US 20060171091 A1 Abstract Servo control using ferromagnetic core material and electrical windings is based on monitoring of winding currents and voltages and inference of magnetic flux, a force indication; and magnetic gap, a position indication. Third order nonlinear servo control is split into nested control loops: a fast nonlinear first-order inner loop causing flux to track a target by varying a voltage output; and a slower almost linear second-order outer loop causing magnetic gap to track a target by controlling the flux target of the inner loop. The inner loop uses efficient switching regulation, preferably based on controlled feedback instabilities, to control voltage output. The outer loop achieves damping and accurate convergence using proportional, time-integral, and time-derivative gain terms. The time-integral feedback may be based on measured and target solenoid drive currents, adjusting the magnetic gap for force balance at the target current. Incorporation of permanent magnet material permits the target current to be zero, achieving levitation with low power, including for a monorail deriving propulsion from the levitation magnets. Linear magnetic approximations lead to the simplest controller, but nonlinear analog computation in the log domain yields a better controller with relatively few parts. When servo-controlled solenoids provide actuation of a pump piston and valves, electronic LC resonance measurements determine liquid volume and gas bubble volume.
Claims(21) 1. A system for controlling movement of a solenoid having a drive coil, an armature capable of movement within the drive coil over a range of movement and a yoke coupled to the drive coil, the system comprising:
a controller that regulates the ongoing movement of the armature within the drive coil in response to a determination of present armature position, and in response to present armature position, provides an input to the drive coil that produces a desired rate of ongoing movement for a predetermined degree of armature position over the range of movement. 2. The system as set forth in 3. The system as set forth in 4. The system as set forth in 5. The system as set forth in 6. The system as set forth in 7. The system as set forth in 8. The system as set forth in 9. The system as set forth in 10. The system as set forth in 11. The system as set forth in 12. The system as set forth in 13. The system as set forth in 14. The system as set forth in 15. The system as set forth in 16. The system as set forth in 17. The system as set forth in 18. The system as set forth in 19. The system as set forth in 20. The system as set forth in 21. A method for controlling movement of a solenoid having a drive coil, an armature capable of movement within the drive coil over a range of movement and a yoke coupled to the drive coil, the method comprising the steps of:
regulating the ongoing movement of the armature within the drive coil in response to a determination of present armature position; and in response to present armature position, providing an input to the drive coil that produces a desired rate of ongoing movement for a predetermined degree of armature position over the range of movement. Description The present application is a continuation of commonly assigned copending U.S. patent application Ser. No. 09/771,892, which was filed on Jan. 30, 2001, by Joseph B. Seale et al., for a SYSTEM AND METHOD FOR SERVO CONTROL OF NONLINEAR ELECTROMAGNETIC ACTUATORS, which is a divisional application of U.S. patent application Ser. No. 08/882,945, which was filed on Jun. 26, 1997, by Joseph B. Seale et al., for a SYSTEM AND METHOD FOR SERVO CONTROL OF NONLINEAR ELECTROMAGNETIC ACTUATORS now U.S. Pat. No. 6,208,497, which was issued on Mar. 27, 2001, and each of which is hereby expressly incorporated by reference. 1. Field of the Invention The present invention relates to systems and methods for controlling the movement of mechanical devices. More particularly, the present invention relates to the servo control of electromagnetic devices. Still more particularly, the present invention relates to the servo control of solenoids using the measurement of position and the approximation of position of the solenoid's armature to regulate movement of that armature. The present invention may be used in a variety of areas where lifting and/or propulsion is desired with minimum energy consumption. 2. Description of the Prior Art A solenoid is a linear motor, inherently capable of efficient conversion of electrical to mechanical energy. In rotary motors, experience teaches that large size favors efficiency, and for a given size motor, the highest efficiency is obtained when there are very close clearances between stator and rotor parts and when operation is at high RPMs. Electrically speaking, a high frequency of magnetic reversals translates into a high rate of transfer of electromagnetic power. At low frequencies, resistive power losses wipe out efficiencies, while at constant magnitudes of peak magnetic flux, higher frequency translates into higher power transfer without significant increase in I There are and will always be solenoids designed for utilitarian binary control operations, e.g., unlocking the downstairs front door: contexts where power efficiency is of minor importance and stroke length is a matter of feasibility and convenience, rather than a matter of efficient motor design. Magnetic steel solenoid parts are typically solid rather than laminated, because eddy current losses in dynamic operation are not a design consideration. Moving from the context of infrequent operation of a door latch to the very frequent operation of a print wire driver in a dot matrix print head, repetitive impact and consequent work hardening of the magnetic steel in a solenoid becomes a serious consideration. Magnetic materials for solenoids should ideally exhibit a low coercive force, i.e. a low inherent resistance to change in magnetic flux. In magnetic steels, low coercive force correlates with a large crystalline structure, attained through high temperature annealing to allow growth of large crystals. Annealed steels are mechanically soft and ductile, and their low-coercive-force property is described as magnetically soft. Repetitive stress and shock break up large crystals in steel, yielding a finer grain structure that is mechanically work-hardened and magnetically harder. Permanent magnets are optimized for high coercive force, or high magnetic hardness: the ability to retain magnetization against external influences. In solenoids, the mechanical work hardening of the steel takes place in a strong magnetizing field, leaving permanent magnetism in the solenoid circuit. The result is to cause the solenoid to stick in its closed position after external current is removed. This is a failure mode for print wire solenoids. A standard approach to keep solenoid parts from sticking is to cushion the landing at full closure, leaving an unclosed magnetic gap, typically through the thickness of the cushion material. This residual gap generates resistance to residual flux after power removal, reducing the tendency of the shuttle to stick closed. Residual magnetic gaps compromise efficiency in two ways: because the most efficient part of the magnetic stroke is approaching full gap closure, where the ratio of force to electric power dissipation is high; and because currents for maintaining extended closure must be made substantially higher to overcome the magnetic resistance of gaps. Prior art techniques for servo control of solenoid motion and, more generally, magnetic actuation, are summarized well in the introductory section of U.S. Pat. No. 5,467,244, issued to Jayawant et al: “The relative position of the object is the separation or gap between the control electromagnet and the object being controlled and in prior art systems is monitored by a transducer forming part of the control signal generator for the feedback loop. Such transducers have included devices which are photocells (detecting the interruption of a light beam by movement of the object); magnetic (comprising a gap flux density measurement, e.g. Hall plate); inductive (e.g. employing two coils in a Maxwell bridge which is in balance when the inductance of the coils is equal); I/B detectors (in which the ratio of the electromagnet coil current and magnetic flux is determined to provide a measure of the gap between electromagnet and object; for small disturbances the division may be replaced by a subtraction); and capacitive (employing an oscillator circuit whose output frequency varies with suspension gap).” Dick (U.S. Pat. No. 3,671,814) teaches magnetic sensing with a Hall sensor. In the succeeding description of “Apparatus for the Electromagnetic Control of the Suspension of an Object” Jayawant et al derive, from a generalized nonlinear electromagnetic model, a linearized small perturbation model for use in magnetic suspension of an object in the vicinity of a fixed target position. Specifically, they make use of what they call “I/B detectors” (see above quote) wherein the ratio of current “I” divided by magnetic field strength “B” provides an approximately linear measure of the magnetic gap. In text to follow, the ratio “I/Φ” will be used in preference to “I/B” since inductive voltage measurements lead to a determination of the total flux “Φ” rather than a local flux density “B.” Specifically, as noted by Jayawant et al, the time derivative “n·dΦ/dt” equals the voltage electromagnetically induced in a winding of n turns linked by the magnetic flux “Φ.” Thus, time integration of the voltage induced in a coil yields a measure of the variation in “Φ,” and additional direct measurement or indirect inference of “I” leads to a determination of the ratio “I/Φ” used to close the servo loop. Where electrical frequency is substantially higher than the frequency associated with solenoid mechanical motion, the ratio “I/Φ” is also the ratio of the time derivatives “(dI/dt)/(dΦ/dt),” so that a measurement of the high frequency change in current slope “dI/dt,” divided by the corresponding measured change in induced voltage across n windings, “V=n·dΦ/dt,” again leads to a measure of position. One recognizes, in this latter ratio measurement, a measure of the inductance of a solenoid. It is well known that inductance can be measured by determining the natural frequency of an LC resonator having a known capacitance “C,” a technique identified in the final part of the quotation from Jayawant et al, above. By either ratio technique, i.e. involving either a time integral of induced voltage or a time derivative of current, one determines position without the use of sensors apart from means to extract measures of current and induced voltage from the coil or coils employed as part of the actuation device. While these relationships are needed building blocks in the conception of the instant invention, they are not an adequate basis for a servo system generating large mechanical motions and correspondingly large changes in solenoid inductance. First, there are limitations to the linearized small-perturbation models taught by Jayawant et al for controlling large solenoid motions. Second, dynamic stability problems would remain even with a more complicated and costly servo implementation using non-linear circuit models, e.g., computing position as the ratio of current/flux and force as the square of flux, instead of Jayawant's tangential linear approximations of the ratio and square law relations. Where solenoid control is based on driving a winding with a voltage V in order to control a position X, the system to be controlled is fundamentally third-order, involving a nonlinear first order system to get from voltage to change in magnetic force (since voltage controls the first derivative of current in an inductive solenoid, and current channel generates force change without significant delay), coupled to a second order system to make the two hops from force to change in velocity and from velocity to change in position. It is understood that servo control over a third order system is prone to instability since phase shifts around the control loop, tending toward 270 degrees at high frequencies, readily exceed 180 degrees over the bandwidth for which control is desired. Phase-lead compensation as taught by Jayawant et al adds 90 degrees of phase margin, bringing at best marginal stability to an efficient electromechanical system. If electromagnetic efficiency is very low, so that resistance R dominates over inductive impedance ωL up to the servo control bandwidth of ω, then the third order nature of the system is not manifest where gain exceeds unity, and phase-lead compensation provides an ample stability margin. An example of such a low-efficiency system is found in Applicant's “Bearingless Ultrasound-Sweep Rotor” system (U.S. Pat. No. 5,635,784), where a combination of extreme miniaturization and lack of a soft ferromagnetic core places the transition from resistive to inductive behavior well into the kilohertz range. For the efficient actuation systems taught in the instant invention, the transition from resistive to inductive impedance can fall below 100 Hz. “Tight” servo control implies a relatively high loop gain over the bandwidth of significant mechanical response implying a loop gain-bandwidth product well in excess of the bandwidth of significant mechanical response. A combination of high efficiency and tight control spell problems for loop stability, for even with single-pole phase lead compensation, minor resonances, e.g., from mechanical flexure, can throw the servo system into oscillation. While Jayawant et al describe closed-loop servo control techniques applicable where perturbations in position from a fixed target position are small, Wielocii (U.S. Pat. No. 5,406,440) describes an open-loop control technique for reducing impact and mechanical bounce in solenoids used in electrical contactors. Prior art actuation had consisted of instantaneously applying to the solenoid winding the full voltage needed to close the contacts under all operating conditions, taking into account manufacturing variations in the spring preload holding the contacts open. The mixed actuation voltage was usually well in excess of the minimum requirement, and the result was actuation with excessive force and resulting severe contact bounce. Wicloch teaches to ramp the solenoid current up slowly so that when the magnetic force is just sufficient to overcome spring preload force and initiate motion, there will be little additional increase in average actuation voltage before the solenoid stroke is complete. Efficient current ramping is accomplished via a switching regulator, which applies a steadily increasing voltage duty cycle to the solenoid winding while winding current recirculates through a diode during intervals between driving voltage pulses. At a sufficiently high switching frequency, the inductance of the solenoid effectively smoothes the current waveform into a ramp. Similar switching regulation is found in preferred embodiments of the instant invention, but with greater control in order to overcome limitations in Wieloch's soft landing design. When a solenoid begins to close, the resulting “back EMF” due to armature motion tends to reduce electric current, in relation to gap, to maintain a constant magnetic flux, with the result that increases in force with gap closure are only moderate. (The simplified model of Jayawant et al, equation 9, implies no change at all for force as a function of gap closure at constant magnetic flux. In the specification below, Eq. 42 corresponds to equation 9 except for the slope function “dx Hurley et al (U.S. Pat. No. 5,546,268) teach an adaptive control device that regulates electric current to follow a predetermined function of the measured solenoid gap, in order to achieve a predetermined pull curve of the electromagnet. Though such a system responds to some of the limitations of Wieloch, it is not readily adaptable to an actuation system that must respond to changing conditions of starting position and the load force curve while achieving quiet, impact-free, efficient operation. Both for controllability and energy efficiency, some solenoids have been designed with a region of operation in which stator and armature components have closely spaced parallel surfaces and the armature moves in-plane through a region of changing overlap, yielding a region of relatively constant actuation force at constant current. Eilertsen (U.S. Pat. No. 4,578,604) teaches such a geometry in a dual-coil device for linear mid-range actuation and a strong holding force at either end of the actuation stroke. Rotary actuation designs accomplish similar linearity properties using rotary overlap of parallel magnetic plates. The touchdown region where magnetic parts close in contact is commonly avoided in servo control contexts. Magnetic characteristics in this region have presumably been considered too nonlinear for practical control. In particular, the region of operation approaching full closure and contact of mating magnetic surfaces presents a very steeply changing inductance and correspondingly steep change in the sensitivity of force to change in coil current. For a solenoid operated below core saturation, the variation in magnetic force “F” with coil current “I” and magnetic gap “x” is described approximately by the proportionality “F∝(I/x) Holding currents or drive voltages for solenoids are commonly set well below the peak currents or voltages needed to get a solenoid moving toward closure. Both drive and holding signal levels must, in open loop systems, be set high enough to insure closure followed by holding under all conditions, including variability in manufacture from unit to unit, including variability of power supply source (e.g., utility line voltage), and including variability in the mechanical load. Closed loop solenoid control offers a way to reduce both drive and holding signals to minimum practical levels. Yet problems with stability and non-linearity inherent to magnetically soft ferromagnetic-core solenoids have impeded the development of servo solenoids, and therefore have prevented the potential efficiency advantages just described. Solenoids have the potential for operating characteristics now associated with efficient motors: quiet impact-free operation, very frequent or continuous motion, and high efficiency at converting electrical energy to mechanical work. Reciprocating power from electricity is traditionally derived from a rotating motor and a cam or crank shaft, yet solenoids have been demonstrated, in the instant invention, to deliver reciprocating power at high efficiency, provided that the solenoid is designed to operate fast, in order to generate rapid changes of magnetic flux in its windings. In many reciprocating power applications, a solenoid with sophisticated control can offer greater simplicity and substantially tighter control than is achieved with a rotary motor and rotary-to-reciprocating motion conversion device. In the realm of control and sensing of external processes via a solenoid, the invention to be disclosed below can be configured to operate as a controller of position and simultaneous sensor of force, or as a controller of force and simultaneous sensor of position, or in an intermediate mode as a source of mechanical actuation with electrically controlled mechanical impedance characteristics, especially of restoration and damping. With rotary motors, such control has involved the use, e.g., of stepper motors used in conjunction with torque or force transducers, or of non-stepper motors used in conjunction with rotary position encoders and possibly torque or force transducers. The following specification will show a solenoid operated as the linear motor to drive a high-efficiency reciprocating pump, while two additional solenoids control the pump's inlet and outlet valves. All three solenoids operate silently and efficiently under servo control. This new system goes beyond objectives described and claimed in Applicant's U.S. Pat. No. 5,624,409, “Variable-Pulse Dynamic Fluid Flow Controller,” a system using valve solenoid actuators that are mechanically similar to the ones described below and that achieve volumetric flow regulation from a pressurized fluid source over a very wide dynamic range of pulse volumes and rates. The system described below replaces the volume measurement device of Applicant's earlier invention with a solenoid that provides active pumping actuation in addition to fluid volume measurement, inferred from the position of the solenoid pump actuator, where that position is determined from measurement of the resonant frequency of the solenoid drive winding with a capacitor. An object of the invention is control of the powered closure of a solenoid to eliminate closure impact and associated noise, efficiency loss, and progressive damage, including damage to the properties of the magnetic materials. Related objects are to eliminate closure impact through two strategies: a low-cost strategy called “launch control;” and a feedback strategy called “servo control.” A further object is to employ servo control for dynamically maintaining a solenoid position in a hovering or levitating mode. A still further object is to employ servo control for smooth opening of a solenoid. Within “launch control” an object is to infer, from current signals and/or induced voltage signals, a parameter to be compared to a threshold function for determining, dynamically, a time to terminate a launch pulse, such that the solenoid gap closes approximately to a target value short of full closure and short of impact. Within mechanical “servo control,” common terminology describes a sense parameter; indicating mechanical response of a servo system; a target parameter to be subtracted from the sense parameter and resulting in an error parameter; PID gain parameters describing three aspects of feedback amplification of the error parameter, namely: Proportional feedback; Integral feedback; and Derivative feedback; and a drive parameter arising from the summation of the P, I, and D feedback components and that determines the actuation output causing the controlled mechanical response. A servo control loop is characterized by a settling time constant, which may be defined by the shortest time interval beyond which an error parameter continues to be reduced by at least a specified ratio below an initial error defined at the start of the time interval. The settling time constant is generally minimized by an optimum combination of proportional and derivative feedback trains. Increasing of the integral feedback gain generally improves long term error reduction while increasing the settling time constant, thus degrading short term settling and, for excessive integral feedback gain, causing instability and oscillation of the servo system. Within this descriptive framework, in the context of sense parameters for servo control, and where the magnetic gap of the solenoid is identified in the instant invention as the parameter to be sensed and controlled, an object is to employ a measure of solenoid current as a sense parameter of the servo loop. It is a related object to exploit the direct electromagnetic interaction between magnetic gap and solenoid current that inclines solenoid current to vary, in the short term and neglecting external influences, in approximate proportion to magnetic gap. It is a further related object to exploit the relationship demanding that, when a servo control loop causes electromagnetic force to balance against a mechanical load force, the result is to establish a solenoid current that necessarily varies in approximate proportion to magnetic gap. Given that, within the context of ongoing servo control, solenoid current is caused to vary in approximate proportion to magnetic gap, both in the short term due to the physics of the electromagnetic interaction, and in a longer term due to the force-balancing properties of the servo loop, it is an object to employ solenoid current as a sense parameter indicative of solenoid magnetic gap, including for servo control. In an alternative embodiment of servo control employing an alternative sense parameter, the actuation output of the servo system is the output of a switching amplifier, which causes the voltage differential across a solenoid coil to switch between two known values with a controlled duty cycle, resulting first in duty cycle control over the coil current as averaged over one or more switching cycles, and resulting second in a measured AC fluctuation of the time derivative of current in the solenoid coil. That AC fluctuation varies monotonically and consistently with the magnetic gap of the solenoid, providing a repeatable measure of that gap. An object, therefore, in a solenoid system driven by a switching amplifier with duty cycle control, is to employ the measured AC fluctuation in current slope as a sense parameter of the servo controller. Total magnetic flux through the solenoid and coils, designated Φ, is a valuable controller parameter related to magnetic force and to determination of magnetic gap, i.e. position. An object of the invention is to determine variation in magnetic flux in a controller by integration of the voltage induced in a coil linked by the solenoid flux. A further related object is to determine absolute flux by initializing the flux integral to zero for an open magnetic gap and when solenoid current is zero. A further related object is to determine induced voltage in the solenoid drive winding by subtracting an estimate of resistive coil voltage from the total voltage across the coil. A still further related object is to measure induced voltage in an auxiliary sense winding, coaxial with and electrically separate from the drive winding. In the context of related drive parameters, sense parameters, and target parameters for servo control, an object is to split a solenoid control servo system functionally into coupled inner and outer loops with distinct drive, sense, and target parameters, and such that the inner loop has a substantially shorter settling time constant than the outer loop. A related object is to establish an outer control loop for which the sense parameter is a measure of position and the drive parameter is a signal related to force. The sensed measure of position may be a solenoid current, or a measured AC variation in a solenoid current slope, or an auxiliary measurement of mechanical position, e.g., via a Hall effect sensor and permanent magnet or an optical sensor and a light source. A further related object is to establish an inner control loop for which the sense parameter is a measure of variation in magnetic flux, and for which the drive parameter of the outer loop defines at least an additive component of the target parameter being compared with the sensed measure of magnetic flux, and for which the drive parameter is a coil-drive voltage. Note that this drive voltage is the actuation output ultimately controlling mechanical motion in the solenoid. A still further related object is to establish an efficient voltage switching oscillation in an amplifier driving a solenoid coil, and to cause the duty cycle of that switching oscillation to vary such that the short-term-average voltage driving the coil is the voltage drive parameter of the inner loop. As a way of simplifying the electronic design of the servo system, an object related to the establishment of a switching oscillation with a controlled duty cycle is to design a controller loop with an intentional short-term instability that gives rise to switching oscillations having the desired characteristics. We recognize that, over periods substantially longer than the time constant defined by the solenoid inductance/resistance ratio L/R, the average voltage applied to a solenoid coil determines the coil current, while inductive effects are “forgotten.” We further recognize that the integral component or PID feedback control is sensitive only to comparatively persistent or long term trends in the input error signal. From these recognitions, it follows that it is possible to substitute voltage or duty cycle for sensed current in the integral component of a PID feedback controller, with similar long-term results, even though settling characteristics will differ. An object is therefore to design controllers based on integral feedback whose sense variable may be drive current or drive voltage or drive duty cycle. For any of these choices of sense variable, the equilibrium magnetic gap established by servo control is dependent on a combination of mechanical load force and the controller target for the sense variable in the integral loop, i.e. the target for current or voltage or duty cycle. In any of these cases, an object of the invention is a controlled solenoid able to pull to near closure and hold there with a practical minimum of electric power. This can be accomplished by setting the bias for zero rate-of-integration at a signal level that is determined in advance to be sufficient to hold the solenoid at a finite gap. The solenoid of the instant invention can include permanent magnet material, so incorporated that a needed range for holding force is obtained, at zero drive coil current, over a corresponding useful range of the solenoid gap. In such a permanent magnet-incorporating embodiment, an object is to set the bias for zero rate-of-integration at or near zero drive coil current, so that except for power transients to compensate for perturbations from equilibrium, the control system achieves solenoid holding with vanishingly small drive power. With or without the inclusion of a permanent magnet, the moving element of the solenoid may be free-floating, in which case an object of the invention is to achieve stable electromagnetic levitation of a free-floating magnetic element. A further related object is to achieve levitation with a minimum of actuation power. In controlling substantial currents to a solenoid winding, there are difficulties and disadvantages to incorporation of a current-sense resistor and associated differential amplification, including the difficulty of having to sense across a resistor whose common mode voltage swing travels outside the power supply range, and including the disadvantage of added power dissipation in the current-sense resistor. The differential voltage output provided by an isolated flux-sense winding, wound coaxial with the power drive winding, carries all the information necessary for the dynamic determination of both current “I” and magnetic flux “Φ” when such a sense winding is used in conjunction with a switching mode drive. It is therefore an object of the invention to employ a sense winding for the determination of both coil current and magnetic flux in a switching mode solenoid controller. From sense coil information, one can derive either the “integral ratio” designated “I/Φ” or the “derivative ratio” designated “(dI/dt)/(dΦ/dt),” or the “derivative difference ratio” designated “Δ(dI/dt)/Δ(dΦ/dt),” any of these three ratios being a measure of effective magnetic gap and therefore a measure of position, for servo control. The integral ratio depends on a determination of absolute flux, as mentioned above and as feasible when the flux integral, as defined by integration of an induced voltage, can be initialized under known zero-flux conditions, e.g. zero flux for an open magnetic gap and a winding current of zero. A further limitation to absolute flux determination is integration drift, which introduces errors in an absolute flux determination if too much time elapses after initialization. Another disadvantage of the integral ratio is the requirement for division. In some embodiments of the instant invention, effective especially for servo control as the magnetic gap approaches close to zero and magnetic flux approaches a constant value that generates a force approaching balance with a constant load force, the denominator of the integral ratio is approximated as a constant, resulting in the use of current “I” as a sense parameter. This approximation fails, leading to an unstable control loop, under conditions of excessive loop gain or for excessively large magnetic gaps. A more robust controller therefore avoids the constant denominator approximation of the integral ratio and either computes the true integral ratio, or makes use of the derivative difference ratio, or makes use of a direct measure of position via an auxiliary sensor. In a switching regulator context, the denominator of the derivative difference ratio, namely Δ(dΦ/dt), is equal to 1/n times the peak-to-peak voltage swing of the switching amplifier output, where “n” is the number of turns in the drive winding. Thus, for a constant drive voltage swing, the denominator of the derivative difference ratio is constant, and the numerator varies in direct proportion to effective magnetic gap. An object, therefore, is to achieve a more robust controller, less prone to instability, by using an accurate measure of either the effective magnetic gap or the true geometric position as the sense parameter of the outer control loop. A related object is to use the ratio of current divided by flux, I/Φ, as the sense parameter for the outer control loop. An alternative related object in a voltage switching servo is to use the peak-to-peak current slope amplitude, “Δ(dI/dt),” or an approximate measure of this current slope amplitude, as the sense parameter of the outer control loop. For operation of a solenoid approaching full magnetic closure, the saw-tooth current waveform resulting from a switching voltage drive becomes very unsymmetric, with short steep rises in current (when a drive voltage is applied) followed by much more gradual decreases in current where current is impeded by only a small resistive voltage and a small drop across a diode or on-site transistor. In this situation, the peak-to-peak current slope amplitude is well approximated by the positive-going current slope designated “{dot over (I)}>0” where the much smaller negative current slope going into the difference “Δ(dI/dt)” is neglected. In controller contexts where sensing and servo control of true mechanical solenoid position is required over extended periods, where the time-integral determination of total magnetic flux will be prone to drift, effective magnetic gap “X” is determined without drift in a switching regulator context by the relation “X=K1·Δ(dI/dt)” as described above, and magnetic force “F” is well approximated in relation to current “I” by the equation “F=K2·(I/X) In systems applications of a servo controlled solenoid, it is sometimes useful to use the solenoid as a precision measurement device, where position of the solenoid armature correlates with a system parameter to be determined, e.g., fluid volume. When a solenoid is designed for good performance in a servo system, e.g., by employing a powder metal or ferrite core to avoid eddy currents that otherwise confuse electromagnetic measurements, and/or by including a flux sense winding in addition to the drive winding, then the solenoid becomes more useful and accurate as a position measurement device. As mentioned above, position, as related to effective magnetic gap, can be measured using any of the three ratios of current over flux, namely the integral ratio, the derivative ratio, or the derivative difference ratio. Yet another way to measure effective magnetic gap and infer position is by measurement of the resonance frequency of a solenoid winding coupled to a capacitor. Since the solenoid is capable of exerting a selectable or variable force while measuring position, it can therefore be used for the quantitative measurement of mechanical compliance. In a fluid-moving system employing solenoid actuation, measurement of position can be used to measure volume, and measurement of mechanical compliance can be used to measure fluid volume compliance, e.g., as an indication and quantitative measure of bubbles present in a substantially incompressible liquid. An object of the invention is therefore to make double use of a solenoid as an actuator and as a position measurement sensor. A related object is to use a solenoid to measure mechanical compliance. A related object in a fluid moving system is to make double use of a solenoid for pumping and fluid volume measurement. A further related object in a fluid moving system is to use a solenoid to measure fluid volume compliance, including as an indication and quantitative measure of bubbles in a liquid. In an application of the invention for developing a sustained magnetic closure force for holding or magnetic bearing or magnetic levitation functions, an object is to combine permanent magnet materials with soft magnetic materials to generate a passive force bias, whereby the controller generates output drive currents that fluctuate about a zero average to correct for deviations from an unstable equilibrium point where steady magnetic force is derived entirely from the bias of the permanent magnet material. A related object is adaptively to seek out the levitating position for which the electric drive current required to hold velocity to zero is a zero drive current, and where non-zero drive current signals are integrated to generate a cumulative bias correction that drives the system toward the balance position calling for zero drive current. In an application of the invention to magnetic levitation and propulsion of a monorail car, an object is to control multiple magnetic lifting modules in a common mode for regulating height of levitation, in a differential mode for regulating tilt, and in a variable-gain traveling wave mode for generating thrust through engagement of traveling magnetic waves with periodic ripples in a track. A related object for minimizing hysteresis and eddy current losses in a track is to generate lifting forces of magnetic attraction from magnetic fields directed mostly vertically and laterally relative to a longitudinal direction of in motion, thereby generating magnetic flux in the track that remains relatively constant during the period of passage of a levitating car. A related object for minimizing lifting power is to combine permanent and soft magnetic materials for generating lift with a reduced or zero-average current to electromagnetic lifting modules. The parameter X defined by X=I/Φ, for solenoid primary winding current I and total flux Φ linking that winding, is called effective magnetic gap and varies approximately in proportion to the geometric gap of a solenoid with a flat-ended pole piece. This effective gap X is used in various solenoid servo controller embodiments, having the advantage of derivation from coil measurements without recourse to auxiliary sensors (e.g., optical encoders or Hall effect devices.) The induced voltage Vi in a winding of n turns is given by Vi=n(dΦ/dt), so time integration of induced voltage yields a measure of variation in Φ. For controllers starting with an open magnetic gap and zero solenoid current, the initial flux is zero, so integration of Vi from a zero initial condition at zero initial flux yields an absolute measure of Φ. Vi in turn can be measured as the voltage differential across a solenoid drive winding, subtracting out the resistive voltage component IR for current I and winding resistance R. Alternatively, Vi can be measured directly from a sense winding wound coaxial with the drive winding, without need to subtract out a resistive voltage. Thus, effective gap X can be determined from a measurement of current and the integral of measurements of induced voltage, starting from an initial condition of zero. In the important situation where a solenoid is converging under servo control to rest at a near-zero value of gap X, where magnetic force is balancing a mechanical load that approaches a limiting force as gap X approaches its final, small value, then flux Φ, the primary determinant of magnetic force, must necessarily approach a constant Φ A servo control loop for operation of a solenoid includes a relatively slow outer loop for regulating magnetic force in order to control the sense parameter X, and a much faster inner loop to vary average output voltage in order to satisfy the force demand of the outer loop. More specifically, magnetic force varies approximately as the square of magnetic flux, i.e. Φ This hierarchy of interacting loops with different speeds splits an inherently difficult-to-control, nonlinear third order controller into a second order linear controller (the outer loop) and a first order nonlinear controller (the inner loop). The rate behavior of the inner loop is approximately linear, since flux Φ is controlled by average output voltage V (averaged over variable-width pulses) and the controlling physical equation is V=n(dΦ/dt), a linear first order equation. The nonlinearity resides in a variable offset or inhomogeneous term, IR, the component of voltage necessary for current I to overcome ohmic resistance R and maintain the current required to produce flux Φ. This inhomogeneous term in the linear controller loop varies more or less in linear proportion to X for constant force, and nonlinearly with respect to required variations in magnetic force. In effect, the inner first-order control loop must respond to a time-varying input target and to a nonlinear time-varying voltage offset in its output (due to resistive voltage drop) in order to drive its input error to zero. Hence, a difficult nonlinear third order controller problem is segmented first by speed, to solve a first order equation rapidly and reduce the remaining control problem from third to second order, and second by confining nonlinearity to the simpler first-order loop, where nonlinearity appears as an innocuous variable offset term. Means for measuring or determining the position parameter X were discussed above. Also mentioned was determination of flux Φ from integration of a measured induced voltage, either directly from a sense winding or with correction for resistive voltage drop from a drive winding. Where control of force is concerned, it is not necessary that the estimation of flux Φ be free from offset or drift with respect to time. The integral component of a PID control loop automatically corrects for offset and gradual drift in the estimation of flux. The control loop may also be designed so that the integration from induced voltage to flux, and from position error to the integral term of the PID controller signal, takes place in the same integrator, whose output is a sum of terms made immune to drift by the action of corrective feedback through the entire servo loop. In controller configurations where estimates of position X include linear terms in both current I and flux Φ, the integral component of the PID loop may be based not on X, but on a correlate of X at equilibrium. For example, for a known range of static weight and/or spring forces at a holding value of X near zero, i.e. hovering at a negligibly small gap after impact-free solenoid closure, both the steady voltage and the steady current required to keep X in the required small range can be determined in advance. The integral control loop uses as its input, therefore, not X, but the voltage or current determined by the faster proportional and derivative components of the control loop. If the steady gap is “wrong” then the operating current and voltage will be off-target. Specifically, if the current and voltage are too high, relative to the target, this indicates that the magnetic gap X is too large, causing an excessive current demand to drive magnetic flux across the gap. Thus, paradoxically, the integral controller must gradually demand still more current, to drive X to a smaller value, so that less current is demanded. The magnetic force at constant current is destabilizing, with a smaller gap giving a greater force to close the gap more. The integral control loop is “unstable” or, specifically, regenerative, responding to an excess current with a rate of increase in current. The regenerative control loop interacts with the destabilizing magnetic property of the gap to give a stable closed loop behavior, as the product of two negative stabilities yields a positive stability. A solenoid adapted for servo control based on sensed electromagnetic parameters is also well adapted for use as a position sensor, based on determination of the reciprocal of inductance, a parameter that is a well-behaved monotonic indicator of solenoid gap. Position sensing is employed in a pumping system for determination of pumped liquid volume and for quantitative determination of air bubbles present in a pumped liquid, as inferred from changes in solenoid position with changes in electromagnetic force. In steady lifting and levitation applications, permanent magnet materials are combined with soft magnetic materials to generate a lifting bias force at zero cost in steady coil power. The principles of servo control and efficient switching-regulator drive taught elsewhere in this Specification are readily adapted to operation with a permanent field bias and to stabilization of an otherwise inherently unstable permanent-magnet suspension system. These principles are extended to levitation and tilt control in a levitated monorail car, whose propulsion is generated by a perturbation in the lifting magnets to generate traveling waves of magnetic field strength that are synchronized to the passage of ripples in the track. In another application of the invention, where real-time closed-loop servo control is not required, knowledge of the known characteristics of the system is embodied in coefficients of a “launch control” apparatus and method, whose goal is to compute, in advance of launch, a pre-programmed sequence of pulses of predetermined starting times and widths, designed to move the solenoid armature, or shuttle, quickly and with a near-minimum of electrical energy consumption, from a starting position to a target finishing position. In systems contemplated here, this pulse sequence begins (and possibly ends) with a single launch pulse of duration designed to bring the solenoid armature to a stop at a target position. If that position is near magnetic closure but short of full closure and an impact click, and if the solenoid is to be held closed electromagnetically, then a pulse sequence follows to gently pull the armature the remaining distance to full gap closure, followed by pulse train at reduced duty cycle to maintain closure. In situations where the starting position is variable or otherwise unknown to the system software before launch time, then the initial position is measured either by electronically connecting a capacitor to a solenoid winding and using one of the resonance methods described earlier in this section, or by using a “probe pulse” from the solenoid driver to provide data adequate to compute a ratio of current/flux, “I/Φ.” The resonant frequency or the current/flux ratio thus determined is used to compute the previously unknown initial position or, more to the point, the parameters necessary to define a launch pulse duration. If the mechanical characteristics of the solenoid and load are well enough known in advance, then the pre-launch data alone is applied to an empirical formula describing the pulse width that will be required. There may be corrective adjustment for measured power supply voltage, as well as for power supply impedance based on measurements from recent launches (which is a significant issue for operation from an unregulated battery supply whose voltage and impedance will change as the battery is progressively discharged.) If the unknown characteristics of the system include parameters that are not readily determined in advance of a launch, e.g., when an unknown effective preload force must be overcome to initiate motion of the solenoid armature from its initial position, then the launch control method includes an on-the-fly correction to the launch pulse duration. In a specific application of the launch controller to pumping with a solenoid-driven piston stroke, the effective preload force is affected by an unknown fluid pressure. Since the pressure is isolated from the solenoid by a valve (passive or active) that remains closed until roughly the moment in launch when the solenoid armature starts to move, the solenoid controller can obtain no advance knowledge of the preload forces that will affect launch. The effect of the preload force will first manliest itself to the system sensors as an advance or delay in progress toward gap closure. This progress is most readily observed in the waveform of current drawn by the solenoid during the launch pulse. Before the armature begins to move, the current waveform will describe an exponential decay from zero upward toward a resistive upper limit of current. Acceleration of the armature toward closure will rapidly curtail and reverse the upward trend in current. At any given instant, the value of current will be less than, equal to, or greater than a predetermined threshold function of time. When the sensed current waveform crosses the threshold function, the drive pulse is terminated and the solenoid coasts to its target. The shape of the threshold function is determined, in advance, to cause the desired outcome, which is generally to have the solenoid armature come to a halt slightly short of full closure and impact. When the armature is expected to have stopped, a pull-in pulse train may be applied to close the remaining gap, or valve closure may prevent the armature from falling back. A comparable threshold function may be defined for another sensed parameter, e.g., the output voltage from a sense winding. The sensitivity of the sense function to incipient armature motion may be enhanced by including time derivative terms of sensed current of induced voltage. In any case, a motion-sensitive sense parameter is compared to a threshold function of time, and the crossing of the parameter and the function causes launch pulse termination at a time predetermined to send the armature to the vicinity of a target. Implementation of the invention summarized above relies on specific quantitative models of solenoid electromechanical dynamics. While parts of these models are to be found scattered among textbooks, the material to follow pulls together the mathematical and formula relationships necessary for the detailed implementation of the apparatus and methods taught. Following a list of the drawings, we begin with fundamental relationships and move forward to applied formulas. The invention description below refers to the accompanying drawings, of which: The mathematical formulas to be derived will be based on a few simplifying assumptions that, in engineering practice, are sometimes realized. It turns out that these assumptions are best realized for a new class of electromagnetic solenoid designs that are optimized for soft landing, as well as for options of two-point and four-point landing control (to be described later). It is difficult to measure electronic parameters adequate for servo control from a solenoid that has a low electromechanical efficiency. It will be seen that transformer-grade ferrites can be used in constructing fast-acting, energy-conserving, quiet solenoids whose electromagnetic characteristics are virtually “transparent” to a dynamic controller, yielding high-quality measures of mechanical position and velocity. The mating faces of existing designs for pot cores, E-E and E-I cores, U-U and U-I cores, are very well adapted for employing these components as electromechanical solenoid parts. The conductivity of iron in conventional solenoids permits eddy currents, which effectively limit the bandwidth for valid determination of position and velocity, as well as the bandwidth for quick closure through the inefficient region near full-open magnetic gap. Core fabrication from sintered powdered iron substantially overcomes these conductivity problems. Poor closure of the flux path further complicates electronic inference of position and velocity for feedback control, while simultaneously compromising electromechanical efficiency. Separating out these issues, then, there are three important assumptions whose relative validity affects both the validity of the mathematical derivations to follow, and the stability and precision achievable (or even go/no-go feasibility) in a soft landing servo system or launch control system: 1) For Fixed Shuttle Position, Solenoid Behaves Like a Linear Inductor. Discussion: This is to validate the textbook inductor energy formula E=(½)I 2) Solenoid has No Memory, so Magnetic Energy “Now”=Function of Electric Current “Now”. Discussion: Two phenomena might invalidate this assumption: magnetic hysteresis, and eddy currents. Referring to assumption #1, concerning nonlinearity, the magnitudes of hysteresis effects are generally smaller than effects of saturation for solenoids operating at comparatively high flux densities (as is inevitable if a solenoid is reasonably compact for the mechanical energy of its stroke.) Thus, air gaps wipe out hysteresis effects in a similar way to wiping out nonlinearity effects, resulting is comparatively “memory-free” magnetic performance. If eddy currents are of sufficient magnitude, they will partially cancel the effect of current flowing into the solenoid leads in a time-dependent manner. The magnetic energy is a function of all currents, including eddy currents. At low frequencies, where magnetic skin depths are larger than the dimensions of conductive solenoid parts, the time constant for dissipation of eddy currents will be shorter than the time constant for change in drive current, and there will be little eddy current buildup. At high frequencies, with shrinking magnetic skin depths, material at skin depth or deeper will be shielded from the coil fields and thus effectively removed from the magnetic circuit, causing degraded performance and poor correlation with the mathematical model to follow. Ferrite solenoids will be effectively immune to eddy current effects. 3) The Distribution of Magnetic Flux Linking the Windings does not Change with Solenoid Position. Discussion: In the derivations below, magnetic flux is treated as a simple scalar quantity with respect to inductance and back-EMF, as if the same flux links every turn in the winding. If the flux distribution is non-uniform, some turns get more flux than others, but the analysis is still valid for being based on an “effective” number of turns, so long as that number is constant. If the flux distribution through the windings changes significantly when the shuttle position changes and alters the magnetic gap, then the effective number of turns could change, violating the modeling assumptions. Furthermore, in designs that employ separate windings for generating and sensing magnetic flux, there may be a somewhat variable relationship between actuation and sensing as flux patterns in space change with changing shuttle position. There will inevitably be some gap-dependent redistribution of flux in the coil or coils, causing minor error in the mathematical model and in the control relationships between drive and sense windings. These issues are believed to be of minor practical importance in a controller context where, for most of the flight path of a solenoid shuttle, only very approximate control is required. As the shuttle approaches the position of full magnetic closure, more precise control of the flight path is required to achieve soft landing, but in that region near magnetic closure, virtually all magnetic flux will be confined in the core material, totally linking drive and sense windings. Deep core saturation will cause greater magnitudes of flux redistribution, introducing error into the analysis for certain operating conditions that push the envelope of solenoid operation. The derivation of the following formulas may be explained by a gedanken experiment: Assume that the solenoid winding is of superconducting wire, so that mathematically we ignore the effects of electrical resistance, which can be reintroduced separately, later. Imagine that, with the solenoid shuttle position fixed, voltage is applied until the current “I” reaches a specified level, at which time the total magnetic energy in the solenoid is
This is the textbook formula for a linear inductor. Now, short the superconducting winding, allowing current to continue circulating with no external connection that would add or remove electrical energy. Theory says that a superconducting surface is an impenetrable barrier to changes in magnetic flux, since induced currents at zero resistance will prevent the flux change. By extension, a superconducting closed loop or shorted winding is an impenetrable barrier against change in the total flux linking the loop, for if flux starts to change incrementally, the flux change will induce a current change in the superconductor that cancels the flux change. With no electrical energy entering or leaving the system through the wires, the sum of magnetic field energy plus mechanical energy must remain constant. Imagine that the solenoid shuttle pulls on an ideal spring that just balances the magnetic force “F” acting on the shuttle. We assume sufficiently slow motions that there is negligible kinetic energy and negligible acceleration force, so that magnetic force matches spring force in magnitude. We define “x” as the coordinate of the solenoid shuttle, such that an increase in “x” corresponds to an increase in the magnetic gap. We will conveniently define “x=0” as the position of full magnetic closure, giving maximum inductance. Magnetic force “F” pulls to close the magnetic gap and reduce “x,” while the equal but opposite spring force pulls to open the magnetic gap and increase “x.” We define “F” as a negative quantity, tending to reduce “x” and close the gap. If the shuttle moves a positive infinitesimal distance “dx,” the spring does work to pull the solenoid more open, so the spring loses energy. Defining “E When a total magnetic flux Φ links n turns of a solenoid coil, voltage across the coil has two expressions:
Setting the right hand terms of Eqs. 3 and 4 equal to each other and integrating with respect to time yields:
Assuming a superconductive shorted winding is equivalent to assuming V With Eq. 6, Eq. 5 implies the constancy of the product I·L through time:
For this special shorted winding condition, substituting Eq. 7 into Eq. 1:
Under these conditions, the differential in magnetic energy from Eq. 8 is:
With no electrical power entering or leaving the system, the sum of magnetic plus mechanical spring energy is a constant, which means the sum of the differentials is zero:
Substituting in Eq. 10 from Eqs. 2 and 9:
Dividing through by the differential in distance, dx, in Eq. 11, and rearranging yields:
Using Eq. 7, we drop the subscripts from “I” and “L” in Eq. 12:
Differentiating both sides of Eq. 7 with respect to x yields the expression:
Solving for dI/dx in Eq. 14 and substituting this expression in Eq. 13 yields:
Observe that “L” is a decreasing function of “x,” so that “F” and “dL/dx” are both negative. Inductance is high when the magnetic gap is closed, so that a small current produces a large magnetic flux. Eq. 15 is based on conservation of energy with an equilibrium force balance and a zero-resistance coil. The expression has general validity, however, under more complicated conditions. Taking the total derivative of Eq. 1 with respect to “x” yields:
Solving Eq. 14 for “dI/dx” in terms of “dL/dx” and substituting into the second term of Eq. 16 gives a negative contribution of “−I Substitution from Eq. 15 now yields:
This serves as a consistency check. Force is negative, or attractive, in a solenoid, always tending to close the magnetic gap and drive positive “x” toward zero, so “−F” is positive. Eq. 18 therefore indicates that total magnetic energy in a solenoid with a shorted superconducting coil is an increasing function of gap. A spring pulling the gap open does work, which is transformed into magnetic energy. Inductance is reduced with increasing gap, but current is increased to keep the product of current and inductance, “IL,” constant (recalling Eq. 7). With constant “IL,” the energy product
Solenoid manufacturers typically publish families of curves showing force as a function of magnetic gap for various coil voltages. These curves bend steeply upward as the gap goes to zero, their slopes being limited at high coil voltages by magnetic saturation. It is common for the magnetic circuits in solenoids to include a significant non-closing net air gap, usually residing partly across an annulus between stator iron and the shuttle, and partly across a cushion or minimum air gap maintained at the end of the shuttle, e.g., by a mechanical stop located away from the critical site of magnetic closure. Experience has shown that allowing uncushioned magnetic parts to impact together generates noise, shock, and some combination of surface damage, work hardening, and magnetic hardening of the material near the impact site. Magnetic hardening results in retention of a permanent magnetic field after the solenoid current is removed, and sticking of the shuttle in its full-closed position. Eliminating air gaps and pushing the design toward full closure of the magnetic flux loop would seem to invite problems of uncontrollable dynamics and a worsening singularity where force tends toward infinity as the gap closes. These appearances are deceptive, being based on steady state relationships among voltage, gap, and force. Dynamically, as a magnetic solenoid gap closes, flux and force tend not to change rapidly, and solenoid current tends to be driven toward zero with closing gap because the solenoid naturally resists abrupt change in total magnetic flux. The alternative to mechanical prevention of impacting closure of a magnetic gap is dynamic electronic control, taking advantage of inherently favorable dynamic properties of the system and employing servo feedback to avoid impact. The optimum physical design of a solenoid changes substantially in light of the possibilities for dynamic electronic control. If there is full magnetic closure, then the point of full mechanical closure becomes virtually identical (typically within tenths or hundredths of a millimeter) with the point of zero magnetic reluctance, so that the target for a zero-impact soft landing controller is readily and consistently identified. With full closure, the holding current needed to keep a solenoid closed under mechanical load becomes almost vanishingly small. If parts mate too well, there can be problems of sticking due to residual magnetic flux at zero coil current, even with undamaged, magnetically soft materials. If needed, a little AC wiggle to the coil current will reliably unstick the shuttle—a function that needs to be automated in the controller implementation. Expanding on a previous statement of definition, the combined strategies of electromagnetic design, including flux-sensing coils as well as drive coils, and including coordinated electromagnetic, mechanical, and electronic design (including analog and digital software parameters) are collectively called “soft landing.” Related to soft landing, as mentioned, are the strategies and designs for two-point landing and four-point landing, which may optionally be combined with soft landing to achieve good electromechanical performance within a simplified and error-tolerant mechanical design. Eqs. 19 and 20 give an approximate model for inductance “L” as a function of gap
Eqs. 19a, 19b, 19c, and 19d, easily derived from Eq. 19, are included here for completeness. Solving first for the effective magnetic gap in terms of inductance:
If inductance “L” is determined from measurement of a resonant frequency,
Eq. 19 is the formula for inductance with a “pillbox” magnetic field, where the magnetic circuit has no magnetic resistance except across a space between parallel circular plates of area “A” and spaced by the distance “x The inverse of Eq. 20 is also useful:
Given electrical measurements to determine either inductance “L” or the radian frequency “ω” that resonates with a known capacitance “C”, then Eq. 19a (from “L”) or 19d (from “ω”) yields a value for “x When actual “x” goes to zero, there is some residual resistance (specifically: reluctance) to the magnetic circuit, associated with small air gaps, with imperfect mating of the stator and shuttle where they close together, and with the large but finite permeability of the ferromagnetic material in the flux path. This resistance is equivalent to a small residual air gap x A valuable approximate formula for force is derived from substituting from Eqs. 19 and 20 into force Eq. 15. First expanding Eq. 15 in terms of “x Differentiating Eq. 19 gives an expression for the first derivative term of Eq. 21:
Differentiating Eq. 20 gives an approximation for the last term of Eq. 22:
Expanding “L” from Eq. 19 in Eq. 23:
It is not useful to show further expansion of Eq. 24, since no simplifications arise to boil the expanded result down to a simpler formula. Since “x Not obvious without numerical computation is that Eq. 25 is a surprisingly good approximation of Eq. 24 over the entire range of the non-dimensional distance parameter “(x+x Dissipation of power in a solenoid coil is I Energy per stroke “E Moving from rate of power dissipation to net energy to accomplish a stroke, if the acceleration of the shuttle is limited by the mass “M” of that shuttle, and if proportional scaling of the moving part is maintained so that “M” varies in proportion to the cube of the characteristic dimension “D,” i.e M∝D Under circumstances where solenoid inertia is the limiting factor for stroke time, such that Eq. 27a is valid, then the Energy dissipated in electrical resistance, “E Eq. 27c expresses the same proportionality as a loss ratio:
Since mass “M” varies as “D If a solenoid drives a load through a lever that provides some ratio of mechanical advantage or disadvantage, so that solenoid stroke length “x” may be varied at will in a design while maintaining a constant curve of force versus stroke position at the load side of the lever, and if the solenoid mass is the limiting factor for acceleration, then the above formulas for “E It is well known that metallic iron and magnetic steel alloys have a substantially higher saturation B-field than ferrites, e.g., about 2.0 Teslas for iron as against about 0.5 Teslas for ferrites, roughly a 4-to-1 advantage. This translates into roughly a 16-to-1 advantage for maximum force at a given size, e.g., a maximum characteristic dimension “D.” Maximization of force, however, is quite different from maximization of efficiency. Eqs. 27 through 28a imply an efficiency advantage to making a solenoid larger that the minimum size dictated by core saturation. Where efficiency optimization drives the solenoid size large enough that saturation will not occur in a ferrite core, then ferrite has the advantage of lower density than iron, implying a quicker stroke. While magnetic core hysteresis loss is a major consideration in transformer design, hysteresis is a very minor issue in solenoid designs, since the magnetic reluctance of the air gap is predominant in controlling the relationship between winding ampere-turns and the field strength that determines force. Thus, sintered powdered iron cores, which are cheaper but more lossy than ferrites in high frequency transformers, perform about as well as ferrites in solenoids at low flux densities while providing a substantially higher saturation field. In the servo control and measurement strategies to be described below, based on measurements of the voltages electromagnetically induced in solenoid windings, the electrical conductivity of solid iron or steel solenoid parts can present substantial problems for accurate determination of solenoid position. These problems are overcome to some degree with higher-resistivity powder metal cores and even more with ferrite cores. Where extremely high acceleration is demanded in a solenoid core, e.g., in moving an automotive engine valve through a prescribed stroke in a time period constrained by high engine RPMs, then iron or powder metal solenoid parts will accelerate faster than ferrite parts due to the higher achievable flux density. The above proportionality optimization equations are based on constant shape of the solenoid pole pieces. When varying taper of the pole faces enters the optimization process, this adds considerable complication to the analysis. For a given size of solenoid and a given stroke energy requirement, use of tapered pole pieces confers little advantage or disadvantage (the particulars depending strongly on the pattern of saturation of inductor material) except where constraints demand a long stroke, in which case tapered pole pieces can offer some advantage. There is some advantage to shaping a solenoid so that most of the magnetic flux path is in the stator, to minimize shuttle mass and thereby minimize the duration of a stroke. Solenoids whose shuttles are cylinders many diameters in length are at a disadvantage for mass minimization. This patent specification will disclose some flatter solenoid geometries that help maximize gap area, minimize moving mass, and in some contexts simplify the task of guiding the motion of the solenoid shuttle, avoiding the traditional bushing design that can suffer from wear problems in high-duty applications. In deriving Eqs. 1 through 26, we conceptually prevented dissipative electrical energy transfer by assuming a resistance-free, shorted coil, thus simplifying the physics. The derivation of Eqs. 27 through 28a, not shown completely above, introduced electrical resistance. The following derivations conceptually permit exchange of electrical energy with the magnetic circuit via coil current and the combination of externally applied voltage and internal voltage drop due to resistance. The inductive voltage of Eq. 4, which promotes change in coil current, is provided by an external drive voltage from which is subtracted a resistive voltage loss:
The resistive voltage drop “I·R” neglects skin effect, which is usually negligible in coil windings at frequencies for which it is possible to overcome the mechanical inertia of a solenoid shuttle and induce significant motion. Skin effect may be significant in metallic alloys of iron and nickel (the primary ferromagnetic components of solenoids), cobalt (the more expensive ferromagnetic element, less likely to find use in solenoids), chromium (an anti-rust alloying component), and the other trace elements commonly appearing in solenoid alloys. Ferrites do not share this problem. High magnetic permeability in a conductive material has the effect of reducing skin depth very substantially, so that skin currents in the shuttle and stator components of a solenoid can transiently shield underlying magnetic material from a coil field and reduce the dynamic response of the solenoid. Reiterating caution number 2 under “SOLENOID PHYSICS AS APPLIED TO THE INVENTION,” the performance analysis that follows will, for some geometries and materials, be overly optimistic concerning the speed of solenoid response and concerning applicability of the methods being derived here for servo control. This author and a colleague have measured solenoids in which change of inductance with shuttle position is dramatic and readily observed over a broad band of frequencies, and other solenoids in which impedance is almost purely resistive in and below the audio frequency band, with shuttle-position-indicating changes in the inductive component of impedance being detectable only with effort at sorting out in-phase and quadrature-phase impedance components. Solenoids in the latter category are not good candidates for the kind of control described herein. Eq. 7, indicating the constancy of the product “I·L,” implies a formula for the partial derivative of current with respect to inductance when x varies. To get the total derivative of current with respect to time, we need to consider the partial derivative with time associated with inductance L and voltage V The partial derivative of current with time is the effect of applied voltage, the familiar expression for fixed inductances:
The partial derivative of current with inductance is derived from Eq. 7:
Substituting Eqs. 31 and 32 into Eq. 30 yields:
Expanding V A finite difference expression equivalent to Eq. 34 as time increment “dt” approaches zero suggests an approach for numerical integration:
Our mathematical description is almost sufficient to simulate the response of a solenoid, so that the understanding gained can be used to design the analog circuit operations and digital methods of a working controller. Eq. 15, defining force as a function of current and inductance, will be needed, as will Eqs. 19 and 20, defining inductance as a function of gap “x,” plus either Eq. 34 or 35 to simulate the changing electric current, and finally an equation for shuttle acceleration, including a description of the mechanical load. One load description is incorporated into Eq. 36, which describes the acceleration of a shuttle of mass “M” driven by magnetic force “F” and by a spring having linear spring rate “K1” and biased from an unstressed shuttle position “x Having developed the tools to model the motion of a solenoid, we require something in addition to exert servo control for soft landing: a method for measuring or inferring shuttle position. An obvious approach taken in past art is to provide an extra transducer to serve solely as a position sensor. It is feasible, however, to infer shuttle position, or a useful smoothly-varying monotonic function of shuttle position, from inductance measurement or inference from related parameters. The parameter “x A pair of readily determined parameters to define “x The reciprocal of “L” is linear with “x As already stated “x An alternative way to determine “x As a solenoid approaches gap closure, current is driven to a small value, so that the resistive component of coil voltage becomes a small fraction of the externally applied voltage. If supply voltage is “Vb” and the positive current slope is designated “{dot over (I)}>0” then Eq. 39 is approximated by:
The relationship expressed by Eq. 39a is exploited in the embodiment of the invention illustrated in As the current waveform in the figure suggests, current immediately after the voltage transient may exhibit overshoot before settling into a more linear slope. Overshoot can be caused by eddy currents in transformer steel transiently lowering the effective inductance. The current slopes to subtract for Eq. 39 should be computed from data taken after transient settling, if possible. Eq. 19 is readily solved for “x Eq. 40 is just a rearrangement of Eq. 19a. The value of magnetic flux “Φ” will need to be determined from data to enable computations described below, whether this value is measured by integrating a sense coil output, or by inference from measured current “I” and reciprocal inductance “1/L” either from Eq. 39 based on AC measurements over pulse widths or from Eq. 19c based on ringing frequency measurements involving a known capacitance in the solenoid circuit. In the AC measurement case, “Φ” comes from “I” and “L” most simply from dividing the sides of Eq. 5 by “n”:
A potential advantage to AC determination of inductance and shuttle position is that the result is valid even if the reference value of flux, Φ, has been lost. This situation could come where soft landing is used not for magnetic closure, but for slowing the shuttle before it impacts a mechanical stop at full-open, e.g., in a device that must operate very quietly. If a solenoid has been kept closed for a long period, flux in relation to current could drift, e.g., with heating of the solenoid. Heat can affect both magnetic permeability and the intimacy of mating of magnetic pole faces, whose alignment or misalignment can be affected by mechanical thermal expansion. The ratio of flux to current is sensitive to both permeability change and very small changes in the nearly-closed magnetic gap. Another more subtle effect is the time dependence of magnetic permeability. It is known that field strength in permanent magnets at constant temperature declines as a function of the logarithm of time over periods from seconds to years. “Soft” ferromagnetic materials have a similar settling behavior under steady magnetomotive force. For soft landing at full open, the “location” of the target in terms of “x A potential disadvantage to AC determination of inductance and position is that in solid metal solenoids (as opposed to ferrite core solenoids or powder metal core solenoids), high frequency inductive behavior is likely to be affected strongly by eddy currents or, to say the same thing, skin effect, which will have the effect of shielding the solenoid winding from the magnetic core, reducing inductance in a frequency-dependent manner that can make position determination impractical. Tracking of net magnetic flux will be much less sensitive to skin effect than AC inductance determination, since flux is a cumulative, or integral, parameter with respect to both drive voltage and shuttle velocity. Correlated with flux is current, which again is a cumulative or integral parameter in an inductive system. Flux and current determinations will be comparatively less perturbed by high-frequency skin effect. An added potential advantage of the cumulative parameter approach is reduced computation, in both digital and analog implementations. Where a solenoid exhibits a high-Q inductance to well above the frequency of a switching controller, a capacitor may be introduced into the circuit to induce high frequency ringing, in which case the ringing frequency may be determined by waveform sampling or by period measurement using appropriate high-pass filtering and a comparator. A sense winding coaxial with the solenoid drive winding provides an easy way to measure either high frequency ringing or a “dΦ/dt” signal for integration to obtain “Φ.” The derivations so far have concentrated on position measurement. The other significant control issue is to simplify dynamic control of force under dramatically changing conditions of current/force response and voltage/current-slope response. In Eqs. 37 and 38, we found that “L” and “x Eq. 42 is exact to the extent that the assumptions outlined earlier are fulfilled, concerning linearity, memory-free response, and consistent flux linkage of the windings. Eq. 38 provides a way to determine, from data, the value of “x The expression in “x,” x Eq. 38 defines “x Now rearranging the right hand side of Eq. 45 slightly and substituting that result into the expression on the far right side of Eq. 43 yields:
While Eq. 46 shows that all the data for computing F comes from flux “Φ” and current “1,” it is useful to substitute back in the expression from the right of Eq. 44, rearranged to express a dimensionless ratio of x's:
The value for “x When magnetic flux is known, force is known approximately, and quite accurately in the gap-closure landing zone. Added information about current yields a correction that makes the force expression accurate everywhere. Concerning well-behaved control relationships, recall Eq. 4, which is repeated here for emphasis:
The inductive component of coil voltage, V If flux were viewed as a type of current, then a solenoid would behave like a linear constant-coefficient “inductor” with respect to “flux current.” Actual electric current is much more complicated, varying as a function of applied voltage and solenoid shuttle position. As Eq. 49 suggests, it is also possible to consider electric current as a dependent variable, determined by a combination of effective shuttle position and total magnetic flux. For setting force in a solenoid, fortunately, it is the “well behaved” magnetic flux parameter whose control is important, so a good approach to servo control is to measure and control flux using relatively simple, constant-coefficient control means, and consider current as a “byproduct” of control, significant only as something that an amplifier must supply as needed to achieve the desired magnetic flux. The demand for current, and for the extra voltage needed to push that current through ohmic coil resistance in order to maintain a prescribed inductive voltage V In a control context, current “I” will have just been measured. Though the “meaning” of “I” in terms of other variables is given by Eq. 49, there is no advantage in substituting the expansion on the right of Eq. 49 into Eq. 50. The controller will be targeting some rate of flux change, “dΦ/dt,” which will set the required inductive voltage V By making the proper choice of measurements and control parameters, soft landing control is reduced to a linear third-order control problem: second order from the double integration from acceleration to position of the shuttle, and moving from second to third order when one adds the integration from voltage to magnetic flux. (If magnetically induced eddy currents are substantial in the time frame of one shuttle flight, this raises the order of the dynamic system from 3 to at least 4, which makes the servo control problem substantially more difficult, and potentially impossible if solenoid coil measurements are the sole source of flux and trajectory information.) Before proceeding with the control discussion, note that Eq. 51 suggests an alternative method for measuring coil current “I” as needed in Eq. 38 to solve for “x To expand upon the controller design, the outer loop of the controller will demand measurements of “I” and “Φ,” from which are computed to a position “x A switching regulator driving a solenoid will typically provide only unipolar pulses, whose widths will become small when the solenoid is closed. If this regulator encounters large and unpredictable load variations, it may find itself requiring negative pulses, to “put on the brakes” and avoid closure impact. A switching method for “regenerative braking” of inductively sustained coil currents, mentioned above, will be shown in greater detail in the next section. Spelling out the above “PID” controller approach in terms of equations in a specific application context, imagine that there is a fixed controller time interval, Δt, at the beginning of which a pulse is fired, preset for an interval t Rewriting the right side of Eq. 51 in terms of pulse width modulation, the controller will be seeking a change in flux, ΔΦ, to get flux up to a target value during one pulse interval Δt. This net flux change per time interval is substituted for the time derivative of flux on the right side of Eq. 51, while the right side of Eq. 52 is substituted for the left side of Eq. 51:
The prescription for ΔΦ will be spelled out below. The controller will require solution of Eq. 53 for the pulse time interval, t Note that the two terms in the parentheses on the right of Eq. 54 have SI units of volt-seconds, and are divided by an on-voltage to give a pulse period in seconds. In the case of a pair of field effect transistors (FETs) switching one end of a solenoid coil between a supply voltage and ground, presuming similar on-resistance for the two FETs, then it is appropriate to include the FET on-resistance as part of the net resistance “R,” in addition to winding resistance, and then set V The value for ΔΦ comes from the most recent determination of flux by measurement, Φ As indicated in Eq. 48, for a magnetic gap approaching zero, force varies roughly as the square of magnetic flux. For a control system in which the landing or holding force to be expected on a given landing is estimated from the force required on recent landings, the controller will establish an end-point value for force or, in practice, the target flux that was required to provide that holding force, Φ In Eq. 56, “G” is the loop gain coefficient, and “τ” is the phase-lead time constant for the derivative controller term. The overall controller method includes repetitive solutions to Eq. 56, with substitution of the result from Eq. 56 into Eq. 55, and from Eq. 55 into Eq. 54, where the pulse interval is set in order to produce the appropriate flux and force. Values for “x Concerning landing point errors, if the estimate used for “Φ 1) the position variable “x 2) the shuttle will land with a “bump” indicated by an abrupt reduction or bounce in “dx In case 1, as successive values of “Φ In case 2, “Φ As a practical matter, there is generally “no hurry” about soft landing. Where touchdown is approached, duty cycle and drive current are very low, so power consumption is near a minimum, whether or not actual mechanical contact is achieved in the solenoid. It is reasonable to contemplate a controller design in which the target landing point is short of actual mechanical closure and the shuttle is caused to hover dynamically for the duration of time that the solenoid is in an “energized” or “on” state. If hovering is maintained, the controller will effectively be measuring time-varying load force. For hovering performance, the controller might reasonably include a slowly-accumulating integral correction to error, which would track changing load and leave the controller initialized to recent load force history for the next launch. The discussion above has concentrated entirely on controller operation approaching a soft landing. At launch, Eq. 54 will generally dictate a pulse interval t The above discussion has been directed toward a controller in which the position variable “X Since force obeys a square-law equation for solenoids, the following linear approximation (also from a Taylor expansion) is useful near a known operating point, and is exploited by Jayawant:
In both formulas, the perturbation differences A-A0 and B-B0 are multiplied by fixed coefficients. When the operating point is predetermined, as in the context described by Jayawant for magnetic levitation with small perturbations from the operating point, then a linear circuit can be used to implement the above quotient and ratio approximations. For continuous levitation, however, there are problems with Jayawant's approach of using the ratio I/Φ where the magnetic flux Φ is determined as the time integral of an induced voltage: specifically, the integral drifts over time. An AC determination of current-change to flux-change is more cumbersome to implement by Jayawant's approaches, requiring the use of a high-frequency carrier and amplitude detection. Furthermore, experience with real solenoids shows that AC eddy currents induced in metal solenoid material cause the measured inductance to deviate substantially from the ideal relationship, exploited by Jayawant, that 1/L indicates position X. An alternative approach offered here, employing I and Φ rather than their derivatives, is to base control not entirely on estimated position, but rather on estimated force in the short term, and average actuation voltage or current in the long term. If a solenoid is subject to a stabilizing mechanical spring force as well as a destabilizing tendency in the electromagnetic force, one can substantially reduce the electromagnetic demobilization by exerting servo control for constant magnetic flux, Φ, as determined by integration of induced voltage. In the short term, solenoid drive voltage is controlled by deviation of flux from a target flux value, which corresponds to a magnetic force in equilibrium with mechanical spring force at a desired final position. To maintain this position, a particular coil current will be required, and long-term deviation of servo-controlled coil current from a target value is taken as an indication that the integral estimate of magnetic flux is drifting. Such drift is eliminated by summing into the flux integrator (or digital accumulator in a digital implementation of the controller) an error signal representing the difference between actual drive winding current or voltage and that target current or voltage associated with the desired position. In the long term, then, the controller stabilizes current or voltage to a target, which only works when the same controller is controlling current to stabilize magnetic flux in the short term. Note that position measurement is absent from this description. If the zero-velocity magnetic flux is on target, or if the long-term average current needed to stabilize flux is on target, then position is on target by inference, based on a knowledge of the system. In a hybrid approach, short term servo control is based on a linear combination of current and flux, as with Jayawant's linear ratio approximation, but long-term control is based on average current or average applied coil voltage, which may in turn be estimated from average pulse duty cycle from zero to a given supply voltage, in the context of a switching regulator. Implementation of this approach will be described in an embodiment of the instant invention. Jayawant's controllers employ linear power amplifiers to actuate the drive coils, an approach which needlessly dissipates substantial power. A switching or Class-D amplifier can give an efficiency improvement, but then the AC signals introduced into the controller circuit must be dealt with. Taking advantage of that situation, embodiments described below are designed intentionally to make the feedback loop go unstable and oscillate, by analogy to a thermostat that maintains a desired temperature within small error by switching its output discontinuity in response to measured error, resulting in a loop that controls duty cycle rather than a continuous analog parameter. This oscillatory control loop approach results in an energy-conservative transformation from DC power at constant voltage into coil power at variable voltage and current. In an oscillatory control loop, AC signal information is present that can be used to advantage for servo control. One use of this information parallels a use employed by Jayawant, where Jayawant applies a known AC voltage amplitude to a coil at high frequency and reads the resulting AC current as a measure of reciprocal inductance and of effective magnetic gap. This approach by itself parallels applicant's use, described under “OBJECTS OF THE INVENTION” as the numerator of the derivative difference ratio, of the quantity Δ(dI/dt), the oscillatory change in current slope. The instant invention, by contrast, derives this quantity from a very robust signal associated with powering the solenoid, without an auxiliary oscillator. In the efficient switching regulator environment taught here, switching noise at constantly varying frequency and duty cycle would mask a small carrier signal such as is taught by Jayawant, but in the new context, the switching noise itself is interpreted as a position-indicating signal. As will be shown, one-sided rectification of switching noise induced in a sense coil can be used to infer solenoid current from a large, robust signal, without reliance on extraction of current information from a current sense resistor, whose voltage differential signal must in the simplest driver topologies be read against a large common-mode voltage swing. We have discussed the achievement of linear servo control, whose outcome is to establish a roughly exponential decay of error, including simple exponential decay and ringing within a decaying exponential envelope. A real solenoid controller has built in slew rate limits that set boundaries to the region of linear behavior and, consequently, the range of applicability of linear control methods. Typically, the solenoid driver amplifier operates between voltage output limits that set the maximum rate at which solenoid current can be increased and decreased. In the most common two-state output controller, the “on” output state drives current toward a maximum while the “off” output state short-circuits the solenoid winding through a transistor, allowing the current to vary and, ultimately, decay, in passive response to resistance and changing magnetic gap. The momentum attained by the solenoid shuttle falls into two categories: mechanical and electromagnetic The mechanical momentum is related to the inertia of the solenoid shuttle and its coupled load. The “electromagnetic momentum” is the natural persistence of the solenoid magnetic field. A controller can be designed to provide braking of electromagnetic momentum if it provides a drive output state that resists the flow of electric current in the solenoid drive winding. A switching controller can provide an output state designated “brake” that slows the flow of current established during the “on” state faster than that flow would slow down in the “off” state. An effective way to provide a “brake” state in a two-transistor output stage, one transistor connecting the output to a supply voltage and the other transistor connecting the output to a ground voltage, is to “tri-state” the output, i.e. to turn both transistors off, and provide a zener clamp diode between the output and ground to limit the inductively-produced voltage swing on the far side of ground potential from the DC supply potential (i.e. a negative swing for a positive supply or a positive swing for a negative supply). A more complicated “H” drive output configuration, familiar to electrical engineers, functions like a double-pole double-throw switch to reverse the solenoid lead connections and allow the inductive current “momentum” of the solenoid to pump current back into the single supply rail in the “regenerative braking” mode mentioned earlier. Notice that regenerative braking can only reduce the electromagnetic “momentum” quickly, removing but not reversing the electromagnetic driving force. This is because electromagnetic force in a solenoid not based on permanent magnets is inherently unipolar, a square law phenomenon, as indicated, e.g., in Eq. 42, whose only controlled term is the square-law term “Φ Where the direction of momentum is specified, i.e. toward full-closure or full-open position, then it is useful to analyze slewing dynamics in terms of energy rather than momentum. Whereas the definitions of mechanical and electromagnetic “momentum” differ, energy is commonly described by the same units (e.g., joules in S.I. units) in both mechanical and electromagnetic contexts, and it is meaningful to speak of the total energy of the solenoid, combining mechanical and electromagnetic terms. At full closure with zero velocity for a solenoid shuttle being pushed open by a mechanical spring, the total energy of the solenoid assembly is the potential energy of the spring. While analysis is possible for any specific nonlinear spring or complex mechanical load including masses, nonlinear springs, and nonlinear dampers, we will restrict ourselves here to the commonplace and useful example of a linear spring and a single lumped mass, described by Eq. 36 as repeated here:
To review, the mechanical spring rate is “K1,” and “x In the simplest control situation, all the constants of Eqs. 36, 59, and 60 are known in advance and can be incorporated into a control method for a specific solenoid. In interesting situations, one or more characteristics of the mechanical load of the solenoid will be unknown at the time of solenoid launch. In a practical solenoid application described later in this paper, the effective total mass “M” and the spring constant “K1” do not vary, but conditions at launch do vary. Specifically, the solenoid pulls on a stroke piston (described later, using a molded plastic “living hinge” or rolling seal rather than a sliding fluid seal) that draws a fluid through a valve, which remains closed before launch time. The pressure of the fluid behind that closed valve is unknown at launch, which amounts to not knowing the force preload on the system and, consequently, the equilibrium value of “x Controller designs and methods meeting the requirements of this pumping application are applicable in more restrictive contexts, e.g., where the full-open start position for the solenoid is fixed but the spring bias resisting solenoid closure to a specified “x Of greater importance in the fluid pump application described above is the freeing of a microprocessor from a solenoid control task to make way for another task. Specifically, the active valve pump embodiment to be described later involves three controlled solenoids, one for piston pumping and two for valve actuation. For economy, all three valves can be made to operate from a single microprocessor controller. The piston solenoid is energized first, to full-on, after which a regular time sequence of samples from a sense winding provide values proportional to “dΦ/dt,” the rate of change of magnetic flux. A running total of these regular samples gives the present flux. Interleaved with sampling and summing of samples of “dΦ/dt” the microprocessor controls the inlet fluid valve solenoid to reach full-open with a soft landing and switch to a low-computation holding mode, e.g., at a predetermined holding duty cycle. The controller then returns its attention to the piston solenoid to determine a cutoff time for achievement of a prescribed “x To summarize the method development task ahead, we seek a launch controller that begins launch with an initially unknown effective spring-balance value “x The specific procedure given below suggests the manner of approaching different but related control problems that will arise in practical situations. A generalized mathematical treatment for control under unknown mechanical conditions would be quite difficult to approach, given the multitude of ways in which practical systems can differ. The analysis below, following relatively quickly from the governing equations given above, represents but one of many variant paths from the governing equations to a control method appropriate for a specific application. With the example to be given, the engineer skilled in the art in this area of control engineering, and schooled in the form of analysis provided in detail in this disclosure, will be able to come up with a control method and/or controller design tailored to the particular application, but falling within the scope of the invention being disclosed herein. In the event that “x By an alternative approach, the initial or open value of “x Having determined “x The values for “I If the launch on-pulse is interrupted by short off-pulses to reveal the changing AC impedance of the coil, an AC equivalent of Eq. 61 is derived from Eq. 39 and expressed in Eq. 62:
In getting from Eq. 39 to Eq. 62, it is assumed that the denominator voltage change ΔV is constant, being primarily the power supply voltage but with corrections, e.g., for the forward drop of a current-recirculating diode. The change in current slope is associated with the switching transition of the driver transistor, e.g. transistor The threshold value for “ε” may be set at a low, practical value, e.g., ε=0.05, so that a combination of circuit noise, quantization error, and arithmetic error will not cause a false trigger. The time delay from the start of the launch pulse to passage of the motion threshold associated with a given “ε,” as determined by the first measurement that satisfies Eq. 61 or 62, is designated simply t If one could extrapolate back from the triggering event at t, to the estimated current where the force balance threshold was crossed, then one could quantify the preload force and, from there, define all the analytic parameters that determine shuttle trajectory as a function of the electrical input. For the pragmatic task of launching the solenoid on a trajectory to a desired maximum closure at x=x x x t Based on these three arguments, one desires a launch pulse period, t The nature of this pulse of width t If the pulse interval is increased by about 3%, the 10% gap residual will be reduced to 0% Returning to Eq. 63, we have postulated an advance measurement of x An obvious method of defining the specific numerical values for Eq. 63 is a combination of empirical measurement and mathematical curve fitting. One begins with an instrumented prototype of the system to be manufactured and controlled. One sets an input bias, e.g., a bias fluid pressure, and experimentally pulses the system until an interval is determined that carries the solenoid from a specified starting position to a specified final position. The determined time intervals are recorded and the test repeated for other input bias values. The resulting data sets define Eq. 63 for a specified, fixed initial position and a specified, fixed final position. A one-dimensional curve fit to the data is obtained and programmed into a controller. If the controller is to be operated with variable initial positions, then the parameter x If the parameter x For defining interpolation over a smooth surface defined by Eq. 63 when two input parameters are free to vary, one approach is multinominal curve fitting. Multinomials become cumbersome even in two domain variables, and much more so in three domain variables, due to the proliferation of cross-product terms at high orders. Interpolation from a two- or three-dimensional table is a relatively easy method for implementing Eq. 63. A hybrid of table interpolation and polynomial curve fitting is to express each coefficient of a polynomial in the variable “t For any of the launch control situations described above, computer simulation may be used at least for a preliminary computational definition of Eq. 63. A curve-fit method derived from computer simulations can be used for designing and evaluating the overall actuation system, including determination of the system's complexity, cost, efficiency, and sensitivity of control to resolution of time and parameter measurements, including the needed bit resolution for analog conversions. Once a system has been computer-designed and built in hardware, the specific parameters for implementation of Eq. 63 may be fine-tuned using empirical data, which will generally be subject to physical phenomena not fully modeled in the computer (e.g., to the viscoelastic properties of a rubber pump diaphragm, which are difficult to predict from a simulation.) Examining potentially simpler methods that accomplish the same purpose as Eq. 63, consider curve By similar reasoning to the above paragraph, a threshold function can be described in relation to induced voltage Vi, instead of current I. This threshold function is illustrated by trace Other threshold functions are readily derived. Consider the example of a threshold function that incorporates the exponential nature of the no-motion induced velocity trace It is noted that the threshold reference function to which (e.g.) the sum “f+τ·df/dt” is compared is a slice of a higher-dimensional function, that slice being cut at a value of x To implement the strategies illustrated in The solenoid windings, including the drive winding Coil Given the computer and interface circuit of Launch control methods and devices are limited in their scope of operation to situations where initial conditions are stable and measurable and where control extends only to a simple trajectory from a starting position to a target. Continuous control is more complicated but much more flexible, allowing for a system about which less is known in advance and, obviously, allowing continuing control, for gap closure, gap opening, and levitation. To identify the “outer” and “inner” feedback loops described more abstractly earlier in this Specification, With too much inner loop gain, the circuit of Instead of using a division circuit to compute the current/flux ratio, I/Φ, we utilize the approximation of Eq. 57 to approximate this ratio as a constant plus two linear terms, a positive term for variation of I about a reference I0, and a negative term for variation of Φ about a reference Φ0. As in The integrator based on amplifier For proportional feedback, the proportional “Prp” signal on resistor The approximations made in going from the servo of For reasons of economy and mechanical simplicity and reliability, earlier circuits have derived all position information from electrical responses of the solenoid winding or windings. Where the solenoid design permits incorporation of a separate position sensor, performance comparable to the relatively complicated “exact” servo of In An example of the mechanical configuration of a Hall effect sensor and permanent magnet is shown as part of The circuit of Examining the circuit in more detail, the solenoid at Pings, or resonant ringing signals, in the resonant circuit consisting of the drive winding and capacitor Unlike solenoid servo circuits of earlier figures, the circuit of As mentioned, the buffered induced voltage signal on Leaving the “current” or “I·R” signal path momentarily and returning to the overall induced signal path, the output of We now consider circuit operation for combinations of logical levels Circuit operation is simplest when the sampled current feedback path via FET The “PID” signals (of Proportional, Integral, and Derivative feedbacks) added to the dynamics of the basic flux servo circuit (described above) include the duty cycle integral (a sort of integral feedback of position error), plus the value of sample current (generating a stabilizing magnetic spring rate) and the time-derivative of sampled current, providing limited levels of approximate velocity damping. The proportional feedback of the current signal via resistor Consider finally events accompanying the setting of “OPEN” on Examining circuit operation, an initial bias from the DAC at the inverting comparator input drives Vd high. Until Vd spikes low, there is no sampled current feedback on traces Magnetic force F Taking the log of both sides of this equation and substituting {dot over (I)}>0 for X yields:
The factors of 1 and 2 for linear and square terms are provided by the ratios of resistors Observe that all the bending in the spring described here is “planar” or “cylindrical,” meaning that local curvature is always tangent to some cylinder whose axis is parallel to the original flat plane. This is in contrast to a flat spiral spring, which is forced to twist with large axial perturbations unless each loop of the spiral makes a full 360 degree arc (or multiple 360 degree arcs) between inner and outer attachments. A thin strip of metal is much stiffer in torsion and in-plane bending than in cylindrical bending. In a flat spiral spring, the initial bending with small departures from a flat plane takes the form of cylindrical bending, since that is the “path of least resistance.” At large axial perturbations, as the cosine of the slope of the spiral arms becomes significantly less than 1.0, the center section of a spiral spring is forced to rotate, which in combination with the axial displacement results in twisting and in-plane bending of the flat spring. The overall result is a nonlinear increase in axial force. By comparison, the spring illustrated here does not tend to rotate with axial displacement and has a significantly larger linear range than a comparable spiral spring. Screw cap The windings for the solenoid are indicated in views The inlet valve solenoid at The pumping and fluid metering action to be described below is similar to the operation of the invention described in Applicant's U.S. Pat. No. 5,624,409, “Variable Pulse Dynamic Fluid Flow Controller,” sharing with that invention the use of valve timing synchronized to the natural periodicity of fluid flow into and out of a container having fluid volume compliance, so that flow can be maximized in a resonant pumping mode, or controlled in very small-volume fluid pulses utilizing a combination of valve timing and fluid inertia to give a non-linear flow regulation affording a very wide dynamic range of delivered pulse volumes. The operation described here shares the fluid volume measurement function described in U.S. Pat. No. 5,624,409, except that in the invention described here, the measurement device doubles as the actuation device, i.e. the solenoid, in an active pump. The system of U.S. Pat. No. 5,624,409 was conceived as a passive metering device reliant on fluid motive force from a pressurized fluid source, unlike the active system described here. When solenoid Bubble detection can proceed at the end of a fluid fill stroke, when both valves are closed, by at least two distinct approaches. By a “static” approach, a high impedance solenoid current source circuit, such as is illustrated in Once inlet fluid is captured under dome Examining fluid cassette The effect of fluid pressures on valve operation in cassette The sequence just described is modified according to desired fluid delivery rate and current progress relative to the time-varying target for total delivered fluid. Long-term cumulative volume is always based on volume difference from just before to just after an inlet stroke, so that uncertainties of long-term drift in the volume estimation are minimized. For a high delivery rate, a maximum volume intake is followed by a maximum volume delivery, each with a flow pulse timed to the natural half-period of oscillatory flow on the inlet or outlet side (unless, e.g., the outlet flow dynamics are more than critically damped, in which case the “ideal” outlet flow pulse interval is less well defined.) For a lower delivery rate, the outlet flow pulse is interrupted by valve closure in early to mid course, before a maximum volume has been delivered, and the fluid energy available from the pump chamber, amounting to spring energy stored in the suspension of In addition to the operating modes just described, a “firehose” operating mode is possible with the hardware of The primary difference between the system of The increased value of effective gap X arising from the inclusion of permanent magnets implies that more coil current must be used to vary the magnetic field than would be required if the permanent magnet material were filled with a high-permeability transformer-type of material. If the major power requirement is for static holding, then using a permanent magnet to offset DC electric power is well worth the sacrifice in AC efficiency. In a magnetic propulsion system to be explained below, however, large AC field variations are employed to effect propulsion, as the steady DC work of lifting is taken over by permanent magnets. To minimize AC power consumption in such an application, the permanent magnet material should be configured, in the geometry, to be thin and spread out over a wide area, so as to offer a low dynamic reluctance to the magnetic path, where reluctance varies as the ratio of length along the magnetic path divided by area. This geometric proportioning, implies that the permanent magnet material will operate at a low permeance coefficient, which is equivalent to saying that the material will experience a high steady demagnetizing H-field. The factor for increased AC current needed to generate a given AC field strength, due to the addition of permanent magnet material, is given very roughly by 1+Pc, where Pc is the steady permanence coefficient at which the permanent magnet operates in the magnetic circuit. The highest energy product for a permanent magnet is obtained at a Pc of about 1.0, implying a doubling of AC current and a quadrupling of AC power for a given AC flux excitation, compared to operation with no permanent magnet. Most permanent magnets are operate at a Pc greater than 1, but in contexts to be described for magnetic levitation and propulsion, values of Pc of Note that the modifications to the circuit of The servo systems described above control one axis of motion. The inherent instability of magnetic alignment has been noted, and a spring suspension system for rigid alignment control has been described. One can correct the alignment of an object by the same techniques used to control position, sometimes with simplifications over the general servo control problem. Consider a solenoid fabricated from standard “E-I” core parts, where the E-core is the stator and the lighter I-core is the armature, drawn to the E-core. As the I approaches the E, any tilt placing one end of the I closer to the E than the other end will cause a concentration of magnetic flux across the narrower gap. For small alignment errors and no core saturation, the destabilizing magnetic/mechanical spring rate is given roughly by the total force of attraction between the E and the I, multiplied by the cube of the distance between the centers of the center and end mating surfaces of the E, and divided by the square of the average gap. This demobilization can overcome very stiff suspensions near closure. Magnetic alignment correction becomes more precise as the gap becomes smaller, with no singularity in the servo loop as the gap approaches zero if the total magnetic force is also under control. Consider an E-core with two pairs of windings: a force drive and force sense winding wound around the center prong of the E, and an alignment drive and alignment sense winding on each end prong of the E, the end windings being wired in series so that current flow is in the opposite rotation sense at either end, as with current going around a figure-8 loop. Thus, after interconnecting the alignment windings, one has a pair of drive leads and a pair of sense leads coming back to the electronic controller, as with an ordinary drive and sense winding. The signal from the sense winding represents the rate of change of flux imbalance between the ends of the E, and the time integral of that signal represents the total flux imbalance. Merely shorting the asymmetry drive winding causes an electromechanical damping of the kind of rotation of the I relative to the E that generates unequal gaps, while shorting a superconductive figure-8 winding around the ends of the E would almost cancel the destabilizing torsional force. The circuit of If conditions at the start of gap closure are nominally symmetric, i.e. when the initial asymmetry is small and unpredictable, then the best guess for the DAC output in While a circuit of the topology of The principles illustrated above find potential applications in heavyweight lifting, eg., of a levitated monorail car suspended below a track. When a long object is suspended from a narrow rail, a two-variable suspension servo is required, to keep the car up and to keep it level from front to back. To provide fore and aft propulsive thrust and braking, the shape of the lower surface of the track is modified to include periodic waves of vertical ripple, varying the height of the track with variation in longitudinal position. Waves of variation in magnetic field strength are generated within electromagnets and their associated control modules arrayed along the length of the car, those waves being caused to travel backwards along the car at a velocity that synchronizes the waves that travel with respect to the car to the stationary vertical ripples in the track, so that a given portion of the track sees a relatively constant magnetic field strength during the passage of the car. Control of the phase and amplitude of the waves in magnetic field strength with respect to the waves of vertical ripple in the track will result in control of thrust or braking. The suspension problem can be approached as two independent servos for the ends of the car, or as a levitation servo for common mode control and a symmetry servo for differential mode control. In either case, individual electromagnetic control and actuation modules, receiving individual flux-target inputs and providing individual position-indicating outputs (or current-indicating outputs, since current required to achieve a given magnetic flux is related to position, or magnetic gap), are controlled as groupings of inputs and outputs. Separate groupings control different degrees of freedom of the motion of the car, e.g., vertical height, fore and aft pitch angle error, and thrust or braking force. A generalized “position” signal associated with a degree of freedom of the motion of the car is represented as a weighted average or weighted sum over a grouping of control and actuation modules. Weighted sums applicable to the geometry of the suspension drawn near the top of With active control of elevation and pitch, the degrees of freedom of lateral translation, yaw, and roll come to be regulated passively. If the fore and aft suspension magnets tend to self-center laterally because of their geometry, then lateral translation and yaw will be passively stable. For an object hanging below a track, gravity controls roll. For high speed operation of a rail car in wind and rounding corners, very effective damping of roll (i.e. of swinging below the rail) can be provided by active aerodynamic fins. Fore/aft position is not controlled in the static sense, being the direction of travel, but thrust and braking may be controlled by synchronization of traveling waves of magnetic flux to the waves of vertical height along the longitudinal dimension of the track, as explained above. To minimize magnetic losses due to hysteresis and eddy currents in the track as the levitated system moves at high speed, the lifting electromagnets preferably generate fields laterally across the track, rather than fore and aft. The lifting electromagnets should abut each other so that their fields merge into a fairly uniform field over a substantial length of track, ideally over the entire length of the magnetic lifting system. The electromagnets cannot readily merge their fields “seamlessly” along the length of the car (although geometries of permanent and soft magnetic materials could greatly smooth the field), for some magnetic separation is required to isolate the different actuation signal strengths of the different magnets. The magnetic field induced in any part of the track goes from zero to a maximum and back to zero just once during the passage of the levitated car. The slight separation of the magnet sections will inevitably cause some ripple in field strength in a given part of the track during passage of the car, but large fluctuations and total field reversals are to be avoided. If the magnetic flux were to travel longitudinally in the track, rather than laterally, then one of two undesirable situations would arise. If there were no flux reversals in a part of the track during the passage of the car, that would imply that all the magnet poles on one end of the car are North, while the poles on the other end of the car are South poles. Then a cross-section perpendicular to the track length of track would have to support the entire magnetic flux that lifts the car, as must the cross-section of the magnetic flux return path through the levitation system on the car. If the magnetic poles on top of the car were to alternate between North and South some number of times along the length of the car, this would cut down on the cumulative buildup of longitudinal flux in the track but would also generate flux reversals in any given portion of the track during the passage of the car. Avoiding the horns of this dilemma, Returning to While the motor sections Calculations for practical vertical gaps to a suspending track (e.g., fluctuating between one and three centimeters) and a practical longitudinal wavelength (e.g., for 250 mile/hour propulsion with a track ripple wavelength on the order of While a propulsion system may bear no relationship to the levitating suspension system, it is advantageous to share the two subsystems in a single magnetic assembly, as now described. Let the bottom surface of the suspending rail include a periodic vertical ripple along the track length, as drawn, e.g., with a wavelength of one-half meter and a peak amplitude of one centimeter with an average suspension gap of two centimeters, thus allowing a one centimeter minimum clearance at the ripple crests (The ripple need not be smooth, but could consist of fine or coarse steps in track height, although coarse steps would generate more vibration harmonics in a motor than would a smooth ripple.) For control purposes, subdivide the signals associated with the magnetic actuation sections The electronic schematic shown in the lower portion of Examining first the differential mode or tilt-control outer feedback loop, position information Xi from The common mode levitation feedback path operates similarly to the differential mode path just described, but lacks the separate channel weighting factors. The Xi signals on bus The propulsion wave feedback path takes the Xi signal on An alternative approach to actuator position sense and flux control weightings, for the thrust/braking degree of freedom, was mentioned above, namely, two sets of periodic sinusoidal and cosinusoidal weightings of position sense and flux control, extending over the entire set of control modules. A sinusoidal set of position weightings then drives a cosinusoidal set of flux control weightings, and a cosinusoidal set of position weightings drives a negative sinusoidal set of flux control weightings (as the derivative of the sine is the cosine and the derivative of the cosine is the negative sine), so that waves of field strength variation along the row of electromagnets are synchronized to slope variations in vertical height of the track in order to produce fore and aft actuation forces for thrust and braking. In addition to the phase-shifted weighted output signals for producing thrust and braking, electromagnetic power can be conserved if the magnetic flux of individual electromagnetic modules is not forced to remain constant, but instead is allowed to vary inversely as the effective time-varying gap (called X or Xeff throughout this Specification) for variations associated with track ripple. In effect, individual control modules should be operated to correct collective errors in height and fore/aft pitch angle, but should not be operated to minimize flux variations tending to occur in individual modules, in the absence of corrective application of AC coil power, due to track ripple. Thus, a two-phase controller generating waves in flux strength, traveling along a row of electromagnetic modules, can be caused to generate in-phase waves in target flux that minimize corresponding in-phase waves of coil current (allowing the field to vary as it depends passively on the interaction of permanent magnets and a time-varying flux gap, as if the drive windings were absent or open-circuited), while simultaneously generating quadrature-phase waves in target flux to generate desired thrust or braking forces. Alternatively, to minimize power squandered on unnecessary compensation for traveling waves of flux strength caused by track ripple, individual electromagnet control modules can be cross-coupled with neighbors so that flux perturbations of certain wavelengths do not cause either corrective current actuation or passively induced currents that would be impeded by electrical resistance and thus cause the kind of damping and energy loss associated with shorting the windings of permanent-magnet motors. The action of such cross-coupling must then be reconciled with control to produce intentional actively-driven waves of magnetic field strength for generation of thrust and braking. It is noted that the wavelength and amplitude of vertical track ripple might be varied along the track length, e.g., to give a greater slope amplitude in regions where large forces will be required for accelerating and decelerating near a stop, or for generating extra thrust to climb grades in the track, or to give a lesser slope amplitude in regions where less thrust or braking is required and where power losses are reduced by a reduction in track ripple slope. If the track is designed for variable ripple wavelength, then the control system over thrust and braking must be capable of adapting its groupings and weightings of control modules in order to adapt to changing track ripple wavelengths. Microprocessor control and DSP (Digital Signal Processor) control components are appropriate tools for implementation of such adaptive control over multiple modules. Finally, various examples from prior art, e.g., Morishita (U.S. Pat. No. 5,477,788), teach a suspension system of springs and dampers to decouple the considerable inertia of the car from the lesser inertia of the levitation magnets. Control problems arise when individual electromagnets are independently suspended. A simpler system attaches all the electromagnets lifting a car to a single rigid frame, which in turn is decoupled from the car by a spring suspension. A mechanical suspension allows the lifting magnetic modules more easily to follow irregularities in the track, allowing the path of the car to be corrected more smoothly and slowly through the suspension. It is recognized that the control system must prevent modules from “fighting” one another “trying” to achieve some unachievable motion, e.g., as prevented by coupling of the modules to a rigid frame. In the scheme illustrated and discussed with reference to In the suspension and control systems described earlier, control of magnetic flux has been preferred to control of current in the inner control loop of a motion control servo, since actuation force is more linearly related to flux (roughly as a square law of flux) than to current (roughly as the square of a ratio of current to inductance). Control of current, like control of flux, shares the advantage over voltage control of generating low phase lag in the servo loop. In the case of multiple magnetic actuators controlling a lesser number of degrees of freedom of a car, and where corrective actuation of modules to compensate for track ripple is undesired, a controller approach is to have individual magnet modules cause current to track a target current, as opposed to causing flux to track a target flux. Magnetic flux information is provided, e.g. from sense coils or Hall effect sensors, by the separate modules, but flux control is achieved at the level of groupings of actuators, rather than for individual actuators. At a higher tier of the system, translational and rotational motion is controlled via control of groupings of flux at an intermediate tier. Thus, a three tier control system controls current and measures flux at the module level, controls patterns of flux and/or force at the intermediate level, and controls position and rotation at the highest level. Such a control system directly avoids wasteful current responses to track ripple at the level of indivdual modules, whereas a two-tier system with flux control at the lower tier relies on corrective compensation going from the group controller to the individual modules. The systems described for solenoid control with soft landing can be applied to the control of automotive valves, resulting in the complete elimination of the cam shaft and mechanical valve lifters. With an automotive valve, one needs quick acceleration and deceleration of the valve, closure of the valve with a minimum of impact, and significant holding force for both open and closed positions. For tight servo control at closure, an advantageous solenoid configuration is normally open, held by spring bias, with mechanical valve closure taking place at a very small magnetic gap, where servo control is at its best precision. The nonlinear control systems of either Referenced by
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