Theory & Practice of Electromagnetic Design of DC Motors & Actuators
George P. Gogue & Joseph J. Stupak, Jr.
FLUX, RELUCTANCE AND PERMEANCE
3.1 Intuitive concepts of flux:
Flux lines may be thought of as being something like thick rubber bands; they pull along their lengths and push outward sideways.
Flux lines leave smooth surfaces of highly permeable material (below saturation) at right angles to the surface and enter the same way. If flux left at a different angle, there would have to be a component of H in the pole material parallel to the surface, as large as the component in air. The component of flux in that direction would then have to be very large because of the high permeability of the material.
At sharp corners, flux leaves or enters on a line bisecting the angle. Flux-density leaving a sharp corner tends to be greater than at nearly smooth surfaces. Flux going around a corner tends to "crowd in" toward the inside of the curve (the rubber-band analogy). Where flux lines are close together, magnetic forces are high and where they are far apart, forces are lower. Flux lines cannot cross each other and must close on themselves into loops. Flux lines in opposite directions at the same point (from two different sources) cancel and flux lines in the same direction add. Flux lines at an angle to each other add vectorially.
3.2 Reluctance and permeance
The basic equation of magnetic circuits stated earlier, Equation (1.15), is:
The reluctance R has not yet been discussed in detail. Its calculation is in general one of the most difficult areas of magnetics. Many magnetics designers avoid the work, relying instead on intuition and experience but these may be misleading. In spite of its complexity, a careful consideration of reluctance and its methods of calculation are essential to successful magnetics design.
Using the magnetics design equations for the circuit of Figure 1.4, with uniform flux-density in the gap and assuming perfect pole material (with infinite permeability) and no leakage of flux, i.e., =kl=1, then:
In the gap, from Equation (1.18)
The reluctance of a uniform gap, without leakage, is therefore:
This result may be compared to the calculation of the electrical resistance of a wire or bar of uniform cross-section,
where is the wire material conductivity, l the length and A is the wire cross-sectional area.
It is often necessary to add reluctances in parallel, since flux in a magnetic circuit may flow in various leakage paths, as well as the useful one. It can be shown, by considering two parallel flux-paths, with flux driven by the same mmf, that the equivalent reluctance for two reluctances in parallel is:
More conveniently, let the permeance P be defined as the reciprocal of reluctance,
Then the formula for combined parallel reluctances, Equation (3.7) becomes:
The combination of reluctances in series, with a common flux-path, results in:
3.3 General formulation of reluctance:
For more complicated reluctance volumes, where the flux lines may not be parallel and where the flux-density varies within the volume, a more general relationship is needed. In Figure 3.1 the volume may be considered as being broken up into flux-tubes. A flux-tube is an imaginary closed wall in space, which is everywhere parallel to the direction of flux at its surface, so that no flux crosses the wall. The cross-section of the flux-tube is chosen small enough that variations in B across the tube may be neglected. The reluctance of the volume is then:
where A is the cross-sectional area perpendicular to the flux-path length l, of which dl is an increment. The reluctances of all the tubes are then summed in parallel.
As an example, the reluctance of an annular gap will be calculated. Such gaps occur in linear actuators, permanent-magnet brushless DC motors and some types of speakers, for example.
Since the flux is radial and is the same at any radius,
The area is:
3.4 Roters' method:
In many practical cases it is very difficult to accomplish the required integrations by analytical means. An approximate and very useful method exists, which seems to have been first used by Herbert C. Roters, in his book Electromagnetic Devices (Reference 12). Although very old, this book is still perhaps the best book ever written on certain aspects of electromagnetic design. The simple but ingenious approach he advocated is usually called Roters' method. Equation (3.5) can be rewritten as follows:
It is not difficult to find a good approximation for the flux-path l by eye. The average cross-sectional area, on the other hand, is very difficult to estimate for complicated three-dimensional shapes. Rotors observed, however, that if the average length were multiplied by the average area, the result should be the real volume of the space in question. The volume of such a space can be found from geometry.
It is then possible to transform Equation (3.16) to:
The new relationship for R' is then Roters' approximation. In practice it is usually within 5% of the actual value and it is rarely as bad as 20% in error. The form given above is not quite the way Roters stated it but is equivalent. He found A from V and l, then used it in Equation (3.16).
Roters then found by experience that even when the volume walls were only approximately parallel to the flux, the approximation still gave excellent results. For example, in Figure 3.2 the corner region (marked 3) cannot be an actual flux-tube since the sharp corners would require an infinite flux-density there. The shape is easy to define geometrically, however and the overall result is found to be a good approximation to the actual reluctance of the gap.
3.5 Numerical calculations of magnetic fields:
The high relative permeability of the ferromagnetic material allows the iron/air interface to be approximated by an equipotential surface. This is true even when the iron is almost saturated. This forms the basis of a number of different approaches to the calculation of the magnetic field in the air regions. The fields are in fact, current sources not potential sources. Following are some of the methods used in plotting electromagnetic fields, both past and present:
1. Curvilinear squares - freehand plotting (Reference 10).
2. Conducting (Teledeltos) paper.
3. Conformal mapping (Schwarz-Christoffel transformations)
4. Numerical methods - finite difference or finite element
The above methods range from the least to the most accurate and from the one that is done freehand to the one that requires a digital computer. With the advent of personal computers and the commercial availability of software, it is appropriate to concentrate on the numerical methods of calculation. The following is some background and typical results from these methods.
3.5.1 Finite difference method:
An analytic iterative technique which may be used either by hand or computer is the method of finite differences. It can be set up on a small personal computer and solved, with very little effort, by persons unskilled in programming.
The vector equivalent of Equation (1.4) is:
These two equations can be combined into the famous vector equation known as Laplace's equation:
This equation describes not only static magnetic fields but also steady flow of incompressible fluids, heat transfer, electrostatics and other areas of importance in engineering. The method to be described will work for problems in any of those fields.
In Cartesian coordinates, Laplace's equation is:
The variable Q is a potential function (a scalar, not a vector), such that its gradient is equal to the vector :
where i, j and k are unit vectors in the x, y and z directions respectively.
This description looks formidable enough but it turns out to have a simple and easily understood interpretation on a grid in space with equal spacings and right-angled corners.
In Figure 3.3, the problem is reduced to two dimensions and the grid is as described. Suppose the potential at each of four points Q(1), Q(2), Q(3) and Q(4) were known, the four points surrounding a point of unknown potential Q(0). We would like to solve for Q(0). The finite-difference equivalent to the partial derivative of Q between Q(1) and Q(0) is:
Then the second partial derivative, at a point halfway between (a) and (b), at (0) is:
In the same manner, we find for the second partial derivative in the vertical (y) direction:
The result states that, for Laplace's equation to hold, the center potential of a close group of five points equally spaced, as shown, is just equal to the average of the potentials of the four surrounding points.
If a volume is laid out on a piece of paper and the potential at each edge point is assigned, then by repeatedly adding up the four potentials surrounding a point, dividing by four and replacing the value at the center point with the result, on either a diamond or a square pattern as may be convenient, the result will converge to a set of numbers describing the field determined by the edge potentials. After convergence, lines of equal potential may be drawn. Lines of flux may then be sketched in, everywhere perpendicular to the potential lines.
The scheme may be implemented on a personal computer by using a spreadsheet. Spreadsheets were written for business applications, of course, not for this sort of thing but most of them, if not all, have the capability to "resolve circular references" by iteration. If one enters a potential at the walls and everywhere else, the formula equivalent to adding up the values in the four surrounding spaces and dividing by four, then the program will just solve each location over and over until it is satisfied with the degree of convergence. When one does this sort of thing by hand, one may use more intelligence and less repetition by using different sizes of squares around the point, different directions of calculation etc. to speed up the result.
It is possible to find the gap reluctance from the potential map. To do this by hand, find the flux crossing any convenient line, by calculating it at each point and integrating (adding up the flux times length):
The original potential difference between poles was known (any convenient value will do; e.g. assigning 100.00 to one pole and 0.00 to the other). The reluctance is then:
To find the reluctance by the spreadsheet program, another method may be more useful. It can be shown (Reference 37) that the reluctance of a volume can be calculated from the following equation:
In the spreadsheet program, a cell is designated to take this sum, calculated from each square in the problem space (after the main iteration is run). From it, the reluctance is found by Equation (3.34) above. Figures 3.4 and 3.5 show typical design problems.
3.5.2 Finite element method:
The feature that makes these methods possible is the ability to calculate a large number of linear algebraic equations that replace the partial differential equations governing a physical system. Most of these equations are second order but linear partial differential equations of the following general form:
The first and third terms are recognizable when compared to the equivalent terms in Equation (3.23). There are, naturally, other types of equation including some that are non-linear, for which several analytical techniques have been developed, (Reference 6).
In these solutions, the fields are discretized into small areas where the nodes of these areas are sufficiently close to each other to minimize the error. These areas can be square, triangular or polygonal. However, triangular areas have been found to give the greatest flexibility when solving complex problems. The result of this discretization is a mesh or lattice and a typical example is shown in Figure 3.6.
The governing equations for this method of analysis are also Equations (3.20) and (3.21). The first, Gauss' law, says that the net gain in flux per unit area (Div B) is always zero. The second, Ampere's law, says that the closed-line integral of H per unit area (curl H) equals the current density of that area.
The solution to the above equations determines the vector potentials and consequently the flux between 2 points.
Other parameters which can be determined from these calculations are:
i. The inductance L from the following:
where is the magnetic energy supplied to the circuit.
ii. The force F from the following:
Programs are now available for use on personal computers to calculate magnetic fields by finite-element methods. Codes using this method of analysis are very long and complicated but the resulting programs are becoming steadily less expensive and easier to use. A model of the problem is built in the computer, i.e. dimensions, material properties, electric currents etc. A triangular mesh is then generated between the points, defining the spaces over which the fields will be approximated. The model-making and mesh-generating methods have been made much easier to use recently. The outer boundary may include "infinite" elements which imitate the effects of an infinitely large empty space around the problem region. Some programs allow entry of non-linear B-H curves, or are able to calculate eddy-current effects.
Studies comparing the finite element and finite difference methods (Reference 13) have shown that the results from these two methods are identical, especially for first order triangular algorithms.