Patent Publication Number: US-11047377-B2

Title: Linear compressor and methods of extension control

Description:
FIELD OF THE INVENTION 
     The present subject matter relates generally to linear compressors, such as linear compressors for refrigerator appliances. 
     BACKGROUND OF THE INVENTION 
     Certain refrigerator appliances include sealed systems for cooling chilled chambers of the refrigerator appliances. The sealed systems generally include a compressor that generates compressed refrigerant during operation of the sealed systems. The compressed refrigerant flows to an evaporator where heat exchange between the chilled chambers and the refrigerant cools the chilled chambers and food items located therein. 
     Recently, certain refrigerator appliances have included linear compressors for compressing refrigerant. Linear compressors generally include a piston and a driving coil. The driving coil receives a current that generates a force for oscillating the piston (i.e., sliding the piston forward and backward within a chamber having a cylinder head). An elastic element, such as a spring, may be provided to aid in such oscillation. During motion of the piston within the chamber, the piston compresses refrigerant. Generally, the force of gas compression acts to push the piston away from the chamber and cylinder head. 
     Motion of the piston within the chamber may be controlled such that the piston does not crash against another component of the linear compressor during motion of the piston within the chamber. Such head crashing can damage various components of the linear compressor, such as the piston or an associated cylinder. Nonetheless, the net positive force of gas compression may act to shift or offset the center of equilibrium for oscillation. Such an offset may cause the elastic element to extend more in one oscillation direction (e.g., a positive direction away from the chamber) than in the opposite oscillation direction (e.g., a negative direction toward the chamber). In some instances, the imbalanced extension of the elastic element and the piston generally may increase the fatigue (e.g., fatigue loading) of certain elements within the linear compressor. Moreover, the rate of part failure may increase and operational life may decrease. 
     Although unbalanced extension and increased fatigue (e.g., through extreme or excessive spring extension) is preferably avoided, it can be difficult to determine a position of the piston and magnitude of displacement within the chamber. For example, a stroke length of the piston is dependent upon a variety of parameters of the linear compressor, and such parameters can vary. In addition, utilizing a sensor to measure the stroke length of the piston can require sensor wires to pierce a hermetically sealed shell of the linear compressor. Passing the sensor wires through the shell provides a path for contaminants to enter the shell. Moreover, utilizing a sensor may present other challenges, such as sensitivity to electrical noise, increased costs, and the potential for sensor failure that may contribute to in failure of the linear compressor. 
     Accordingly, it would be useful to provide a linear compressor and method of operation for addressing one or more of the above-identified issues. In particular, a linear compressor and method for minimizing uneven extension (e.g., extreme or excessive spring extension) and part fatigue would be especially advantageous. 
     BRIEF DESCRIPTION OF THE INVENTION 
     Aspects and advantages of the invention will be set forth in part in the following description, or may be obvious from the description, or may be learned through practice of the invention. 
     In one exemplary aspect of the present disclosure, a method of operating a linear compressor is provided. The method may include supplying a time varying voltage to a motor of the linear compressor; determining an uneven fatigue condition at the linear compressor; and applying a limiting force at the motor in a negative axial direction during a portion of the supplying step in response to the determining step. 
     In another exemplary aspect of the present disclosure, a method of operating a linear compressor is provided. The method may include supplying a time varying voltage to the motor of the linear compressor; determining an uneven fatigue condition at the linear compressor; and directing a negative direct current (DC) voltage to the motor to induce a limiting force at the motor in the negative axial direction during a portion of the supplying step in response to the determining step. 
     These and other features, aspects and advantages of the present invention will become better understood with reference to the following description and appended claims. The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and, together with the description, serve to explain the principles of the invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A full and enabling disclosure of the present invention, including the best mode thereof, directed to one of ordinary skill in the art, is set forth in the specification, which makes reference to the appended figures. 
         FIG. 1  is a front elevation view of a refrigerator appliance according to an exemplary embodiment of the present disclosure. 
         FIG. 2  is schematic view of certain components of the exemplary refrigerator appliance of  FIG. 1 . 
         FIG. 3  provides a perspective view of a linear compressor according to an exemplary embodiment of the present disclosure. 
         FIG. 4  provides a side section view of the exemplary linear compressor of  FIG. 3 . 
         FIG. 5  provides an exploded view of the exemplary linear compressor of  FIG. 4 . 
         FIG. 6  provides a flow chart illustrating a method for operating a linear compressor according to an exemplary embodiment of the present disclosure. 
         FIG. 7  provides a flow chart illustrating a method for operating a linear compressor according to another exemplary embodiment of the present disclosure. 
         FIG. 8  provides a flow chart illustrating a method for operating a linear compressor according to an additional exemplary embodiment of the present disclosure. 
         FIG. 9  provides a flow chart illustrating a method for operating a linear compressor according to a further exemplary embodiment of the present disclosure. 
         FIG. 10  provides a flow chart illustrating a method for operating a linear compressor according to a still further exemplary embodiment of the present disclosure. 
         FIG. 11  provides exemplary movement plot of an experimental linear compressor model. 
         FIG. 12  illustrates a method for operating a linear compressor according to yet another example embodiment of the present disclosure. 
         FIG. 13  provides a simplified schematic view of circuit wired in a first direction according to an example embodiment of the present disclosure. 
         FIG. 14  provides a simplified schematic view of a circuit wires in a reversed direction according to an example embodiment of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     Reference now will be made in detail to embodiments of the invention, one or more examples of which are illustrated in the drawings. Each example is provided by way of explanation of the invention, not limitation of the invention. In fact, it will be apparent to those skilled in the art that various modifications and variations can be made in the present invention without departing from the scope or spirit of the invention. For instance, features illustrated or described as part of one embodiment can be used with another embodiment to yield a still further embodiment. Thus, it is intended that the present invention covers such modifications and variations as come within the scope of the appended claims and their equivalents. 
       FIG. 1  depicts a refrigerator appliance  10  that incorporates a sealed refrigeration system  60  ( FIG. 2 ). It should be appreciated that the term “refrigerator appliance” is used in a generic sense herein to encompass any manner of refrigeration appliance, such as a freezer, refrigerator/freezer combination, and any style or model of conventional refrigerator. In addition, it should be understood that the present subject matter is not limited to use in appliances. Thus, the present subject matter may be used for any other suitable purpose, such as vapor compression within air conditioning units or air compression within air compressors. 
       FIG. 2  is a schematic view of certain components of refrigerator appliance  10 , including a sealed refrigeration system  60  of refrigerator appliance  10 . A machinery compartment  62  contains components for executing a known vapor compression cycle for cooling air. The components include a compressor  64 , a condenser  66 , an expansion device  68 , and an evaporator  70  connected in series and charged with a refrigerant. As will be understood by those skilled in the art, refrigeration system  60  may include additional components, e.g., at least one additional evaporator, compressor, expansion device, and/or condenser. As an example, refrigeration system  60  may include two evaporators. 
     Within refrigeration system  60 , refrigerant flows into compressor  64 , which operates to increase the pressure of the refrigerant. This compression of the refrigerant raises its temperature, which is lowered by passing the refrigerant through condenser  66 . Within condenser  66 , heat exchange with ambient air takes place so as to cool the refrigerant. A fan  72  is used to pull air across condenser  66 , as illustrated by arrows A C , so as to provide forced convection for a more rapid and efficient heat exchange between the refrigerant within condenser  66  and the ambient air. Thus, as will be understood by those skilled in the art, increasing air flow across condenser  66  can, e.g., increase the efficiency of condenser  66  by improving cooling of the refrigerant contained therein. 
     An expansion device (e.g., a valve, capillary tube, or other restriction device)  68  receives refrigerant from condenser  66 . From expansion device  68 , the refrigerant enters evaporator  70 . Upon exiting expansion device  68  and entering evaporator  70 , the refrigerant drops in pressure. Due to the pressure drop and/or phase change of the refrigerant, evaporator  70  is cool relative to compartments  14  and  18  of refrigerator appliance  10 . As such, cooled air is produced and refrigerates compartments  14  and  18  of refrigerator appliance  10 . Thus, evaporator  70  is a type of heat exchanger which transfers heat from air passing over evaporator  70  to refrigerant flowing through evaporator  70 . 
     Collectively, the vapor compression cycle components in a refrigeration circuit, associated fans, and associated compartments are sometimes referred to as a sealed refrigeration system operable to force cold air through compartments  14 ,  18  ( FIG. 1 ). The refrigeration system  60  depicted in  FIG. 2  is provided by way of example only. Thus, it is within the scope of the present subject matter for other configurations of the refrigeration system to be used as well. 
       FIG. 3  provides a perspective view of a linear compressor  100  according to an exemplary embodiment of the present disclosure.  FIG. 4  provides a side section view of linear compressor  100 .  FIG. 5  provides an exploded side section view of linear compressor  100 . As discussed in greater detail below, linear compressor  100  is operable to increase a pressure of fluid within a chamber  112  of linear compressor  100 . Linear compressor  100  may be used to compress any suitable fluid, such as refrigerant or air. In particular, linear compressor  100  may be used in a refrigerator appliance, such as refrigerator appliance  10  ( FIG. 1 ) in which linear compressor  100  may be used as compressor  64  ( FIG. 2 ). As may be seen in  FIG. 3 , linear compressor  100  defines an axial direction A, a radial direction R, and a circumferential direction C. Linear compressor  100  may be enclosed within a hermetic or air-tight shell (not shown). The hermetic shell can, e.g., hinder or prevent refrigerant from leaking or escaping from refrigeration system  60 . 
     Turning now to  FIG. 4 , linear compressor  100  includes a casing  110  that extends between a first end portion  102  and a second end portion  104 , e.g., along the axial direction A. Casing  110  includes various static or non-moving structural components of linear compressor  100 . In particular, casing  110  includes a cylinder assembly  111  that defines a chamber  112 . Cylinder assembly  111  is positioned at or adjacent second end portion  104  of casing  110 . Chamber  112  extends longitudinally along the axial direction A. Casing  110  also includes a motor mount mid-section  113  and an end cap  115  positioned opposite each other about a motor. A stator, e.g., including an outer back iron  150  and a driving coil  152 , of the motor is mounted or secured to casing  110 , e.g., such that the stator is sandwiched between motor mount mid-section  113  and end cap  115  of casing  110 . Linear compressor  100  also includes valves (such as a discharge valve assembly  117  at an end of chamber  112 ) that permit refrigerant to enter and exit chamber  112  during operation of linear compressor  100 . 
     A piston assembly  114  with a piston head  116  is slidably received within chamber  112  of cylinder assembly  111 . In particular, piston assembly  114  is slidable along a first axis A 1  within chamber  112 . The first axis A 1  may include a negative axial direction A(−) and a positive axial direction A(+), and may be substantially parallel to the axial direction A. Thus, piston assembly  114  may alternately slide or oscillate, e.g., the piston head  116 , in the negative axial direction A(−) and the positive axial direction A(+). During sliding of piston head  116  within chamber  112 , piston head  116  compresses refrigerant within chamber  112 . As an example, from a top dead center position (i.e., top dead center point), piston head  116  can slide within chamber  112  towards a bottom dead center position (i.e., bottom dead center point) along the negative axial direction A(−), i.e., an expansion stroke of piston head  116 . When piston head  116  reaches the bottom dead center position, piston head  116  changes directions and slides in chamber  112  along the positive axial direction A(+) back towards the top dead center position, i.e., a compression stroke of piston head  116 . It should be understood that linear compressor  100  may include an additional piston head and/or additional chamber at an opposite end of linear compressor  100 . Thus, linear compressor  100  may have multiple piston heads in alternative exemplary embodiments. 
     Linear compressor  100  also includes an inner back iron assembly  130 . Inner back iron assembly  130  is positioned in the stator of the motor. In particular, outer back iron  150  and/or driving coil  152  may extend about inner back iron assembly  130 , e.g., along the circumferential direction C. Inner back iron assembly  130  extends between a first end portion  132  and a second end portion  134 , e.g., along the axial direction A. 
     Inner back iron assembly  130  also has an outer surface  137 . At least one driving magnet  140  is mounted to inner back iron assembly  130 , e.g., at outer surface  137  of inner back iron assembly  130 . Driving magnet  140  may face and/or be exposed to driving coil  152 . In particular, driving magnet  140  may be spaced apart from driving coil  152 , e.g., along the radial direction R by an air gap AG. Thus, the air gap AG may be defined between opposing surfaces of driving magnet  140  and driving coil  152 . Driving magnet  140  may also be mounted or fixed to inner back iron assembly  130  such that an outer surface  142  of driving magnet  140  is substantially flush with outer surface  137  of inner back iron assembly  130 . Thus, driving magnet  140  may be inset within inner back iron assembly  130 . In such a manner, the magnetic field from driving coil  152  may have to pass through only a single air gap (e.g., air gap AG) between outer back iron  150  and inner back iron assembly  130  during operation of linear compressor  100 , and linear compressor  100  may be more efficient than linear compressors with air gaps on both sides of a driving magnet. 
     As may be seen in  FIG. 4 , driving coil  152  extends about inner back iron assembly  130 , e.g., along the circumferential direction C. Driving coil  152  is operable to move the inner back iron assembly  130  along a second axis A 2  during operation of driving coil  152 . The second axis A 2  may be substantially parallel to the axial direction A and/or the first axis A 1 . As an example, driving coil  152  may receive a current from a current source (not shown) in order to generate a magnetic field that engages driving magnet  140  and urges piston assembly  114  to move along the axial direction A in order to compress refrigerant within chamber  112  as described above and will be understood by those skilled in the art. In particular, the magnetic field of driving coil  152  may engage driving magnet  140  in order to move inner back iron assembly  130  along the second axis A 2  and piston head  116  along the first axis A 1  during operation of driving coil  152 . Thus, driving coil  152  may slide piston assembly  114  between the top dead center position and the bottom dead center position, e.g., by moving inner back iron assembly  130  along the second axis A 2 , during operation of driving coil  152 . 
     A piston flex mount  160  is mounted to and extends through inner back iron assembly  130 . A coupling  170  extends between piston flex mount  160  and piston assembly  114 , e.g., along the axial direction A. Thus, coupling  170  connects inner back iron assembly  130  and piston assembly  114  such that motion of inner back iron assembly  130 , e.g., along the axial direction A or the second axis A 2 , is transferred to piston assembly  114 . Piston flex mount  160  defines an input passage  162  that permits refrigerant to flow therethrough. 
     Linear compressor  100  may include various components for permitting and/or regulating operation of linear compressor  100 . In particular, linear compressor  100  includes a controller (not shown) that is configured for regulating operation of linear compressor  100 . The controller is in, e.g., operative, communication with the motor, e.g., driving coil  152  of the motor. Thus, the controller may selectively activate driving coil  152 , e.g., by supplying current to driving coil  152 , in order to compress refrigerant with piston assembly  114  as described above. 
     The controller includes memory and one or more processing devices such as microprocessors, CPUs or the like, such as general or special purpose microprocessors operable to execute programming instructions or micro-control code associated with operation of linear compressor  100 . The memory can represent random access memory such as DRAM, or read only memory such as ROM or FLASH. The processor executes programming instructions stored in the memory. The memory can be a separate component from the processor or can be included onboard within the processor. Alternatively, the controller may be constructed without using a microprocessor, e.g., using a combination of discrete analog and/or digital logic circuitry (such as switches, amplifiers, integrators, comparators, flip-flops, AND gates, field programmable gate arrays (FPGA), and the like) to perform control functionality instead of relying upon software. 
     Linear compressor  100  also includes a spring assembly  120 . Spring assembly  120  is positioned in inner back iron assembly  130 . In particular, inner back iron assembly  130  may extend about spring assembly  120 , e.g., along the circumferential direction C. Spring assembly  120  also extends between first and second end portions  102  and  104  of casing  110 , e.g., along the axial direction A. Spring assembly  120  assists with coupling inner back iron assembly  130  to casing  110 , e.g., cylinder assembly  111  of casing  110 . In particular, inner back iron assembly  130  is fixed to spring assembly  120  at a middle portion  119  of spring assembly  120 . 
     During operation of driving coil  152 , spring assembly  120  supports inner back iron assembly  130 . In particular, inner back iron assembly  130  is suspended by spring assembly  120  within the stator or the motor of linear compressor  100  such that motion of inner back iron assembly  130  along the radial direction R is hindered or limited while motion along the second axis A 2  is relatively unimpeded. Thus, spring assembly  120  may be substantially stiffer along the radial direction R than along the axial direction A. In such a manner, spring assembly  120  can assist with maintaining a uniformity of the air gap AG between driving magnet  140  and driving coil  152 , e.g., along the radial direction R, during operation of the motor and movement of inner back iron assembly  130  on the second axis A 2 . Spring assembly  120  can also assist with hindering side pull forces of the motor from transmitting to piston assembly  114  and being reacted in cylinder assembly  111  as a friction loss. 
       FIG. 6  illustrates a method  600  for operating a linear compressor according to an exemplary embodiment of the present disclosure. Method  600  may be used to operate any suitable linear compressor. For example, method  600  may be used to operate linear compressor  100  ( FIG. 3 ). Thus, method  600  is discussed in greater detail below with reference to linear compressor  100 . Utilizing method  600  various mechanical and electrical parameters or constants of linear compressor  100  may be established or determined. For example, method  600  may assist with determining or establishing a spring constant of spring assembly  120 , a motor force constant of the motor of linear compressor  100 , a damping coefficient of linear compressor  100 , a resistance of the motor of linear compressor  100 , an inductance of the motor of linear compressor  100 , a moving mass (such as mass of piston assembly  114  and inner back iron assembly  130 ) of linear compressor  100 , etc. Knowledge of such mechanical and electrical parameters or constants of linear compressor  100  may improve performance or operation of linear compressor  100 , as will be understood by those skilled in the art. 
     At step  610 , an electrical dynamic model for the motor of linear compressor  100  is provided. Any suitable electrical dynamic model for the motor of linear compressor  100  may be provided at step  610 . For example, the electrical dynamic model for the motor of linear compressor  100  may be 
     
       
         
           
             
               
                 d 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 i 
               
               
                 d 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 t 
               
             
             = 
             
               
                 
                   v 
                   a 
                 
                 
                   L 
                   i 
                 
               
               - 
               
                 
                   
                     r 
                     i 
                   
                   ⁢ 
                   i 
                 
                 
                   L 
                   i 
                 
               
               - 
               
                 
                   α 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     x 
                     . 
                   
                 
                 
                   L 
                   i 
                 
               
             
           
         
       
     
     where
         v a  is a voltage across the motor of linear compressor  100 ;   r i  is a resistance of the motor of linear compressor  100 ;   i is a current through the motor of linear compressor  100 ;   α is a motor force constant;   X is a velocity of the motor of linear compressor  100 ; and   L i  is an inductance of the motor of linear compressor  100 .       

     The electrical dynamic model for the motor of linear compressor  100  includes a plurality of unknown constants. In the example provided above, the plurality of unknown constants of the electrical dynamic model for the motor of linear compressor  100  includes the resistance of the motor of linear compressor  100  (e.g., the resistance of driving coil  152 ), the inductance of the motor of linear compressor  100  (e.g., the inductance of driving coil  152 ), and the motor force constant. Knowledge or accurate estimates of such unknown constants can improve operation of linear compressor  100 , e.g., by permitting operation of linear compressor  100  at a resonant frequency without head crashing and/or while preventing part fatigue (e.g., extreme or excessive part fatigue loading). 
     At step  610 , the electrical dynamic model for the motor of linear compressor  100  may also be solved for a particular variable, such as di/dt in the example provided above. Thus, as an example, the electrical dynamic model for the motor of linear compressor  100  may be provided in parametric form as 
     
       
         
           
             Φ 
             ⁢ 
             
               = 
               Δ 
             
             ⁢ 
             
               W 
               ⁢ 
               
                   
               
               ⁢ 
               
                 θ 
                 e 
               
             
           
         
       
       
         
           
             
               
                 where 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 W 
               
               ⁢ 
               
                 = 
                 Δ 
               
               ⁢ 
               
                 [ 
                 
                   
                     
                       
                         v 
                         a 
                       
                     
                     
                       
                         - 
                         i 
                       
                     
                     
                       
                         - 
                         
                           x 
                           . 
                         
                       
                     
                   
                 
                 ] 
               
             
             ; 
             
               
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   θ 
                   e 
                 
               
               ⁢ 
               
                 = 
                 Δ 
               
               ⁢ 
               
                 
                   [ 
                   
                     
                       
                         
                           1 
                           
                             L 
                             i 
                           
                         
                       
                       
                         
                           
                             r 
                             i 
                           
                           
                             L 
                             i 
                           
                         
                       
                       
                         
                           ∝ 
                           
                             L 
                             i 
                           
                         
                       
                     
                   
                   ] 
                 
                 . 
               
             
           
         
       
     
     However, di/dt is difficult to accurately measure or determine. Thus, a filtering technique may be used to account for this signal and provide a useable or implementable signal. In particular, the electrical dynamic model for the motor of linear compressor  100  may be filtered, e.g., with a low-pass filter, to account for this signal. Thus, a filtered electrical dynamic model for the motor of linear compressor  100  may be provided as
 
Φ f     W   f θ e .
 
     In alternative exemplary embodiments, the electrical dynamic model for the motor of linear compressor  100  may be solved for {dot over (x)} at step  610 . Thus, the electrical dynamic model for the motor of linear compressor  100  may be provided in parametric form as 
             Φ   ⁢     =   Δ     ⁢     W   ⁢           ⁢     θ   e                 where               Φ   ⁢     =   Δ     ⁢     [     di   dt     ]       ;                 W   ⁢     =   Δ     ⁢     [           v   a           -   i           -     di   dt             ]       ;   and                 θ   e     ⁢     =   Δ     ⁢       [           1   ∝             r   i     ∝             L   i     ∝           ]     .           
Again, the electrical dynamic model for the motor of linear compressor  100  may be filtered, e.g., to account for di/dt.
 
     At step  620 , each unknown constant of the plurality of unknown constants of the electrical dynamic model for the motor of linear compressor  100  is estimated. For example, a manufacturer of linear compressor  100  may have a rough estimate or approximation for the value of each unknown constant of the plurality of unknown constants of the electrical dynamic model for the motor of linear compressor  100 . Thus, such values of the each unknown constant of the plurality of unknown constants of the electrical dynamic model for the motor of linear compressor  100  may be provided at step  620  to estimate each unknown constant of the plurality of unknown constants of the electrical dynamic model for the motor of linear compressor  100 . 
     At step  630 , the motor (e.g., driving coil  152 ) of linear compressor  100  is supplied with a time varying voltage, e.g., by the controller of linear compressor  100 . Any suitable time varying voltage may be supplied to the motor of linear compressor  100  at step  630 . For example, the time varying voltage may have at least two frequencies components at step  630  when the electrical dynamic model for the motor of linear compressor  100  is solved for di/dt. Thus, the time varying voltage may be
 
 v   a ( t )= v   0 [sin(2π f   1   t )+sin(2π f   2   t )]
 
     where
         v a  is a voltage across the motor of linear compressor  100 ;   f 1  is a first frequency; and   f 2  is a second frequency.
 
The first and second frequencies f 1 , f 2  may be about the resonant frequency of linear compressor  100 . In particular, the first and second frequencies f 1 , f 2  may be just greater than and just less than the resonant frequency of linear compressor  100 , respectively. For example, the first frequency f 1  may be within five percent greater than the resonant frequency of linear compressor  100 , and the second frequency f 2  may be within five percent less than the resonant frequency of linear compressor  100 . In alternative exemplary embodiments, the time varying voltage may have a single frequency at step  630 , e.g., when the electrical dynamic model for the motor of linear compressor  100  is solved for {dot over (x)}. When the time varying voltage has a single frequency at step  630 , the gas force of fluid within linear compressor  100  may be incorporated within the model for the motor of linear compressor  100 .
       

     A time varying current through the motor of linear compressor  100  may also be determined, e.g., during step  630 . An ammeter or any other suitable method or mechanism may be used to determine the time varying current through the motor of linear compressor  100 . A velocity of the motor of linear compressor  100  may also be measured, e.g., during step  630 . As an example, an optical sensor, a Hall effect sensor or any other suitable sensor may be positioned adjacent piston assembly  114  and/or inner back iron assembly  130  in order to permit such sensor to measure the velocity of the motor of linear compressor  100  at step  630 . Thus, piston assembly  114  and/or inner back iron assembly  130  may be directly observed in order to measure the velocity of the motor of linear compressor  100  at step  630 . In addition, a filtered first derivative of the current through the motor of linear compressor  100  with respect to time may also be measured or determined, e.g., during step  630 . Accordingly, the values or filtered values of W may be measured during step  630 . To permit such measuring, step  630  and the measurements described above may be conducted prior to sealing the motor of linear compressor  100  within a hermetic shell. 
     At step  640 , an error between a measured variable (e.g., di/dt or {dot over (x)}) of the electrical dynamic model at a first time and an estimated variable of the electrical dynamic model at the first time is calculated. For example, an estimate of θ e , {circumflex over (θ)} e , is available, e.g., from step  620 . An error between θ e  and {circumflex over (θ)} e  may be given as
 
{tilde over (θ)} e   θ e −{circumflex over (θ)} e .
 
However, θ e  may be unknown while Φ f  is known or measured. Thus, a related error signal may be used at step  640 . The related error signal may be given as
 
{tilde over (Φ)} f   Φ f −{circumflex over (Φ)} f .
 
The related error signal along with W f  may be used to update {circumflex over (θ)} e , as described in greater detail below.
 
     At step  650 , the estimate for each unknown constant of the plurality of unknown constants of the electrical dynamic model for the motor of linear compressor  100  are repeatedly updated at each time after the first time in order to reduce the error between a measured variable of the electrical dynamic model at each time after the first time and an estimated variable of the electrical dynamic model at each time after the first time. In particular, an adaptive least-squares algorithm may be utilized in order to drive the error between the measured value for the electrical dynamic model at each time after the first time and the estimated variable of the electrical dynamic model at each time after the first time towards zero. In particular, the Adaptive Least-Squares Update Law ensures that
 
{tilde over (θ)} e ( t )→0 as  t →∞:
 
                     θ   ^     .     e     ⁢     =   Δ     ⁢       -     k   e       ⁢         P   e     ⁢     W   f   T     ⁢       Φ   ~     f         1   +       γ   e     ⁢     W   f     ⁢     P   e     ⁢     W   f   T               ,         
{circumflex over (θ)} e (t 0 ) is estimated, e.g., at step  620 .
 
     where P e (t)∈  is the covariance matrix 
     
       
         
           
             
               
                 
                   P 
                   . 
                 
                 e 
               
               ⁢ 
               
                 = 
                 Δ 
               
               ⁢ 
               
                 
                   - 
                   
                     k 
                     e 
                   
                 
                 ⁢ 
                 
                   
                     
                       P 
                       e 
                     
                     ⁢ 
                     
                       W 
                       f 
                       T 
                     
                     ⁢ 
                     
                       W 
                       f 
                     
                     ⁢ 
                     
                       P 
                       e 
                     
                   
                   
                     1 
                     + 
                     
                       
                         γ 
                         e 
                       
                       ⁢ 
                       
                         W 
                         f 
                       
                       ⁢ 
                       
                         W 
                         f 
                         T 
                       
                     
                   
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   P 
                   e 
                 
                 ⁡ 
                 
                   ( 
                   
                     t 
                     0 
                   
                   ) 
                 
               
               = 
               
                 
                   ρ 
                   e 
                 
                 ⁢ 
                 
                   I 
                   3 
                 
               
             
           
         
       
     
     where k e , γ e , ρ e ∈  are constant gains. 
     From {circumflex over (θ)} e , estimates of each unknown constant of the plurality of unknown constants of the electrical dynamic model for the motor of linear compressor  100  may be given as 
                 α   ^     =         θ   ^       e   3           θ   ^       e   1           ,       R   ^     =         θ   ^       e   2           θ   ^       e   1           ,       L   ^     =     1       θ   ^       e   1                 
when the electrical dynamic model for the motor of linear compressor  100  is solved for di/dt at step  610  or
 
                 α   ^     =     1       θ   ^       e   1           ,       R   ^     =         θ   ^       e   2           θ   ^       e   1           ,       L   ^     =         θ   ^       e   3           θ   ^       e   1                 
when the electrical dynamic model for the motor of linear compressor  100  is solved for {dot over (x)} at step  610 .
 
     Generally, initial estimate provided for the electrical motor parameters of linear compressor  100  may be off an actual or previously determined value. However, the experimental electrical motor parameter estimates converge to the previously determined values over time. 
     With the unknown constants of the electrical dynamic model for the motor of linear compressor  100  suitably estimated, a final estimate for each unknown constant of the plurality of unknown constants of the electrical dynamic model for the motor of linear compressor  100  may be saved within the controller of linear compressor  100 . The saved constant values may be used to facilitate efficient and/or proper operation of linear compressor  100 . In particular, knowledge of the constants of the electrical dynamic model for the motor of linear compressor  100  may assist with operating linear compressor  100  at a resonant frequency while avoiding head crashing. 
     As discussed above, method  600  may also provide estimates of the mechanical parameters or constants of linear compressor  100 . Thus, method  600  may also include providing a mechanical dynamic model for linear compressor  100 . Any suitable mechanical dynamic model for linear compressor  100  may be provided. For example, the mechanical dynamic model for linear compressor  100  may be 
     
       
         
           
             
               F 
               m 
             
             = 
             
               
                 i 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 
                   
                     M 
                     α 
                   
                   ⁢ 
                   
                     x 
                     ¨ 
                   
                 
                 + 
                 
                   
                     C 
                     α 
                   
                   ⁢ 
                   
                     x 
                     . 
                   
                 
                 + 
                 
                   
                     K 
                     α 
                   
                   ⁢ 
                   x 
                 
               
             
           
         
       
     
     where
         M is a moving mass of linear compressor  100 ;   α is a motor force constant;   {umlaut over (x)} is an acceleration of the motor of linear compressor  100 ;   C is a damping coefficient of linear compressor  100 ;   {dot over (x)} is a velocity of the motor of linear compressor  100 ;   K is a spring stiffness of linear compressor  100 ; and   x is a position of the moving mass of linear compressor  100 .       

     The mechanical dynamic model for linear compressor  100  includes a plurality of unknown constants. In the example provided above, the plurality of unknown constants of the mechanical dynamic model of linear compressor  100  includes a moving mass of linear compressor  100  (e.g., a mass of piston assembly  114  and inner back iron assembly  130 ), a damping coefficient of linear compressor  100 , and a spring stiffness of linear compressor  100  (e.g., a stiffness of spring assembly  120 ). Knowledge or accurate estimates of such unknown constants can improve operation of linear compressor  100 , e.g., by permitting operation of linear compressor  100  at a resonant frequency without head crashing and/or while preventing part fatigue (e.g., extreme or excessive part fatigue loading). 
     The mechanical dynamic model for linear compressor  100  may also be solved for a particular variable, such as i(t) in the example provided above. Thus, as an example, the electrical dynamic model for the motor of linear compressor  100  may be provided in parametric form as 
     
       
         
           
             Ψ 
             ⁢ 
             
               = 
               Δ 
             
             ⁢ 
             
               Y 
               ⁢ 
               
                   
               
               ⁢ 
               
                 θ 
                 m 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               Ψ 
               ⁢ 
               
                 = 
                 Δ 
               
               ⁢ 
               
                 [ 
                 i 
                 ] 
               
             
             ; 
           
         
       
       
         
           
             
               Y 
               ⁢ 
               
                 = 
                 Δ 
               
               ⁢ 
               
                 [ 
                 
                   
                     
                       
                         x 
                         ¨ 
                       
                     
                     
                       
                         x 
                         . 
                       
                     
                     
                       x 
                     
                   
                 
                 ] 
               
             
             ; 
             and 
           
         
       
       
         
           
             
               θ 
               m 
             
             ⁢ 
             
               = 
               Δ 
             
             ⁢ 
             
               
                 
                   [ 
                   
                     
                       
                         
                           M 
                           ∝ 
                         
                       
                       
                         
                           C 
                           ∝ 
                         
                       
                       
                         
                           K 
                           ∝ 
                         
                       
                     
                   
                   ] 
                 
                 T 
               
               . 
             
           
         
       
     
     However, {umlaut over (x)} is difficult to accurately measure or determine. Thus, a filtering technique may be used to account for this signal and provide a measurable variable. In particular, the mechanical dynamic model for linear compressor  100  may be filtered, e.g., with a low-pass filter, to account for this signal. Thus, a filtered electrical dynamic model for the motor of linear compressor  100  may be provided as
 
Ψ f     Y   f θ m .
 
Each unknown constant of the plurality of unknown constants of the mechanical dynamic model for linear compressor  100  may also be estimated, and the motor (e.g., driving coil  152 ) of linear compressor  100  may be supplied with a time varying voltage, e.g., in the manner described above for steps  620  and  630 .
 
     An error between a measured variable of the mechanical dynamic model at the first time and an estimated variable of the mechanical dynamic model at the first time may also be calculated. For example, an estimate of θ m , {circumflex over (θ)} m , is available as discussed above. An error between θ m  and {circumflex over (θ)} m  may be given as
 
{tilde over (θ)} m   θ m −{circumflex over (θ)} m .
 
However, θ m  may be unknown while Ψ f  is known or measured. Thus, a related error signal may be used. The related error signal may be given as
 
{tilde over (Ψ)} f   Ψ f −{circumflex over (Ψ)} f .
 
The related error signal along with Y f  may be used to update θ m , as described in greater detail below.
 
     The estimate for each unknown constant of the plurality of unknown constants of the mechanical dynamic model for linear compressor  100  are repeatedly updated at each time after the first time in order to reduce the error between a measured variable of the mechanical dynamic model at each time after the first time and an estimated variable of the mechanical dynamic model at each time after the first time. In particular, an adaptive least-squares algorithm may be utilized in order to drive the error between the measured value for the mechanical dynamic model at each time after the first time and the estimated variable of the mechanical dynamic model at each time after the first time towards zero. In particular, the Adaptive Least-Squares Update Law ensures that
 
{tilde over (θ)} m ( t )→0 as  t →∞:
 
                     θ   ^     .     m     ⁢     =   Δ     ⁢       -     k   m       ⁢         P   m     ⁢     Y   f   T     ⁢       Ψ   ~     f         1   +       γ   m     ⁢     Y   f     ⁢     P   m     ⁢     Y   f   T               ,         
{circumflex over (θ)} m (t 0 ) is estimated.
 
     where P m (t)∈  is the covariance matrix 
     
       
         
           
             
               
                 
                   P 
                   . 
                 
                 m 
               
               ⁢ 
               
                 = 
                 Δ 
               
               ⁢ 
               
                 
                   - 
                   
                     k 
                     m 
                   
                 
                 ⁢ 
                 
                   
                     
                       P 
                       m 
                     
                     ⁢ 
                     
                       Y 
                       f 
                       T 
                     
                     ⁢ 
                     
                       Y 
                       f 
                     
                     ⁢ 
                     
                       P 
                       m 
                     
                   
                   
                     1 
                     + 
                     
                       
                         γ 
                         m 
                       
                       ⁢ 
                       
                         Y 
                         f 
                       
                       ⁢ 
                       
                         Y 
                         f 
                         T 
                       
                     
                   
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   P 
                   m 
                 
                 ⁡ 
                 
                   ( 
                   
                     t 
                     0 
                   
                   ) 
                 
               
               = 
               
                 
                   ρ 
                   m 
                 
                 ⁢ 
                 
                   I 
                   3 
                 
               
             
           
         
       
     
     where k m , γ m , ρ m ∈  are constant gains. 
     From {circumflex over (θ)} m  and the estimate of the motor force constant from step  650 , estimates of each unknown constant of the plurality of unknown constants of the mechanical dynamic model for linear compressor  100  may be given as
 
 {circumflex over (M)}={circumflex over (α)}{circumflex over (θ)}   m     1     ,Ĉ={circumflex over (α)}{circumflex over (θ)}   m     2     ,{circumflex over (K)}={circumflex over (α)}θ   m     3   .
 
     With the unknown constants of the mechanical dynamic model for linear compressor  100  suitably estimated, a final estimate for each unknown constant of the plurality of unknown constants of the mechanical dynamic model for linear compressor  100  may be saved within the controller of linear compressor  100 . The saved constant values may be used to facilitate efficient and/or proper operation of linear compressor  100 . In particular, knowledge of the constants of the mechanical dynamic model for linear compressor  100  may assist with operating linear compressor  100  at a resonant frequency while avoiding head crashing and/or preventing part fatigue (e.g., extreme or excessive part fatigue loading). 
       FIG. 7  illustrates a method  700  for operating a linear compressor according to another exemplary embodiment of the present disclosure. Method  700  may be used to operate any suitable linear compressor. For example, method  700  may be used to operate linear compressor  100  ( FIG. 3 ). Thus, method  700  is discussed in greater detail below with reference to linear compressor  100 . Utilizing method  700 , a stroke length of the motor of linear compressor  100  may be established or determined. Knowledge of the stroke length of the motor of linear compressor  100  may improve performance or operation of linear compressor  100 , as will be understood by those skilled in the art. 
     At step  710 , an electrical dynamic model for the motor of linear compressor  100  is provided. Any suitable electrical dynamic model for the motor of linear compressor  100  may be provided at step  710 . For example, the electrical dynamic model for the motor of linear compressor  100  described above for step  610  of method  600  may be used at step  710 . The electrical dynamic model for the motor of linear compressor  100  may also be modified such that 
     
       
         
           
             
               di 
               dt 
             
             = 
             
               
                 
                   v 
                   a 
                 
                 
                   L 
                   i 
                 
               
               - 
               
                 
                   
                     r 
                     i 
                   
                   ⁢ 
                   i 
                 
                 
                   L 
                   i 
                 
               
               - 
               f 
             
           
         
       
       
         
           
             
               where 
               ⁢ 
               
                   
               
               ⁢ 
               f 
             
             = 
             
               
                 α 
                 
                   L 
                   i 
                 
               
               ⁢ 
               
                 
                   x 
                   . 
                 
                 . 
               
             
           
         
       
     
     At step  720 , the motor (e.g., driving coil  152 ) of linear compressor  100  is supplied with a time varying voltage, e.g., by the controller of linear compressor  100 . Any suitable time varying voltage may be supplied to the motor of linear compressor  100  at step  720 . As an example, the motor (e.g., driving coil  152 ) of linear compressor  100  may be supplied with a time varying voltage in the manner described above for step  630  of method  600 . A time varying current through the motor of linear compressor  100  may also be determined, e.g., during step  720 . An ammeter or any other suitable method or mechanism may be used to determine the time varying current through the motor of linear compressor  100 . 
     At step  730 , a back-EMF of the motor of linear compressor  100  is estimated, e.g., during step  720 . The back-EMF of the motor of linear compressor  100  may be estimated at step  730  using at least the electrical dynamic model for the motor of linear compressor  100  and a robust integral of the sign of the error feedback. As an example, the back-EMF of the motor of linear compressor  100  may be estimated at step  730  by solving
 
 {circumflex over (f)} =( K   1 +1) e ( t )+∫ t     0     t [( K   1 +1) e (σ)+ K   2  sgn( e (σ)] d σ−( K   1 +1) e ( t   0 )
 
     where
         {circumflex over (f)} is an estimated back-EMF of the motor of linear compressor  100 ;   K 1  and K 2  are real, positive gains; and   e=ï−i and e=f−{circumflex over (f)}; and   sgn is the signum or sign function.       

     At step  740 , a velocity of the motor of linear compressor  100  is estimated. The velocity of the motor of linear compressor  100  may be estimated at step  740  based at least in part on the back-EMF of the motor from step  730 . For example, the velocity of the motor of linear compressor  100  may be determined at step  740  by solving 
     
       
         
           
             
               
                 x 
                 . 
               
               ^ 
             
             = 
             
               
                 
                   L 
                   i 
                 
                 α 
               
               ⁢ 
               
                 f 
                 ^ 
               
             
           
         
       
     
     where
         {dot over ({circumflex over (x)})} is an estimated velocity of the motor of linear compressor  100 ;   α is a motor force constant; and   L i  is an inductance of the motor of linear compressor  100 .
 
The motor force constant and the inductance of the motor of linear compressor  100  may be estimated with method  600 , as described above.
       

     At step  750 , a stroke length of the motor of linear compressor  100  is estimated. The stroke length of the motor of linear compressor  100  may be estimated at step  750  based at least in part on the velocity of the motor from step  740 . In particular, the stroke length of the motor of linear compressor  100  may be estimated at step  750  by solving 
     
       
         
           
             X 
             = 
             
               
                 
                   
                     L 
                     i 
                   
                   α 
                 
                 ⁢ 
                 
                   ∫ 
                   
                     
                       f 
                       ^ 
                     
                     ⁢ 
                     dt 
                   
                 
               
               = 
               
                 
                   
                     x 
                     ^ 
                   
                   initial 
                 
                 + 
                 
                   
                     x 
                     ^ 
                   
                   ⁡ 
                   
                     ( 
                     t 
                     ) 
                   
                 
               
             
           
         
       
     
     where {circumflex over (x)} is an estimated position of the motor of linear compressor  100 . 
     It should be understood that steps  720 ,  730 ,  740  and  750  may be performed with the motor of linear compressor  100  sealed within a hermitic shell of linear compressor  100 . Thus, method  700  may be performed at any suitable time during operation of linear compressor  100  in order to determine the stroke length of the motor of linear compressor  100 , e.g., because moving components of linear compressor  100  need not be directly measured with a sensor. Knowledge of the stroke length of the motor of linear compressor  100  may assist with operating linear compressor  100  efficiently and/or properly. For example, such knowledge may assist with adjusting the time varying voltage supplied to the motor of the linear compressor  100  in order to operate the motor of linear compressor  100  at a resonant frequency of the motor of linear compressor  100  without head crashing and/or while preventing part fatigue (e.g., extreme or excessive part fatigue loading), etc., as will be understood by those skilled in the art. 
       FIG. 8  illustrates a method  800  for operating a linear compressor according to an additional exemplary embodiment of the present disclosure. Method  800  may be used to operate any suitable linear compressor. For example, method  800  may be used to operate linear compressor  100  ( FIG. 3 ). Thus, method  800  is discussed in greater detail below with reference to linear compressor  100 . Utilizing method  800 , a position of the motor of linear compressor  100  when the motor of linear compressor  100  is at a top dead center point may be established or determined. Knowledge of the motor of linear compressor  100  at the top dead center point may improve performance or operation of linear compressor  100 , as will be understood by those skilled in the art. 
     At step  810 , a mechanical dynamic model for linear compressor  100  is provided. Any suitable mechanical dynamic model for linear compressor  100  may be provided. For example, the mechanical dynamic model for linear compressor  100  described above for method  600  may be used at step  810 . As another example, the mechanical dynamic model for linear compressor  100  may be
 
 F   m   =αi=M{umlaut over (x)}+C{dot over (x)}+K ( x   avg   −x   0 )+ F   gas  
 
     where
         M is a moving mass of linear compressor  100 ;   α is a motor force constant;   {umlaut over (x)} is an acceleration of the motor of linear compressor  100 ;   C is a damping coefficient of linear compressor  100 ;   {dot over (x)} is a velocity of the motor of linear compressor  100 ;   K is a spring stiffness of linear compressor  100 ;   x is a position of the moving mass of linear compressor  100 ; and   F gas  is a gas force.
 
Solving for acceleration, the mechanical dynamic model for linear compressor  100  may be given as
       

     
       
         
           
             
               x 
               ¨ 
             
             = 
             
               
                 
                   
                     - 
                     
                       C 
                       M 
                     
                   
                   ⁢ 
                   
                     x 
                     . 
                   
                 
                 - 
                 
                   
                     K 
                     M 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         x 
                         avg 
                       
                       - 
                       
                         x 
                         0 
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     α 
                     M 
                   
                   ⁢ 
                   i 
                 
                 + 
                 
                   
                     1 
                     M 
                   
                   ⁢ 
                   
                     F 
                     gas 
                   
                 
               
               = 
               
                 
                   
                     α 
                     M 
                   
                   ⁢ 
                   i 
                 
                 + 
                 
                   
                     f 
                     x 
                   
                   ⁡ 
                   
                     ( 
                     t 
                     ) 
                   
                 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               
                 f 
                 x 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   M 
                 
                 ⁢ 
                 
                   F 
                   gas 
                 
               
               - 
               
                 
                   C 
                   M 
                 
                 ⁢ 
                 
                   x 
                   . 
                 
               
               - 
               
                 
                   K 
                   M 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       x 
                       avg 
                     
                     - 
                     
                       x 
                       0 
                     
                   
                   ) 
                 
               
               + 
               
                 
                   α 
                   M 
                 
                 ⁢ 
                 
                   i 
                   . 
                 
               
             
           
         
       
     
     At step  820 , the motor (e.g., driving coil  152 ) of linear compressor  100  is supplied with a time varying voltage, e.g., by the controller of linear compressor  100 . Any suitable time varying voltage may be supplied to the motor of linear compressor  100  at step  820 . As an example, the motor (e.g., driving coil  152 ) of linear compressor  100  may be supplied with a time varying voltage in the manner described above for step  630  of method  600 . At step  830 , a time varying current through the motor of linear compressor  100  may also be determined, e.g., during step  820 . In particular, a current to the motor of linear compressor  100  may be measured at step  830  when the motor of linear compressor  100  is at a bottom dead center point. Thus, a velocity of the motor of linear compressor  100  may be zero or about (e.g., within about a tenth of a meter per second) zero when the current to the motor of linear compressor  100  is measured at step  830 . A voltmeter or any other suitable method or mechanism may be used to determine the current through the motor of linear compressor  100 . 
     At step  840 , an acceleration of the motor of linear compressor  100  is estimated, e.g., during step  820 . The acceleration of the motor of linear compressor  100  may be estimated at step  840  using at least the mechanical dynamic model for linear compressor  100  and a robust integral of the sign of the error feedback. As an example, the acceleration of the motor of linear compressor  100  may be estimated at step  840  by solving 
                 x   ¨     ^     =         α   M     ⁢   i     +         f   ^     x     ⁡     (   t   )               
with f x  being given as
 
 {circumflex over (f)} =( k   1 +1) e ( t )+∫ t     0     t [( k   1 +1) e (σ)+ k   2  sgn( e   x (σ)] d σ−( k   1 +1) e ( t   0 )
         and where
           {umlaut over ({circumflex over (x)})} is an estimated acceleration of the motor of linear compressor  100 ;   k 1  and k 2  are real, positive gains; and   e x ={dot over (x)}−{circumflex over ({dot over (x)})} and s x =ė x +e x .   
               

     At step  850 , a position of the motor of linear compressor  100  when the motor of the linear compressor  100  is at the bottom dead center point is determined. The position of the motor of linear compressor  100  when the motor of linear compressor  100  is at the bottom dead center point may be estimated at step  850  based at least in part on the current to the motor of linear compressor  100  from step  830  and the acceleration of the motor from step  840 . For example, the position of the motor of linear compressor  100  when the motor of linear compressor  100  is at the bottom dead center point may be estimated at step  850  by solving 
     
       
         
           
             
               x 
               BDC 
             
             = 
             
               
                 
                   α 
                   K 
                 
                 ⁢ 
                 
                   i 
                   BDC 
                 
               
               - 
               
                 
                   M 
                   K 
                 
                 ⁢ 
                 
                   
                     x 
                     ¨ 
                   
                   BDC 
                 
               
             
           
         
       
     
     where
         α is a motor force constant;   K is a spring stiffness of linear compressor  100 ;   i BDC  is the current to the motor of linear compressor  100  at the bottom dead center point;   M is a moving mass of linear compressor  100 ; and   {umlaut over (X)} BDC  is the acceleration of the motor at the bottom dead center point.
 
The motor force constant, the spring stiffness of linear compressor  100  and the moving mass of linear compressor  100  may be estimated with method  600 , as described above.
       

     At step  860 , a position of the motor of linear compressor  100  when the motor of linear compressor  100  is at the top dead center point is determined. The position of the motor of linear compressor  100  when the motor of linear compressor  100  is at the top dead center point may be estimated at step  860  based at least in part on the position of the motor of linear compressor  100  when the motor of linear compressor  100  is at the bottom dead center point from step  850  and a stroke length of the motor of linear compressor  100 . For example, the position of the motor of linear compressor  100  when the motor of linear compressor  100  is at the top dead center point may be estimated at step  860  by solving
 
 x   TDC   =x   BDC   −SL  
         where SL is the stroke length of the motor of linear compressor  100 .
 
The stroke length of the motor of linear compressor  100  may be estimated with method  700 , as described above.
       

     It should be understood that steps  820 ,  830 ,  840 ,  850  and  860  may be performed with the motor of linear compressor  100  sealed within a hermitic shell of linear compressor  100 . Thus, method  800  may be performed at any suitable time during operation of linear compressor  100  in order to determine the position of the motor of linear compressor  100  when the motor of linear compressor  100  is at the top dead center point, e.g., because moving components of linear compressor  100  need not be directly measured with a sensor. Knowledge of the position of the motor of linear compressor  100  when the motor of linear compressor  100  is at the top dead center point may assist with operating linear compressor  100  efficiently and/or properly. For example, such knowledge may assist with adjusting the time varying voltage supplied to the motor of the linear compressor  100  in order to operate the motor of linear compressor  100  at a resonant frequency of the motor of linear compressor  100  without head crashing and/or while preventing part fatigue (e.g., extreme or excessive part fatigue loading), etc., as will be understood by those skilled in the art. 
       FIG. 9  illustrates a method  900  for operating a linear compressor according to a further exemplary embodiment of the present disclosure. Method  900  may be used to operate any suitable linear compressor. For example, method  900  may be used to operate linear compressor  100  ( FIG. 3 ). Thus, method  900  is discussed in greater detail below with reference to linear compressor  100 . Utilizing method  900 , imbalances within linear compressor  100  (e.g., extreme or excessive extension of the spring assembly  120 ) may be notably reduced and fatigue (e.g., fatigue loads) may be advantageously limited, thereby improving reliability, performance, or operation of linear compressor  100 . 
     At step  910 , an electrical dynamic model for the motor of linear compressor  100  is provided. Any suitable electrical dynamic model for the motor of linear compressor  100  may be provided at step  910 . For example, the electrical dynamic model for the motor of linear compressor  100  described above for step  610  of method  600 , step  710  of method  700 , and/or step  810  for method  800  may be used at step  910 . As another example, the mechanical dynamic model for linear compressor  100  may be
 
 F   m   =αi=M{umlaut over (x)}+C{dot over (x)}+K ( x−L   0 )− F   gas  
 
where
         M is a moving mass of linear compressor  100 ;   α is a motor force constant;   {umlaut over (x)} is an acceleration of the motor of linear compressor  100 ;   C is a damping coefficient of linear compressor  100 ;   {dot over (x)} is a velocity of the motor of linear compressor  100 ;   K is a spring stiffness of linear compressor  100 ;   x is a position of the moving mass of linear compressor  100 ;   L 0  is a natural equilibrium point of linear compressor; and   F gas  is a gas force.       

     At step  920 , the motor (e.g., driving coil  152 ) of linear compressor  100  is supplied with a time varying voltage, e.g., by the controller of linear compressor  100 . Any suitable time varying voltage may be supplied to the motor of linear compressor  100  at step  920 . As an example, the motor (e.g., driving coil  152 ) of linear compressor  100  may be supplied with a time varying voltage in the manner described above for step  630  of method  600 . A time varying current through the motor of linear compressor  100  may also be determined, e.g., during step  920 . For instance, a time varying current may be determined in the manner described above for step  830  of method  800 . Additionally or alternatively, an ammeter or any other suitable method or mechanism may be used to determine the time varying current through the motor of linear compressor  100 . 
     At step  930 , an uneven fatigue condition (i.e., condition at which uneven fatigue is possible or likely to occur) may be determined at linear compressor  100 . For instance, an imbalance for oscillation of piston assembly  114  (e.g., affecting spring assembly  120 ) may be determined. Such an imbalance may be indicative of or indicated by extreme or excessive spring extension for spring assembly  120 . 
     An example of an imbalance is illustrated generally at  FIG. 11 . In particular,  FIG. 11  illustrates an exemplary movement plot of an experimental linear compressor model, e.g., taken during steps  920  and  930 . As may be seen in  FIG. 11 , the movement or oscillation of piston assembly  114  may be plotted as a sinusoidal wave wherein x corresponds to piston assembly  114  position (i.e., relative to the chamber  112 ). Thus, the position at which x=0 is understood to correspond to the base portion of chamber  112  (e.g., a cylinder head). As shown, the sinusoidal wave is defined across one or more strokes of the piston assembly  114 . Thus, the sinusoidal wave may be formed from one or more sinusoidal cycles defined by movement (e.g., of piston head  116 ) from a midpoint to a top dead center point, to a bottom dead center point, and back to the midpoint. The position x mid  is the actual midpoint of the sinusoidal wave. In other words, x mid  is the midpoint of stroke length (i.e., Δx SL ) between bottom dead center (i.e., x BDC ) and top dead center (i.e., x TDC ). In a free-floating or ideal system, piston assembly  114  would naturally oscillate about its equilibrium point L 0  (i.e., x mid =L 0 ). However, pressure within the chamber  112  (e.g., F gas ) moves x mid  upward in the positive axial direction A(+). In other words, extension of the piston assembly  114  in the positive axial direction A(+) is greater than extension in the negative axial direction A(−). An imbalanced extension (i.e., Δx ext ) may thus be determined. In some such embodiments, Δx ext  is calculated as
 
Δ x   ext     x   BDC   −L   0  or Δ x   ext     x   TDC   +Δx   SL   −L   0 .
 
     Returning to  FIG. 9 , in some embodiments, step  930  includes determining an axial movement threshold (e.g., axial movement toward BDC) has been exceeded at the piston assembly  114  as it reciprocates or oscillates. Such a determination may include measuring or estimating a contemporary axial movement value (e.g., as instantaneous value of Δx ext  or ΔSL for a particular stroke of the piston assembly  114 ; as an average value of Δx ext  or ΔSL for a predetermined period; or as another suitable value). For instance, measuring or estimating a contemporary axial movement value may include estimating the stroke length of the motor of linear compressor  100  with method  700 , as described above. The axial movement threshold may be a predetermined value (e.g., stored within the controller). In some such embodiments, the contemporary axial movement value is compared directly to the axial movement threshold. A determination that the axial movement threshold is exceeded may thus indicate an undesirable fatigue loading has occurred or is likely to occur. Generally, the determination that the axial movement threshold has been exceeded may occur during an initial portion of step  920 . In turn, step  920  may continue to supply the time varying voltage during and after step  930 , e.g., such that the piston assembly  114  continues to reciprocate after the axial movement threshold has been exceeded. 
     In additional or alternative embodiments, step  930  includes determining a pressure threshold has been exceeded at the piston assembly  114  as it reciprocates or oscillates. Such a determination may include measuring or estimating a contemporary pressure or force value within linear compressor  100  (e.g., as a voltage value utilizing the method  700  or, alternatively, as another suitable value). The pressure threshold may be a predetermined value (e.g., stored within the controller). In some such embodiments, the contemporary pressure value is compared directly to the pressure threshold. A determination that the pressure threshold is exceeded may thus indicate an undesirable fatigue loading has occurred or is likely to occur. Generally, the determination that the pressure threshold has been exceeded may occur during an initial portion of step  920 . In turn, step  920  may continue to supply the time varying voltage during and after step  930 , e.g., such that the piston assembly  114  continues to reciprocate after the pressure threshold has been exceeded. 
     At step  940 , a limiting force may be applied at the motor of the linear compressor  100  in response to a determination of the uneven spring condition (i.e., in response to step  930 ). In particular, the limiting force may be applied against the piston assembly  114  in the negative axial direction A(−) while the time varying voltage continues to be applied to the motor (i.e., during at least a portion of the continued step  920 ). In some embodiments, the limiting force of step  940  is induced by a supplemental direct current (DC) voltage to the motor (e.g., at linear compressor  100 ). Thus, step  940  may include directing a DC voltage to the motor. As induced by a negative DC voltage, the limiting force is thus applied in the negative axial direction A(−). Advantageously, the limiting force may adjust the midpoint of stroke length (x mid ) downward [i.e., in the negative axial direction A(−)] and toward the natural equilibrium (L 0 ). In some embodiments, the limiting force can prevent or restrict the linear compressor from continuing to exceed the axial movement threshold. In other words, the limiting force may be sufficient to restrict axial movement (e.g., toward BDC) below the axial movement threshold (e.g., predetermined axial movement threshold). 
     In certain exemplary embodiments, the DC voltage of step  940  may be directed continuously or constantly after the determination is made at step  930 . Thus, the negative DC voltage may be a constant voltage that is applied during both the positive axial movement and negative axial movement of the piston assembly  114 . Moreover, the negative DC voltage may be applied across a plurality of sinusoidal cycles (i.e., strokes) of the piston assembly  114  as it travels between bottom dead center (x BDC ) and top dead center (x TDC ). Notably, directing a constant DC voltage may preserve the existing harmonics for the sinusoidal motion within linear compressor  100 . 
     In additional or alternative exemplary embodiments, the DC voltage of step  940  may be directed intermittently after the determination is made at step  930 . 
     As another example, the intermittent DC voltage may be applied according to a set amplitude skew. In particular, the amplitude skew may increase the amplitude of sinusoidal motion for the linear compressor  100  in the negative axial direction A(−). The amplitude skew is applied across a plurality of sinusoidal cycles (i.e., strokes) of the piston assembly  114  as it travels between bottom dead center (x BDC ) and top dead center (x TDC ). Thus, the amplitude skew may increase half-cycle amplitude in the negative axial direction A(−), e.g., such that half-cycle amplitude in the negative axial direction A(−) (e.g., amplitude of movement below L 0 ) is greater than half-cycle amplitude in the positive axial direction A(+) (e.g., amplitude of movement above L 0 ). 
     As another example, the intermittent DC voltage may be applied according to a set phase skew. In particular, the phase skew may increase the wavelength of sinusoidal motion for the linear compressor  100  in the negative axial direction A(−). The phase skew is applied across a plurality of sinusoidal cycles (i.e., strokes) of the piston assembly  114  as it travels between bottom dead center (x BDC ) and top dead center (x TDC ). Thus, the phase skew may increase half-cycle wavelength in the negative axial direction A(−), e.g., such that half-cycle wavelength in the negative axial direction A(−) (e.g., wavelength or time of movement below L 0 ) is greater than half-cycle wavelength in the positive axial direction A(+) (e.g., wavelength or time of movement above L 0 ). 
     In some embodiments, the method  900  may continue after applying a limiting force at step  940  to adjust or correct the limiting force applied at the motor (e.g., as step  920  continues). For instance, the method  900  may further include evaluating whether the uneven fatigue condition is present after applying the limiting force (e.g., by repeating step  930 ). 
     If the uneven fatigue condition is present, the limiting force may be increased (i.e., in response to an evaluation that the uneven fatigue condition is present). Optionally, the limiting force may be increased by a predetermined amount. For instance, the directed DC voltage may be increased by a predetermined voltage value. In other words, the magnitude of the directed DC voltage may be increased by the predetermined voltage value, such that the increased value has an absolute value that is greater than the original directed DC voltage. If the directed DC voltage is characterized as a negative value, the predetermined voltage value must also be characterized as a negative value. Increasing the magnitude of the directed DC voltage may thus increase the limiting force. In some such embodiments, the directed DC voltage is progressively indexed (e.g., such that the magnitude of the directed DC voltage is increased incrementally according to a predetermined feedback loop). Thus, the predetermined voltage value may be an index value. The method  900  may repeatedly evaluate whether the uneven fatigue condition is present and increase the directed DC voltage until one or more evaluations are made that the uneven fatigue condition is not present. 
     If the uneven fatigue condition is not present, the limiting force may be decreased (i.e., in response to an evaluation that the uneven fatigue condition is not present). Optionally, the limiting force may be decreased by a predetermined amount. For instance, the directed DC voltage may be decreased by a predetermined voltage value. In some such embodiments, the directed DC voltage is progressively indexed (e.g., decreased incrementally according to a predetermined feedback loop). Thus, the predetermined voltage value may be an index value. The method  900  may repeatedly evaluate whether the uneven fatigue condition is present and decrease the directed DC voltage until the directed voltage reaches zero or one or more evaluations are made that the uneven fatigue condition is present. 
     Turning now to  FIG. 12 , a method  1200  is illustrated for operating a linear compressor according to yet another exemplary embodiment of the present disclosure. Method  1200  may be used to operate any suitable linear compressor, such as linear compressor  100  ( FIG. 3 ). Moreover, it is understood that the entirety (or a portion) of the method  1200  may be utilized as part of, or as an alternative to, any of the above-described methods. In particular, the method  1200  may be utilized for selectively supplying or directing a DC voltage as a time varying voltage is supplied to the motor of linear compressor  100 . As described above (e.g., with respect to the method  900 ), the DC voltage may induce a limiting force in response to a determination of the uneven spring condition. 
     With respect to  FIG. 12 , the DC voltage is indicated as a variable value at V dc . The time varying voltage is indicated at V ac . A resulting applied voltage function for the combined DC voltage (V dc ) and time varying voltage (V ac ) is indicated at V(t), which controls a duty cycle generator to the motor. As discussed above, a value for a measured or estimated contemporary extension imbalance (i.e., distance between a natural equilibrium point and bottom dead center) is indicated at Δx ext . An axial movement threshold (e.g., for extension imbalance) is indicated at ext lim . An index value for the DC voltage is indicated at ΔV dc . An index limit for the combined DC voltage (V dc ) may be provided in some embodiments. For instance, a lower index limit, such as 0 (e.g., as shown at  FIG. 12 ) may be provided. Additionally or alternatively, although not shown in  FIG. 12 , an upper index limit (e.g., between 2 Volts and 5 Volts) may be provided. An index rate (e.g., between 0.25 second and 1.5 seconds) is indicated at T EC , such that a delay in the combined DC voltage (V dc ) is indicated at Z −TEC . 
     As illustrated, at a determination may be made whether the contemporary extension imbalance (Δx ext ) exceeds the axial movement threshold (ext lim ). If the contemporary extension imbalance (Δx ext ) within method  1200  does exceed the axial movement threshold (ext lim ), the DC voltage (V dc ) is indexed higher (e.g., from a starting value of 0). In particular, the DC voltage (V dc ) is increased by the index value (ΔV dc ). Moreover, the DC voltage (V dc ) is combined as a negative value with the time varying voltage (V ac ) to form the voltage function [V(t)]. If the contemporary extension imbalance (Δx ext ) continues to exceed the axial movement threshold (ext lim ), the DC voltage (V dc ) may be repeatedly increased by the index value (ΔV dc ). Moreover, the repeated increases may occur at the index rate (T EC ) until the DC voltage (V dc ) exceeds the index limit (e.g., upper index limit) or the contemporary extension imbalance (Δx ext ) no longer exceeds the axial movement threshold (ext lim ). 
     If the contemporary extension imbalance (Δx ext ) within method  1200  does not exceed the axial movement threshold (ext lim ), the DC voltage (V dc ) is indexed lower (e.g., from a starting value above 0). In particular, the DC voltage (V dc ) is decreased by the index value (ΔV dc ). Moreover, the DC voltage (V dc ) is combined as a negative value with the time varying voltage (V ac ) to form the voltage function [V(t)]. If the contemporary extension imbalance (Δx ext ) remains below the axial movement threshold (ext lim ), the DC voltage (V dc ) may be repeatedly decreased by the index value (ΔV dc ). The repeated decreases may occur at the index rate (T EC ) until the DC voltage (V dc ) reaches the lower index limit (e.g., 0) or the contemporary extension imbalance (Δx ext ) exceeds the axial movement threshold (ext lim ). 
       FIG. 10  illustrates a method  1000  for operating a linear compressor according to a still further exemplary embodiment of the present disclosure. Method  1000  may be used to operate any suitable linear compressor. For example, method  1000  may be used to operate linear compressor  100  ( FIG. 3 ). Thus, method  1000  is discussed in greater detail below with reference to linear compressor  100 . Utilizing method  1000 , a polarity or wiring direction for the motor of the linear compressor  100  may be known. Knowledge of the polarity of the motor of linear compressor  100  may assist with operating linear compressor  100  efficiently and/or properly. For example, such knowledge may assist with adjusting the time varying voltage supplied to the motor of the linear compressor  100  in order to operate the motor of linear compressor  100  at a resonant frequency of the motor of linear compressor  100  without head crashing and/or while preventing part fatigue (e.g., excessive part fatigue loading), etc., as will be understood by those skilled in the art. In certain embodiments, such knowledge may advantageously assist with directing a supplemental negative force to the motor, e.g., in order to reduce imbalances and fatigue (e.g., fatigue loading) within the motor of linear compressor  100 . 
     At step  1010 , an electrical dynamic model for the motor of linear compressor  100  is provided. Any suitable electrical dynamic model for the motor of linear compressor  100  may be provided at step  1010 . For example, the electrical dynamic model for the motor of linear compressor  100  described above for step  610  of method  600 , step  710  of method  700 , step  810  for method  800 , and/or step  910  for method  900  may be used at step  1010 . 
     At step  1020 , the motor (e.g., driving coil  152 ) of linear compressor  100  is supplied with a time varying voltage, e.g., by the controller of linear compressor  100 . Any suitable time varying voltage may be supplied to the motor of linear compressor  100  at step  1020 . As an example, the motor (e.g., driving coil  152 ) of linear compressor  100  may be supplied with a time varying voltage in the manner described above for step  630  of method  600 . A time varying current through the motor of linear compressor  100  may also be determined, e.g., during step  1020 . For instance, a time varying current may be determined in the manner described above for step  830  of method  800 . Additionally or alternatively, an ammeter or any other suitable method or mechanism may be used to determine the time varying current through the motor of linear compressor  100 . 
     At step  1030 , a first acceleration of the motor of the linear compressor  100  may be estimated (e.g., during a portion of step  1020 ). In particular, the estimation may account for acceleration when the motor is at a bottom dead center position (i.e., {umlaut over (x)} BDC ). In some embodiments, step  1030  includes calculating an acceleration value (e.g., first acceleration value) from the mechanical dynamic model of step  1010 . For instance, an estimated value may be calculated as discussed above, e.g., according to step  840  of method  800 . In additional or alternative embodiments, step  1030  includes calculating a slope value for velocity (e.g., a graphed velocity of the piston assembly  114 ) at the bottom dead center position, as would be understood in light of the present disclosure. 
     At step  1040 , a second acceleration of the motor of the linear compressor  100  may be estimated (e.g., during a portion of step  1020 ). In particular, the estimation may account for acceleration when the motor is at a top dead center position (e.g., within the same sinusoidal cycle as step  1030 ) (i.e., In some embodiments, step  1040  includes calculating an acceleration value (e.g., second acceleration value) from the mechanical dynamic model of step  1010 . For instance, an estimated value may be calculated as discussed above, e.g., according to step  840  of method  800 . In additional or alternative embodiments, step  1040  includes calculating a slope value for velocity (e.g., a graphed velocity of the piston assembly  114 ) at the top dead center position, as would be understood in light of the present disclosure. 
     At step  1050 , the first acceleration is compared to the second acceleration. In particular it may be determined if the magnitude of the first acceleration (i.e., absolute value of the first acceleration—|{umlaut over (x)} BDC |) is greater than the magnitude of the second acceleration (i.e., absolute value of the second acceleration—|{umlaut over (x)} TDC |). In some such embodiments, the magnitude of the first acceleration is directly compared to the magnitude of the second acceleration. In alternative embodiments, the magnitude of the first acceleration may be compared to a modified value of the second acceleration. For instance, the second acceleration may be modified (e.g., multiplied or divided by) a set margin of error (i.e., σ). Thus, step  1050  may permit a determination of whether the first acceleration is either greater than the sum of the second acceleration and the set margin of error [i.e., if |{umlaut over (x)} BDC |&gt;|{umlaut over (x)} TDC |*(1+σ)]. In certain embodiments, the set margin of error is five percent or greater (e.g., 5%, 10%, 15%, etc.). 
     At step  1060 , it is determined whether the assumed polarity is correct based on the comparison. For instance, the assumed polarity may be determined to be incorrect (i.e., not correct) if the magnitude of the first acceleration exceeds the magnitude of the second acceleration by at least a certain amount (e.g., a set value or, alternatively, a relative value). A comparatively large deviation may indicate that the assumed polarity is incorrect, while a comparatively small deviation may indicate that the assumed polarity is correct. In some such embodiments, determining that the assumed polarity is correct includes determining that the first acceleration diverges from the second acceleration by less than the set margin of error [e.g., determining |{umlaut over (x)} BDC |≤|{umlaut over (x)} TDC |*(1+σ)]. In other words, if the magnitude of the first acceleration is less than or equal to magnitude of the second acceleration plus the set margin of error, the assumed polarity may be determined to be correct. By contrast, in such embodiments, determining that the assumed polarity is not correct includes determining that the first acceleration diverges from the second acceleration by at least the set margin of error [e.g., determining |{umlaut over (x)} BDC |&gt;|{umlaut over (x)} TDC |*(1+σ)]. In other words, if the magnitude of the first acceleration is greater than magnitude of the second acceleration plus the set margin of error, the assumed polarity may be determined to be incorrect. 
     Turning briefly to  FIGS. 13 and 14 , simplified schematic views of a wired circuit are illustrated. As shown, an inverter may be wired or connected in electrical communication with the linear compressor (e.g., linear compressor  100 — FIG. 3 ) in a first direction ( FIG. 13 ) or an opposite second direction ( FIG. 14 ). The two directions are generally understood to provide a time varying voltage at opposite polarities. Generally, the first direction may be assumed. However, the second direction may be, e.g., inadvertently provided during assembly, thereby reversing the polarity of the system. 
     Thus, returning to  FIG. 10 , step  1060  may determine whether a system has been wired in the first direction or in the second direction. If the assumed polarity is determined to be correct or otherwise verified, the method  1000  may continue to step  1072 . By contrast, if the assumed polarity is determined to be incorrect, the method  1000  may continue to step  1074 . 
     At step  1072 , the method  1000  generally includes permitting continued operation of the motor for the linear compressor  100 . In particular, the time varying voltage initiated at  1020  may be sustained or perpetuated. Thus, step  1072  may include continuing to supply the motor of the linear compressor  100  with the initial time-varying voltage at the assumed polarity (e.g., until operation of the linear compressor  100  is completed or otherwise ended). 
     At step  1074 , the method  1000  includes adjusting or changing the voltage to the motor of the linear compressor  100 . In particular, the initial time varying voltage is halted at step  1074 . Thus, the motor of the linear compressor  100  may be at least temporarily prevented from continuing to operate (e.g., oscillate piston assembly  114 ). Optionally, step  1074  may include supplying the motor of the linear compressor  100  with a new time varying voltage (e.g., until operation of the linear compressor  100  is completed or otherwise ended). The new time varying voltage will be provided at a reversed polarity from the assumed polarity of the initial time varying voltage. In some such embodiments, the new time varying voltage is equal (e.g., in amplitude and wavelength) to the initial time varying voltage. 
     This written description uses examples to disclose the invention, including the best mode, and also to enable any person skilled in the art to practice the invention, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the invention is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they include structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims.