Theory & Practice of Electromagnetic Design of DC Motors & Actuators
George P. Gogue & Joseph J. Stupak, Jr.
5.1 Magnetizing Requirements:
Magnets are normally shipped unmagnetized from the manufacturer and are magnetized either in place, after assembly or, if necessary, just before assembly. There are a number of reasons why this is preferred over shipping them already magnetized. Large masses of charged magnets may be dangerous, capable of pinching or crushing limbs accidentally caught between them. They may affect navigation instruments, wipe out data on magnetic tape or disks or destroy delicate machinery (such as watches). They may pick up magnetically permeable dirt, which is then difficult to clean off (several turns of masking tape, wound face-out on the hand, helps to clean off this sort of dirt). The magnets may even crush themselves, or pull tools out of a worker's hands several feet away, in extreme cases. It may be very difficult to assemble a device with already magnetized parts, because the forces caused make alignment and clamping (to set adhesives) difficult. There are restrictions on the shipment of charged magnets.
To magnetize or charge a permanent magnet, it is necessary to achieve a magnetic field high enough to completely saturate the magnet everywhere, in the pattern required for the pole locations in the magnet. The time required is extremely brief. Once the required coercive force is reached in the magnet, domain reversal occurs in less than a hundredth of a microsecond (100 nanosec), which is negligibly short for magnetization purposes.
Although some non-electric methods of magnetization exist, especially for the older low-coercivity materials, almost all magnetization today is done by generating a very short pulse of a very high electric current, perhaps a few milliseconds long, with currents of a few hundred to over 100,000 amps. The electrical pulse is then used to cause a brief but very strong magnetic field. The electric pulse is usually caused by storing up electric current in a bank of capacitors at high voltage and then suddenly discharging the capacitors through an electronic switch. These electronic assemblies, called magnetizers or chargers, are general-purpose devices and are relatively expensive. They usually permit adjusting the discharge voltage over some range, continuously or by steps, to accommodate different requirements. The pulse is then applied to a magnetizing fixture. The fixture might be as simple as a coil of wire, or a "c" framed structure for straight-through magnetizing, or it might be a complicated arrangement of wire or copper bar, laminated iron poles and support structure. The fixture is often designed for use with a specific magnet and it must also be designed for use with the intended pulse generator.
The magnetizer/fixture combination must meet the following requirements:
1. Sufficient coercivity (and therefore enough electric current) must be supplied to completely magnetize the part. This condition places a lower limit on the value of current times turns.
2. Most of the energy stored in the capacitors is converted into heat within a few milliseconds in the coil. The time is too brief for significant amounts of heat to escape to the surroundings. The coil must have enough mass that the resulting temperature rise is kept low enough to avoid overheating which could destroy the insulation. In some cases the coil (or part of it) could even be vaporized.
3. Rejection of heat from the fixture as a whole must be high enough to allow cycling at an economical rate (if the fixture is intended for production use).
4. Eddy currents in the fixture or magnet itself must not be high enough to prevent complete and even magnetization of the part, or overheat the fixture.
5. The fixture must be strong enough mechanically to take the high forces generated by the magnetic pulse without damage. It must be constructed with enough accuracy to locate the magnet poles to within the required tolerances. It must restrain the magnet and support it well enough that the part will not be broken by the magnetic forces, and yet must not fit so tightly that thermal expansion might jam or break the part. Fast and easy loading and unloading of the part must be provided for.
5.2 Current versus time in an ideal magnetizer:
The linear idealized description of a charger/magnetizer combination is of only limited use in predicting the behavior of actual magnetizers but is a good starting point. In Figure 5.1, an idealized electromagnetic circuit is shown consisting of a capacitor of fixed value C, an inductance of fixed inductance L and a fixed resistance R. The voltages across each circuit element are:
The sum of these voltages around the circuit loop must equal zero,
This is the well-known second-order homogeneous linear differential equation with constant coefficients. Solutions are of the form , where b may be real, imaginary or complex.
The solutions may be presented in various ways. Perhaps the most useful approach is to define two new constants, derived from the circuit values:
If < 1, the system is said to be underdamped. In an underdamped state, the current oscillates, surging back and forth between the capacitor and inductance, as shown in Figure 5.1. Because of energy dissipation as heat by the resistor, the peak current amplitude decreases with each swing. This behavior would be highly undesirable in a magnetizer, since even a relatively small reversal of current could remagnetize some of the material in the reverse direction. If 5% were remagnetized backward, it would have the effect of canceling the magnetization of a similar volume, for a loss of 10% overall.
Older magnetizer designs have mercury-filled tubes called ignitrons for switches. These tubes may conduct in the reverse direction for a short time before extinguishing. Manufacturers' data indicates that peak reverse current under some conditions could rise to as much as 40% of the forward current before the tube shuts off. Blocking diodes are sometimes but not always included in these designs. If no blocking diode exists in a particular ignitron charger, then the fixture should be designed with enough resistance that the combination is overdamped, to avoid ringing.
If the current is suddenly cut off by a blocking diode, the large inductance of the fixture will cause the voltage across it to rise without limit, until current is able to flow by some means to dissipate the stored energy. This may be by arcing through the insulation at many thousands of volts, if no other path is available. A diode is connected in parallel with the fixture, called a freewheeling diode, as shown in Figure 5.1. It blocks current from the first pulse but then allows current to circulate around and back through the fixture, when the drive circuit is cut off. The stored energy is then dissipated in the fixture resistance. It is sometimes suggested that an underdamped magnetizer fixture does not have to dissipate all the energy in the capacitors and thus will operate cooler than an overdamped circuit, for the same capacitor voltage. This is not true, as the energy is dissipated in the fixture resistance anyway. The underdamped circuit may however, have a higher peak current for the same charging voltage.
Solving Equation (5.10) for the underdamped case,
Peak current is found by setting di/dt = 0 and solving for the time for peak current, then substituting this time back into the solution:
If = 1, the circuit is said to be critically damped and has just enough resistance to prevent overshooting. For this condition:
Peak current is again found by solving for the time at which di/dt=0, and then substituting it into the formula for current.
Current versus (voltage/resistance) E/R is plotted as a function of damping coefficient in Figure 5.2.
Another possibility is that of R = 0. The situation cannot exist, of course, in a real magnetizer but is useful as a limiting case for low resistance. If the circuit resistance is zero, the solution is:
5.3 Real Magnetizers:
Unfortunately, many real magnetizers do not behave in a manner which would be required for the linear analysis results to have much validity. Older magnet materials could be saturated at coercivities low enough that the permeability of the pole material could be neglected (taken as infinity). The newer, more powerful materials take far higher fields to magnetize at coercivities well above the saturation of any pole material. For a material (mild steel) which requires 25,000 Oe to fully magnetize, the pole steel will hard saturate at 20,500 G. The remaining 4,500 G (to obtain 25,000 Oe in the gap) requires much more mmf (current times turns) to achieve than the 20,500 G obtained up to saturation.
The copper windings heat up during the firing pulse. The resistance of copper is a function of temperature and can rise as much as 30% during the pulse. The temperature dependence of the resistance of copper is:
The total resistance imposed on the circuit is the sum of the fixture and source resistances. For some fixtures, the fixture resistance is high enough that the source resistance has little effect. Manufacturers of chargers (magnetizers) rarely if ever publish information about source resistance. If the fixture is designed with very low resistance however, the source resistance becomes dominant.
The largest component of the source resistance is usually the ESR
(equivalent series resistance) of the capacitor bank. Unfortunately for calculation purposes, this resistance is a function of the rate of change of current. Figure 5.3 shows a typical curve of ESR versus frequency for the type of capacitor usually found in magnetizers (a large aluminum-foil electrolytic). These components have wide manufacturing tolerances in capacitance (e.g. +75%, -10%) and change capacitance with time, use and temperature.
The charger itself will have some inductance and the amount might be deliberately increased to limit rate of current buildup in the electronic switch (which could cause it to fail, if excessive).
The material to be magnetized affects the circuit. It absorbs some energy from the system as it is magnetized. Magnetization does not occur everywhere at once, because some of the magnetic domains are harder to coerce than others. The process follows a curve like that shown in Figure 5.4, which is for a ferrite. The magnetizing curve for Samarium-cobalt 2-17 has an extra bump in the middle, as shown in that figure. As the field builds up, some energy is absorbed by the magnet, slowing down the rate of current increase. From then on, the field is increased by the new contribution from the partially magnetized part. The increased field will change the saturation level of the pole material and thus the current rate-of-change.
As the current changes at a rapid rate, circular voltage fields are induced in the pole material, in the magnet itself and in the surrounding structures, if they are electrically conductive. Eddy currents are set up, which affect the current-time history. To the source, the load seems to momentarily have more resistance and less inductance.
The cumulative effect of these differences from ideal behavior may be considerable. Figure 5.5 shows an actual time-current plot of a charger/fixture combination with a magnet in place, initially uncharged (the solid line).
The peak current calculated by the idealized formulas, using values for fixture resistance and inductance (measured at very low signal level) and the known capacitance of the charger, gave a result of 9073 amps. The actual measured peak was 5150 amps.
A computer program which included eddy current effects, pole-material nonlinearity, magnetizing effects in the magnet as it was charged and source ESR fared much better (dashed line), predicting 5416 amps (5% too high). The offset between peak times for the measured versus the calculated curves may be due to the fact that the current had to rise to a considerable value before the digital oscilloscope used in the measurement could detect it and trigger. The sudden change in slope seen in both the computer-predicted and measured curves is the result of the steel pole material coming out of saturation.
The worst nonlinearity of the various factors complicating magnetizer design is the problem of pole saturation. If hand methods must be used, or for an approximation early in a design, the magnetizer behavior may be considered as the sum of two parts: as a magnetizer with poles of infinite permeability, up to the point of saturation and as a device with air-core coils (i.e. pole material with a relative permeability of 1) for current above that required for saturation (Reference 16).
It is sometimes very difficult to magnetize a particular part with a given charger. In the design of a difficult fixture, such as might be the case to magnetize a small ring of high-energy material with a large number of poles, the available volume for both winding and pole material has an upper boundary. Within some limits, most of the variables under the control of the designer reduce to only three, the charging voltage, the wire cross-sectional area and the wire length. The charger capacitance is probably fixed and eddy currents are suppressed as much as possible. Number of turns becomes a function of the wire length.
These three variables, voltage E, area A and length l, may be represented as three dimensions in space, as shown in Figure 5.6. The limits imposed on the design may then be shown as limiting surfaces in EAl space. The resulting volume cut out by the surfaces (provided, of course, that there is a remaining volume) represents all possible designs which meet the required conditions.
In Figure 5.6, the upper voltage and minimum voltage at which the charger is able to operate are marked. Another possible upper voltage limit would be the insulation limit, if it were lower than the maximum voltage the charger could put out.
There is some minimum wire length l which represents one strand of wire threading the minimum path required for magnetization. There is a maximum possible wire area, for one turn (often a fraction of a complete turn) around each pole. These limits cut out a rectangle, extending to infinity along l.
Another limit is the maximum allowable winding volume, V = Al. A curve of Al = (a constant) is a hyperbola in the A-l plane, and a vertical, curved sheet in EAl space (Figure 5.7a).
The energy stored in the capacitors when they are charged is:
To prevent this energy from overheating the wire, a lower limit is imposed on wire volume,
This limit surface is shown in Figure 5.7b. The purpose of the fixture/magnetizer is, of course, to magnetize the part, which requires some minimum magnetomotive force,
In a simple resistive electric circuit, the current is determined by Ohm's law,
In a circuit containing capacitance and inductance, the current cannot reach this value but can still be related to the resistance by:
where is a function of the damping ratio . The resistance R of a wire is a function of the area and length,
The current i can then be represented as
Since the number of turns is related to length (approximately) linearly and mmf = ni,
This surface is a rectangular parabola in the AE plane
One possible resulting shape for the volume cut out by these limits is shown in Figure 5.8 representing all allowed solutions. Not all included points are actually realizable however. Unless special wire is made, the allowable wire areas are limited to standard AWG sizes (and perhaps half-sizes if available). Wire length is not a continuous function since only integral multiples are possible. Some chargers are continuously variable, while others are adjustable only in steps. A few are not adjustable at all. Given the limit volume, it is then possible to define an optimum solution, depending on what is meant by optimum in a particular situation. If minimum instantaneous heating is intended, then the curve of least temperature of the shape
shown in Figure 5.7b which just touches the limit volume, is chosen. On the other hand, if the sharpest possible transitions are needed, then the extreme intersection of a curve of the type shown in Figure 5.9 with the volume is required. Maximum overall cycle rate occurs at minimum voltage, within the volume.
5.5 Other Considerations in Magnetizer Design:
Wire of AWG (American Wire Gage) #14, which is 0.066" in diameter (single build insulation) and larger sizes are difficult to form because of stiffness. If wire of more area is required, it may be better to use several smaller strands in parallel. When the highest possible density is required, commercially available wire of square cross-section may be used. Even machined and formed copper bar (subsequently annealed) may be used. Eddy currents in magnetizing fixtures may seriously limit performance if not controlled. One method is to use laminations, thin sheets of steel with a coat of electrical lacquer (one side only to reduce buildup) for insulation. The lacquer layer may be very thin, because the driving voltages are low. The required lamination thickness may be estimated by calculating the "skin effect". An electromagnetic wave impinging on the surface of a semi-infinite conducting material penetrates into the material with intensity decreasing with depth, repelled by eddy currents. The equivalent depth to which the field would penetrate, if all the currents were confined in a uniform layer to that depth, is called the skin depth . If a lamination exposed to the frequency used in the calculation is two or less skin depths in thickness, then eddy currents on one side of the sheet would meet and destructively interfere with currents on the other side to some extent; this design limit is found to be generally acceptable in practice. The depth of the skin is:
The permeability depends on the flux-density in the pole material at some representative time. For fixtures to be driven into saturation and mild steel, = 200 has given satisfactory results (corresponding to about B = 17 kG). Some fixture designers advocate using no metallic pole material at all for fixtures exposed to very high flux levels, because of eddy currents. For fields up to 30,000 Oe however, steel laminations still seem to be useful.
5.6 Forces on conductors and coils:
If a current-carrying wire is in a hole or deep slot in a magnetically permeable material such as steel and the pole material is not magnetically saturated, almost all of the magnetic forces generated are exerted on the pole material and very little on the wire. Where the wire extends beyond the end of the pole material, such as at the ends of coil turns, or on freestanding coils in air, high forces may be induced in the wire. In magnetizing fixtures, wire loop ends may be severely deformed if not well secured. For conductors near pole surfaces, the forces generated pull the wire into the pole surface and spread out the loop. Wires in slots are pulled deeper into the slots (provided significant eddy currents are not present).
In an air-core coil of wire subjected to a strong pulse of electric current, the forces produced tend to crush the coil inward axially, while spreading out each loop radially. For an idea of the size of these forces, an approximate calculation may be done for a 2 inch diameter coil (assumed to be about 4 inches long) intended to magnetize high-energy materials at 30 kilo Oersted.
In order to reach a field of 30 kOe at the edges of the useful volume of the coil (about the middle section of the coil length), the field will have to be somewhat higher in the center and it will also vary slightly across the diameter. It will be assumed that an average field strength of 40 kOe (40 kilo Gauss flux-density) will be present across the diameter, halfway along the axis of the coil (Figure 4.1c).
If the coil should be crushed inward by a very small length dl, the energy relations at the ends will be unchanged but the volume A dl will disappear. The energy stored in this volume is:
The work done in contracting by this distance, on the ends, is:
The term 1/(2) in convenient mixed engineering units is 0.571x10 (lb/inch Gauss). Solving for a diameter of 2 inches:
F = 2900 lb (crushing inward axially)
To calculate the hoop stress in the coil, let us assume a coil wall thickness t = 0.5 inch, over a region of axial length h (assumed small compared to the length).
For a small (virtual) change of width across a rigid-body cut, in a particular direction x perpendicular to the axis, a change of volume will occur,
For t = 0.5 inch, r = 1 inch and B = 40 kG, the hoop stress is about 1850 psi.
When current flows in wires which are near other conductors, eddy currents are induced in the other bodies. These currents generate magnetic forces which oppose the fields set up by the original currents, and the conductors repel each other. The same currents may at the same time cause attractive magnetic fields in the same bodies (or in bodies attached firmly to them) if poles are induced in magnetically permeable parts. How these forces combine depends on both the absolute level of current and on its frequency or rate of change. For example, in one particular type of linear induction motor, a set of coils (operated in three phases) around a laminated steel pole structure faces a track consisting of solid, unlaminated steel covered with a layer of copper, with a gap between them. Exciting the coils with three-phase ac current results in a thrust at right angles to the gap. At low frequencies, the laminated part and the track are also strongly attracted to each other, and at 60 Hz the force of attraction is about 10 times the thrust. At higher frequencies, however, the attraction is reduced and above 120 Hz, the track and laminated part repel each other, while still maintaining a linear thrust.
5.7 Winding patterns:
After an air-core coil, the simplest fixture shape is perhaps the "c" frame, used for straight-through, single-pole pair magnetizing, as shown in Figure 5.10. If the windings are placed near the pole faces, as shown, more useful flux will cross the gap than if the windings are in the middle (shown dashed). For a design like this, with a relatively large number of turns, the effects of a fractional turn is probably unimportant. For critical designs (in which fixture resistance must be minimized, the space is restricted and each pole having a small number of turns) the actual turns "count" is important and a uniform pattern producing an even field is needed. A minimum-length pattern is shown in Figure 5.11 (b), which gives about 3/4 turn per pole. This pattern may result in substantial unevenness of field. If two conductors of equal area and length, with a total area equal to the original wire, are used in parallel instead, as shown in Figure 5.11 (c), the electrical characteristics are unchanged but the magnetization is more even.
When a series of adjacent poles are wound with opposite polarities by a single wire, the number of turns per pole will not be integral. It is easy to make a systematic error in winding such a pattern, which can badly affect the evenness of magnetization, for small total turns count.