**Voice-coil
actuators**

**George P. Gogue
Joseph J. Stupak, Jr.**

**G2 Consulting,
Beaverton, OR 97005
**

Voice-coil actuators are electromagnetic devices which produce accurately controllable forces over a limited stroke with a single coil or phase. They are also often called linear actuators, a name also used for other types of motors. A related form is the swing-arm actuator, which is used to rotate a load through a limited angle (usually 30 degrees or less).

These
devices are capable of
extremely high accelerations (more than 20 times gravitational
acceleration, or "g"), and great positioning accuracy when suitably
controlled (one millionth of an inch or better). A properly designed
device may have a settling time (the time required for structural
vibration to settle down to below the measurement threshold after a
high-acceleration move) of two milliseconds or less.

The major use of this type of actuator is in computer peripheral disk drives. They are also used in shaker tables, lens focusing, medical equipment, laser-cutting tools and elsewhere.

The
name "voice coil" derives from
the origins of the earliest designs, which were scaled up from the
drive mechanisms of cone speakers. That form, shown in Figure 1,
continued in use for some twenty years but is rarely seen today. About
1971, a newer form appeared, using a shorter coil and longer magnet
section (Figure 2). It had a much smaller, lighter coil, and was
capable of higher acceleration, shorter settling time, and better
linearity of performance. Since that time a great many improvements
have been discovered, and new developments continue to occur.

The
design of these deceptively
simple-looking devices is more challenging than it might seem at first.
Almost any design can be made to work, but not very well. Second-order
effects immediately become important and must be dealt with early in
the design.

**Basic
principles**

If a conductor (wire) carrying electric current is placed in a magnetic field, as shown in Figure 3, a force is generated on the wire at right angles to both the direction of current and magnetic flux:

F = force vector

i = electric current

dl = a differential element of length of the conductor

B = magnetic flux density vector (magnetic induction)

In convenient engineering units:

F = reaction force, lb

B = field strength, G

L = conductor (wire) length, inches

i = current, A

If the magnetic field in the gap of the linear actuator shown in Figure 2 is 3000 G and the coil average diameter is 2 inch with 200 turns, then the conductor length L is:

L = 2 inches x 200 (turns) = 1257 inches

The force constant (lb force per amp of current) is:

= F/i = 0.5771 x (lb/G-A-in) x 1257 inch x 3000 G

= 2.18 lb/A

The term "force constant" is in common use but it should probably be replaced with "force-current ratio", since in practice it is hardly constant but varies with current amount and direction, coil speed and position, and rate of change of current.

As
soon as the coil begins to
move, in response to the generated force, a voltage is induced in it,
caused by its motion in a magnetic field. This voltage, called back
electromotive force or "back emf", is proportional to speed, field
strength and current, and is in a direction to oppose motion. It
reduces the voltage across the coil, lowering the current and the rate
of acceleration. The back emf is of magnitude:

As the motor accelerates, if its range of travel were unlimited, it would eventually approach a speed at which the emf just equalled the applied voltage. At this point no current would flow, since the voltages would just cancel, and acceleration would drop to zero. The maximum speed which the driven load cannot exceed or quite reach (unless affected by other forces) is called the terminal velocity . If the coil supply is voltage-limited (rather than current-limited) its rate of acceleration will be the highest just as it starts and then will decrease as it picks up speed.

A similar emf is built up over the length of any moving conductor in the magnetic field. Thus if the coil assembly contains electrically conductive parts, eddy currents will be generated which will cause a drag or retarding force proportional to speed, degrading motor performance. The coil is the most critical element in the design and all other components are scaled from it. Design tradeoffs involving square versus round wire and copper versus aluminum wire are important to maximize peak acceleration and minimize motion time. Heat dissipation will ultimately limit voice coil motor design.

If metal must be used inside the magnetic gap for a moving part, the conductive path may be broken by a non-conductive joint, such as an axial slot filled with epoxy. If the moving conductor is in a fringing magnetic field however, such as that occurring at the outer edges of the magnets, eddy currents will still be induced in the conductive part in spite of axial cuts.

A special shorted turn configuration is used to overcome winding inductance and to get the high performance coil in motion. The methods that are used to size this important structure are beyond the scope of this paper.

**Static
magnetic circuit**

A permanent magnet has two important parameters, the magnetic induction B (which for engineering purposes is generally measured in Gauss) and the field intensity H, measured in Oersteds.

The magnetic flux crossing an area A is defined as:

The field intensity H integrated over a path l equals the magnetomotive force (mmf) over that length:

The reluctance of a magnetic circuit (a closed loop of flux, driven by an mmf) is:

The description may be thought of as analogous to Ohm's law of electrical circuits:

R = E/i

The flux is analogous to current, mmf to voltage, and the reluctance (written with a script R) to resistance.

The
definitions of the units of
Gauss and Oersted have been set up so that the reluctance of free space
(which is very nearly the same as air) is one G per Oe. If at a point
in space the induction is one G, then the magnetic intensity there is
one Oe.

Figure
4 Magnetizing B-H Curve

Figure 5 Demagnetizing B-H Curve

Typical B-H curves for mild steel and a good grade of ferrite (ceramic) magnet material are given in Figures 4 and 5, respectively. They are given for different quadrants of the graph, as is the usual practice, because the steel pole piece is used in a way that causes some opposition to flux flow, whereas the permanent magnet actually produces the flux. The polarities of H are opposed for these two cases.

Magnetic flux is defined as being limited to loops which close on themselves (no loose ends), so that the integral of B.dA over one area crossing the flux loop is the same as the integral over another area crossing the same set of flux lines.

In the example of Figure 2, if B is 3000 G everywhere at the coil at the average radius of 1.00 inch, then the area is:

All of this flux must flow into the center pole, back into the rear plate and out again into the steel outer shell to the magnets. If the center pole had a radius of 0.8 inch (1.6 inch diameter), the cross-sectional area normal to flux flow would be:

At the back of the center pole all the flux which entered from the magnets must pass through this area to reach the back plate. The flux-density or magnetic induction there is:

From the B-H curve for mild steel (Figure 4) it can be seen that the material saturates at about 20,500 G. Additional flux above this level would encounter a reluctance just as if it were in air. The maximum value for the flux-density at the back of the pole used by some voice-coil motor designers, is 18,500 G. The relative permeability of the steel in relation to air at this flux-density is still about 125. The calculated value of 18,700 G will actually be reduced somewhat by flux-leakage to the center pole from the outer shell at the open front face. It should, of course, be checked later by calculation of the various reluctances of different flux-paths. The thicknesses of the outer shell and back plate may be similarly computed, from the total flux and cross-sectional areas.

A first estimate of the magnet thickness may be found next. Suppose the outside radius of the coil were to be 1.12 inch, and 0.04 inch clearance to the magnet wall was decided on. The gap from center pole to magnet face is thus 1.12 + 0.04 - 0.8 = 0.36 inch (where the center pole radius is 0.8 inch) A ferrite magnet similar to that described by Figure 5 will be used. As the flux moves radially from the center pole to the magnets, the magnetic induction B must decrease as the radius increases to keep B x A constant. The radius to a point halfway through the magnet thickness (estimated to be 0.6 inch) is as follows:

From the B-H curve for the magnet, we can see that the value of H corresponding to B = 2055 G is about 1700 Oe.

For a field of 3000 G at the coil, the coercive force will be 3000 Oe and the approximate mmf required across the gap will be:

The length through the magnet must then be (about):

Of course, the above is not highly accurate. The assumption was made that all the "drop" in mmf was experienced in the gap (none in the steel pole pieces) and that sufficient accuracy could be obtained by replacing the variables B and H by average values in both the gap and magnet. The values of B and H were assumed also to be independent of axial position. From the approximate values, progressively more accurate calculations may be made by replacing local values of B and H with better ones, based on the last iteration. Along any closed loop of flux:

The integral of H.dl through the magnet is equal to the integral outside of the magnet, along the flux path,(and is of opposite sign).

There are other computation methods that can be utilized to compute voice coil winding and permanent magnet performance. A reluctance model of the voice coil actuator structure will allow one to compute the soft iron losses, the permanent magnet field and the winding field values in order to compute actuator performance. The simulation method will yield exact results which may not be much improvement over the piece-wise integration method.

Finite element magnetic computations are also being used in increasing volume. Again, the various non-linearities of the soft iron and permanent magnet materials will be properly entered into the finite element program. One can expect better results at the expense of large amounts of set up and computer processing time.

**Actuator
dynamics**

It may be that the designer of a voice-coil motor has available a complete specification, covering such parameters as force constant, allowable non-linearity, maximum coil resistance, rate of change of current and servo performance, heat dissipation, settling time for mechanical vibration, duty cycle information at maximum required performance etc. However, this information may not be available. It may be in a form which is inconsistent, allowing for no possible realization, or it might require a motor so bulky and expensive that the design is unreasonable. Particularly in the case of disk drive voice-coil motors, the requirements of the system may be stated in a manner that requires a great deal of calculation before the information needed for design can be obtained. In addition, the actual detailed calculation of the dynamics of a state-of-the-art system, tuned for optimal performance with a nonlinear controller can be very complicated. In these circumstances it is very useful to have an approximate method available for initial estimates and as a check.

For disk drive applications, an actuator is required to start at any position within a range of motion and halt at any other location. The target location is known before the motion (called a "seek") is begun. The load is almost purely inertial. Once the new location is reached, a settling time is required before the device stops vibrating to a point that the read or write operation at that location may proceed. The goal of the design is not to minimize the time for a single operation but rather the overall time of a large number of operations. It would be possible for example,

to reach a single location and stop in a very short time and to have heated up the coif during the seek to an extent that the next seek cannot be made quickly without overheating and destroying the coil.

One way to achieve these goals is to set a maximum velocity (cutoff speed) less than Vt. When the cutoff speed is reached during acceleration, current is turned off and the load is allowed to coast.

A
convenient choice for the cutoff
speed is the speed which will result in a velocity/time history for the
longest seek in which acceleration occurs for one-sixth of the total
distance coasting for four-sixths of the distance, and deceleration for
the final one sixth of the distance. This is show in Figures 6a and 6b.

It can be shown that with the above and a number of simplifying assumptions, the average access time for a random mix of all locations and distances is one half the time for the longest seek. The distance for this average seek is one third of the maximum seek. Because of this, the method is often referred to as the one-third-stroke approximation. The average seek is the one which just reaches cutoff speed at the moment current is reversed. Constant acceleration and deceleration, at the same rate, is assumed. From this model of motion the cutoff speed is:

where:

d(max) : the distance of the longest stroke (seek)

: cutoff velocity

a : acceleration

In the above model, the reduction of force (and thus acceleration) by back emf is ignored. Limits on the controller's ability to change current rapidly are not considered nor are several other factors. Nevertheless, the approximate method often gives surprisingly good results when compared to much more complicated dynamics calculations or actual test data.

One
other control scheme sometimes
used for larger inertial loads deserves mention. It is the same as
above for the acceleration and coast portions of the trajectory but the
deceleration is proportional to distance from the target. This is shown
in Figure 6c. In this way strain energy stored by structural
deformation due to inertial forces is gradually reduced, greatly
shortening the settling time. If the driven load has relatively low
frequencies of vibration or damping, overall access time can be reduced
even though the actual time-to-target is slightly increased by this
means.