Method for and control system with piston amplitude recovery for free-piston machines

Abstract

A method and apparatus for detecting the displacement amplitude of an armature of a linear motor or alternator that is drivingly coupled to a load or prime mover. The method and apparatus require only three inputs all derived from the input terminals of the linear motor or alternator: (1) the voltage measured across the linear motor terminals; (2) the current consumed by the linear motor; and (3) the phase between the voltage and current. The three inputs are sensed at the terminals of the linear motor or alternator and used to perform mathematical calculations in the microcomputer of a control system or controller. The mathematical calculations are based on equivalent circuits that are modifications of the equivalent circuit for the linear motor or alternator. The detected displacement amplitude can be used by a controller to limit the displacement amplitude of the armature to prevent collisions.

Claims

1. A method for detecting the displacement amplitude X of a reciprocating armature of an alternating current, linear motor or linear alternator for controlling the reciprocation of the armature, the motor or alternator having a stator including a coil with a pair of coil terminals and wound around a ferromagnetic core with air gaps, the motor or alternator also having magnets fixed to the armature and positioned for reciprocating through the gaps, the motor or alternator also having a motor constant α, an equivalent resistance R.sub.dc representing resistive losses in the coil, an equivalent resistance ΔR.sub.external representing induced losses in structures surrounding the motor or alternator and an equivalent resistance R.sub.s representing losses due to eddy currents and hysteresis induced by motion of the magnets with the coils open circuited, wherein the method comprises: (a) sensing the amplitude of alternating voltage V.sub.appl at the coil terminals; (b) sensing the amplitude of alternating current I through the coil terminals; (c) sensing the phase angle ϕ.sub.pf between said voltage and said current; (d) calculating I.sub.m according to the equation $I_m=I-\frac{V_{\mathrm{appl}}}{R_s}$ (e) calculating (V.sub.ind).sub.I according to the equation
(V.sub.ind).sub.I=V.sub.appl cos Ø.sub.pf−(R.sub.dc+ΔR.sub.external)I.sub.m (f) calculating (V.sub.ind).sub.L according to the equation
(V.sub.ind).sub.L=V.sub.appl sin Ø.sub.pf−LωI.sub.m (g) calculating the displacement amplitude X according to the equation $X=\frac{1}{\mathrm{\omega \alpha }}\sqrt{{\left(V_{\mathrm{ind}}\right)}_I^2+{\left(V_{\mathrm{ind}}\right)}_L^2}$ wherein ω is the radian frequency of V.sub.appl.

2. The method according to claim 1 and further comprising: (a) determining a motor or alternator armature displacement amplitude X.sub.c at which the armature or a structure fixed to the armature collides with a stationery structure; (b) selecting and storing a motor or alternator armature displacement amplitude X.sub.max that is less than X.sub.c; (b) repeatedly performing said calculation of the displacement amplitude X; and (c) limiting the displacement amplitude of the motor or alternator to maintain a displacement amplitude X≤X.sub.max.

3. A method for detecting the displacement amplitude X of a reciprocating armature of an alternating current, linear motor or linear alternator for controlling the reciprocation of the armature, the motor or alternator having a stator including a coil with a pair of coil terminals and wound around a ferromagnetic core with air gaps, the motor or alternator also having magnets fixed to the armature and positioned for reciprocating through the gaps, the motor or alternator also having a motor constant α, an equivalent resistance R.sub.ac representing resistive losses in the coil and losses due to eddy currents and hysteresis induced by motion of the magnets with the coils open circuited, and an equivalent resistance ΔR.sub.external representing induced losses in structures surrounding the motor or alternator, wherein the method comprises: (a) sensing the amplitude of alternating voltage V.sub.appl at the coil terminals; (b) sensing the amplitude of alternating current I through the coil terminals; (c) sensing the phase angle ϕ.sub.pf between said voltage and said current; (d) calculating (V.sub.ind).sub.I according to the equation
(V.sub.ind).sub.I=V.sub.appl cos Ø.sub.pf−(R.sub.ac+ΔR.sub.external)I (e) calculating (V.sub.ind).sub.L according to the equation
(V.sub.ind).sub.L=V.sub.appl sin Ø.sub.pf−LωI (f) calculating the displacement amplitude X according to the equation $X=\frac{1}{\mathrm{\omega \alpha }}\sqrt{{\left(V_{\mathrm{ind}}\right)}_I^2+{\left(V_{\mathrm{ind}}\right)}_L^2}$ wherein ω is the radian frequency of V.sub.appl.

4. The method according to claim 3 and further comprising: (a) determining a motor or alternator armature displacement amplitude X.sub.c at which the armature or a structure fixed to the armature collides with a stationery structure; (b) selecting and storing a motor or alternator armature displacement amplitude X.sub.max that is less than X.sub.c; (b) repeatedly performing said calculation of the displacement amplitude X; and (c) limiting the displacement amplitude of the motor or alternator to maintain a displacement amplitude X≤X.sub.max.

5. A freezer comprising a free piston Stirling cooler mechanically linked to and driven by a reciprocating armature of an alternating current, linear motor, the linear motor having a stator including a coil with a pair of coil terminals and wound around a ferromagnetic core with air gaps, the motor also having magnets fixed to the armature and positioned for reciprocating through the gaps, the motor also having a motor constant α, an equivalent resistance R.sub.dc representing resistive losses in the coil, an equivalent resistance ΔR.sub.external representing induced losses in structures surrounding the motor and an equivalent resistance R.sub.s representing losses due to eddy currents and hysteresis induced by motion of the magnets with the coils open circuited, the freezer having a motor controller comprising: (a) a voltage sensor connected to the coil terminals and configured for sensing the amplitude of an alternating voltage V.sub.appl at the coil terminals; (b) a current sensor connected is series with a coil terminal and configured for sensing the amplitude of an alternating current I through the coil terminals; (c) a phase detector circuit connected to the coil terminals and configured for sensing the phase angle ϕ.sub.pf between said voltage and said current; (d) a microcomputer circuit configured for (i) calculating I.sub.m according to the equation $I_m=I-\frac{V_{\mathrm{appl}}}{R_s}$ (ii) calculating (V.sub.ind).sub.I according to the equation
(V.sub.ind).sub.I=V.sub.appl cos Ø.sub.pf−(R.sub.dc+ΔR.sub.external)I.sub.m (iii) calculating (V.sub.ind).sub.L according to the equation
(V.sub.ind).sub.L=V.sub.appl sin Ø.sub.pf−LωI.sub.m (iv) calculating the displacement amplitude X according to the equation $X=\frac{1}{\mathrm{\omega \alpha }}\sqrt{{\left(V_{\mathrm{ind}}\right)}_I^2+{\left(V_{\mathrm{ind}}\right)}_L^2}$ wherein ω is the radian frequency of V.sub.appl and (v) limiting the displacement amplitude X of the motor to less than a selected and stored maximum displacement amplitude.

6. A freezer comprising a free piston Stirling cooler mechanically linked to and driven by a reciprocating armature of an alternating current, linear motor, the linear motor having a stator including a coil with a pair of coil terminals and wound around a ferromagnetic core with air gaps, the motor also having magnets fixed to the armature and positioned for reciprocating through the gaps, the motor also having a motor constant α, an equivalent resistance R.sub.ac representing resistive losses in the coil and losses due to eddy currents and hysteresis induced by motion of the magnets with the coils open circuited, and an equivalent resistance ΔR.sub.external representing induced losses in structures surrounding the motor or alternator, the freezer having a motor controller comprising: (a) a voltage sensor connected to the coil terminals and configured for sensing the amplitude of an alternating voltage V.sub.appl at the coil terminals; (b) a current sensor connected is series with a coil terminal and configured for sensing the amplitude of an alternating current I through the coil terminals; (c) a phase detector circuit connected to the coil terminals and configured for sensing the phase angle ϕ.sub.pf between said voltage and said current; (d) a microcomputer circuit configured for (i) calculating (V.sub.ind).sub.I according to the equation
(V.sub.ind).sub.I=V.sub.appl cos Ø.sub.pf−(R.sub.ac+ΔR.sub.external)I (ii) calculating (V.sub.ind).sub.L according to the equation
(V.sub.ind).sub.L=V.sub.appl sin Ø.sub.pf−LωI (ii) calculating the displacement amplitude X according to the equation $X=\frac{1}{\mathrm{\omega \alpha }}\sqrt{{\left(V_{\mathrm{ind}}\right)}_I^2+{\left(V_{\mathrm{ind}}\right)}_L^2}$ wherein ω is the radian frequency of V.sub.appl and (iv) limiting the displacement amplitude X of the motor to less than a selected and stored maximum displacement amplitude.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) FIG. 1 is a view in axial section of a linear motor that is a component part of a coupled linear motor driving a free piston Stirling cooler that represent an embodiment of the invention.

(2) FIG. 2 is a diagrammatic illustration of the linear motor of FIG. 1.

(3) FIG. 3 is a circuit schematic diagram of an equivalent circuit for a linear alternator or for a linear motor such as those illustrated in FIGS. 1 and 2.

(4) FIG. 4 is a circuit schematic diagram of an equivalent circuit for a linear alternator or for a linear motor and is a simplified modification of the circuit of FIG. 3 and is a basis for an explanation of equations used for embodiments of the invention.

(5) FIG. 5 is a circuit schematic diagram of an equivalent circuit for a linear alternator or for a linear motor and is a further simplified modification of the circuit of FIG. 4 and is a basis for an explanation of equations used for embodiments of the invention.

(6) FIG. 6 is a phasor diagram showing relationships of electrical parameters for the equivalent circuits of FIG. 3-5.

(7) FIG. 7 is a graph of magnet displacement amplitudes that were recovered in accordance with to the invention as based on the two equivalent circuits of FIGS. 4 and 5 versus a measured amplitude.

(8) FIG. 8 illustrates an apparatus for applying the invention in a control circuit for a free piston Stirling cooler.

(9) In describing the preferred embodiment of the invention which is illustrated in the drawings, specific terminology will be resorted to for the sake of clarity. However, it is not intended that the invention be limited to the specific term so selected and it is to be understood that each specific term includes all technical equivalents which operate in a similar manner to accomplish a similar purpose. For example, the word connected or terms similar thereto are often used. They are not limited to direct connection, but include connection through other circuit elements where such connection is recognized as being equivalent by those skilled in the art. In addition, circuits are illustrated which are of a type which perform well known operations on electronic signals. Those skilled in the art will recognize that there are many, and in the future may be additional, alternative circuits which are recognized as equivalent because they provide the same operations on the signals.

DETAILED DESCRIPTION OF THE INVENTION

(10) In the following description, magnet amplitude, piston amplitude and armature amplitude are all displacement amplitude, are synonymous and have the identical magnitude (value). The term “ideal” has the meaning that is associated with the practice of forming an equivalent circuit from lumped, idealized circuit elements as well known to those skilled in the art of circuit analysis.

(11) Recognizing that the amplitude of the magnet is described by a practically linear system and that a linear motor itself is practically linear, it is possible to form an equivalent circuit from which the magnet amplitude can be extracted for a given applied voltage and current. From the voltage and current and their phase relationship to each other, it is then possible to extract the ideal induced voltage of the motor. This ideal induced voltage (RMS or peak) is proportional to the velocity amplitude of the magnet and, since the frequency is known, the amplitude may be computed directly. The algorithm for doing so is only a few lines of code and can be calculated after each cycle or after several cycles since the machine does not respond quickly to a load change.

(12) The following description begins with a description of the contents of the Figures and is followed by an analysis describing the basis for the mathematical relationships used in the invention.

(13) FIG. 1 shows a section view of a typical linear motor of the type devised by Redlich as used in free-piston machines. FIG. 2 is a schematic or diagrammatic view of the linear motor that is represented in FIG. 1 using the same reference numbers in both figures. Referring to FIGS. 1 and 2, a toroidal coil winding 10 within an iron core 12 creates a magnetic field across a radial air gap 14 that is perpendicular to the faces of the core 12 at the boundary of the air gap 14. A reciprocating armature 11 has a ring 16, which is made up of radially polarized rectangular permanent magnets 18 (or a ring permanent magnet) within the air gap 14. The magnets 18 are moved axially within the iron core 12 in response to an alternating voltage 15 (FIG. 2) applied to the winding terminals 13 (FIG. 2) to drive current through the coil 10. The coil 10 is shown in the outer iron part 20 of the core 12. Alternatively, it is often convenient to place the winding within the inner iron part 22 of the core 12.

(14) FIG. 3 shows an equivalent circuit for a linear motor that is disclosed in the prior art by Redlich, Unger and van der Walt. R.sub.s represents losses due to eddy currents and hysteresis induced by the motion of the magnets with the coil windings open-circuited. R.sub.dc represents the Ohmic loss in the coil and ΔR.sub.external represents induced losses in the surrounding structure of the motor. L is the inductance of the motor mainly due to the coil windings. V.sub.ind is the open circuit voltage induced by the magnet motion and V.sub.appl is the applied terminal voltage. I and I.sub.m are the terminal and motor currents, respectively.

(15) Because FIGS. 3, 4 and 5 are equivalent circuits comprised of lumped ideal circuit elements, the only terminals that are available for a real linear alternator or motor are the terminals of the voltage source V.sub.appl. Those are the alternator or motor terminals that are referred to in this description. Persons skilled in the art will recognize that there are techniques known to those skilled in the art for obtaining, such as from laboratory testing, the values of the constants representing the resistance and inductance circuit elements of the components of the equivalent circuits. The described resistive losses for prior art linear motors are well known. The values of these constants are stored in the microcomputer of the controller and can be used for subsequent data processing.

(16) FIG. 4 shows a modification of the circuit of FIG. 3 that provides a practical equivalent circuit on which the invention is based. The equivalent circuit of FIG. 4 provides a close approximation to the equivalent circuit in FIG. 3. This modification is based on an observation that, in practice, the voltage across R.sub.s is much greater than the voltage across R.sub.dc+ΔR.sub.external. This provides a simplification as follows:
Voltage across R.sub.s≈V.sub.app

(17) Therefore, to a good approximation, the practical equivalent circuit of FIG. 4 is applicable. This circuit modification permits a simplification of the calculation of the displacement amplitude.

(18) FIG. 5 shows an equivalent circuit that is a further simplification. In the FIG. 5 equivalent circuit all the induced losses due to magnet motion are bundled into the R.sub.ac equivalent circuit resistance. Here no energy dissipation occurs with the terminals open-circuited, which leads to some additional inaccuracy. For the equivalent circuit of FIG. 5 I.sub.m=I.

(19) FIG. 6 shows the phasor relationship of the voltages and motor current. The motor current is the current I.sub.m that flows through the windings of coil 10. X is the displacement amplitude of the magnets, V.sub.ind is in phase with the velocity of the magnets and hence in quadrature with the displacement of the magnets. Resistive voltages are in phase with the current I.sub.m and inductive voltages are in quadrature with the current I.sub.m.

(20) FIG. 7 shows a graph comparing the magnet displacement amplitude X that was recovered according to the invention based on equations developed from the two equivalent circuits of FIGS. 4 and 5 versus the actual measured displacement amplitude. FIG. 7 illustrates how closely the displacement amplitude X that is detected (recovered) according to the invention approximates the actual measured displacement amplitude. The line 24 shows the actual measured displacement amplitude. The line 26 (dotted) shows the displacement amplitude X recovered (detected) according to the invention based on equations developed from the FIG. 4 equivalent circuit. The line 28 (dotted) shows the displacement amplitude X recovered (detected) according to the invention based on equations developed from the FIG. 5 equivalent circuit.

(21) FIG. 8 shows the displacement amplitude recovery method embodied in a controller for a free-piston Stirling cooler to obtain a signal representing piston displacement amplitude X for use in a feedback control system. Most of the circuit arrangement shown in FIG. 8 is known in the prior art. The prior art components are described first and in a summary manner because they are known in the prior art to persons of ordinary skill in the art. A free piston Stirling cooler 30 absorbs heat from a target 32 such as an ultracold freezer. The Stirling cooler 30 has a linear motor 34 constructed internally within its back space. The component parts of the linear motor 34 are the same as described in connection with FIGS. 1 and 2 and therefore the same reference numbers are applied to those component parts. The piston 36 of the Stirling cooler is fixed to the magnets 18 of the linear motor 34 which drive the piston 36 in reciprocation within a cylinder 37 as a result of the alternating voltage applied to the coil 10. As well known in the free piston Stirling machine field, pressure variations within the Stirling cooler cause the displacer 38 to reciprocate. The reciprocations of the displacer 38 pushes the working gas within the cylinder 37 periodically in alternate directions through a regenerator 40. Heat is conducted from the target to the upper cold end of the Stirling cooler 30 and pumped by the periodic motion of the working gas combined with the periodic pressure variations within the cylinder from the upper cold end of the Stirling cooler 30 to a heat exchanger 42 from which the heat is transferred to the ambient atmosphere. The periodic motion of the working gas combined with the periodic pressure variations within the cylinder cause heat to be pumped out from the upper cold end of the Stirling cooler 30 and the target 32 to a heat exchanger 42 from which the heat is transferred to the ambient atmosphere. Stirling coolers also have a driver 44 that applies an alternating voltage to the coil 10 of the linear motor 34. The amplitude of that drive voltage is controlled by a controller 46. Under steady state operating conditions, following a start-up sequence, negative feedback control principles are applied to maintain the target temperature at a set point temperature. For that purpose a temperature sensor 43 applies a signal representing the target temperature to a summing junction 62 to which a signal representing the set-point temperature is also applied. The error between the two causes the driver 44 to vary the voltage applied to the linear motor in a manner to maintain the target temperature at the set point temperature. All this is accomplished according to long-established control principles and typically is performed by digital processing.

(22) In order to practice the invention, the piston displacement amplitude must be computed according to the mathematical operations to be described below. The piston displacement amplitude is recovered from electrical parameters that are sensed at the coil terminals 13, which are available to the driver 44, as illustrated at block 48. The recovered (detected) piston displacement amplitude X is applied to a summing junction 50 to which a set point maximum displacement amplitude X.sub.max is also applied. The error signal from the summing junction 50 is then applied to the controller which limits the displacement amplitude so that it does not exceed X.sub.max.

(23) Of course, as known to those skilled in the technology, all the summing junction operations, the voltage limiting operations and the performance of calculations according to equations are typically done by digital processing within the controller. A modern controller includes a microcomputer. The term “microcomputer” is used to identify a computing circuit of the type commonly used in the prior art to perform computing operations for control circuits. It is not intended to be directed to its alternative meaning of a desktop, laptop or other form of user interactive computer that includes peripheral equipment such as monitors, keyboards and a mouse. Because the processing is performed by the controller's microcomputer, the physical sensing devices and circuits that are electrically connected to the coil terminals for sensing coil voltage and current can transfer their I, and V outputs directly to the controller where calculation of the phase (power factor) and all the other mathematical calculations can be performed.

(24) Turning now to an analysis showing the foundation of the invention, a fundamental aspect of linear motor theory is that the induced voltage, V.sub.ind is directly proportional to the magnet axial velocity ωX, where ω is the frequency in radians per second and X is the displacement amplitude. For a Stirling cooler driven by a linear motor, the frequency ω is the frequency of the AC voltage driving the linear motor and typically the resonant frequency of the coupled machines. The proportionality constant α, commonly known in the art as the motor constant, with units Volts seconds per meter or Newtons per Amp that are numerically identical, allows the induced voltage to be written:
V.sub.ind=αωX  (1)

(25) Equation (1) shows that, by obtaining or inferring V.sub.ind, and knowing α and ω, it is possible to obtain the magnet and hence the armature displacement amplitude X.

(26) FIG. 1 shows a typical linear motor where the armature 11 is caused to oscillate axially with an amplitude X. The magnets 18, which are radially polarized, are attached to and form a part of the armature 18. The circumferential coil 10 winding provides an alternating field in the outer and inner iron core components 20 and 22 respectively. The coil 10 has electrical terminals 13. When an alternating voltage is applied to the terminals 13 of coil 10, the stator field generated by the coil 10 crosses the air gaps 14 between the iron core components 20 and 22. The field within the air gaps 14 will interact with the magnet edge currents of the magnets 18, which are essentially circumferential, to provide the force that will move the magnets 18, and hence the armature 11 in oscillatory reciprocation in the axial (vertical in FIGS. 1 and 2) direction. This is shown schematically in FIG. 2 where all the elements have been identified by the same numbers as in FIG. 1. The applied alternating voltage at terminals 13 is V.sub.appl.

(27) In the equivalent circuit for the linear motor that is shown in FIG. 3, R.sub.s accounts for the eddy current and hysteresis losses associated with magnet motion with the coil 10 open-circuited. The DC resistance of the coil 10 is accounted for by R.sub.dc while losses associated with induced currents in the supporting structure are represented by ΔR.sub.external. The inductance L is mainly associated with the coil 10 windings but includes small contributions from other sources as well, e.g. the stator iron core components 20 and 22. When an alternating voltage V.sub.appl is applied at terminals 13, current I flows. The induced open-circuit voltage associated with magnet motion is V.sub.ind.

(28) In the analysis that follows, peak values are used. It is recognized that RMS values will function just as well except that the factor √{square root over (2)} would need to be applied to the voltages and currents in the following equations.

(29) From circuit analysis applied to the equivalent circuit of FIG. 3, it is possible to obtain the induced voltage V.sub.ind in terms of the terminal voltage and current as follows:

(30) $\begin{array}{cc}{V}_{\mathrm{ind}}=V_{\mathrm{appl}}\left(1+j\frac{\omega L}{R_s}\right)-I\left[j\omega L\left(1+\frac{R_{\mathrm{dc}}+\Delta R_{\mathrm{external}}}{R_s}\right)+R_{\mathrm{dc}}\right]& \left(2\right)\end{array}$

(31) Where the j denotes the imaginary term √{square root over (−1)}.

(32) While it is certainly possible to use Equation (2) to obtain the induced voltage V.sub.ind from the terminal voltage V.sub.appl and current I, it would require computational power that is not always conveniently available and cost effective in a small on-board microcomputer.

(33) The practical equivalent circuit of FIG. 4 is a very close approximation to the equivalent circuit of FIG. 3 because the voltage across R.sub.s is much greater than the voltage across R.sub.dc+ΔR.sub.external. Therefore, Rs may be moved from its position in FIG. 3 to its position in FIG. 4, which is shunted directly across the terminals 13 of the coil 10 of the linear motor. The current I.sub.m can then be expressed as:

(34) $\begin{array}{cc}{I}_m=I-\frac{V_{\mathrm{appl}}}{R_s}& \left(3\right)\end{array}$

(35) All other elements remain the same.

(36) Referring to FIG. 6, the voltage-current phasor diagram for the motor, I.sub.m is known from equation (3) and the voltage across the motor is V.sub.appl for the practical equivalent circuit.

(37) The power factor pf, like the voltage V.sub.appl and the current I can be measured by one of several conventional devices, circuits and methods that are well known in the prior art. The simplest and most common method used in the prior art for determining power factor is to compute real power (or true power) and compute apparent power. The quotient of these two is the power factor.

(38) From the power factor pf, we obtain the phase angle Ø.sub.pf between V.sub.appl and I.
Ø.sub.pf=cos.sup.−1(pf)  (4)

(39) Looking at the voltage-current relationship shown by the phasors in FIG. 6, and bearing in mind that V.sub.m=V.sub.appl, components of V.sub.ind can be computed. One component is in the I direction and one component is normal to the I direction (i.e. the quadrature direction of the voltage across an inductor as a function of current I). As seen graphically in FIG. 6 and using equation (3) these components are computed by:
(V.sub.ind).sub.I=V.sub.appl cos Ø.sub.pf−(R.sub.dc+ΔR.sub.external)I.sub.m (component in I direction)  (5)
and
(V.sub.ind).sub.L=V.sub.appl sin Ø.sub.pf−LωI.sub.m (comp. in LωI.sub.m, normal to I.sub.m direction)  (6)

(40) Using equation (1) and the theorem of Pythagoras, the magnet (armature) amplitude follows from the magnitude of the components of V.sub.ind:

(41) $\begin{array}{cc}X=\frac{1}{\mathrm{\omega \alpha }}\sqrt{{\left(V_{\mathrm{ind}}\right)}_I^2+{\left(V_{\mathrm{ind}}\right)}_L^2}& \left(7\right)\end{array}$

(42) This is an accurate result but will vary to some degree with changes in motor constants due to temperature effects and/or other non-linearities that may, of course, be accounted for with appropriate and known functional relationships.

(43) Thus, knowing the phasor relationship of voltage and current in addition to basic motor parameters, it is possible to extract (or recover) the magnet amplitude. This non-invasive means to determine amplitude is useful in control systems.

(44) For the simple equivalent circuit of FIG. 5, Im=I and, as mentioned, the induced losses due to magnet motion are handled entirely by R.sub.ac. The circuit of FIG. 5 has slightly more severe assumptions than the circuit of FIG. 4 and they introduce some additional error. For the circuit of FIG. 5, the two voltage components of V.sub.ind would be given by:
(V.sub.ind).sub.I=V.sub.appl cos Ø.sub.pf−(R.sub.ac+ΔR.sub.external)I (component in I direction)  (8)
(V.sub.ind).sub.L=V.sub.appl sin Ø.sub.pf−LωI (component in LωI direction, normal to I direction)  (9)

(45) The computed components from equations (8) and (9) are then applied in equation 7 to obtain the magnet amplitude.

(46) The recovered amplitudes of the practical equivalent circuit of FIG. 4 and the simple equivalent circuit of FIG. 5 are shown plotted against measured values of magnet amplitude in FIG. 7. The recovered amplitude correlation for the practical equivalent circuit of FIG. 4 is shown at 52 in FIG. 7 and for the simple equivalent circuit of FIG. 5 is shown at 54 in FIG. 7. The practical equivalent circuit (FIG. 4) has better correlation than the simple equivalent circuit (FIG. 5) though for some applications, this difference may not be significant.

(47) The magnet amplitude recovery method of the invention is easily integrated into a control system for linear machinery. FIG. 8 shows how the magnet amplitude recovery method may be used to obtain the piston amplitude in a free-piston Stirling cooler of the beta-type used to control the temperature of target 32. The displacer 38 moves sympathetically to the piston 36 motions, so control of the piston motions will control the displacer motions. The linear motor 34 drives piston 36 by its mechanical connection to the magnets 18. The amplitude of the piston motions must be controlled so that the moving parts do not collide with the mechanical stops 56 or 58. In the case of stop 58, the displacer 38 would collide with the interior top of the machine if driven to an excessive amplitude by the piston 36. The difference in the set-point temperature 60 and the target temperature obtained by temperature sensor 43, both applied to the summing junction 62, provides an error signal to the controller 46. The controller 46 sets the voltage for the driver 44 that provides the input to the linear motor 34. The driver 44 extracts the linear motor terminal current, voltage and phase between the voltage and current. These values are communicated to a microcomputer which processes the magnet amplitude recovery equations described above. Consequently, the control system operates according to the following logic:

(48) 1. If the target temperature is higher than the set point temperature, increase piston amplitude by increasing voltage (increases cooling capacity). If the target temperature is lower than the set point temperature, reduce piston amplitude by reducing voltage.

(49) 2. Recover piston amplitude.

(50) 3. Compare recovered piston amplitude to a stored set point maximum amplitude.

(51) 4. If the recovered amplitude is greater than the set point maximum amplitude, reduce the piston amplitude by reducing voltage until piston amplitude is no greater than the set point maximum amplitude.

LIST OF EQUATION CONSTANTS AND VARIABLES AND FIGURE REFERENCE NUMBERS

(52) α is the motor constant in Volts seconds per meter or Newtons per Amp ω is the machine operating angular frequency in radians per second X is the displacement amplitude of the armature and its component parts. R.sub.S represents losses due to eddy currents and hysteresis induced by motion of the magnets reciprocating in the gap with the windings open-circuited. R.sub.dc represents the Ohmic loss in the coil. ΔR.sub.external represents induced losses in the surrounding structure of the motor. R.sub.ac represents the induced losses due to magnetic motion lumping together the losses of R.sub.S and R.sub.dc. L is the inductance of the motor mainly due to the coil windings. V.sub.ind is the open circuit voltage generated by the magnet motion (i.e. induced in the coil). (V.sub.ind).sub.I is the phasor component of V.sub.ind in the I direction. (V.sub.ind).sub.L is the phasor component of V.sub.ind in the L direction. V.sub.appl is the applied terminal voltage. I and I.sub.m are the terminal and motor currents, respectively. jωL is the impedance of L 10 linear motor coil 11 reciprocating armature 12 iron core 13 coil terminals 14 iron core air gap 15 voltage applied to coil 16 ring support structure 18 permanent magnets (side by side segments or continuous ring) 20 outer part of core 22 inner part of core 24 actual measured displacement amplitude 26 recovered displacement amplitude [FIG. 4, equations (3)-(7)] 28 recovered displacement amplitude [FIG. 5, equations (4), (8), (9) and (7)] 30 free piston Stirling cooler 32 target (e.g. freezer) 34 linear motor (FIG. 8) 36 Stirling cooler piston 37 Stirling cooler cylinder 38 Stirling cooler displacer 40 Stirling cooler regenerator 42 Stirling cooler heat exchanger 43 temperature sensor 44 driver circuit 46 controller 48 piston amplitude X recovery 50 displacement amplitude summing junction 52 recovered amplitude correlation for the equivalent circuit of FIG. 4 54 recovered amplitude correlation for the equivalent circuit of FIG. 5 56 mechanical stop at which Stirling cooler piston can collide. 58 mechanical stop at which Stirling cooler displacer can collide 60 set-point temperature 62 summing junction for temperature control

(53) This detailed description in connection with the drawings is intended principally as a description of the presently preferred embodiments of the invention, and is not intended to represent the only form in which the present invention may be constructed or utilized. The description sets forth the designs, functions, means, and methods of implementing the invention in connection with the illustrated embodiments. However, the same or equivalent functions and features may be accomplished by different embodiments that are also intended to be encompassed within the spirit and scope of the invention and that various modifications may be adopted without departing from the invention or scope of the following claims.