ACTUATORS: INSIGHT INTO THE DESIGN
Joseph J. Stupak, Jr.
Beaverton, OR 97005
Voice-coil actuators are a special form of electric motor, capable of moving an inertial load at extremely high accelerations (more than 20 times the rate of acceleration of gravity "g" at the Earth's surface) and relocating it to an accuracy of millionths of an inch over a limited range of travel. Motion may be in a straight line (linear actuators) or in an arc (rotary, or swing-arm actuators). After completing a motion (called a "seek" in the computer peripheral memory industry) the moving parts must stop vibrating very quickly. This period, called the settling time, may be a few milliseconds or less.
The first voice-coil actuators resembled scaled-up loudspeaker mechanisms in construction, from which they were named. These early designs (as shown in Figure 1) had a very short gap with high field strength (perhaps 12 kG) and a long coil. They were replaced in the early 70's with motors in which the magnetic fields were much longer and at lower intensity (typically 2-6 kG), an example of which is shown in Figure 2. The coils of this design are much shorter and remain entirely within the gap region. The moving mass is less and is capable of higher acceleration with less settling time. Coil resistance is lower and linearity is improved.
A flat motor configuration, shown in Figure 3, has increased moving mass and electrical resistance but is easy and inexpensive to build. Both flux direction and current direction are reversed on opposite sides of the coil, resulting in forces which sum in the same direction. The same idea is often used in swing-arm actuators, using either flat (Figure 4a) or curved (Figure 4b) coils. It is also possible in a rotary mechanism to use a coil or coils which have axes pointed in the direction of travel, like the linear actuator of Figure 2.
When electric current flows in a conducting wire which is in a magnetic field, as shown in Figure 6, a force is produced on the conductor which is at right angles to both the direction of current and magnetic field,
F = force on the conductor
dl = a differential length of conductor
i = electric current
B = magnetic flux-density vector
If the conductor (wire) is at right angles to the direction of travel and magnetic field then the force in the travel direction, in usual US engineering units, is:
F = force, lbf
B = flux-density at the coil, G
l = conductor length, inches
i = electric current, amps
The design of voice-coil motors for accurate performance is far more challenging than it appears at first. The very simple linear relationship above is complicated in practice by variations in the static magnetic field, flux-leakage, nonlinearities in the B-H curve of the pole steel, field variations caused by DC coil current, other effects caused by the rate of change of flux, effects on the drive electronics caused by coil motion, changing resistance due to heating, changing inductance and other problems.
The ratio of force produced to coil current is usually called the "force constant" but it would be more accurate to refer to it as the force-current ratio because it is significantly affected by many factors.
In US engineering units,
= back EMF, volts
B = field strength, Gauss
l = conductor length, inches
v = coil velocity, inch/sec
If a fixed supply voltage is applied across the coil of a VCM (voice-coil motor) it will accelerate until the back EMF just equals the supply voltage (if the allowed travel is long enough). The speed at which the two voltages are equal is called the terminal velocity Vt. The coil cannot exceed this speed without a higher applied voltage. In practice the coil is usually limited to a lower speed, called the cutoff velocity Vc, by the controls. If the controls should fail, however, the coil may approach this speed as it accelerates into the "soft" end stop. A limit on Vt may be imposed by the space available for deceleration, by the maximum allowed deceleration rate to avoid damage (e.g. to the recording heads) and by the stop force/distance characteristics.
When voltage is suddenly applied to the coil of a voice-coil actuator, the resulting mmf (amps x turns) either increases or decreases the overall magnetic field. The magnitude of the mmf is, in engineering units:
i = current (A)
n = number of turns in the coil
A delay is experienced in the buildup of the stored energy, represented above by the current i. The effect may be described as circuit inductance although it is somewhat nonlinear. If a sleeve of conductive material (copper) is added concentric to the coil but fixed to the magnet/pole structure, eddy currents are induced in it by changes in coil current. These currents cause magnetic fields which oppose the original changes of field and thus to a large extent cancel the apparent inductance. The result is a much faster, more linear response. The penalty, on the other hand, is increased gap width which in turn decreases B and the force-current ratio. In addition, the coil appears to have an increased resistance (reflected from the shorted turn). Heating is increased, although the extra heat is generated in the shorted turn, rather than in the coil. The shorted turn is usually placed inside the coil, on the center pole, and running its full length. It is possible, in some designs, to place it outside and surrounding the coil instead. The thickness of the shorted turn can be analytically determined, by methods too lengthy to be considered here. If placed inside the coil, it may be less than 70% as thick as the coil, as a rough rule-of-thumb.
The shorted turn acts like a shorted one-turn secondary of a transformer, as shown in Figure 2. The circuit is equivalent to that shown in Figure 7, with symbol meanings as follows:
= coil resistance (steady state)
= shorted-turn resistance
n = number of coil turns
= leakage inductance
= magnetizing inductance
= signal frequency (radians/sec)
Z = coil complex impedance
The equivalent circuit of Figure 7 has an impedance of:
The amplitude of the frequency response of the circuit is shown in Figure 8. The shorted turn resistance , decreases at lower frequencies owing to electrical conduction in the iron behind the turn and increases at high frequencies owing to skin effects.
Static magnetic circuit:
Magnetic lines of flux must close on themselves so that any flux entering a closed surface must also leave it. The flux is equal to the flux-density or magnetic induction B integrated over an area A:
Then, over a closed volume:
If the flux-density (measured in Gauss) is uniform over areas and , where (m) refers to the magnet at its neutral magnetic plane and (g) refers to the gap (at the coil), and using (l) to denote flux "leakage", i.e. flux which takes paths other than the useful ones through the coil, then:
Where = a leakage factor, equal to or greater than 1
Along any closed path, the line integral of mmf (magnetomotive force) must equal the current flowing through the enclosed area. If there is no current (i.e. a static magnetic circuit) the integral must equal zero:
This integral can be thought of as occurring in several parts, with one through the magnet, one through the gap (which includes all nonmagnetic regions including the coil and shorted turn as well as any open volume) and one or more in the pole pieces:
Assuming that the permeability of the pole material is essentially constant (although not necessarily from place to place), H is constant within the magnet and in the gap. If the field strength in the pole pieces is proportional to them, then:
In the gap, is related to by the permeability of free space, which is very nearly the same for air, copper, aluminum, plastics, etc. In engineering units, of course, the definitions of Gauss (the measure of B) and Oersteds (the unit of H) have been adjusted so that the permeability of space is one Gauss per Oersted:
= magnetic induction or flux-density, Gauss
= magnetic coercivity, Oersteds
= 1 Gauss/Oersted
In the magnet, B is related to H by the B-H curve which is supplied by the manufacturer. The B-H curve of one magnet material is shown in Figure 9.
These equations, which are now restated, are sometimes referred to as the magnetic design equations:
The above relationships are well-known and widely used in magnetics design. It is reasonable to use them for first-order calculations in the steel pole regions of a linear motor. However, they are somewhat too simplified to use in determining the magnetic operating point, or the flux-density (B) at the coil (and thus the flux/current ratio) by direct application in a single calculation. By assuming that the flux is purely radial (no leakage) and that the B-H curve is a straight line in the region of interest (a good assumption, for "square loop" materials like ferrites, samarium-cobalt and neodymium-iron) and that the magnetic reluctance may be modeled as a constant , an analytic solution may be found.
C = a constant
S = slope of the B-H curve =
= the value of H at the point on the H-axis where the upper part of the B-H curve meets it when extended in a straight line.
= radius at the outside of the magnet
= radius at the inside of the magnet
= radius at the inner edge of the effective gap (outer radius of the center pole)
From the B-H curve, it can be seen that, on the upper slanted part of the curve:
Substituting and solving,
Another method is to apply the design formulas (15-18) to the magnet and gap to get an approximate value. The flux paths are then divided up into segments, using this initial guess as a starting point. For each segment, H and B.A is found and summed.
Repeated adjustments are then made iteratively to B and H to drive the error sums toward zero. This latter approach may appear crude but it permits inclusion iteratively of the nonlinear effects of the pole steel, flux-leakage paths, effects of coil current, etc. Since the material properties are probably not known to an accuracy of better than a few percent, convergence to an error of 1% or less is probably sufficient. The B-H curve of mild steel is shown in Figure 10.
The coil is the most critical element of the design, and all the other components are scaled from it. A small decrease in coil size may permit a considerable reduction in cost, size and overall weight. For these reasons it is best that the motor design not be limited to standard AWG (American Wire Gage) sizes of wire. It is entirely possible to have special wire made to any dimensions required at a cost which is not prohibitive for either design or production. Heat is the ultimate limiter of VCM performance, so the wire should be in a fully annealed state. Cold-worked wire may have as much as 10% or more greater electrical resistance than annealed wire of the same size. The cross-section should be rectangular, with rounded edges. Rectangular wire is somewhat more difficult to handle but has a much better "packing factor" (the ratio of copper to total coil volume).
It is sometimes asserted that the division of number of turns, width and thickness of wire in the coil is unimportant and may be left until late in the design. It can be shown, in fact, that many important motor parameters depend only on total coil volume, provided that packing factor and input power are constant. As wire size decreases, however, the minimum thickness of insulation which can be applied and still result in even, reliable coverage of the wire does not decrease as fast. At small sizes the insulation occupies a significantly larger share of cross-sectional area than it does for larger ones.
The designer is usually faced with the choice between using aluminum or copper for the coil wire. Aluminum has about half the conductivity of copper but only weighs about a third as much. Its increased volume may be an advantage since it would produce a longer coil with increased surface area for better cooling. On the other hand increased coil length may raise the settling time because the longer coil will have a lower first frequency of vibration. Motor length will be increased and there are other structural design consequences. Aluminum wire is harder to terminate and has greater thermal expansion. The only method known to the authors to determine which is best for a particular set of design specifications is to do a preliminary design on paper for each and compare them.
The coil must be carefully considered from a structural point of view or it may come apart in service. It is subjected to high forces, great accelerations, elevated temperatures and rapid thermal cycling.
Space permits only a brief outline of some of the thermal considerations in coil design. Since heating will be the ultimate limiter of the motor's performance, the coil's ability to reject heat is of considerable importance. Although some heat will be conducted away through the bobbin, the major path of heat rejection is from the coil surface into the surrounding air. The coil internal thermal resistance will probably be found to be insignificant compared with the thermal resistance of the boundary layer of air at the surface of the coil.
h = heat transfer coefficient
L = coil length
k = thermal conductivity of air
U = air velocity past the coil
= kinematic viscosity of air
= thermal diffusivity of air
= specific heat of air at constant pressure
= coefficient of dynamic viscosity
If the coil is in forced convection with laminar flow [N(Re)<400,000] and if N(Pr)>0.6 (valid for air) then:
From the Nusselt number N(Nu) the heat transfer coefficient h may be calculated. If the flow is fully turbulent or is in transition, other relationships using these same three numbers must be used.
Even under conditions of no current flowing in the coil, flux across the gap of a voice-coil motor of the simple shape shown in Figure 2 is not uniform for a number of reasons. Near the ends of the magnet gaps the field is diminished because the flux spreads out into the space beyond the magnets. Near the front (outer end) the open face of the center pole provides a leakage path from the outer shell around the magnets. The field is retarded less (because of the leakage) near the front than it is further in. The steel in the center pole carries very little flux near the front and has little reluctance. The flux-density and the reluctance per unit length build up deeper in toward the back of the motor.
When a steady DC current is passed through the coil, a magnetic field is set up which adds to or reduces the static field (depending on the directions of current and field) increasing or decreasing the degree of saturation of the center pole steel. The paths followed by coil-generated flux are not the same as for the static field. Significant leakage across the gap occurs and it is different for various positions, current level and direction.
It is possible to greatly alleviate these problems by careful magnetics design. If the center pole is "hollowed out" from the face, flux-leakage across it can be greatly reduced. If the hole has a shape such that the cross-sectional area of the remaining steel increases linearly with distance inward (a parabola) then the pole steel will all be at about the same permeability. If the gap thickness is varied, becoming wider near the back of the motor, the static flux-paths can be adjusted to have equal reluctances. The coil should complete its stroke entirely within the gap region with small flux guard regions of magnet overlapping at each end.
Direct, analytic solutions of this equation are possible for certain shapes but actual configurations are nearly always too complicated to solve. Instead, Roter's method (an approximate, non-iterative calculation, Ref. 1) is often used for preliminary work. Computer methods (by finite element) are becoming readily available at moderate cost for the personal computer owner but are at present only two-dimensional.
Considerable engineering judgment is therefore still required. Three-dimensional programs of sufficient capacity are offered by a number of sources for mainframe computer use. Another numerical method, finite difference, is also applicable to this problem but is not in wide use. This method is simple enough to be used by almost anyone either by hand or with the aid of a personal computer. Mathematical modeling using a personal computer is also a widely used approach which has had good results (Ref. 2) .
An actuator designer would like to have, as a starting point for his design, information such as the required current, duty cycle, maximum forces and accelerations, stroke etc. Instead, he is often given an average access time, carriage mass, distances allowed for deceleration in a soft crash and other system performance information, from which the parameters of actuator performance must be derived. The required mathematics to do this transformation may be quite lengthy and difficult.
Provided that the intended control scheme meets certain conditions, an approximate method exists (Ref. 3) which gives surprisingly good agreement with much more complete calculations and with measured performance. It is useful for approximate, first estimates of performance, for reviewing specifications for consistency and as a check for calculations.
The procedure is known as the Third-Stroke method. The effects of damping, friction and even of back EMF are ignored. For the last simplification to be reasonable, the cutoff speed should be limited to less than about 65% or so of the terminal velocity. Acceleration and deceleration are assumed to be constant and equal (except for direction, of course). The cutoff speed is set so that when the VCM coil starts at one extreme end of its travel, accelerating toward the other end, cutoff speed is reached when 1/6 of the total travel is reached. If, at that point, current is reversed and deceleration is begun, the coil will stop after traveling a total of 1/3 of the total stroke (in acceleration plus deceleration). It can be shown that the time for this seek (motion) is equal to the average access time. The time required to make the longest seek, from one extreme end to the
= cutoff speed
a = acceleration (or deceleration) rate
= maximum seek or travel distance
= average access time
= maximum access time
"Average" access time in this case means the time (not including settling times) required to make a large number of seeks of every length, divided by the number of seeks. It is assumed that for each seek, any starting point or ending point on the length of possible travel is equally likely.
1. R.C. Roters, "Electromagnetic Devices", John Wiley & Sons, 1941
2. H.D. Chai & R. Lissner, "Voice-coil Motors for Disc Storage Applications", IMCSD, June 1988
3. F.R. Hertrich, "Average Motion Times of Positioners in Random Access Devices", IBM Journal, March 1965