fringe and the edge fluxes. An ideal flux is calculated under the assumption that the pole surfaces are parts of unlimited equipotential surfaces of simple geometric forms and that the field configuration has a corresponding simple geometric form, e.g. a rectilinear field between parallel plane surfaces, a cylindrical field between inclined plane surfaces, also between circular-cylindrical surfaces, etc. Formulas for the calculation o f these permeances can be developed from the similar theory of electrostatic fields. The fringe permeances take account o f the additional 323
S. Berglund /Permeance in magnetic circuits
324
flux emitted from the ideal pole surface, caused by the field bulging out in the neighbourhood of the ideal surfaces limiting contour. The edge flux is emitted from the edge surfaces surrounding the ideal pole surface. It is shown that the fringe flux and permeance are substantially located in the immediate neighbourhood of the main limiting contour. The total permeance in an air gap may therefore be calculated as the sum of one ideal permeance plus the fringe and edge permeances, the latter proportional to the piecemeal lengths of the limiting edges and edges surfaces. The corners must also be regarded. The manual gives formulas for ideal permeances and their derivatives for various forms of air gap, e.g. air gaps between plane, cylindrical, conical or spherical polar surfaces and limited by rectangular, elliptical or circular contours. For the permeance between two spheres it is assumed that there exists somewhere an (almost) plane equipotential surface, and that this plane is so situated that the resulting permeance to both spheres has an extreme value. Cylindrical or spherical surfaces may be used as substitutes in the calculations for more complicated bodies, provided the minimum distance is not changed. The fringe and edge permeances are evaluated according to the theory of complex functions and conformal mapping including Schwartz-Christoffel's transformation. The results can be given in explicit formulas for each part of the contour. As an example here are some useful formulas for the permeances along the width b of a rectangular box with length
I
I,
Fig. 2. Conformal mapping of the flux cells.
g, height h and with distance a to an unlimited parallel plane surface. Ideal: A i = l.Lo(g/a)b.
(1)
The flux cells of fig. 1 are transformed into the halfcylindrical cells of fig. 2 and a characteristical value p is calculated: P=
r
\-a-/
The fringe permeance is Apr = Ap - A i and it has a limit value when g > 2a: Apr~ - / ~ - I I + P + I / - 2 /'v~+--i b v 2~/p l n v p - 1 -
In 4P I p- 1 '
(3)
When g < 2a the Apr is smaller and the calculation is similar to (5) and (6) below. Vertical edge surface: Aql/b = (//o/rr) In p .
(4)
Sometimes also the back surface should be considered:
Aq2/b = (lao/rr) ln(r/p),
,I
^. Fig. 1. Flux cells around a rectangular pole piece.
(5)
S. Berglund / Permeance in magnetic circuits
3 25
obtained or will become too complicated, a numerical derivation method is used, based on a Taylor's series expression (for logarithmic values).
I
r 3. Computer
Fig. 3. Cylindrical segment added at a corner.
where r is determined by g_ 1 ----x/(r a ff~/p × in
lh+a - p ) ( r - 1) + - - n a
p-1
1 ( X / ~ - 1) + rx/~-p-p) 2 + - In n r(p - 1)
(6)
In the three-dimensional case it is necessary to go along the whole contour adding fringe, edge (and back) permeances for each of the four sides surrounding the ideal air gap. A cylindrical segment should also be included for each corner (see fig. 3), which has the approximate extra permeance: A h = P-O X 0.4 c~h ,
(7)
with a = n/2 for a rectangle. We can find for an air gap with magnetomotance ~a and permeance Aa that the force is: 1 a2 dAa/dX force = --~gr
For calculations with a computer, most of the formulas can be directly programmed and used as routines, as can the single steps o f reducing a network including a solution for bridge connections. Leak permeances between parallel branches may often be treated as concentrated at a few branch points with sufficient accuracy in the final results. When iron parts carry considerable flux, their permeances are flux dependent and iterative calculations of the whole circuit are needed, which the computer, once programmed, can do very fast. An approximate analytical form of the hysteresis curve, adaptable for different types of soft iron, has been found and used. However, a simple expression cannot usually cover the whole curve, so it is divided into a few parts. What is then left to the programmer is to analyze and build up the actual magnetic circuit from its dimensional parameters. To aid this we have preferred a simple programming language with full interactive possibilities (ICL JEAN). This has favored the direct use of such programs to optimize magnetic circuits under construction. The calculated values have shown astonishingly good agreement with measured forces for tested relays.
(8)
so derivative values of the permeances are needed for the calculation of forces or torques acting on a movable armature. When a direct derivate formula cannot be
Reference [1] S. Enger, Dissertation, KTH, Stockholm (1977).
THE PERMEANCE IN MAGNETIC CIRCUITS Stig BERGLUND Telefonaktiebolaget L.M. Ericsson, S-126 25 Stockholm, Sweden
The field distribution of a magnetic circuit, and the forces acting on a movable armature, is always possible to solve when all permeances and their derivatives are known. An airgap can be seen as an ideal permeance surrounded by fringe and edge permeances, and accurate formulas have been found by conformal mapping. A suitable computer program is indicated.
1. Method
bridge into a cell with known character. Enger recommends a flux potential diagram in logarithmic scale as in good aid to perform the solution of a network. When an implicit (exact) solution is impossible, a preliminary flux distribution can be assumed and the necessary adjustments are then rather easily found from the diagram - a graphical solution. Another way is to perform such adjustments numerically, by recalculating the flux distribution of the circuit several times using a computer, until a nearly exact solution is found, satisfying the conditions o f the whole network. This has now been successfully done and is used for optimization in our miniature relay construction. This is an example o f a circuit type where the influence of even small permeances must be considered.
A magnetic device presents a three-dimensional field problem of intricate nature. But it is shown by Enger [1], that the problem can be solved with sufficient accuracy by a systematic approach. He has put together a lot of experience so that his paper can be used as a manual for calculating: 1) The distribution of magnetic flux and magnetomotances in magnetic apparatus when permeances in all parts are known; 2) Mechancial forces in an air gap, when the permeance function is known; 3) The permeance in various forms of air gaps as the sum of an ideal permeance coupled parallel to fringe and edge permeances. The principle used is to split up the entire field of a magnetic device into flux tubes and these in turn into flux cells, limited by flux surfaces and equipotential surfaces. Thus the whole three-dimensional field can be transformed into a compound net i.e. a net composed of cells coupled in parallel and in series. Cells may be passive or active (containing magnetomotance). The relationship between flux and magnetic potential in each cell is given by the permeance, and is linear in air cells and usually non-linear in ironfilled cells. The number of necessary cells will not be too large. This makes a solution possible, as all compound nets are hierarchic structures of branches and may be solved by starting at the branches of lowest degree in the hierarchy. If the field contains bridges i.e. if cross connections exist between parallel branches, methods are given for the evaluation of such a
2. Permeances The magnetic flux in an airgap may be seen as built up by three parts: an ideal flux, the fringe and the edge fluxes. An ideal flux is calculated under the assumption that the pole surfaces are parts of unlimited equipotential surfaces of simple geometric forms and that the field configuration has a corresponding simple geometric form, e.g. a rectilinear field between parallel plane surfaces, a cylindrical field between inclined plane surfaces, also between circular-cylindrical surfaces, etc. Formulas for the calculation o f these permeances can be developed from the similar theory of electrostatic fields. The fringe permeances take account o f the additional 323
S. Berglund /Permeance in magnetic circuits
324
flux emitted from the ideal pole surface, caused by the field bulging out in the neighbourhood of the ideal surfaces limiting contour. The edge flux is emitted from the edge surfaces surrounding the ideal pole surface. It is shown that the fringe flux and permeance are substantially located in the immediate neighbourhood of the main limiting contour. The total permeance in an air gap may therefore be calculated as the sum of one ideal permeance plus the fringe and edge permeances, the latter proportional to the piecemeal lengths of the limiting edges and edges surfaces. The corners must also be regarded. The manual gives formulas for ideal permeances and their derivatives for various forms of air gap, e.g. air gaps between plane, cylindrical, conical or spherical polar surfaces and limited by rectangular, elliptical or circular contours. For the permeance between two spheres it is assumed that there exists somewhere an (almost) plane equipotential surface, and that this plane is so situated that the resulting permeance to both spheres has an extreme value. Cylindrical or spherical surfaces may be used as substitutes in the calculations for more complicated bodies, provided the minimum distance is not changed. The fringe and edge permeances are evaluated according to the theory of complex functions and conformal mapping including Schwartz-Christoffel's transformation. The results can be given in explicit formulas for each part of the contour. As an example here are some useful formulas for the permeances along the width b of a rectangular box with length
I
I,
Fig. 2. Conformal mapping of the flux cells.
g, height h and with distance a to an unlimited parallel plane surface. Ideal: A i = l.Lo(g/a)b.
(1)
The flux cells of fig. 1 are transformed into the halfcylindrical cells of fig. 2 and a characteristical value p is calculated: P=
r
\-a-/
The fringe permeance is Apr = Ap - A i and it has a limit value when g > 2a: Apr~ - / ~ - I I + P + I / - 2 /'v~+--i b v 2~/p l n v p - 1 -
In 4P I p- 1 '
(3)
When g < 2a the Apr is smaller and the calculation is similar to (5) and (6) below. Vertical edge surface: Aql/b = (//o/rr) In p .
(4)
Sometimes also the back surface should be considered:
Aq2/b = (lao/rr) ln(r/p),
,I
^. Fig. 1. Flux cells around a rectangular pole piece.
(5)
S. Berglund / Permeance in magnetic circuits
3 25
obtained or will become too complicated, a numerical derivation method is used, based on a Taylor's series expression (for logarithmic values).
I
r 3. Computer
Fig. 3. Cylindrical segment added at a corner.
where r is determined by g_ 1 ----x/(r a ff~/p × in
lh+a - p ) ( r - 1) + - - n a
p-1
1 ( X / ~ - 1) + rx/~-p-p) 2 + - In n r(p - 1)
(6)
In the three-dimensional case it is necessary to go along the whole contour adding fringe, edge (and back) permeances for each of the four sides surrounding the ideal air gap. A cylindrical segment should also be included for each corner (see fig. 3), which has the approximate extra permeance: A h = P-O X 0.4 c~h ,
(7)
with a = n/2 for a rectangle. We can find for an air gap with magnetomotance ~a and permeance Aa that the force is: 1 a2 dAa/dX force = --~gr
For calculations with a computer, most of the formulas can be directly programmed and used as routines, as can the single steps o f reducing a network including a solution for bridge connections. Leak permeances between parallel branches may often be treated as concentrated at a few branch points with sufficient accuracy in the final results. When iron parts carry considerable flux, their permeances are flux dependent and iterative calculations of the whole circuit are needed, which the computer, once programmed, can do very fast. An approximate analytical form of the hysteresis curve, adaptable for different types of soft iron, has been found and used. However, a simple expression cannot usually cover the whole curve, so it is divided into a few parts. What is then left to the programmer is to analyze and build up the actual magnetic circuit from its dimensional parameters. To aid this we have preferred a simple programming language with full interactive possibilities (ICL JEAN). This has favored the direct use of such programs to optimize magnetic circuits under construction. The calculated values have shown astonishingly good agreement with measured forces for tested relays.
(8)
so derivative values of the permeances are needed for the calculation of forces or torques acting on a movable armature. When a direct derivate formula cannot be
Reference [1] S. Enger, Dissertation, KTH, Stockholm (1977).