Theory & Practice of Electromagnetic Design of DC Motors & Actuators
George P. Gogue & Joseph J. Stupak, Jr.
Beaverton, OR 97007
Thousands of years ago it was noticed that certain rocks were attracted to iron by some mysterious means. There were deposits of such rock in the area known as Magnesia, in what is now Turkey, and from the name they came to be called magnets. They were also called lodestones, which means "journey stone". If a needle was rubbed on a lodestone and then floated on a piece of wood in water or hung on a string, it would point in a North-South direction. The end which pointed North was called a north-seeking pole, or just a north pole. Since opposite poles attract, it can be seen that the earth's geographic north pole is a magnetic south pole!
If one is unsure where the north pole of a magnet is, the old experiment can still be used, by hanging the magnet on a string and watching how it turns. Avoid nearby automobiles and steel belt buckles. The suspension must have very little torsional stiffness, because the magnet can't exert much torque (a piece of tape is too stiff).
Magnetism was investigated by scientific methods long before electricity was discovered, and "unit poles" were used to describe them. Later, electrical units were introduced, as the interrelationships between them became understood. The metric system came into being, with still different units, and went through several revisions before arriving at its present form (called SI). Today there is a wild jumble of units in common use, which makes it difficult for newcomers to the field. In this seminar, the units presented will be those normally used in engineering in the US today. Moreover, industrial measuring devices indicate in those units and most printed engineering data is given in them. They are somewhat mixed between English, "old" metric and new metric. This is unscientific, but practical. Because of the very great extent of material to be covered in a very short time, there will be little opportunity to show the many elegant mathematical proofs with which the field abounds. They are important, but had to be set aside in order to show as much of the material which is directly useful for design as possible. The interested investigator should then be able to find and study these mathematical developments in the references.
In the early studies of magnetism, it was felt that something was flowing in and out of the magnet poles and it came to be called
flux. The same thing was thought about electricity and in that case it was true. Magnetic flux, however, does not exist, in the physical sense of the motion of matter. It certainly isn't concentrated into "lines", of course, even though we draw lines representing flux flow. The fact that there is a unit of flux called a "line" doesn't help get the concept across. Iron filings on a surface above a magnet tend to gather into paths somewhat randomly, aligned in the direction of flux flow. Nonetheless, flux is not divided into lines.
Even though flux does not exist, it is useful to pretend that it does. There are a number of different "models" or means of description by which magnetics may be considered. They are all somewhat artificial, but are helpful in understanding and predicting magnetic behavior. The older "unit pole" model is one example. It was proposed hundreds of years ago and physicists have been looking for a single magnetic pole (as distinguished from a pole pair) ever since, without much success. This seminar will generally follow the "flux" model. Later on the "potential field" model will be discussed briefly. None of these can be said to be more "correct" than the others; they are just different ways to consider the same thing, and in any given situation, one approach may be easier to understand than another. Each leads to the same conclusion, with different degrees of difficulty.
1.2 Magnetic circuits and the design equations:
A simple electric circuit is shown in Figure 1.1 with a battery, conducting wire and two resistors and . The current in the circuit is determined by the well-known Ohm's law:
In a magnetic circuit, flux is defined as the integral of the magnetic induction or flux-density B times a differential of area normal (at right angles) to the direction of B. The unit of B in US engineering practice is the Gauss (G). The equivalent SI unit, the Tesla, is equal to 10,000 Gauss and is also sometimes used. The induction B is the quantity measured by Gauss meters, and is the magnetic property useful in calculating forces on electric wires, steel pole pieces etc. It is a vector, with a direction as well as strength. If the Gauss meter probe is normal to the flux direction, it will indicate the strength of B; at any other angle it will show a lower value. If turned over, it will show the same value (hopefully!) with sign reversed.
In the electric circuit of Figure 1.1, the same current flowed
through both and . In the magnetic circuit of Figure 1.2, idealized with no leakage flux (that is, all the flux goes through the gaps), the flux crossing gap (1) is , the B's being magnetic induction (assumed constant at every point in the gap) and the A's being the cross-sectional area. The same flux must cross the second gap.
where the integral is taken over a closed surface in space.
In Figure 1.3, part of the flux crossing surface (1) doesn't get to the gap (2), but instead "leaks" by another path (3). Assuming that B is uniform across each of the cross-sectional areas,
If is in a fixed proportion to , this may be written as:
where is a leakage constant, equal to or greater than 1.
In the electric circuit, the sum of the voltages around the circuit must equal zero, i.e.:
A magnet has another basic property at each point called the coercivity or field strength H, measured in Oersted (Oe) in US engineering units. The magnetomotive force acting over a length is defined as the integral of H (i.e., the component of H which is parallel to the path increment dl) times the path length dl, over the entire length l:
Just as the sum of voltages around the electric circuit equals zero, the sum of increments of magnetomotive force around any closed loop in a magnetic circuit, including electric contributions, must equal zero. For the time being, we will assume that there are no electric currents present, therefore:
(the integral being taken around any closed path, not crossed by electric current)
In the magnetic circuit of Figure 1.2, the mmf across the magnet (H times the magnet thickness l) must equal (with sign reversed) the sum of the two mmf's (H.l) across the gaps, if we assume that the pole pieces are perfectly permeable, and therefore present no resistance at all to flux flow. This is a reasonably good first-order assumption, because iron or steel pole pieces at flux-densities well below saturation, conduct flux thousands of times better than free space (or air, which is very nearly the same). Breaking up the closed loop into sections,
Again assuming that the H's are constant along each length increment,
In Figure 1.4, there is only one gap but now the pole material is assumed to be less than perfectly conductive; it has a magnetomotive "drop" across it:
If the pole material can be considered to have a constant permeability independent of flux-density, then the pole mmf drop will be proportional to the mmf across the gap and the relationship may be written as:
The constant is the reluctance coefficient and is equal to or greater than 1. The part of it which is greater than 1 represents the pole losses. In actual magnetic circuits, is unlikely to be much greater than 1; a typical value might be 1.05.
The magnetomotive force over a length is comparable to electromotive force, or voltage, across an element of an electric circuit.
Just as resistance R in an electric circuit is defined as the ratio of current to voltage, a property of magnetic circuits exists, called reluctance, the ratio of mmf to flux. It is written with a script R to distinguish it from electrical resistance:
Compared with Ohm's law, magnetic reluctance is similar to electric resistance, mmf is similar to EMF (voltage) and flux is similar to electric current i.
The definitions of the units of B (Gauss) and H (Oersted) have been arranged so that the permeability of free space is 1 Gauss per Oersted.
This does NOT mean that B and H are the same thing. In air, they are fundamentally different. Carrying the electric analogy a little further, it is as if the definition of resistance were changed so that the resistance of pure water across some volume (say, opposite faces of a 1 inch cube) were one ohm (it isn't, of course).
Inside a magnet or in pole material, the ratio of B to H is not a constant but varies with B in a manner which depends on the material, its temperature and its previous magnetic history. The B-H curves for magnetic materials are supplied by the manufacturers.
All materials have some magnetic effect but for most materials the effect is extremely small, their relative permeability (compared to free space) being 1 to within a few parts per million.
The basic magnetic equations which have been discussed so far may be assembled into two equivalent sets:
The other set of equations are widely used in engineering and may be referred to as the
Magnetic design equations:
Another useful relationship may be derived from the above:
The magnetic design equations, perhaps with the addition of Equation (1.24) above and the B-H curves for magnet and pole materials, are all that is necessary for first-order design of most static magnetic circuits. An example may help to make the process clearer.
1.3 Sample calculation of magnetic flux-density in a gap:
From Equation (1.24) and Figure 1.5:
Therefore, from the magnet B-H curve,
In the gap, B = H numerically; the above calculations are in reasonably good agreement (about 1%), especially since the magnet tolerances on B and H may be as wide as +/-7%. The value of may now be checked. An "average" flux path in the mild steel pole pieces is approximated as shown in Figure 1.6 below:
The maximum flux-density in the steel will be:
We may estimate that about 1/3 of the path will be at this density (0.95 in) and the rest (1.91 in) at 2,000 Gauss. From the steel BH curve, H at 12,000 G is about 5 Oe and at 2,000 G it is about 1.2 Oe. The mmf drop in each pole is then:
mmf/pole pc = (0.95 in X 5 Oe) + (1.91 in X 1.2 Oe) = 7.0 Oe-in
The ratio of drop in the poles to that in the gap is then:
The constant represents the ratio of total mmf (gap plus poles) to
the drop in the gap, so our computed value is:
This difference is too small to materially affect the result. If further accuracy were required, of course, additional iterations could be made to improve the agreement.
1.4 The B-H curves of permanent magnet materials:
In the last example, a B-H curve for the permanent magnet material was needed, representing the relationship within the magnet, at every point, between the coercivity H and the flux-density B. The coercivity is a measure of the magnet's ability to "push" the flux through the resistance (i.e., reluctance) outside the magnet, in the magnetic circuit. Coercivity may also be caused by passing electric current through a coil of wire, as will be discussed later in detail. If an unmagnetized sample of magnetic material is placed in a test fixture in which H can be controlled (by means of current through a coil) and B measured, the state of the material will start out at the origin, as shown in Figure 1.7. As H is increased, B will also gradually increase. following path (o) to (a). As H becomes larger, B increases steeply, then less so and finally levels out to a slope of 1 Gauss/Oersted, which is the same that would result if the material were not present. The magnet is then said to be saturated. If H is now decreased, the curve moves down along another line, with a higher B than before, at points below saturation. When H is reduced to zero in the magnet, B remains at a high value (b), labeled (the subscript "r" is for remanence). The magnet has been permanently magnetized. If H is now increased in the negative direction, the curve moves from the first into the second quadrant of the graph, with positive B supplied by the magnet against a negative external coercivity H. This is the normal operating state of the magnet, supplying flux into a reluctance load. In many of the newer magnet materials, the slope of B to H is constant over some range, at a rate of only slightly more than 1 Gauss per Oersted. This slope is the magnet recoil permeability. After H is sufficiently negative, the curve begins to drop off (point c) and then falls precipitously (d). The region of the downward bend is called the knee of the curve. At the knee and beyond, the magnet is partially magnetized in the reverse direction. The curve crosses the horizontal axis at a point labeled (the "c" is for coercive). Further small increases in H in the negative direction cause large negative changes in B, until the slope again levels off to 1 Gauss/Oersted. The magnet is then saturated in the reverse direction. Relaxing H toward zero traces a curve which is an inverted and reversed copy of the upper curve. The line crosses the vertical axis at . It then moves into the fourth quadrant from the third, along a line with the slope of the recoil permeability (parallel to the line in the second quadrant), then bending upward in a second knee as the magnet remagnetizes in the forward direction again. The curve rises almost vertically to slope over into the upper line in the saturation region again, completing the curve. If H is now cycled back and forth between values large enough to saturate the magnet in both directions, the B-H point in the material will retrace the curve over and over. The curve so described, not including the inside part of the line before initial saturation, is called the major loop. If instead of increasing H
negatively to saturation, it had remained on the straight part of the curve, as shown in Figure 1.7, and if H were then returned toward zero again, the point of B versus H would remain on the major loop. This is the usual behavior of a magnet in operation. On the other hand, if H had been increased negatively somewhat more, moving the B-H point around the knee of the curve, then when H returns toward zero the line does not follow the major loop, but instead moves inward, along a line with the same slope as the major loop, that is, at the recoil permeability slope. The B-H point is now inside the main curve, on what is called a minor loop. There are an infinite number of possible but only one major loop at a given temperature. If the temperature changes, however, the loop may move outward and change shape, depending on the material.
In Figure 1.8, a set of B-H curves are shown versus temperature for a fairly strong ceramic magnet (barium/strontium ferrite) of the type sometimes called M8. The shapes of the magnet and air-gap in a magnetic circuit, with some contribution also from the pole material and shape, determine the B/H load line, as was seen in the example. If the B/H line were at (a) as shown and the magnet cooled from 20°C (68°F) to perhaps -20°C (-4°F), perhaps in shipping, or in cold-weather use, the field in the gap would get stronger (B would increase). On returning to room temperature, the magnet would again be at its original state. If the B/H line had been at (b), however, the knee of the curve would have moved through it, moving the operating point from (i) to (j), Now when the magnet warmed up again, the operating point would move inward, to (k), on a minor loop. It is thus possible for magnets to partially demagnetize themselves due to temperature effects. Use of the magnetic field while cold, such as by turning on a motor, pulling a magnet from an attached piece of steel, and so on would make the demagnetization worse.
1.5 Excursions of the operating point:
The place where the B/H slope line crosses the B-H curve is called the operating point. It represents the particular state of B and H in the magnet at a specific point, whereas the curve itself represents all possible states. If the magnet is completely magnetized everywhere and is in a uniform magnetic field, and if the magnet is at the same temperature everywhere, then the operating point represents every point in the magnet but this is not always the case.
In considering the effects of temperature, the B/H line remained fixed and the B-H curve moved. For most operating conditions, however, the opposite happens; the B-H curve remains fixed and the B/H load line moves. In order to produce almost any useful result from a magnetic device, something must react against the magnetic field, moving the B/H line and the operating point along the curve. Attracting a metal part, or turning a rotor, has the effect of changing the gap dimensions; the B/H line changes slope but continues to pass through the origin. It thus rotates about the origin. On the other hand, if a current-carrying wire is introduced into the gap, a magnetomotive force is caused and an offset to H is created: the B/H line remains at the same slope but shifts sideways. These variations of the B/H line are both shown in Figure 1.8. The effect of either type of change is to move the operating point on the B-H curve between two limiting positions. The magnetic circuit designer must ensure that the operating point stays above the knee of the curve for all design conditions of temperature and operating states, at all points within the magnet. If this is impossible then it is probably best to deliberately "knock down" the magnet, somewhat, by applying an opposing coercive force (current in a coil) or other means, moving its state to a minor loop. The magnet will be less powerful but at least it will not change unexpectedly in service.
A few magnet materials, e.g. Alnico 8, do not have a straight-line region of the B-H curve, as shown in Figure 1.9. If consistent behavior is required in service, they need to be stabilized by moving them to a minor loop, where a straight-line segment of the curve exists. If this is not done, any excursion at all downward on the curve by the operating point will result in decreased field in the gap.
The ferrite material sometimes called M7 and certain forms of samarium-cobalt have curve knees which are actually below the horizontal axis. These materials cannot be accidentally demagnetized by normal temperatures, or by being taken out of pole structures into open space.
1.6 Energy product and maximum energy product:
If the units of B and H are multiplied together, the result is found to be equivalent to energy. It can be shown that the magnetic energy stored in a volume, whether it contains free space, a permanent magnet, or pole material, is equal to:
One consequence of this relationship is that it can be immediately seen that almost all the energy in a magnetic circuit (outside of the magnet itself) is stored in the air-gaps and surrounding space, not in the pole pieces. The induction B is the same in the gap as in the pole parts facing it but because of the high permeability of the poles, H is thousands of times less there.
Curves of constant B x H may be drawn on the B-H curve. These lines are hyperbolas, symmetric about the 45 degree diagonal. The factor (1/2) is dropped for simplicity, since for purposes of comparison, a constant multiplier will not change the result. Some of the hyperbolas do not reach the B-H curve and others intersect it twice. One hyperbola just touches the B-H curve at a single point. The value of (B x H) represented by that curve is called the maximum energy product (MEP) of the magnet material. It is a measure of the greatest energy per unit volume which the magnet material can deliver to the gap, regardless of magnetic circuit shape; so it is a useful comparison parameter. Two different magnet materials of the same volume with differently shaped B-H curves could cause the same field strength in a particular gap shape, if they had the same energy product (ignoring the effects of leakage and pole reluctance). The magnets would in general have different shapes.
The point of maximum energy product on the B-H curve is often taken as the starting point in design, since it represents an approximation to the most efficient use of the magnet material. Excursions about the operating point and temperature effects must also be considered however, along with the location of the magnet knee, to avoid demagnetization. It is often the case that a greater amount of flux is required than can be obtained at the maximum energy product, even if the magnet could be operated safely there; so a higher point on the curve is chosen. When the magnet is immediately beside the gap there is no pole to concentrate the flux. If in addition the magnet is curved and if the outer surface is operating near the MEP, the regions further in must be at a higher flux-density. This will keep the total flux constant.
1.7 Intrinsic and normal B-H curves:
The set of curves described so far are the so-called normal curves. These are the curves to be used for magnetics design. Another set, called the intrinsic curves, are usually published along with them and are used for scientific purposes (Figure 1.10). The intrinsic curves are plots of the same data shown in the normal curves, plus an additional factor of (). The intrinsic curve represents the entire coercive force of the magnet, including that necessary to pass the flux through the magnet itself. The intrinsic curve passes through ; as does the normal curve. Elsewhere in the second quadrant, the intrinsic curve is above the normal curve.
1.8 Magnetic forces on permeable materials:
In Figure 1.11, one pole is free to move into the gap. Let us assume that the field in the gap is uniform and there is no leakage flux elsewhere. From Equation (1.18):
Then Equation (1.25) may be rewritten as:
A virtual change of energy in the gap is caused by an incremental change of gap width dl.
Since work, or energy, equals force times distance,
That is, the force on a pole of permeable material is equal to the integral of a constant times B squared (times a directional vector) times area, over its surface. For a highly permeable material not in saturation, the direction of force will be very nearly normal to the surface. In US engineering units, the relationship is:
The above is referred to as a Maxwell force.
Soft steel or iron saturates at about 20,500 Gauss. Substituting for B in Equation (1.29) we get:
The force is proportional to the square of the flux-density and can, therefore, be quickly estimated for a particular value of flux-density from the value of F above.
The resultant force on the part may actually be less than that calculated above because of the opposing force on the other side of the part. This is true in the case where flux travels through the part without changing its direction or density. The relationship in Equation (1.29) is not limited to steel. Other permeable materials can be used and lower values of force would result, corresponding to their saturation flux-density. See Reference 36 for further analysis of this subject.
For a small thin plate of highly permeable material such as steel, in a large uniform magnetic field oriented perpendicular to the field (since B must be the same on both sides) the forces cancel and the net force on the plate is zero. In a fringing magnetic field, however (field strength decreasing in the direction of plate thickness) the field will be different on opposite sides of a plate of finite width and a net force will result. In some real magnetic circuits, the permeable pole part is large enough compared to the field source that the field is significantly warped by it, perhaps to the point that the field on the side away from the source has such a low field that it may be ignored.
Magnetic fields are used in attracting magnetically permeable objects which are small enough that their fields have a negligible effect on the overall field. An example of this is the separation of nails from wood chips. The attraction force exerted on the small part is not a function of the absolute field strength. It is actually a function of the rate of change of field strength over the length of the part. If a permeable object such as a carbon steel cube were placed in an air-gap where a uniform magnetic field exists, the cube would not be pulled in any direction since it does not experience any force. If the cube were to be moved near the edge of the gap, the non-uniform field would pull it further in from the edge. In a uniform field, the flux crossing the opposing sides of a part, in line with the direction of flux, remains the same. The integral of is the same for both faces of the part but in opposing directions. The forces at both sides, therefore, cancel each other.
A net force is possible only if the field varies over the length of the part in such a way that the flux at one face is different from that at the face opposite to it. Since flux lines are continuous, they must leave the part through some other face. The force exerted on an electrical conductor in a uniform field will be discussed in Section 4.1. If the permeable part is not symmetrical about its own axis, rotational force can be exerted on it.