Optimal Design of
Joseph J. Stupak, Jr.
Beaverton, OR 97005
Permanent magnets are used in large numbers in brushless DC motors, actuators, sensors, instruments, and other electromagnetic devices. Magnetic materials are rapidly becoming more powerful and the devices which use them are constantly being redesigned into smaller sizes, with equal or better performance. The reduced dimensions of the magnets, along with increased number of poles, in materials of very high coercivity, make the magnetizing process much more difficult. Given a particular magnetizer (electric pulse generator) it may be impossible to properly charge a given magnet with a specified pole pattern, even though the equipment is capable of supplying the required energy. In this article, methods will be considered which lead to a strategy for defining an "optimal" magnetization process for a given permanent magnet when charged by a particular pulse generator, depending on the competing needs of cycle rate, sharpness of magnetic transitions, etc. and show when no solution is possible. A different magnetizer, or modifications to the existing magnetizer, might then be needed.
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 usually 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. The pulse width can be a few milliseconds long and the currents 100 to over 100,000 amps. The electrical pulse is then used to create a brief but very strong magnetic field. This pulse is usually produced by storing up electric charge in a bank of capacitors at high voltage and then rapidly discharging the capacitors through an electronic switch. These electronic assemblies, called magnetizers or chargers, are general-purpose devices and are fairly 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 the windings of 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 supports. The fixture is often designed for use with a specific magnet and it must also be designed to match the characteristics of the charger.
The magnetizer/fixture combination must meet the following requirements:
i. Sufficient coercivity must be supplied to completely magnetize the part. This condition places a minimum requirement on the product of current and turns (ni).
ii. Most of the energy stored in the capacitors is converted into heat in the coil within a few milliseconds. 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.
iii. 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 use at production rates).
iv. Eddy-currents in the fixture or magnet itself must not be high enough to prevent complete and even magnetization of the part or to overheat the fixture.
v. 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; 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.
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 la, an idealized electromagnetic circuit is shown, consisting of a capacitor of fixed value C, a fixed inductance L and a fixed resistance R.
The voltages across each circuit element are:
Figure 1 Magnetizing circuit with freewheeling diode
The sum of these voltages around the circuit loop must equal zero,
Differentiating to get rid of the integral and dividing by L,
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:
( is the undamped natural frequency of the system, radians/sec) and
The variable is a dimensionless damping factor. Substituting these new constants into the equation,
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 lb. Because of energy dissipation as heat by the resistor, the peak current decreases with each swing. This behavior would be highly undesirable in a magnetizer, because even a relatively small reversal of current could remagnetize some of the material in the reverse direction.
Older magnetizer designs may 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 peak forward current, before the tube shuts off. Blocking diodes are sometimes 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 (at many thousands of volts) through the insulation, if no other path is available. A diode is connected in parallel with the fixture, called a freewheeling diode , as shown in Figure lc to provide this path. The diode blocks current from the first pulse but then allows current to circulate 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 magnetizing 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, however, may have a higher peak current for the same charging voltage. Solving equation 9 for the underdamped case ( < 1),
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.
The ratio of current to (voltage/resistance) or E/R is plotted as a function of damping coefficient in Figure 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:
Unfortunately, the behavior of many real magnetizer-fixture pairs is substantially different from that predicted by linear analysis. Older magnet materials could be saturated at coercivities low enough that the permeability of the pole material could be ignored (taken as infinity). The newer, more powerful materials require far higher fields for magnetization at coercivities well above the saturation of any pole material. For a material which requires 25,000 Oe to fully magnetize, the pole steel will hard saturate at 20,500 G (for mild steel). The remaining 4,500 G (to obtain 2,500 Oe in the gap) requires much more mmf (current X 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:
= resistance at temperature
= resistance at temperature +
= temperature rise, degrees C
The total resistance imposed on the circuit is the sum of the fixture and source resistances. The resistance of some fixtures is high enough that the source resistance has little effect. Manufacturers of chargers (magnetizers) rarely 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 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 (+75%, -10%, for example) 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 fail at excessive currents).
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 4 , which is for a ferrite material. Curves for other materials may have different shapes, such as that for Samarium-Cobalt 2-17 in the same 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 charged magnet. 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 the materials are electrically conductive, eddy-currents are set up in them affecting 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 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 resistance and inductance measured (at very low signal level) in the fixture and the known capacitance of the charger, gave a result of 9073 amps. The actual measured peak was 5150 amps. A computer program fared much better (dashed line in Figure 5) predicting 5416 amps (5% too high). The program takes into account eddy-current effects, pole-material nonlinearity, magnetizing effects in the magnet as it was charged, and source ESR. 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 to make the measurement could detect it and trigger. The sudden change in slope seen in both the computer-predicted and measured curves (in the lower right of the 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 as an approximation early in a design, the magnetizer behavior may be considered in two stages. First, up to the point of saturation, it is treated as a magnetizer with poles of a high and fixed (average) permeability. Second, as a device having air-core coils (i.e. pole material with a relative permeability of 1) for currents above that required for saturation,
It is sometimes very difficult, or perhaps impossible, to magnetize a particular part with a given charger, in the design of a difficult fixture, such as for magnetizing 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 bound, 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 far 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 6. The limits imposed on the design may then be shown as limiting surfaces in EAl space. The resulting volume defined by the surfaces (provided, of course, that there is a remaining volume) represents all possible designs which meet the required conditions.
In Figure 6, the maximum and minimum voltages at which the charger is able to operate are marked. Another possible maximum 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 also a maximum possible wire area, for one turn (sometimes 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 = A x l. A curve of A x l = (a constant) is a hyperbola in the A-l plane, and a vertical, curved sheet in EAl space (Figure 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 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 wire material conductivity
The current i can then be represented as:
Since the number of turns is related to length (approximately) linearly, and mmf = n i,
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 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 may be set to a peak charging voltage which is continuously variable, while others are adjustable only in steps. A few are not adjustable at all.
Given the limit volume, i.e. the positive volume of space cut out by the various limiting surfaces, 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 7b (which just touches the limit volume) is chosen. On the other hand, if the sharpest possible magnetic transitions are needed, then the intersection of a curve of the type shown in Figure 9 with the volume is chosen. Maximum overall cycle rate occurs at minimum voltage within the volume.