motor design and power electronics

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Course Contents

CHAPTER 1: MAGNETIC PRINCIPLES1.1 Introduction1.2 Magnetic circuits and the design equations1.3 Sample calculation of magnetic flux design in a gap1.4 The B-H curves of PM materials1.5 Excursions of the operating points

1.6 Energy product and maximum energy product1.7 Intrinsic and normal B-H curves1.8 Magnetic forces on permeable materials

CHAPTER 2: MAGNETIC MATERIALS2.1 Magnetically hard (PM) materials2.2 Magnetically soft materials

CHAPTER 3: FLUX, RELUCTANCE AND PERMEANCE3.1 Intuitive concept of flux3.2 Reluctance and permeance3.3 General formulation of reluctance3.4 Roter's method3.5 Numerical calculations of magnetic fields3.5.1 Finite difference method3.5.2 Finite element method

CHAPTER 4: ELECTROMAGNETICS4.1 Force and emf generation4.2 Transformer operation4.3 Instruments of magnetics

CHAPTER 5: MAGNETIZING OF PERMANENT MAGNETS5.1 Magnetizing requirements5.2 Current vs. time in an ideal magnetizer5.3 Real magnetizers5.4 Optimization

5.5 Other considerations5.6 Forces on conductors and coils5.7 Winding patterns

CHAPTER 6: MOTOR DYNAMICS6.1 Force production6.1.1 Forces between a conductor and steel6.2 Energy considerations6.2.1 The force equation6.3 Torque balance equation6.3.1 Dynamic determination of torque6.3.2 Torque development by 2 fields

CHAPTER 7: MOTOR DESIGN7.1 Introduction7.2 Overall dimensions7 3 Th ti i it

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7.4 Magnet performance7.5 Design features7.6 Motor winding7.7 Winding connections7.8 Motor characteristics7.9 Loss calculation7.10 Armature reaction and demagnetization7.11 Acceleration

7.12 Designing with computers

CHAPTER 8: MANUFACTURING CONSIDERATIONS8.1 Introduction8.2 Laminations8.2.1 Die punching8.2.2 Chemical etching8.2.3 Laser cutting8.3 Stator stack8.4 Winding8.5 Magnet magnetization8.6 Bearing assembly

CHAPTER 9: ELECTRONIC CONTROLLERS9.1 Types of drives9.2 Speed control9.3 Sensorless control

CHAPTER 10: STEPPER MOTORS10.2 Torque characteristics10.3 Electromagnetic principles10.4 Stepper design tips

CHAPTER 11: ACTUATORS11.1.2 Basic principles11.1.3 Shorted turn11.1.4 Equivalent circuit11.1.5 Static magnetic circuit11.1.6 Coil construction11.1.7 Improving linearity11.1.8 Actuator dynamics11.2 Solenoids11.2.1 Introduction11.2.2 First order force calculation11.2.3 Idealized model11.2.4 Bobbin and winding11.2.5 Packing factor11.2.6 Gap location11.2.7 Plunger face shape11.2.8 Remanence and sticking

11.2.9 The plunger-wall flux crossing region11.2.10 Solenoid drive circuit consideration11.2.11 Solenoids operating against springs11.2.12 Constant force variable position solenoid11.2.13 Solenoid actuation speed11.2.14 Some other solenoid types11.2.14.1 AC solenoids11.2.14.2 Rotary solenoids11.2.15 Testing of solenoids11.3 Linear multiphase motors11.4 Other actuators

BIBLIOGRAPHY

LIST OF SYMBOLS

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CHAPTER

MAGNETIC PRINCIPLES

1 Introduction:

housands of years ago it was noticed that certain rocks were attracteo iron by some mysterious means. There were deposits of such rock in

rea known as Magnesia, in what is now Turkey, and from the name they

ame to be called magnets. They were also called lodestones, which meaourney stone". If a needle was rubbed on a lodestone and then floaten a piece of wood in water or hung on a string, it would point in a

orth-South direction. The end which pointed North was called aorth-seeking pole, or just a north pole. Since opposite poles attract

t can be seen that the earth's geographic north pole is a magnetic soole!

f one is unsure where the north pole of a magnet is, the old experime

an still be used, by hanging the magnet on a string and watching how urns. Avoid nearby automobiles and steel belt buckles. The suspension

ust have very little torsional stiffness, because the magnet can't exuch torque (a piece of tape is too stiff).

agnetism was investigated by scientific methods long before electricias discovered, and "unit poles" were used to describe them. Later,

lectrical units were introduced, as the interrelationships between thecame understood. The metric system came into being, with stillifferent units, and went through several revisions before arriving at

ts present form (called SI). Today there is a wild jumble of units inommon use, which makes it difficult for newcomers to the field. In th

eminar, the units presented will be those normally used in engineerinn the US today. Moreover, industrial measuring devices indicate in th

nits and most printed engineering data is given in them. They areomewhat mixed between English, "old" metric and new metric. This is

nscientific, but practical. Because of the very great extent of matero be covered in a very short time, there will be little opportunity t

how the many elegant mathematical proofs with which the field aboundshey are important, but had to be set aside in order to show as much o

he material which is directly useful for design as possible. Thenterested investigator should then be able to find and study these

athematical developments in the references.

n the early studies of magnetism, it was felt that something was flown and out of the magnet poles and it came to be called

lux. The same thing was thought about electricity and in that case itas true. Magnetic flux, however, does not exist, in the physical sens

f the motion of matter. It certainly isn't concentrated into "lines",ourse, even though we draw lines representing flux flow. The fact tha

here is a unit of flux called a "line" doesn't help get the concept

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cross. Iron ngs on a sur ace a ove a magnet ten to gat er ntoaths somewhat randomly, aligned in the direction of flux flow.

onetheless, flux is not divided into lines.

ven though flux does not exist, it is useful to pretend that it does.

here are a number of different "models" or means of description by whagnetics may be considered. They are all somewhat artificial, but areelpful in understanding and predicting magnetic behavior. The older

unit pole" model is one example. It was proposed hundreds of years agnd physicists have been looking for a single magnetic pole (as

istinguished from a pole pair) ever since, without much success. Thiseminar will generally follow the "flux" model. Later on the "potentia

ield" model will be discussed briefly. None of these can be said to bore "correct" than the others; they are just different ways to consid

he same thing, and in any given situation, one approach may be easiernderstand than another. Each leads to the same conclusion, with

ifferent degrees of difficulty.

2 Magnetic circuits and the design equations:

simple electric circuit is shown in Figure 1.1 with a battery,onducting wire and two resistors and . The current in the circuit

etermined by the well-known Ohm's law:

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n a magnetic circuit, flux is defined as the integral of the magnetic

nduction or flux-density B times a differential of area normal (at ringles) to the direction of B. The unit of B in US engineering practic

s the Gauss (G). The equivalent SI unit, the Tesla, is equal to 10,00auss and is also sometimes used. The induction B is the quantity

easured by Gauss meters, and is the magnetic property useful inalculating forces on electric wires, steel pole pieces etc. It is a

ector, with a direction as well as strength. If the Gauss meter probeormal to the flux direction, it will indicate the strength of B; at a

ther angle it will show a lower value. If turned over, it will show tame value (hopefully!) with sign reversed.

n the electric circuit of Figure 1.1, the same current flowed

hrough both and . In the magnetic circuit of Figure 1.2, idealize

ith no leakage flux (that is, all the flux goes through the gaps), thlux crossing gap (1) is , the B's being magnetic induction (assume

onstant at every point in the gap) and the A's being the cross-sectiorea. The same flux must cross the second gap.

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here the integral is taken over a closed surface in space.

n Figure 1.3, part of the flux crossing surface (1) doesn't get to th

ap (2), but instead "leaks" by another path (3). Assuming that B isniform across each of the cross-sectional areas,

f is in a fixed proportion to , this may be written as:

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here is a leakage constant, equal to or greater than 1.

n the electric circuit, the sum of the voltages around the circuit muqual zero, i.e.:

magnet has another basic property at each point called the coercivitr field strength H, measured in Oersted (Oe) in US engineering units.

he magnetomotive force acting over a length is defined as the integraf H (i.e., the component of H which is parallel to the path increment

) times the path length dl, over the entire length l:

ust as the sum of voltages around the electric circuit equals zero, tum of increments of magnetomotive force around any closed loop in a

agnetic circuit, including electric contributions, must equal zero. Fhe time being, we will assume that there are no electric currentsresent, therefore:

the integral being taken around any closed path, not crossed by electurrent)

n the magnetic circuit of Figure 1.2, the mmf across the magnet (H tihe magnet thickness l) must equal (with sign reversed) the sum of the

wo mmf's (H.l) across the gaps, if we assume that the pole pieces areerfectly permeable, and therefore present no resistance at all to flulow. This is a reasonably good first-order assumption, because iron o

teel pole pieces at flux-densities well below saturation, conduct fluhousands of times better than free space (or air, which is very nearl

he same). Breaking up the closed loop into sections,

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gain assuming that the H's are constant along each length increment,

n Figure 1.4, there is only one gap but now the pole material is assuo be less than perfectly conductive; it has a magnetomotive "drop"

cross it:

f the pole material can be considered to have a constant permeability

ndependent of flux-density, then the pole mmf drop will be proportiono the mmf across the gap and the relationship may be written as:

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he constant is the reluctance coefficient and is equal to or great

han 1. The part of it which is greater than 1 represents the pole

osses. In actual magnetic circuits, is unlikely to be much greaterhan 1; a typical value might be 1.05.

he magnetomotive force over a length is comparable to electromotive

orce, or voltage, across an element of an electric circuit.

ust as resistance R in an electric circuit is defined as the ratio ofurrent to voltage, a property of magnetic circuits exists, called

eluctance, the ratio of mmf to flux. It is written with a script R toistinguish it from electrical resistance:

ompared with Ohm's law, magnetic reluctance is similar to electricesistance, mmf is similar to EMF (voltage) and flux is similar to

lectric current i.

he definitions of the units of B (Gauss) and H (Oersted) have been

rranged so that the permeability of free space is 1 Gauss per Oersted

his does NOT mean that B and H are the same thing. In air, they areundamentally different. Carrying the electric analogy a little furthe

t is as if the definition of resistance were changed so that theesistance of pure water across some volume (say, opposite faces of a

nch cube) were one ohm (it isn't, of course).

nside a magnet or in pole material, the ratio of B to H is not a

onstant but varies with B in a manner which depends on the material, emperature and its previous magnetic history. The B-H curves for

agnetic materials are supplied by the manufacturers.

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ll materials have some magnetic effect but for most materials the effs extremely small, their relative permeability (compared to free spac

eing 1 to within a few parts per million.

he basic magnetic equations which have been discussed so far may be

ssembled into two equivalent sets:

heoretical equations:

he other set of equations are widely used in engineering and may beeferred to as the

agnetic design equations:

nother useful relationship may be derived from the above:

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he magnetic design equations, perhaps with the addition of Equation

1.24) above and the B-H curves for magnet and pole materials, are allhat is necessary for first-order design of most static magnetic

ircuits. An example may help to make the process clearer.

3 Sample calculation of magnetic flux-density in a gap:

rom Equation (1.24) and Figure 1.5:

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herefore, from the magnet B-H curve,

n the gap, B = H numerically; the above calculations are in reasonabl

ood agreement (about 1%), especially since the magnet tolerances on Bnd 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 ashown in Figure 1.6 below:

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he maximum flux-density in the steel will be:

e may estimate that about 1/3 of the path will be at this density (0.n) and the rest (1.91 in) at 2,000 Gauss. From the steel BH curve, H

2,000 G is about 5 Oe and at 2,000 G it is about 1.2 Oe. The mmf dropach pole is then:

mf/pole pc = (0.95 in X 5 Oe) + (1.91 in X 1.2 Oe) = 7.0 Oe-in

he ratio of drop in the poles to that in the gap is then:

he constant represents the ratio of total mmf (gap plus poles) to

he drop in the gap, so our computed value is:

his difference is too small to materially affect the result. If furthccuracy were required, of course, additional iterations could be made

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mprove t e agreement.

4 The B-H curves of permanent magnet materials:

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n the last example, a B-H curve for the permanent magnet material was

eeded, representing the relationship within the magnet, at every poinetween the coercivity H and the flux-density B. The coercivity is a

easure of the magnet's ability to "push" the flux through the resistai.e., reluctance) outside the magnet, in the magnetic circuit.

oercivity may also be caused by passing electric current through a cof wire, as will be discussed later in detail. If an unmagnetized samp

f magnetic material is placed in a test fixture in which H can beontrolled (by means of current through a coil) and B measured, the st

f the material will start out at the origin, as shown in Figure 1.7. is increased, B will also gradually increase. following path (o) to

a). As H becomes larger, B increases steeply, then less so and finallevels out to a slope of 1 Gauss/Oersted, which is the same that wouldesult if the material were not present. The magnet is then said to be

aturated. If H is now decreased, the curve moves down along anotherine, with a higher B than before, at points below saturation. When H

educed to zero in the magnet, B remains at a high value (b), labeled

the subscript "r" is for remanence). The magnet has been permanently

agnetized. If H is now increased in the negative direction, the curve

oves from the first into the second quadrant of the graph, with positsupplied by the magnet against a negative external coercivity H. This the normal operating state of the magnet, supplying flux into aeluctance load. In many of the newer magnet materials, the slope of B

is constant over some range, at a rate of only slightly more than 1auss per Oersted. This slope is the magnet recoil permeability. After

s sufficiently negative, the curve begins to drop off (point c) and talls precipitously (d). The region of the downward bend is called the

nee of the curve. At the knee and beyond, the magnet is partiallyagnetized in the reverse direction. The curve crosses the horizontal

xis at a point labeled (the "c" is for coercive). Further smallncreases in H in the negative direction cause large negative changes

until the slope again levels off to 1 Gauss/Oersted. The magnet ishen saturated in the reverse direction. Relaxing H toward zero tracesurve which is an inverted and reversed copy of the upper curve. The l

rosses the vertical axis at . It then moves into the fourth quadran

rom the third, along a line with the slope of the recoil permeability

parallel to the line in the second quadrant), then bending upward in econd knee as the magnet remagnetizes in the forward direction again.

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e curve r ses a most vert ca y to s ope over nto t e upper ne nhe saturation region again, completing the curve. If H is now cycled

ack and forth between values large enough to saturate the magnet in birections, the B-H point in the material will retrace the curve over

ver. The curve so described, not including the inside part of the linefore initial saturation, is called the major loop. If instead of

ncreasing H

egatively to saturation, it had remained on the straight part of theurve, as shown in Figure 1.7, and if H were then returned toward zero

gain, the point of B versus H would remain on the major loop. This ishe usual behavior of a magnet in operation. On the other hand, if H h

een increased negatively somewhat more, moving the B-H point around tnee of the curve, then when H returns toward zero the line does not

ollow the major loop, but instead moves inward, along a line with theame slope as the major loop, that is, at the recoil permeability slop

he B-H point is now inside the main curve, on what is called a minoroop. There are an infinite number of possible but only one major loop

given temperature. If the temperature changes, however, the loop mayove outward and change shape, depending on the material.

n Figure 1.8, a set of B-H curves are shown versus temperature for a

airly strong ceramic magnet (barium/strontium ferrite) of the typeometimes called M8. The shapes of the magnet and air-gap in a magneti

ircuit, with some contribution also from the pole material and shape,etermine the B/H load line, as was seen in the example. If the B/H li

ere at (a) as shown and the magnet cooled from 20C (68F) to perhaps20C (-4F), perhaps in shipping, or in cold-weather use, the field i

he gap would get stronger (B would increase). On returning to room

emperature, the magnet would again be at its original state. If the Bine had been at (b), however, the knee of the curve would have movedhrough it, moving the operating point from (i) to (j), Now when theagnet warmed up again, the operating point would move inward, to (k),

minor loop. It is thus possible for magnets to partially demagnetizehemselves due to temperature effects. Use of the magnetic field while

old, such as by turning on a motor, pulling a magnet from an attachediece of steel, and so on would make the demagnetization worse.

5 Excursions of the operating point:

he place where the B/H slope line crosses the B-H curve is called theperating point. It represents the particular state of B and H in the

agnet at a specific point, whereas the curve itself represents allossible states. If the magnet is completely magnetized everywhere and

n a uniform magnetic field, and if the magnet is at the same temperatverywhere, then the operating point represents every point in the mag

ut this is not always the case.

n considering the effects of temperature, the B/H line remained fixed

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n t e B-H curve move . For most operat ng con t ons, owever, t epposite happens; the B-H curve remains fixed and the B/H load line

oves. In order to produce almost any useful result from a magneticevice, something must react against the magnetic field, moving the B/

ine and the operating point along the curve. Attracting a metal part,urning a rotor, has the effect of changing the gap dimensions; the B/

ine changes slope but continues to pass through the origin. It thusotates about the origin. On the other hand, if a current-carrying wir

s introduced into the gap, a magnetomotive force is caused and an offo H is created: the B/H line remains at the same slope but shiftsideways. These variations of the B/H line are both shown in Figure 1.

he effect of either type of change is to move the operating point on -H curve between two limiting positions. The magnetic circuit designe

ust ensure that the operating point stays above the knee of the curveor all design conditions of temperature and operating states, at all

oints within the magnet. If this is impossible then it is probably beo deliberately "knock down" the magnet, somewhat, by applying an

pposing coercive force (current in a coil) or other means, moving its

tate to a minor loop. The magnet will be less powerful but at least iill not change unexpectedly in service.

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few magnet materials, e.g. Alnico 8, do not have a straight-line reg

f the B-H curve, as shown in Figure 1.9. If consistent behavior isequired in service, they need to be stabilized by moving them to a mi

oop, where a straight-line segment of the curve exists. If this is noone, any excursion at all downward on the curve by the operating poin

ill result in decreased field in the gap.

he ferrite material sometimes called M7 and certain forms ofamarium-cobalt have curve knees which are actually below the horizont

xis. These materials cannot be accidentally demagnetized by normalemperatures, or by being taken out of pole structures into open space

6 Energy product and maximum energy product:

f the units of B and H are multiplied together, the result is found te equivalent to energy. It can be shown that the magnetic energy stor

n a volume, whether it contains free space, a permanent magnet, or poaterial, is equal to:

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ne consequence of this relationship is that it can be immediately seehat almost all the energy in a magnetic circuit (outside of the magne

tself) is stored in the air-gaps and surrounding space, not in the po

ieces. The induction B is the same in the gap as in the pole partsacing it but because of the high permeability of the poles, H ishousands of times less there.

urves of constant B x H may be drawn on the B-H curve. These lines aryperbolas, symmetric about the 45 degree diagonal. The factor (1/2) i

ropped for simplicity, since for purposes of comparison, a constantultiplier will not change the result. Some of the hyperbolas do noteach the B-H curve and others intersect it twice. One hyperbola just

ouches the B-H curve at a single point. The value of (B x H) represen

y that curve is called the maximum energy product (MEP) of the magnetaterial. It is a measure of the greatest energy per unit volume whichhe magnet material can deliver to the gap, regardless of magnetic

ircuit shape; so it is a useful comparison parameter. Two differentagnet materials of the same volume with differently shaped B-H curves

ould cause the same field strength in a particular gap shape, if theyad the same energy product (ignoring the effects of leakage and pole

eluctance). The magnets would in general have different shapes.

he point of maximum energy product on the B-H curve is often taken as

he starting point in design, since it represents an approximation to ost efficient use of the magnet material. Excursions about the operatoint and temperature effects must also be considered however, along w

he location of the magnet knee, to avoid demagnetization. It is oftenhe case that a greater amount of flux is required than can be obtaine

t the maximum energy product, even if the magnet could be operatedafely there; so a higher point on the curve is chosen. When the magne

s immediately beside the gap there is no pole to concentrate the fluxf in addition the magnet is curved and if the outer surface is operat

ear the MEP, the regions further in must be at a higher flux-density.

his will keep the total flux constant.

7 Intrinsic and normal B-H curves:

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he set of curves described so far are the so-called normal curves. Thre the curves to be used for magnetics design. Another set, called th

ntrinsic curves, are usually published along with them and are used fcientific purposes (Figure 1.10). The intrinsic curves are plots of t

ame data shown in the normal curves, plus an additional factor of (

he intrinsic curve represents the entire coercive force of the magnet

ncluding that necessary to pass the flux through the magnet itself. Tntrinsic curve passes through ; as does the normal curve. Elsewhere

he second quadrant, the intrinsic curve is above the normal curve.

8 Magnetic forces on permeable materials:

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n Figure 1.11, one pole is free to move into the gap. Let us assume t

he field in the gap is uniform and there is no leakage flux elsewhererom Equation (1.18):

hen Equation (1.25) may be rewritten as:

virtual change of energy in the gap is caused by an incremental

hange of gap width dl.

ince work, or energy, equals force times distance,

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hat is, the force on a pole of permeable material is equal to the

ntegral of a constant times B squared (times a directional vector) tirea, over its surface. For a highly permeable material not inaturation, the direction of force will be very nearly normal to the

urface. In US engineering units, the relationship is:

he above is referred to as a Maxwell force.

oft steel or iron saturates at about 20,500 Gauss. Substituting for Bquation (1.29) we get:

he force is proportional to the square of the flux-density and can,herefore, be quickly estimated for a particular value of flux-density

rom the value of F above.

he resultant force on the part may actually be less than that calcula

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ove ecause o t e oppos ng orce on t e ot er s e o t e part. Ts true in the case where flux travels through the part without changi

ts direction or density. The relationship in Equation (1.29) is notimited to steel. Other permeable materials can be used and lower valu

f force would result, corresponding to their saturation flux-density.ee Reference 36 for further analysis of this subject.

or a small thin plate of highly permeable material such as steel, in

arge uniform magnetic field oriented perpendicular to the field (sincust be the same on both sides) the forces cancel and the net force on

he plate is zero. In a fringing magnetic field, however (field strengecreasing in the direction of plate thickness) the field will be

ifferent on opposite sides of a plate of finite width and a net forceill result. In some real magnetic circuits, the permeable pole part i

arge enough compared to the field source that the field is significanarped by it, perhaps to the point that the field on the side away fro

he source has such a low field that it may be ignored.

agnetic fields are used in attracting magnetically permeable objects

hich are small enough that their fields have a negligible effect on tverall field. An example of this is the separation of nails from woodhips. The attraction force exerted on the small part is not a functio

f the absolute field strength. It is actually a function of the rate hange of field strength over the length of the part. If a permeable

bject such as a carbon steel cube were placed in an air-gap where aniform magnetic field exists, the cube would not be pulled in any

irection since it does not experience any force. If the cube were to oved near the edge of the gap, the non-uniform field would pull it

urther in from the edge. In a uniform field, the flux crossing the

pposing sides of a part, in line with the direction of flux, remains ame. The integral of is the same for both faces of the part but

pposing directions. The forces at both sides, therefore, cancel eachther.

net force is possible only if the field varies over the length of thart in such a way that the flux at one face is different from that at

he face opposite to it. Since flux lines are continuous, they must lehe part through some other face. The force exerted on an electrical

onductor in a uniform field will be discussed in Section 4.1. If the

ermeable part is not symmetrical about its own axis, rotational forcean be exerted on it.

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CHAPTER

MAGNETIC MATERIALS

1 Magnetically hard (permanent magnet) materials:

efore the 1930's, the only available magnet material other thanodestone was hardened steel. Since steel with a high carbon contentardened by heat-treatment would retain its magnetism, whereas soft or

ild steel with a low carbon content would not, magnetic materials camo be called "hard" or "soft" depending on whether they would retain aermanent magnetic field or not. The names "hard" and "soft" as aescription of magnetic properties have remained, even though some modagnetically soft materials are very hard mechanically, and someelatively soft materials are magnetically hard.

ardened steel has a very high (high flux) but very little coercivit

), as shown in Figure 1.9. Reluctance to flux flow (in the surroundi

pace) is less for a long, thin rod which is axially magnetized than fshorter, thicker part. In order to avoid self-demagnetization owing hape, i.e. a B/H slope line behind the knee of the curve, magnets hade long and thin, like the traditional compass needle. In the early930's, various magnetic materials were found which had lower , but

reater coercivity and maximum energy product than steel. The mostuccessful of these were the alnico magnets. The name "alnico" is derirom the constituent metals aluminum, nickel and cobalt. Whereas a steagnet might need to have an aspect ratio (length to width) of 50:1 tovoid losing its field in air, an alnico magnet might be safe atomewhere between 3:1 and 10:1, depending on the type used and its

ntended service. By comparison, newer materials such as ferrites,amarium-cobalt, and neodymium-iron have such high resistance toemagnetization from shape effects in air that sometimes personsxperienced in only these materials have never encountered the effect o not realize that the hazard exists. It is nonetheless possible toartially demagnetize some of these materials by shape effect, e.g. bysing large, thin magnets magnetized through the thickness, or byadially magnetizing a long, thin-walled tube with few poles. The B/Hlopes resulting in magnets as a function of several shapes andength:width ratio are shown in Figure 2.1 (Ref. 41). If the slope

esults in the operating point being behind the knee of the curve for articular material intended, then that magnet may work well in aagnetic circuit (with the magnet magnetized in place, or transferrednto the circuit with the aid of "keeper" flux shunts); however, if thagnet is taken out for even an instant and then replaced, the field ihe gap will be less than before.

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lnico is a hard, brittle material, which has to be ground to shape iflose tolerances on dimensions are required. Ceramic (ferrite) magnetsre extremely hard, so much so that they are cut only very slowly byungsten-carbide or silicon-carbide. They must be ground or cut to shaith diamond tools.

amarium-cobalt magnets are also difficult to cut and can be made onlyimited sizes and shapes (in solid form). Neodymium-iron when cut,roduces a powder which is easily ignited by the heat caused by the tohich makes the process hazardous.

errite, samarium-cobalt and neodymium-iron are also available in powdorm, which is either pressed to shape with a binder, or molded with aarrier plastic. Bonded magnets made by either process are less powerfhan the solid forms of the same material but can be made into a muchider range of shapes at lower cost. The final part may be relativelyigid, bonded with materials like nylon 6-6, or relatively flexible,onded with rubber.

ome of the bonded materials are isotropic (having the same magneticroperties in all directions) and are machineable. Some bonded productnd most solid magnet materials have a preferred direction ofagnetization (but with either N-S or S-N alignment allowed). Resistano magnetization in other directions may be extremely high. For examplhe coercive force required to completely magnetize a particular ferrin the preferred direction is about 10,000 G. The same material was no

ffected at all when exposed to a field of 20,000 G in a transverseirection and it reportedly requires 100,000 G to magnetize it in thatirection.

he properties of permanent magnet materials are given in Table 2.1.hese are typical values only, since the actual properties vary betweeanufacturers.

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otes on the above table:

All materials have a preferred direction of magnetization unless nos isotropic.

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T erma expans on or some mater a s s non- sotrop c. I two gurre given, "P" data is for direction parallel to preferred direction oagnetization and "T" data is for directions transverse to the preferrirection of magnetization.

Data are from various sources, of unknown accuracy and are given asxamples only. Design data should be obtained from the manufacturer.xcept for temperatures, data are at 20C (68F).

f magnetic materials are heated above a characteristic temperaturealled the Curie temperature, they suddenly lose all magnetism. If there then cooled to room temperature, they are found to be completelyemagnetized but otherwise unaffected. It may not be possible to use tethod to demagnetize bonded magnets because the required temperature igh enough to destroy the bonding material but it is an excellent wayemagnetize solid magnet materials without coating or adhesives. For saterials it may be advisable to heat the parts in an inert atmosphererevent corrosion.

errite magnets sometimes continue to lose small amounts of powder froheir surfaces. They are brittle and easily chipped. Samarium-cobalt mccasionally spall off tiny bits of material. In order to avoid possibontamination and to help protect against surface and edge damage,agnets are sometimes coated. The surfacing materials used are oftenpoxies of polyurethane and the coatings may be as thin as a fewen-thousandths of an inch (0.0002) up to perhaps five thousandths of nch (0.005). Ferrites may also develop cracks in manufacturing which ot affect their magnetic performance but reduce their strength andesistance to spalling. The cracks may be filled with epoxy. Some magn

aterials are subject to corrosion and oxidation which a coating mayrevent.

number of other magnetic materials are known, besides the onesiscussed. Some of them have been in volume production in the past butre rarely used today, because other materials are less expensive or hetter properties. Among them are cunife (copper-nickel-iron), which ce machined and cold-worked; cunico (copper-nickel-cobalt); silmanalsilver-nickel-aluminum), a material with great resistance toemagnetization; and vicalloy (iron-cobalt-vanadium), which isachineable.

eferences 1-9 are recommended for further reading on permanent magnetnd their properties.

2 Magnetically soft materials:

agnetically soft materials have a very narrow B-H curve and have veryittle remanence, which means that after an applied magnetic field isemoved, very little flux remains in the circuit. The behavior of thesaterials is variable and significant over such a wide range that thei

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-H curves are usua y p otte w t sem - og sca es, as s own n F gur2 (H plotted on the horizontal scale, logarithmically). Since theaterials are reluctance rather than coercive force sources, B and H aoth plotted as positive (in the first quadrant). There being very litr no difference between the rising and falling curves, the plot is aingle line. Plots are normally made of B versus H; occasionally plotsermeability versus B or H are also made. Maximum permeability does

sually occur at the beginning of the curve (very low H) but somewhere

he middle. For very low-carbon steel, maximum permeability may occur pproximately 7000 G and may be 3000 or so; initial permeability (at 2auss) is about 200. Permeability in this case means total B divided botal H. Differential permeability, on the other hand, is the ratio ofhange of B to change of H, which is the slope of the B-H curve.ifferential permeability is of interest when small variations (perhapaused electrically) are imposed over a steady magnetic field.

here is a big difference between choosing a pole material to carry aaximum amount of flux (high saturation flux-density) and in choosing

or high permeability. To specify high permeability, the maximumoercivity must also be known. For very low coercivity (to perhaps 0.0ersted or so) a material like Supermalloy may be best, with aermeability at this level as high as 200,000. This alloy saturates atess than 8,000 G however. For high flux-density, Vanadium Permandur iuperior. Its initial permeability is only about 1,100 but it saturatet 23,000 G (compared to 20,500 G for low-carbon steel).

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igure 2.2: B-H Curves

otes on above Figure:

The curves shown are intrinsic. The flux-densities shown represent ncrease in field strength caused by the material, not the total fieldo get the normal curve (used for design), add the value of B found fr

he curve to the value of (H times 1 Gauss/oersted). For example, foranadium-Permandur at 1000 Oe, the curve shows 22,600 G. The flux-densn an immediately adjacent gap would be 23,600 G.

The data shown is plotted from information from various sources, ans of unknown accuracy. It is given for illustrative purposes only.esign data should be obtained from manufacturer.

The materials are shown in their highest state of permeability. Somaterials require special heat-treatment to reach their best magnetic

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ropert es, an may e a verse y a ecte su sequent mac n ng anorming is done. The 3% silicon-iron sheet is anisotropic, and bestagnetic properties are obtained in only one direction.

n order to help suppress electrical eddy currents which slow downagnetic field strength changes and waste energy, high electricalesistance is desirable. Adding silicon to iron (up to 5%) increases tlectrical resistance but reduces the flux saturation density. It alsoakes the material brittle and hard to form or machine. Silicon-iron iidely used in transformers, electronic chokes and electrical motorsRefs. 42 & 44).

any of the magnetic alloys require heat-treatment (annealing) to obtahe best magnetic properties. The effect is reduced or lost by machinir forming however, so it must be repeated after these operations. Somaterials require heat-treatment in a hydrogen atmosphere to obtain thest properties, a somewhat dangerous process. If this is not possiblereatment may be done in other gas mixtures, producing results which aot optimal but are greatly improved over the untreated state (Referen

1).

esides iron, the metals nickel and cobalt are ferromagnetic and highlermeable. Pure nickel saturates at 6080 Gauss which makes it useful occasion for calibration purposes.

ickel plating can be either magnetic or non-magnetic, depending on thpplication process. Magnetic nickel is often used to protect pole sterom corrosion, with minimum added reluctance.

he 300 Series stainless steels (302, 303, 304, 316 and others) which

ontain significant amounts of nickel (8% to 22%), as well as chrome (o 24%) are usually austinitic in structure and are not considered to agnetic. A magnet is sometimes used in machine shops to test stainlesteel versus other steels. The 300 Series stainless materials may becoagnetically permeable as the result of cold-working as, for example, appen to cold-forged bolts. Certain special types (329, 355) may beither austinitic or martensitic(and therefore magnetically impermeablr permeable) depending on heat treatment.

he 400 Series stainless steels contain chrome but little or no nickel

hey are martensitic, hardenable by heat-treatment and magneticallyermeable. Because they can be hardened, 400 Series stainless steels aften used for tools, instruments and bearing surfaces.

he B-H curves of magnetic materials change with temperature to varyinegrees, with flux density and permeability reducing with increasedemperature. Certain iron alloys containing a high percentage of nickearound 30%) have permeabilities that are very strongly affected byemperature. For example, one material changes its permeability by abofactor of 2 between -40F and 200F, at 46 Oe coercive force. The

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ar at on o permea ty s a so a unct on o coerc v ty; t s samelloy increases its permeability at an even greater rate over the sameange, at a low coercivity (0.2 Oe). It is possible to use these alloyo compensate for variations in the strength of permanent magnets withemperature. The magnet is designed to produce more flux than is requin the gap and the extra flux is shunted around the gap with a sectionhe compensator material. If the temperature increases, the magnet outill decrease but less flux will be passed by the shunt. With proper

esign, the resulting flux in the gap can be made to be very nearlyonstant over a wide range of temperature.

lthough the B-H curves of pole materials are usually drawn as a singline, there is in fact a small difference between the rising and falliurves. The area traced out in B-H space represents energy lost perycle, which is called hysteresis loss. Although the loss per cycle isery small, the power dissipated as heat over time, in devices subjecto constant cycling (such as motors and transformers) can be significaddy current losses in these materials are also important and the twodded together are termed core loss. The amount of core loss, in unitsatts per pound, is supplied by the manufacturers of these materials, function of frequency (f), lamination thickness (t) and maximumlux-density (B). An empirical formula for the core loss in poleaterials is:

he first term represents the hysteresis loss and the coefficient x isalled the Steinmetz coefficient (after Charles Proteus Steinmetz, whoiscovered the relationship). It varies from about 1.5 to 2.5, dependin the material; 1.6 is often used. The second term represents eddyurrent loss. The constants and depend on the material. If the po

oss is known for several different conditions, the constants can beound and then the power losses at other frequencies, materialhicknesses and flux-densities can be calculated.

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CHAPTER

FLUX, RELUCTANCE AND PERMEANCE

1 Intuitive concepts of flux:

lux lines may be thought of as being something like thick rubber bandhey pull along their lengths and push outward sideways.

lux lines leave smooth surfaces of highly permeable material (below

aturation) at right angles to the surface and enter the same way. Iflux left at a different angle, there would have to be a component of n the pole material parallel to the surface, as large as the componenn air. The component of flux in that direction would then have to beery large because of the high permeability of the material.

t sharp corners, flux leaves or enters on a line bisecting the angle.lux-density leaving a sharp corner tends to be greater than at nearlymooth surfaces. Flux going around a corner tends to "crowd in" towardhe inside of the curve (the rubber-band analogy). Where flux lines ar

lose together, magnetic forces are high and where they are far apart,orces are lower. Flux lines cannot cross each other and must close onhemselves into loops. Flux lines in opposite directions at the sameoint (from two different sources) cancel and flux lines in the sameirection add. Flux lines at an angle to each other add vectorially.

2 Reluctance and permeance

he basic equation of magnetic circuits stated earlier, Equation (1.15s:

he reluctance R has not yet been discussed in detail. Its calculationn general one of the most difficult areas of magnetics. Many magneticesigners avoid the work, relying instead on intuition and experience hese may be misleading. In spite of its complexity, a carefulonsideration of reluctance and its methods of calculation are essentio successful magnetics design.

sing the magnetics design equations for the circuit of Figure 1.4, wi

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n orm ux- ens ty n t e gap an assum ng per ect po e mater a w

nfinite permeability) and no leakage of flux, i.e., =kl=1, then:

n the gap, from Equation (1.18)

he reluctance of a uniform gap, without leakage, is therefore:

his result may be compared to the calculation of the electricalesistance of a wire or bar of uniform cross-section,

here is the wire material conductivity, l the length and A is the wi

ross-sectional area.

t is often necessary to add reluctances in parallel, since flux in a

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agnet c c rcu t may ow n var ous ea age pat s, as we as t e usene. It can be shown, by considering two parallel flux-paths, with fluriven by the same mmf, that the equivalent reluctance for twoeluctances in parallel is:

ore conveniently, let the permeance P be defined as the reciprocal ofeluctance,

hen the formula for combined parallel reluctances, Equation (3.7)ecomes:

he combination of reluctances in series, with a common flux-path,esults in:

3 General formulation of reluctance:

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or more complicated reluctance volumes, where the flux lines may not arallel and where the flux-density varies within the volume, a moreeneral relationship is needed. In Figure 3.1 the volume may beonsidered as being broken up into flux-tubes. A flux-tube is an

maginary closed wall in space, which is everywhere parallel to theirection of flux at its surface, so that no flux crosses the wall. Thross-section of the flux-tube is chosen small enough that variations across the tube may be neglected. The reluctance of the volume is th

here A is the cross-sectional area perpendicular to the flux-path len

of which dl is an increment. The reluctances of all the tubes are thummed in parallel.

s an example, the reluctance of an annular gap will be calculated. Suaps occur in linear actuators, permanent-magnet brushless DC motors aome types of speakers, for example.

ince the flux is radial and is the same at any radius,

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he area is:

4 Roters' method:

n many practical cases it is very difficult to accomplish the requirentegrations by analytical means. An approximate and very useful methoxists, which seems to have been first used by Herbert C. Roters, in hook Electromagnetic Devices (Reference 12). Although very old, this bs still perhaps the best book ever written on certain aspects of

lectromagnetic design. The simple but ingenious approach he advocatedsually called Roters' method. Equation (3.5) can be rewritten asollows:

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t is not difficult to find a good approximation for the flux-path l byye. The average cross-sectional area, on the other hand, is veryifficult to estimate for complicated three-dimensional shapes. Rotorsbserved, however, that if the average length were multiplied by theverage area, the result should be the real volume of the space inuestion. The volume of such a space can be found from geometry.

t is then possible to transform Equation (3.16) to:

he new relationship for R' is then Roters' approximation. In practices usually within 5% of the actual value and it is rarely as bad as 20n error. The form given above is not quite the way Roters stated it b

s equivalent. He found A from V and l, then used it in Equation (3.16

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oters then found by experience that even when the volume walls were opproximately parallel to the flux, the approximation still gavexcellent results. For example, in Figure 3.2 the corner region (marke cannot be an actual flux-tube since the sharp corners would requirenfinite flux-density there. The shape is easy to define geometricallyowever and the overall result is found to be a good approximation to ctual reluctance of the gap.

5 Numerical calculations of magnetic fields:

he high relative permeability of the ferromagnetic material allows thron/air interface to be approximated by an equipotential surface. This true even when the iron is almost saturated. This forms the basis oumber of different approaches to the calculation of the magnetic fieln the air regions. The fields are in fact, current sources not potentources. Following are some of the methods used in plottinglectromagnetic fields, both past and present:

Curvilinear squares - freehand plotting (Reference 10).

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Con uct ng Te e e tos paper.

Conformal mapping (Schwarz-Christoffel transformations)

Numerical methods - finite difference or finite element

he above methods range from the least to the most accurate and from tne that is done freehand to the one that requires a digital computer.ith the advent of personal computers and the commercial availability

oftware, it is appropriate to concentrate on the numerical methods ofalculation. The following is some background and typical results fromhese methods.

5.1 Finite difference method:

n analytic iterative technique which may be used either by hand oromputer is the method of finite differences. It can be set up on a smersonal computer and solved, with very little effort, by personsnskilled in programming.

he vector equivalent of Equation (1.4) is:

hese two equations can be combined into the famous vector equation kns Laplace's equation:

his equation describes not only static magnetic fields but also steadlow of incompressible fluids, heat transfer, electrostatics and other

reas of importance in engineering. The method to be described will woor problems in any of those fields.

n Cartesian coordinates, Laplace's equation is:

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he variable Q is a potential function (a scalar, not a vector), suchhat its gradient is equal to the vector :

here i, j and k are unit vectors in the x, y and z directionsespectively.

his description looks formidable enough but it turns out to have aimple and easily understood interpretation on a grid in space with eqpacings and right-angled corners.

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n Figure 3.3, the problem is reduced to two dimensions and the grid is described. Suppose the potential at each of four points Q(1), Q(2),(3) and Q(4) were known, the four points surrounding a point of unknootential Q(0). We would like to solve for Q(0). The finite-differencequivalent to the partial derivative of Q between Q(1) and Q(0) is:

hen the second partial derivative, at a point halfway between (a) andb), at (0) is:

n the same manner, we find for the second partial derivative in theertical (y) direction:

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e resu t states t at, or Lap ace s equat on to o , t e centerotential of a close group of five points equally spaced, as shown, isust equal to the average of the potentials of the four surroundingoints.

f a volume is laid out on a piece of paper and the potential at eachdge point is assigned, then by repeatedly adding up the four potentiaurrounding a point, dividing by four and replacing the value at theenter point with the result, on either a diamond or a square pattern ay be convenient, the result will converge to a set of numbersescribing the field determined by the edge potentials. Afteronvergence, lines of equal potential may be drawn. Lines of flux mayhen be sketched in, everywhere perpendicular to the potential lines.

he scheme may be implemented on a personal computer by using apreadsheet. Spreadsheets were written for business applications, ofourse, not for this sort of thing but most of them, if not all, have apability to "resolve circular references" by iteration. If one enterotential at the walls and everywhere else, the formula equivalent to

dding up the values in the four surrounding spaces and dividing by fohen the program will just solve each location over and over until it atisfied with the degree of convergence. When one does this sort ofhing by hand, one may use more intelligence and less repetition by usifferent sizes of squares around the point, different directions ofalculation etc. to speed up the result.

t is possible to find the gap reluctance from the potential map. To dhis by hand, find the flux crossing any convenient line, by calculatit at each point and integrating (adding up the flux times length):

he original potential difference between poles was known (any convenialue will do; e.g. assigning 100.00 to one pole and 0.00 to the otherhe reluctance is then:

o find the reluctance by the spreadsheet program, another method may ore useful. It can be shown (Reference 37) that the reluctance of a

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o ume can e ca cu ate rom t e o ow ng equat on:

n the spreadsheet program, a cell is designated to take this sum,alculated from each square in the problem space (after the mainteration is run). From it, the reluctance is found by Equation (3.34)bove. Figures 3.4 and 3.5 show typical design problems.

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5.2 Finite element method:

he feature that makes these methods possible is the ability to calcul

large number of linear algebraic equations that replace the partialifferential equations governing a physical system. Most of thesequations are second order but linear partial differential equations ohe following general form:

he first and third terms are recognizable when compared to thequivalent terms in Equation (3.23). There are, naturally, other typesquation including some that are non-linear, for which several analytiechniques have been developed, (Reference 6).

n these solutions, the fields are discretized into small areas where odes of these areas are sufficiently close to each other to minimize rror. These areas can be square, triangular or polygonal. However,

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r angu ar areas ave een oun to g ve t e greatest ex ty w enolving complex problems. The result of this discretization is a mesh attice and a typical example is shown in Figure 3.6.

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he governing equations for this method of analysis are also Equations3.20) and (3.21). The first, Gauss' law, says that the net gain in fl

er unit area (Div B) is always zero. The second, Ampere's law, says the closed-line integral of H per unit area (curl H) equals the currenensity of that area.

he solution to the above equations determines the vector potentials aonsequently the flux between 2 points.

ther parameters which can be determined from these calculations are:

The inductance L from the following:

here is the magnetic energy supplied to the circuit.

i. The force F from the following:

rograms are now available for use on personal computers to calculateagnetic fields by finite-element methods. Codes using this method ofnalysis are very long and complicated but the resulting programs areecoming steadily less expensive and easier to use. A model of theroblem is built in the computer, i.e. dimensions, material propertieslectric currents etc. A triangular mesh is then generated between the

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o nts, e n ng t e spaces over w c t e e s w e approx matehe model-making and mesh-generating methods have been made much easieo use recently. The outer boundary may include "infinite" elements whmitate the effects of an infinitely large empty space around the probegion. Some programs allow entry of non-linear B-H curves, or are ablo calculate eddy-current effects.

tudies comparing the finite element and finite difference methodsReference 13) have shown that the results from these two methods aredentical, especially for first order triangular algorithms.

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CHAPTER

ELECTROMAGNETICS

1 Force and emf generation:

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f electric current flows in a long straight wire, an odd magnetic fie

s set up in a circle around the wire (no poles), as shown in Figure1a. The strength of the field is:

Biot-Savart law)

here:

= current in the wire, amperes

= coercive force at a point (amps/meter)

= radius from the wire to the point, meters

f the wire is wrapped into a long cylinder, the field inside the coil

for an infinitely long coil) is

he magnetomotive force produced over a length of this coil is:

here n = total number of turns in the coil

n convenient engineering terms,

Oersted-inch = 2.0213 amp-turns (4.4)

f an electron is injected into a magnetic field, with its path at rig

ngles to the field, a force is exerted on the electron, at right anglo its direction of travel and to the field, proportional to the strenf the field, the charge on the electron and its velocity. If instead

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s ng e e ectron, a stea y current s passe t roug a w re, w c smagnetic field, a force is created on the wire:

n US engineering units, for a wire perpendicular to the magnetic fielith the resultant force perpendicular to both the field and the wire

i.e., sideways on the wire),

t seems at first as if there were two different magnetic forces at wone in line with the magnetic flux and proportional to the square of t

ield, and the other at right angles to the field and proportional to he two are, in fact, a single force, in spite of their apparentifferences.

steady magnetic field does not affect the voltage or current in a wit rest, in or near it. If the magnetic field changes with time howeve

circular electric potential (EMF, or voltage) is caused around the p

f the flux. If a wire surrounds the path, voltage will be caused in ind if the wire forms a closed loop, electric currents will be inducedt. This induced voltage and current (called eddy currents, in some

ircumstances) can be very useful or very troublesome, depending onircumstances. The current is in a direction so as to cause a magnetic

ield, which opposes the change, in the original field which caused itf it is able to flow, the result is to slow down or reduce the rate ourrent change (either increase or decrease).

he voltage caused by changing flux linking a coil of number of turns s:

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s before, a wire moving at right angles to a magnetic field shows theame effect in a somewhat disguised form (Figure 4.2). The wire "cuts"

lux lines at a rate related to its speed, length and field strength, back EMF is caused in it. For a wire perpendicular to the surroundinagnetic field, moving at right angles to it and to its length (i.e.,

ideways), the induced voltage is:

f a wire is free to move and is initially at rest in a magnetic fieldnd a fixed voltage is imposed across it, the wire will initiallyccelerate at a rate determined by its mass, its resistance and the fo

rom Equation (4.6). As its speed increases however, a back EMF will bnduced in it. The back EMF opposes the applied voltage and with less

oltage across the resistance, less current flows. The rate ofcceleration decreases. Eventually if the magnet is long enough, the w

ill approach the speed at which the back EMF caused by its motion jusquals the applied voltage. At this speed, called the terminal speed,

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urt er acce erat on s poss e, as s own n F gure . . I t e spee isturbed by outside forces, causing either an increase or decrease, t

lectromagnetic system will act to oppose the change. Referring to Fig4, if electric current is caused in a wire which links a magnetic

ircuit, the magnetic field will change with the mmf (amp-turns) causey the current. The magnetic field contains stored energy, which had t

ome from the electric circuit. The energy is removed by slowing down urrent due to back EMF.

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he constant L is called the inductance. Inductance is a magnetic effeelated to the permeance (or 1/reluctance) by:

he permeance of the gap (which is where almost all of the energy is

tored, if the circuit is not saturated) is from Equations (3.8) and

3.11):

or this reason, it is often convenient to use units of which are

onsistent with electrical units. In this form:

s an example, the inductance of an electromagnetic circuit with 28urns, having a gap 0.25" wide X 4" X 4" on a side, ignoring leakage,

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he calculation is only approximate because we have ignored leakage flf the turns are wound close to the gap, the approximation will be fai

ood, since the gap width is relatively small compared to itsross-sectional dimensions.

he relationship holds regardless of the nature of the magnetic

ircuit but, of course, P is dependent on the permeability of the pole

aterial. In the example, it was assumed that the permeability of the

teel in the surrounding structure was high enough that it could beeglected (which is equivalent to infinite permeability). If the magnetructure is driven into saturation however, (either entirely, or onlyertain locations of dense flux), the permeability and thus the

nductance will change, and no longer will be well represented by aingle constant.

2 Transformer operation:

he emphasis of these notes is primarily on motors and actuators but i

onetheless seems appropriate to add a brief description of electrical

nductors and transformers. Those components are used to affect electrircuits, rather than to produce useful force and motion but the

rinciples involved are often encountered in actuator and motor designn the past, the analysis of some types of motor was done by regarding

hem as rotating transformers.

n an electric circuit, such as that shown in Figure 5.1, capacitors a

omething like the electrical analog of mechanical springs. They storelectric charge and release it later. An inductor, on the other hand,

cts like the electric equivalent of a mass. It resists the change of

urrent through it, independent of the amount of current, as a massesists acceleration from rest, or deceleration once it is moving. Theoltage across an inductor of fixed inductance L is:

his energy equation assumes, however, that L (and, therefore, P or R)

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constant. I a core mater a s use , t e mater a w saturate a oome coil current value, bringing about a dramatic reduction in P and

t much lower current, core materials begin to become non-linear, and s no longer a constant. Since inductance often must operate at

elatively high frequencies, eddy current behavior becomes verymportant. Coil materials should therefore, have high electrical

esistance as well as low hysteresis. To meet these requirements, highaturation density often has to be sacrificed.

n inductor limits current flow for an ac signal by storing up energy

uring part of the cycle and retrieving it later in the cycle. Theoltage E and current I are out of phase by some angle. When the phase

ngle equals 90, power flows into the inductor during one part of theycle and an equal amount flows back to the source during the next par

esulting in (almost) no dissipated power. This is how a transformer wn open secondary avoids waste power even though it has low internal

esistance and is connected across the power line.

simple transformer is shown in Figure 4.4b. The shape shown is easy

nderstand but is not a likely one for actual use, because it would haigh flux-leakage. An alternating electrical current flows in the

rimary circuit of turns . An alternating magnetomotive force mmf =

s then set up and flux flows through the magnetic loop in response.

ince is varying sinusoidally, mmf is varying in the same way and

herefore, is an approximately sinusoidal function of time. Since

inks each turn on the primary as well as the secondary winding, a bac

mf or voltage is caused, which is the same for each turn in both

oils. In order for the system to be at equilibrium:

e., the output voltage is proportional to the input voltage and to turns ratio.

f we ignore the small losses in the transformer core and windings, thower into the primary circuit must be equal to the power out of the

econdary circuit:

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t can be seen therefore, that the output current is proportional to tnput current times the reciprocal of the turns ratio:

n practice, the actual turns ratio is increased somewhat from the

esired ratio of voltage out/voltage in. This is done to account for toltage "droop" caused by the resistance drop iR across the secondaryinding at rated current.

he first step in the design of a transformer is the design of a magne

ircuit which is capable of blocking the primary voltage with a currenlow small enough that the power lost is acceptable. An appropriate

umber of turns and wire size is chosen for the primary coil. The numbf turns of the secondary is then determined to provide the required

utput voltage. of course, the "window" in the magnetic circuit must barge enough to permit both windings.

he waveform of voltage input to the primary winding is usuallyinusoidal. The time intergral of the voltage, in volt-seconds, reache

aximum as the voltage decreases to zero.

e., the flux which must pass through the magnetic circuit is equal

his time integral:

weber = 1 volt-sec = 10,000 Gauss-meter = 15.5 M Gauss-in

approximately)

n order to avoid flux-leakage in a single-phase power transformer, thrimary and secondary coils are wound on top of each other. The design

-phase transformers and the various methods of connecting them is beyhe scope of these notes (see Reference 14).

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ulse transformers are electromagnetic sensors which give an output

oltage proportional to electric current passing through them. Since tely on d /dt for this voltage, they cannot operate with dc current, a

he fidelity of the output voltage depends on the rate of change or

requency of the linking current.

he primary is just a single turn and the current to be sensed passestraight through the magnetic circuit only once. An integral of mmf

round this current-carrying wire is the same for any complete path. Tmf causes a voltage in each turn of the toroidal winding about the co

herefore, the current in the secondary is less than that in the primay this ratio. For example, if the current to be sensed is 100 A and

here are 200 turns on the toroid, the current in the sensor winding we 500 mA.

resistor, usually of small value, is connected across the twoensor-coil leads. The voltage measured across this resistor is theutput. It is proportional to the current in the main circuit but is

solated electrically from it and scaled as required by the number ofurns and the sensor resistor value.

3 Instruments for magnetics:

f an electron moves in a magnetic field, a force is exerted on it at

ight angles to the direction of motion. In 1879, Dr. Edwin Halliscovered that if an electric current were passed through a flat strif conductive material (he used gold) in a magnetic field, the electro

ould be forced to one side, creating a voltage difference between thewo sides (Figure 4.5).

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or the same current, electrons must travel faster in a poor conductorhan in a good one and the force (and thus the voltage) is proportionao speed. Modern Hall sensors use silicon, not for its switching and

ontrol capabilities but just as a high-resistance material. A typicalall sensor of the linear variety may have a resistance of 1000 ohms o

o across the reference current leads and perhaps a little less acrosshe sense leads. If 10 V is connected across the reference leads, caus

current of 10 mA, the sense leads may indicate 50 mV when the sensorlaced in a 100 G field. With Hall sensors which have no internal

rocessing, the direction of reference current may be reversed, revershe polarity of the output. The active site is very small, perhaps 0.0

nch on a side, so the field is measured nearly at a point. Only theomponent of B which is normal to the surface is measured, so care mus

e taken to align the sensor properly to the field. Proper alignmentauses the maximum reading at that point. The values given here are ono convey an approximate idea of what may be expected; actual componen

ary widely from these numbers. Another type of Hall sensor haslectronic processing built in and might contain, for example, a

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onstant-current source, a sw tc w t ysteres s an an output r verith relatively high current capability (Reference 15). The voltage

annot be reversed across this type of device, of course.

all type Gauss meters are relatively easy to use but the making of

ccurate measurements is complicated by several factors. The earth'sagnetic field varies in strength and direction from place to place; iverages about 0.4 Gauss. In a room containing electric lights and oth

lectrically-powered equipment and walls carrying electric wires, theield may be higher. The presence of tools, desk parts, lamps and fili

abinets of steel may warp and concentrate the stray field enough toause reading errors of tens of Gauss.

all sensors have a slight offset, indicating a reading with no magnetield present, and the offset may vary with time and temperature. The

ffset is caused by slight inaccuracies in manufacturing. Gauss metersre usually provided with a shielding tube and an offset adjustment to

llow for zero correction. They may also have a means to adjust theeference current (the current through the Hall sensor which is deflec

y the field to cause the output), if it should drift.

nother problem is the scale adjustment. It is a common experience amoorkers in magnetics to place a Hall probe at a marked location, with

ide of the probe flat against the surface. The flux is known to beormal to the surface. A reading is made. As a check, the probe is tur

ver, in the same location, and the second reading is found to beomewhat different. The operator is then unsure as to which to use, or

hether to take an average. A means is needed to check the output at anown high reading, near that which the operator is trying to make. So

agnet companies sell standard magnets for this purpose. However, theperator finds that he gets different readings at different places in upposedly uniform reference gap. Some of these devices are variable w

emperature. If one compares several of the standard magnets fromifferent vendors, they may not agree.

t is possible to generate a magnetic field, the strength of which isnown from basic, traceable standards up to perhaps 100 Gauss. One way

o this is with a Helmholtz coil, which is really two coils with axesligned, separated by the distance of one coil radius. For higher fiel

owever, the coils overheat and the attainable field is too low to givood check on measurements made at thousands of Gauss.

hile a Hall-effect Gauss meter reads field strength nearly at a point

lux meter indicates total flux over an area. A search coil is used, wne or more turns. The coil may be handmade on the spot to conform to

agnet or region being measured. A flux meter reads the time integral oltage induced across the coil when it is introduced into a field, or

ithdrawn from it. If the coil is turned over in place, the reading isoubled. Some flux meters are difficult to read because they drift wit

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CHAPTER

MAGNETIZING OF PERMANENT MAGNETS

1 Magnetizing Requirements:

agnets are normally shipped unmagnetized from the manufacturer and aragnetized either in place, after assembly or, if necessary, just befo

ssembly. There are a number of reasons why this is preferred over

hipping them already magnetized. Large masses of charged magnets may angerous, capable of pinching or crushing limbs accidentally caughtetween them. They may affect navigation instruments, wipe out data on

agnetic tape or disks or destroy delicate machinery (such as watches)hey may pick up magnetically permeable dirt, which is then difficult lean off (several turns of masking tape, wound face-out on the hand,

elps to clean off this sort of dirt). The magnets may even crushhemselves, or pull tools out of a worker's hands several feet away, ixtreme cases. It may be very difficult to assemble a device with alreagnetized parts, because the forces caused make alignment and clampin

to set adhesives) difficult. There are restrictions on the shipment oharged magnets.

o magnetize or charge a permanent magnet, it is necessary to achieve agnetic field high enough to completely saturate the magnet everywher

n the pattern required for the pole locations in the magnet. The timeequired is extremely brief. Once the required coercive force is reachn the magnet, domain reversal occurs in less than a hundredth of aicrosecond (100 nanosec), which is negligibly short for magnetization

urposes.

lthough some non-electric methods of magnetization exist, especially he older low-coercivity materials, almost all magnetization today isone by generating a very short pulse of a very high electric current,

erhaps a few milliseconds long, with currents of a few hundred to ove00,000 amps. The electrical pulse is then used to cause a brief but vtrong magnetic field. The electric pulse is usually caused by storing

lectric current in a bank of capacitors at high voltage and thenuddenly discharging the capacitors through an electronic switch. Theslectronic assemblies, called magnetizers or chargers, areeneral-purpose devices and are relatively expensive. They usually per

djusting the discharge voltage over some range, continuously or byteps, to accommodate different requirements. The pulse is then applieo a magnetizing fixture. The fixture might be as simple as a coil of

ire, or a "c" framed structure for straight-through magnetizing, or iight be a complicated arrangement of wire or copper bar, laminated iroles and support structure. The fixture is often designed for use witpecific magnet and it must also be designed for use with the intended

ulse generator.

he magnetizer/fixture combination must meet the following requirement

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Sufficient coercivity (and therefore enough electric current) must

upplied to completely magnetize the part. This condition places a lowimit on the value of current times turns.

Most of the energy stored in the capacitors is converted into heatithin a few milliseconds in the coil. The time is too brief forignificant amounts of heat to escape to the surroundings. The coil mu

ave enough mass that the resulting temperature rise is kept low enoug

o avoid overheating which could destroy the insulation. In some caseshe coil (or part of it) could even be vaporized.

Rejection of heat from the fixture as a whole must be high enough t

llow cycling at an economical rate (if the fixture is intended forroduction use).

Eddy currents in the fixture or magnet itself must not be high enouo prevent complete and even magnetization of the part, or overheat th

ixture.

The fixture must be strong enough mechanically to take the high forenerated by the magnetic pulse without damage. It must be constructedith enough accuracy to locate the magnet poles to within the required

olerances. It must restrain the magnet and support it well enough thahe part will not be broken by the magnetic forces, and yet must not fo tightly that thermal expansion might jam or break the part. Fast anasy loading and unloading of the part must be provided for.

2 Current versus time in an ideal magnetizer:

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he linear idealized description of a charger/magnetizer combination if only limited use in predicting the behavior of actual magnetizers b

s a good starting point. In Figure 5.1, an idealized electromagneticircuit is shown consisting of a capacitor of fixed value C, annductance of fixed inductance L and a fixed resistance R. The voltage

cross each circuit element are:

he sum of these voltages around the circuit loop must equal zero,

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his is the well-known second-order homogeneous linear differentialquation with constant coefficients. Solutions are of the form

here b may be real, imaginary or complex.

he solutions may be presented in various ways. Perhaps the most usefupproach is to define two new constants, derived from the circuit valu

f < 1, the system is said to be underdamped. In an underdamped stat

he current oscillates, surging back and forth between the capacitor a

nductance, as shown in Figure 5.1. Because of energy dissipation as hy the resistor, the peak current amplitude decreases with each swing.

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s e av or wou e g y un es ra e n a magnet zer, s nce even elatively small reversal of current could remagnetize some of theaterial in the reverse direction. If 5% were remagnetized backward, i

ould have the effect of canceling the magnetization of a similar voluor a loss of 10% overall.

lder magnetizer designs have mercury-filled tubes called ignitrons fowitches. These tubes may conduct in the reverse direction for a short

ime before extinguishing. Manufacturers

motor design and power electronics

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