Method and system to measure torque per unit current as a function of angle in hard disk drive actuators

A method and apparatus for calculating a torque constant, K.sub.t, of an actuator for a computer disk drive by passing the actuator through a magnetic field and measuring the induced change in potential. A stepper motor moves the coil through a magnetic field (either by rotation or linear movement), and a fluxmeter measures changes in flux. The measurements are synchronized using an encoder which commands a digital multimeter to sample the fluxmeter. By passing the actuator in both a forward and a reverse direction, the contribution based on voltage integrator drift is reduced. This method and system are not influenced by the torque from mechanical effects such as friction or torque from the current carrying leads.

BACKGROUND OF THE INVENTION
 1. Field of the Invention
 A method and apparatus for analyzing magnets and/or actuator coils in
 magnetic head assemblies for magnetic disk drives, and specifically a
 method and apparatus for measuring the torque per unit current generated
 by magnets as a function of position or angle in actuator coils in hard
 disk drives.
 2. Discussion of the Background
 Hard disk drives contain disk media (i.e., platters on which data is
 written and from which the data is read), actuators which move the head
 over the spinning disk media, and drive electronics which control the
 positioning of the actuator over the disk media. Radial actuators use a
 pivot to swing the head in an arcing path, and linear actuators move the
 head in a linear path. UGIMAG, Inc, the assignee of the present
 application, manufactures permanent magnets and magnet assemblies for use
 in actuators. As shown partially in FIGS. 1A and 1B, in a voice coil motor
 (VCM) type electromagnetic rotary actuator, the actuator includes at least
 one permanent magnet 84, having north and south polarizations, and an
 actuator coil 50 with a number, N, of windings, which generates a magnetic
 field when current is applied to the windings. Often the permanent magnet
 84 is mounted on a metal mounting plate 82 to form a plate assembly 80B.
 The plate 82 is typically made of steel or another highly magnetizable
 material. The actuator may also contain a second plate assembly 80A on the
 opposite side of the actuator coil 50 as compared to the plate assembly
 80B. In FIG. 1B, the plate assembly 80A is not illustrated to provide a
 clearer view of the actuator coil 50 with reference to the polarity of the
 magnet 84. Based on the magnitude and direction of the current in the
 actuator coil 50, the actuator coil 50 is biased toward one side or the
 other of the permanent magnet, producing a torque on the actuator arm. The
 torque or force is produced by the interaction of the current in the coil
 with the magnet 84 of the plate assemblies 80A and 80B. This is known as
 the Lorentz force.
 In rotary actuators, since the actuator coil 50 is mounted on an actuator
 arm near the arm's pivot point, the opposite end of the actuator arm that
 supports the disk drive's read-write head is moved in the opposite
 direction over the disk media. The current is supplied by a servo-control
 system in response to commands from a computer control system. The torque
 is applied to the assembly to move the heads from track to track over the
 recording surface. Currents are supplied that accelerate and decelerate
 the angular motion in order to achieve the desired motion.
 One of the design parameters of an actuator system is the torque produced
 on the actuator arm per unit current. For proper control of the position,
 this torque per unit current, often called the torque constant K.sub.t,
 should be nearly constant over the usable angular range of the actuator.
 For rapid repositioning of the read-write heads, the torque constant
 should be high enough so that the necessary torque can be applied with
 currents available from the servo-controller. In a practical actuator, the
 torque constant K.sub.t is not constant. Often K.sub.t is maximum near the
 center of the angular range and decreases as the arm approaches either
 limit. Designers specify the properties of the function K.sub.t (.theta.)
 where .theta. is the angular position of the actuator arm referenced to a
 suitable origin.
 A variety of methods have been used to test manufactured magnet assemblies
 for quality control purposes. Most of the methods use very simple Hall
 probe measurements of the induction in the gap of the magnet assembly at a
 few selected locations. Sometimes, additional locations are tested to
 indicate possible errors in the position of the magnets or in the location
 of the transition zone of bi-polarized magnets. Static flux measurements
 have also been used to measure the induction in the gap of the magnet
 assembly.
 These known techniques compare measurements to the theoretically determined
 induction at selected points or areas when using flux measurements, but
 the techniques do not correlate to the properties of the function K.sub.t
 (.theta.). There have been several attempts to measure torque directly
 with a torque transducer or indirectly by measuring the force on a
 transducer at a certain position on the actuator arm when a known current
 is applied to the coil. While these systems work, they require complex
 alignment, almost continuous calibration, and do not isolate the magnetic
 contribution to the torque from mechanical effects such as friction or
 torque from the current carrying leads. The most complex systems use two
 force transducers and can measure the torque at only three angles in about
 one minute.
 One known system measures K.sub.t (.theta.) indirectly by analyzing the
 signals in the disk drive servo controller. The actuator assembly is
 mounted in the disk drive housing which is mounted on a rotary stage. This
 allows the measurements to be carried out as a function of angle.
 Calibration is done by comparing the measurement at one angle to the
 torque produced by a standard mass. The test takes about forty-five
 minutes. The initial alignment of the disk drive housing and actuator
 requires about fifteen minutes. The system further requires an expensive
 signal analyzer. This system also requires a complex interface to the disk
 drive servo electronics and is very slow.
 A second torque measurement system is available from Vibrac Corporation of
 Amherst, N.H. The system measures torque from an applied current as a
 function of angle by using a torque transducer. A measurement over an
 actuator's range of motion takes up to thirty minutes. A third torque
 measurement system, the M-15 Universal Torque Tester, is available from
 Measurement Research, Inc. (MRI) of San Fernando, Calif. This system
 automatically measures torque in the clockwise and counterclockwise
 directions due to an applied current derived from force measurements. To
 measure over the entire range takes about two minutes.
 In the second and third systems, contributions from friction and forces on
 current terminals are indistinguishable from the magnetic component in the
 torque data, and these systems are difficult to set up and operate.
 Calibration is difficult, with the results being operator-dependent and
 incompatible with standards from the National Institute for Standards and
 Technology (NIST). Further, because of the testing time required for these
 systems, these systems are inappropriate for even medium to large scale
 production volumes.
 SUMMARY OF THE INVENTION
 It is an object of the present invention to address at least one
 disadvantage of known torque measurement systems for actuators of hard
 disk drives.
 It is an object of the present invention to provide a more accurate,
 reliable and faster measurement system with NIST traceability.
 These and other objects of the invention are addressed by a method and
 system for measuring the torque per unit current as a function of angle or
 as a function of position for actuators for hard disk drives by using
 purely magnetic measurements. For radial actuators, the actuator coil is
 attached to a stepper motor which swings the actuator coil forward and
 backward through a magnetic field in order to measure changes in magnetic
 flux throughout the range of motion. For linear actuators, a motorized
 linear stage drives the actuator coil forward and backward through a
 magnetic field. Changes in flux are measured by a flux meter through a
 digital multimeter. This technique is independent of any constant offset
 in the fluxmeter. Also, by using data from both forward and reverse
 measurements, errors from drift are reduced. This technique is suitable
 for either testing coils with known magnets or for testing magnets with
 known coils.

DESCRIPTION OF THE PREFERRED EMBODIMENTS
 Turning now to the drawings, in which like reference numerals designate
 identical or corresponding parts throughout the several views, FIG. 2 is a
 schematic illustration of a system for measuring the torque per unit
 current in the limit of zero current (i.e., the torque constant K.sub.t)
 as a function of position or angle in actuators in hard disk drives. It is
 to be understood that the system of FIG. 2 is provided as an example, and
 various aspects of the present invention can also be advantageously
 utilized in other systems which utilize flux measuring devices. An
 actuator coil 50 to be analyzed is mounted on a mounting bracket attached
 to a stepper motor 78 to drive the actuator coil 50 through a range of
 motion corresponding to operating conditions inside a hard disk drive. In
 one embodiment, the movement is performed in a magnetic field created by a
 plate assembly 80A and 80B. In a second embodiment, either plate assembly
 80A or plate assembly 80B is omitted. The present invention is useable in
 configurations where the polarizations are created by either (1) separate
 magnets or (2) a single magnet with two polarizations on the same magnet.
 For the first embodiment using two plate assemblies either four separate
 magnets 84 or two bi-polarized magnets 84 are utilized and attached to two
 separate mounting plates 82. For the second embodiment two separate
 magnets or one bi-polarized magnet are used and attached to a single
 mounting plate 82.
 The movement of the stepper motor 78 is controlled by an indexer 70 and
 tracked by an encoder 75. In the embodiment shown in FIG. 2, a trigger
 signal is sent from the encoder 75 to a sampling control unit, such as a
 digital multimeter (DMM) 65. In an alternate embodiment, the output of the
 encoder is sent to the indexer 70 which generates the trigger signal and
 outputs the triggre signal to the DMM 65. The DMM 65 reads integrated
 voltage measurements from a fluxmeter 60 which is attached to the actuator
 coil 50. The data obtained from the DMM 65 is passed (e.g., transferred
 using an IEEE-488-, parallel- or serial-interface) to a computer 100. The
 computer 100 is also connected to the indexer 70.
 For illustrative purposes only, the actuator coil 50 will be assumed to be
 in a magnetic field in an assembly. The magnet generating the field has
 remanent induction B.sub.r =1.2T and has magnetic length 2.54 mm. Standard
 electromagnetic theory allows this to be modeled by an equivalent current
 of 2400 amperes on the edge of the magnet volume. The actuator coil 50 has
 number of turns=250 and a current of 0.3 ampere producing an equivalent
 total current of 75 amperes. With these parameters, the effective current
 of the actuator coil 50 would be 3% of the equivalent current on the edge
 of the magnet volume and would produce a change in the magnetization by
 0.15%. In typical use, the actuator coil 50 has a range of -18 degrees to
 +18 degrees, and this range can be broken down into steps (i.e.,
 inter-sample distances) of 0.2 degrees. This provides an array of over 200
 values which are used to calculate K.sub.t (.theta.).
 Generally, the present invention utilizes a relationship describing force
 on a rigid current carrying conductor in a static externally applied field
 caused by at least one plate assembly 80A or 80B, or by both plate
 assemblies 80A and 80B. The rigid coil is the actuator coil, and the
 measured flux is the total flux through the coil due to the externally
 applied field. Background on such forces is given in Julius Stratton's
 Electromagnelic Theory, published by McGraw-Hill Book Company, New York,
 1941, in the section entitled "2.14. Magnetic Energy of Stationary
 Currents," the contents of which are incorporated herein by reference.
 Equation (7) in Stratton is defined as:
 ##EQU1##
 which relates 1) the work done on the rigid coil as the inner product of
 the total magnetic force due to the externally applied field and 2) the
 virtual displacement to the current in the coil multiplied by the change
 in flux due to the virtual displacement. Since
 .delta.W=F.multidot..delta.r, the force per unit current on the rigid coil
 is given by:
 ##EQU2##
 That is, the magnitude of the force per unit current in the direction of
 the displacement is the change in the flux due to the displacement. This
 is the general result for linear motion.
 The theory for the calculation of the force per unit current in the
 direction of angular displacement is described hereafter with relation to
 FIG. 11. In FIG. 11, conventional terminology is used which follows the
 terminology of Stratton. Two current contours (e.g., current carrying
 filaments) are defined as C.sub.1 and C.sub.2. A first corresponding
 surface S.sub.1 is defined as a regular surface spanning the contour
 C.sub.1. A second corresponding surface S.sub.2 is defined to span
 C.sub.2, but in such a way as to pass through C.sub.1 and coincide with
 S.sub.1. When the two contours are rotated .delta..theta. about an axis n,
 .delta.r=r.times.n.delta..theta.. The torque is expressed by:
 .tau.=r.times.f=r.times.(Ids.times.B).
 Consequently, the work done by the torque .tau. on the element ds for a
 rotation .delta..theta. about the axis n is given by:
 ##EQU3##
 Therefore, the work performed by the torque integrated around the entire
 path C.sub.1 is given by:
 ##EQU4##
 In order to calculate the angularly varying torque constant K.sub.t
 (.theta.), it is possible to measure angularly varying flux,
 .PHI.(.theta.). The torque constant K.sub.t is defined as the torque per
 unit current in the limit of zero current and is given by:
 ##EQU5##
 In hard disk drive actuators, .PHI. is a function of .theta., and .theta.
 is a function of time. Therefore, the time-varying flux in a coil causes a
 potential v(t) to appear across the terminals according to the equation:
 ##EQU6##
 where .omega. is the angular velocity. From that equation, K.sub.t may
 therefore be rewritten as:
 ##EQU7##
 However, it is impractical to determine K.sub.1 this way since t, v(t),
 .theta.(t), and .omega.(t) must be determined and correlated according to
 this method. Instead, the equation for v(t) can be rewritten such that
 v(t) dt=-d.PHI.. By integrating both sides with respect to time from
 t.sub.0 to t.sub.1, we get the equation:
 ##EQU8##
 Since -(.PHI..sub.1 -.PHI..sub.0) is independent of time and angular
 velocity, only the time integral of the induced potential v(t) needs to be
 measured at predetermined angles. This provides flux as a function of
 angle and can be measured by fluxmeters (i.e., integrating voltmeters).
 FIG. 3 is a simplistic view of how a first torque produced by one side
 branch of the actuator coil 50 moving within a magnetic field B. The total
 torque actually is given by the sum of the torque contributions of both
 side branches. In the other side branch, the directions of the field and
 the magnetic field are both reversed, producing a second torque of equal
 magnitude and direction to the first torque. As shown in FIG. 3, the
 actuator coil 50 is placed in a magnetic field B and rotated through a
 change in angle, .DELTA..theta.. The method and apparatus do not require
 that the magnetic field B be constant. As long as B does not vary with
 time but only position, the method and apparatus properly measure the flux
 relative to position. For simplicity, the remainder of the discussion
 assumes that B is constant. During angular movement, the area of a sweep
 is given by:
 ##EQU9##
 This change in area produces a change in flux given by:
 ##EQU10##
 Therefore, the change in flux per change in angle is given by:
 ##EQU11##
 As the actuator coil 50 moves angularly, a change in force given by: dF=NIB
 dr is produced, where N is a number of coils and I is the applied current
 . This produces a change in torque given by: d.tau.=r dF=NIBr dr. By
 integrating with respect to the length of the coil, torque is given by:
 ##EQU12##
 Consequently, the torque constant K.sub.t is given by:
 ##EQU13##
 To calculate these relationships for an actuator coil 50, the present
 invention utilizes a computer system 100 illustrated in FIG. 4. The
 computer system 100 has a housing 102 which houses a motherboard 104 which
 contains a CPU 106, memory 108 (e.g., DRAM, ROM, EPROM, EEPROM, SRAM and
 Flash RAM), and other optional special purpose logic devices (e.g., ASICs)
 or configurable logic devices (e.g., GAL and reprogrammable FPGA). In
 addition, the computer system can have analog-to-digital (A/D) inputs 126
 for receiving information from various analog detectors. The computer also
 has a communication port 128 for communicating with other computers. The
 computer 100 further includes plural input devices, (e.g., a keyboard 122
 and mouse 124), and a display card 110 for controlling a monitor 120. In
 addition, the computer system 100 includes a floppy disk drive 114; other
 removable media devices (e.g., compact disc 119, tape, and removable
 magneto-optical media); and a hard disk 112, or other fixed, high density
 media drives, connected using an appropriate device bus (e.g., a SCSI bus
 or an Enhanced IDE bus). Although compact disc 119 is shown in a CD caddy,
 the compact disc 119 can be inserted directly into CD-ROM drives which do
 not require caddies. Also connected to the same device bus or another
 device bus as the high density media drives, the computer 100 may
 additionally include a compact disc reader 118, a compact disc
 reader/writer unit or a compact disc jukebox. In addition, a printer also
 provides printed copies of important information related to the process,
 including: raw flux measurements; corrected flux measurements; fitted,
 corrected flux measurements; and graphs of any of these measurements.
 The computer system further includes at least one computer readable medium.
 Examples of such computer readable media are compact discs 119, hard disks
 112, floppy disks, tape, magneto-optical disks, PROMs (EPROM, EEPROM,
 Flash EPROM), DRAM, SRAM, etc. Stored on any one or on a combination of
 the computer readable media, the present invention includes software for
 controlling both the hardware of the computer 100, the indexer 70, and the
 DMM 65; and for enabling the computer 100 to interact with a human user.
 Such software may include, but is not limited to, device drivers,
 operating systems and user applications, such as development tools. Such
 computer readable media further includes a computer program, according to
 the present invention, for measuring the angularly varying torque
 constant. A user interface such as LabWindows CVI 4.0 from National
 Instruments is adaptable to customer data, analysis, and storage
 requirements.
 The computer 100 can serve as a remote computer, and can allow an operator
 to "log on" from a remote location and to initiate measurements or collect
 data from previous measurements. In an embodiment where testing is
 automatically performed (e.g., by automatically loading new actuator coils
 to test with a known plate assembly, or by automatically loading new plate
 assemblies to test with known actuator coils), a log of test results are
 kept for each new component tested. In another embodiment, the computer
 100 provides data to remote computers without having a user "log on." In
 this embodiment, the computer 100 provides responses to queries using a
 stateless query mechanism, such as the Hypertext Transfer Protocol (HTTP)
 queries. In this embodiment, the responses are in the form of Hypertext
 Markup Language (HTML) documents, and may include text, graphics or a
 combination of text and graphics.
 To use the above relationships to determine the torque constant, the
 present invention uses a multi-step process: (1) connect a measuring
 device (i.e., fluxmeter 60) to the actuator coil 50 and move the magnet
 assembly 80A and 80B over the coil, (2) check drift rate and adjust if
 required, (3) check offset and zero if too large, (4) move coil to a
 position so that the plate assemblies 80A and 80B can be moved into place,
 (5) move the plate assemblies 80A and 80B into place, (6) rotate the coil
 to the minimum angle as shown in FIG. 5A, (7) arm the measuring device
 control unit (e.g., digital multi-meter 65) (8) rotate the coil to the
 maximum angle, as shown in FIG. 5B, while the programmed pulses from the
 indexer 70 trigger the measuring device control unit to read from the
 fluxmeter 60, (9) rotate the coil to the minimum angle while the
 programmed pulses from the indexer 70 trigger the measuring device control
 unit to read from the fluxmeter 60, and (10) move the coil to a position
 to allow removal of the plate assemblies 80A and 80B. If less precise
 measurements (i.e, measurements with more drift) are acceptable, the
 measuring step (9) can be omitted.
 In order to provide accurate measurements, drift measurements must be
 removed from the flux measurements. Temperature-dependent, constant
 voltages are always present in circuits due to potentials generated at
 junctions of dissimilar metals. However, any change temperature dependent
 change causes a disturbing constant voltage change since the induced
 voltage being measured is very small--typically on the order of
 microvolts. Existing commercial fluxmeters have an adjustment to
 compensate for drift, but this adjustment is too coarse for accurate
 positioning. In practice, signals causing drift are constant over typical
 measurement intervals. These constant input signals result in changes in
 the measured flux that are linear in time with a slope determined by the
 input signal.
 The data collected in steps (8) and (9) are used to calculate
 .PHI.(.theta.) point-by-point and to calculate the derivative with respect
 to the rotational angle, .theta., such that K.sub.t =d.PHI./d.theta.. In a
 preferred embodiment, the step of calculating .PHI.(.theta.)
 point-by-point further includes a method for correcting for drift in the
 fluxmeter 60. The measurement of the present invention records extra data
 to calculate and remove the drift. All that is required is that the drift
 rate be adjusted to be small to minimize the very small errors from the
 assumption of constant drift rate. To do this, the drift rate is adjusted
 by compensating for constant potential on the terminals of the fluxmeter.
 An operator observes the rate of change of the reading with time and
 adjusts the fluxmeter to minimize this rate. In steps (8) and (9), the
 computer 100 records pairs of values, i.e., (1) the angle of that actuator
 coil 50 as measured by the encoder 75 and (2) the flux measurement at that
 angle. FIG. 6A is a graph showing the measurements taken in step (8)
 (i.e., in the forward scan) and in step (9) (i.e., in the reverse scan).
 Although the measurements are shown with reference to time, the time is
 convertible to angle, with the midpoint of the time axis corresponding to
 the maximum angle, .theta..sub.max. The data from step (8) and the data
 from step (9) are stored separately. To correlate the data from step (9)
 with the data from step (8), the data from step (9) is reversed, creating
 two flux measurements per stored angle. FIG. 6B is a graph of the
 correlated data after reversal. It should be noted that another reason
 that this technique works is that any constant offset in the measured flux
 is removed when taking the derivative of the flux with respect to the
 rotational angle of the actuator.
 A series of corrected average values between the forward and reverse scans
 are calculated according to the equation:
EQU corrected(.theta.)=(forward(.theta.)+reverse(.theta.))2,
 and shown by the dashed line in FIG. 6B. From the averaged values, the
 drift signal can be determined by taking the difference point-by-point
 between the average and the two scans, i.e., the forward and reverse
 passes of steps (8) and (9), respectively. The averaged data is nearly
 linear for a typical actuator assembly, and the slope is either positive
 or negative depending on coil connections, magnet polarity, and the
 definition of angle direction. To correlate two sets of average data
 measurements for two different actuator coils or plate assemblies, the
 average of all averaged points is subtracted from each averaged point.
 This creates averaged values which are comparable since the constant
 potential of the set of measurements is removed.
 In order to determine K.sub.t, the derivative of the data with respect to
 the angle measured in radians is determined. If first differences of the
 data are calculated, the torque constant is subject to measurement noise.
 Using interpolation functions gives the same result. Instead, the present
 invention takes more data samples than required to model the magnetic
 field and fits the data to an interpolating polynomial of sufficiently
 high order that all real structure is present. Another factor in producing
 a good fit is to fit to polynomials that are orthogonal over the angle
 range of test. This results in fit parameters that decrease rapidly as the
 order of the polynomial is increased. This works because the main feature
 of the data is the linear term which allows the torque to be constant over
 the angle range. When using additional parameters no longer reduces the
 fit error, the real features have been captured and the remaining error is
 attributable to noise.
 Having determined a fit polynomial, the derivative of the fit polynomial is
 then taken. This procedure smooths the data, retaining all the physical
 information but removing the noise. Such a procedure is described in
 "Interpolation of Hall probe calibration data," Light Source Note LS-207,
 by D. W. Carnegie, in cooperation with the Argonne National Laboratory,
 Argonne, Ill. Jul. 23, 1992, incorporated herein by reference. FIG. 7 is a
 graph of fit data versus flux. By examining a graph of the residuals of
 the fit, a visual indication of the quality of the fit is obtained.
 Numerically the fit can be determined by calculating the sum of the
 squares of the errors of the fit. For higher order fit polynomials, the
 sum of the squares of the errors decreases. Experimental testing shows
 that a polynomial of order 11 to 15 is adequate. A least squares fit is
 often ill-conditioned and sensitive to round off during computation.
 Instead, a numerically stable fit algorithm is used to perform this fit,
 for example using a singular value decomposition algorithm. The torque
 constant K.sub.t is then computed directly using the fit coefficients in
 the derivative of the fit polynomial. If the flux is measured in
 volt-seconds and the angle is in radians, the torque constant (i.e., the
 torque per unit current in the limit of zero current) is in units of
 newton-meter/ampere. FIG. 8 is a graph of torque/current versus angle
 using the fit polynomial. Together, FIGS. 7 and 8 show that the method of
 the present invention is very reproducible.
 To make this procedure traceable to NIST standards, two primary instruments
 are used with calibration traceable to NIST standards. The first is an
 HP-3458A digital multimeter. The second is an HP-53132A universal counter.
 Thus, both the volt and the second are traceable. These instruments are
 used to calibrate volt-second generators which are in turn used to
 calibrate fluxmeters in the plant. Calibration is done by external
 companies as would be appreciated by one of ordinary skill in the art.
 The remaining source of variation is the coil itself. When repeatedly using
 the same coil, the results are reproducible and traceable. Coil variations
 in a design can often allow .+-.3 turns in a coil of 260 turns. This
 causes a .+-.1% variation from coil to coil when measuring the same magnet
 assembly. FIG. 9 is a graph comparing eight different coils which are
 swept through an angular range of -0.3 radians to 0.3 radians. Although
 changing actuator coils can introduce error into the testing of plate
 assemblies, plate assemblies can be accurately measured by consistently
 using a single coil or a few representative coils. Finally, FIG. 10 shows
 the results of five tests using the same actuator but different plate
 assemblies. FIG. 10 shows that flux versus angular for different plate
 assemblies varies widely and that proper classification through individual
 testing will provide more accurate positioning.
 Just as the present invention can analyze rotary actuators, so too can it
 analyze linear actuators. Similar to the movement of a rotary actuator
 shown in FIG. 3, FIG. 12 shows the movement of a linear actuator through a
 magnetic field, B. Although the magnetic field is shown as a uniform field
 projecting out of the page inside the dashed lines, the magnetic field may
 also be position-dependent. FIG. 12 further illustrates an idealized zero
 magnetic field outside the dashed lines. Based on a movement of .delta.x,
 the resulting force can be calculated using the relations:
EQU dF=I dl.times.B
EQU dF=IBdl
EQU F=IBL
 Using the force equation, an amount of work due to the displacement
 .delta.x is calculated according to:
EQU F.multidot..delta.x=F .delta.x=IBL .delta.x=I[B .delta.A]=I.delta..PHI..
 Accordingly, the force per unit current in the direction of the
 displacement is the change in the flux due to the linear displacement as
 given by:
 ##EQU14##
 To move the linear actuator through a magnetic field, either the coil is
 linearly driven through a fixed magnetic field, or a magnetic assembly is
 driven above, below, or around a fixed coil. The motion is achieved
 through the use of a motorized linear stage or the like whose position is
 tracked by an indexer and a linear encoder, similar to FIG. 2. The linear
 encoder provides both position data and trigger signals to the system as
 described above. The resulting sampled data is flux as a ftinction of
 position. By calculating the derivative of the flux data with respect to
 the displacement, the system obtains the force constant, i.e., the force
 per unit current in the direction of the displacement.
 The method and apparatus of the present invention are directly related to
 automating the testing of plate assemblies 80a and 80B and actuator coils
 50. In an automated tester, the metal mounting plates 82 include alignment
 pins for aligning a magnet 84 on the mounting plate 82. The magnet 82 is
 placed over the pins either by an operator or automatically. The plate
 assembly 80A then is moved into position for testing, either manually or
 automatically after sensing the presence of the magnet 84. When the
 operator or the automatic control system senses that the plate assembly
 80A and the actuator coil 50 are properly positioned, either visually or
 using sensors (e.g., trip switches), respectively, the flux measurements
 are taken automatically under computer control. After testing, the magnet
 84 is separated from the mounting plate 82, either manually or
 automatically. The actuator coils or magnets 82 are then conveyed to a
 holding/shipping area for transport. The data concerning the coils or
 magnets may either be conveyed with the corresponding part or conveyed
 separately. The conveyance also may be automated with either mechanical
 belts or air conveyance systems. The system likewise can adjust the height
 or location of the magnets or actuator coils on a multi-purpose testing
 device to simulate any one of several usage configurations.
 As a result of using the coil of an actuator under test to sense an induced
 voltage (rather than generating a magnetic field and moving an actuator
 arm as in known systems), the present invention has many advantages over
 known systems. A complete measurement of K.sub.t (.theta.) over the whole
 range (e.g., 200 measurements in 0.2 degree increments) is very fast,
 taking only two seconds. In fact, the faster the measurement is taken, the
 more accurate the result. Also, the measurements for K.sub.t (.theta.) are
 repeatable at each given .theta. with an accuracy of 0.1%, as compared to
 2-3% for known systems. Further, the measurements are traceable to NIST
 standards by relying on the calibration of two electronic instruments
 which themselves can be calibrated to traceable standards. Further, the
 alignment of the system of the present invention is correct to 0.1
 degrees, a substantial improvement over known systems which have errors in
 the range of 1% to 2%. Also, no current is applied to the system which
 would require a further calibration. The torque measurements are likewise
 (1) independent of the effects and influences of all external forces
 (e.g., pivot friction and lead tension), and (2) independent of vibrations
 in the testing environment which would adversely affect force transducers.
 The setup procedure ensures that reliable angle values can be used to
 measure such characteristics as roll-off at defined angles, the effects of
 magnet displacement, transition zone misplacement, and the effects of
 design changes in steel plates and the magnet outline.
 Obviously, numerous modifications would be evident to one of ordinary skill
 in the art in light of the teachings of the present application. The scope
 of protection for the present application is limited only by the scope of
 the attached claims.