Abstract:
An embodiment of a circuit for driving an under-damped system comprises first and second signal generators. The first generator is operable to generate a first drive signal. And the second generator is operable to receive the first drive signal and a second drive signal, and to generate from the first and second drive signals a system drive signal having a first amplitude for a first duration and having a second amplitude after the first duration, the system drive signal operable to cause the under-damped system to operate in a substantially damped manner. Either or both of the first and second generators may be programmable such that one may adjust the response of any under-damped system by generating an appropriate drive signal instead of by physically modifying the system itself. In another embodiment, an under-damped system is caused to oscillate at a damped frequency having a first phase, and is also caused to oscillate at substantially the damped frequency having a second phase such that the oscillation at the first phase substantially cancels the oscillation at the second phase. Such embodiments may allow one to realize a faster settling time without slowing down the response time of an under-damped system.

Description:
PRIORITY CLAIM 
     The instant application claims priority to Chinese Patent Application No. 200911000096.7, filed Dec. 31, 2009, which application is incorporated herein by reference in its entirety. 
     SUMMARY 
     An embodiment of a circuit for driving an under-damped system includes first and second signal generators. The first generator is operable to generate a first drive signal. And the second generator is operable to receive the first drive signal and a second drive signal, and to generate from the first and second drive signals a system drive signal having a first amplitude for a first duration and having a second amplitude after the first duration, the system drive signal operable to cause the under-damped system to operate in a substantially damped manner. Either or both of the first and second generators may be programmable such that one may adjust the response of any under-damped system by generating an appropriate drive signal instead of by physically modifying the system itself. 
     In another embodiment, an under-damped system is caused to oscillate at a damped frequency having a first phase, and is also caused to oscillate at substantially the same damped frequency having a second phase, such that the oscillation at the first phase substantially cancels the oscillation at the second phase. 
     Such embodiments may allow one to realize a faster settling time without slowing down the response time of an under-damped system. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram of an embodiment of a spring-loaded electromagnetic lens assembly. 
         FIG. 2  is a plot of the step response of an embodiment of the assembly of  FIG. 1 . 
         FIG. 3A  is a plot of an embodiment of a drive waveform that may improve the response of an embodiment of the lens assembly of  FIG. 1  by reducing the overshoot/undershoot and settling time of the assembly. 
         FIG. 3B  is a plot of the response of an embodiment of the lens assembly of  FIG. 1  when driven with the drive waveform of  FIG. 3A . 
         FIG. 4A  is a plot of another embodiment of a drive waveform that may improve the response of an embodiment of the lens assembly of  FIG. 1  by reducing the overshoot and settling time of the lens assembly. 
         FIG. 4B  is a plot of the components of the response of an embodiment of the lens assembly of  FIG. 1  when driven by the drive waveform of  FIG. 4A . 
         FIG. 4C  is a plot of the overall response (i.e., sum of the response components) of an embodiment of the lens assembly of  FIG. 1  when driven by the drive waveform of  FIG. 4A . 
         FIG. 5  is a diagram of an embodiment of a drive circuit for driving an embodiment of the lens assembly of  FIG. 1  with an embodiment of the waveform of  FIG. 4A . 
         FIG. 5A  is a diagram of an embodiment of a programmable signal generator of  FIG. 5 . 
         FIG. 6  is a diagram of another embodiment of a drive circuit for driving an embodiment of the assembly of  FIG. 1  with an embodiment of the waveform of  FIG. 4A . 
         FIG. 7  is a diagram of a system that may incorporate an embodiment of the lens assembly of  FIG. 1  and an embodiment of a drive circuit of  FIGS. 5 and 6 . 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  is a diagram of an embodiment of a spring-loaded electromagnetic lens assembly  10 , which includes a focusable lens  12 . The lens assembly  10  may be part of a system such as a camera or video recorder (not shown in  FIG. 1 ). An example of such a lens assembly is shown in U.S. Pat. No. 7,612,957, which is incorporated by reference. 
     In addition to the lens  12 , the lens assembly  10  includes a spring  14  for forcing the lens toward a reference location  16 , and includes a coil  18  and at least one permanent magnet  20  (two magnets shown in  FIG. 1 ), which cooperate to force the lens away from the reference location. Because one may model the lens assembly  10  as a second-order spring-mass system, a damper  22  is included in  FIG. 1  to represent a level of damping inherent in the lens assembly—the damper is not an actual component of the lens assembly, but represents the combination of all of the sources of damping, such as the stiffness of the spring  14  and any friction that the lens  12  may encounter as it moves. The coil  18  may include an electrical resistance, an indication of which is omitted from  FIG. 1 . Furthermore, the combination of the spring  14 , coil  18 , and at least one magnet  20  is sometimes referred to as a spring-loaded voice-coil motor (vcm). 
     In operation, the system (e.g., a camera) in which the lens assembly  10  is installed determines (e.g., with an auto-focus circuit not shown in  FIG. 1 ) a position of the lens  12  for proper focusing, and generates and sends a corresponding drive voltage V coil  to the coil  18 . Because the lens  12  is to “step” from its current position to a focus position, the system “steps” the drive voltage V coil  such that when V coil  is applied across the coil  18 , the resulting current I coil  through the coil is sufficient to move the lens to, and thereafter maintain the lens at, the focus position. 
     For example, assume that the lens  12  is currently in a position x 1  relative to the reference location  16 , and that the focus position x 2  (the position of the lens  12  shown in  FIG. 1 ) is farther away from the reference location  16  than is x 1 . 
     Therefore, to move the lens  12  from x 1  to x 2 , the system substantially steps the voltage V coil  across the coil  18  from a starting coil voltage V coil =V 1  to an ending coil voltage V coil =V 2  that is higher than V 1 . 
     Because the coil  18  acts as an inductor, although the current I coil  through the coil increases, it does not step from one value to another, at least not as quickly as the voltage V coil  from V 1  to V 2 . But over a period of time that depends, e.g., on the inductance of the coil  18 , the drive voltage V 2  does cause the coil current I coil  to increase from a starting current I coil =I 1  to an ending current I coil =V 2 /R coil =I 2  that is greater than I 1 , where R coil  is the resistance of the coil  18  (R coil  is omitted from  FIG. 1 ). 
     As the coil current I coil  increases, the magnitude of the magnetic field generated by the coil  18  increases, and this increasing coil magnetic field interacts with the magnetic field generated by the magnets  20  so as to increase the net magnetic force F magnetic  in a direction away from the reference location  16 . 
     As F magnetic  increases and becomes larger than the spring force F spring , it causes the lens  12  to move away from the reference location  16 . 
     Therefore, F magnetic  continues to increase, and the lens  12  continues to move, until I coil  reaches I 2 , at which time F magnetic  reaches its steady-state value and the lens  12  approximately attains the position x 2 , which is the position of the lens  12  shown in  FIG. 1 . 
     The lens  12  remains approximately in the position x 2  as long as V coil  and I coil  remain at V 2  and I 2  respectively. 
     Still referring to  FIG. 1 , alternate embodiments of the lens assembly  10  are contemplated. For example instead of pulling the lens  12  toward the reference location  16 , the spring  14  may push the lens away from the reference location; and instead of forcing the lens away from the reference location, the combination of the coil  18  and magnets  20  may force the lens toward the reference location 
     Referring to  FIG. 2 , a potential problem with the lens assembly  10  is described. 
       FIG. 2  is a plot of the position of the lens  12  versus time for an under-damped embodiment of the lens assembly  10  of  FIG. 1 , where the voltage V coil  across the coil  18  is theoretically stepped instantaneously from V coil =V 1  to V coil =V 2 . Although an actual instantaneous stepping of V coil  from V 1  to V 2  may be impractical or impossible, a theoretical stepping of V coil  from V 1  to V 2  allows one to examine the step response of the lens assembly  10 , and to thus gain an insight into the operation of the lens assembly when V coil  is increased from V 1  to V 2  in a relatively short time. 
     Referring to  FIGS. 1 and 2 , if the lens assembly  10  is under-damped (i.e., the damping level is relatively low) and the coil voltage V coil  is quickly increased from V 1  to V 2  to move the lens  12  from x 1  to x 2 , then the lens  12  overshoots x 2 , and oscillates around x 2  in an exponentially decaying manner for a period of time. A step response of a system where the system initially overshoots, and then oscillates about, a final steady-state value (here position x 2 ) may be called an under-damped step response. For example, as shown in  FIG. 2 , an embodiment of the lens  12  may initially overshoot x 2  by approximately 80% of the difference between x 1  and x 2 , and the time required for the amplitude of this oscillation to settle to 10% of the difference between x 1  and x 2  may be approximately 150 milliseconds (ms). A similar analysis also applies of V 1  is greater than V 2  and x 1  is greater than x 2 , in which case the lens  12  may undershoot its desired new focus position x 2 . 
     Unfortunately, a relatively poor under-damped response such as shown in  FIG. 2  may render the lens assembly  10  of  FIG. 1  unsuitable for some applications. For example, an embodiment of the lens assembly  10  with the under-damped response of  FIG. 2  may be unsuitable for use in a pocket-digital-camera application because the assembly takes too long to move the lens  12  between steady-state focus positions. 
     One technique for reducing the overshoot/undershoot and settling time of a second-order system such as the lens assembly  10  is to increase the damping level of the assembly. But increasing the damping level may increase the power required to move the lens  12  from one position to another, and this increased power may be unsuitable for some applications, such as an application where the lens assembly  10  is battery powered. 
     Another technique for reducing the overshoot/undershoot and settling time of a second-order system such as the lens assembly  10  is described in conjunction with  FIGS. 3A and 3B . 
       FIG. 3A  is a plot of an embodiment of the coil voltage V coil  with which a system may drive the coil  18  of  FIG. 1 . 
       FIG. 3B  is a plot of the response of an embodiment of the lens assembly  10  of  FIG. 1  when the coil  18  is driven with V coil  of  FIG. 3A . 
     Referring to  FIG. 3A , instead of changing the coil  18  drive voltage V coil  as quickly as possible, the system in which the lens assembly  10  is installed may lengthen the time over which it changes V coil . For example, the system may increase V coil  from V 1  to V 2  in a series of steps having substantially the same voltage magnitude and substantially the same duration. 
     But referring to  FIG. 3B , even increasing the time over which the system changes the drive voltage V coil  may fail to render an embodiment of the lens assembly  10  suitable for some applications. Although increasing the time over which V coil  is changed may reduce the amplitude of the overshoot/undershoot, it may also increase the settling time to an unsuitable level. That is, increasing the time over which V coil  is increased/decreased to achieve a suitable level of overshoot/undershoot in the lens assembly  10  may undesirably increase the settling time of the lens assembly to an unsuitable level. 
     Referring to  FIGS. 4A-4C , another technique for reducing the overshoot/undershoot amplitude and the settling time of an under-damped embodiment of the lens assembly  10  (or of any other under-damped second-order system) is described. 
       FIG. 4A  is a plot of an embodiment of a multi-component voltage V coil  with which a system may drive the coil  18  of  FIG. 1 . 
       FIG. 4B  is a plot of the respective responses of an embodiment of the lens assembly  10  of  FIG. 1  to the V coil  components of  FIG. 4A . 
       FIG. 4C  is a plot of the overall response of an embodiment of the lens assembly  10  of  FIG. 1  to V coil  of  FIG. 4A . 
     Referring to  FIGS. 4A-4C , in general, an embodiment of the technique is to drive a second-order system (e.g., the lens assembly  10  of  FIG. 1 ) with a multi-component drive signal such that the decaying oscillations caused by each of the drive-signal components substantially cancel one another. That is, once the system reaches a desired new position, it substantially remains there. Consequently, the effective settling time is approximately equal to the time required for the system to first reach the new position. Furthermore, the system may reach the new position more quickly than if the system&#39;s physical level of damping were increased to reduce overshoot/undershoot 
     For example, referring to  FIGS. 1 and 4A , a system that incorporates the lens assembly  10  may change the drive voltage V coil  across the coil  18  in two component steps  30  and  32  having respective amplitudes A 1  and A 2 , where the second component step  32  begins approximately a time T/2 after the first component step  30 , where T is the period at which each of the component steps causes the lens assembly to oscillate, and wherein A 1 +A 2 =G=V 2 −V 1 . 
     Referring to FIGS.  1  and  4 A- 4 B, the V coil  components  30  and  32  of  FIG. 4A  set up respective oscillation components  34  and  36  in the lens assembly  10 , where the components  34  and  3  substantially cancel one another. The components  34  and  36  have the same oscillation frequency f d , but have different respective steady-state amplitudes D 1  and D 2  such that x 2 =++D 2 . But because the V coil  component  32  is applied to the coil  18  approximately a time T/2 (T=1/f d ) after the V coil  component  30  is applied to the coil, the oscillation component  36  is shifted by approximately 180° relative to the oscillation component  34 . Therefore, the oscillations (i.e., the overshoot and undershoot portions) of the component  36  substantially cancel the oscillations of the component  34  starting at approximately time T/2. For example, the peaks of the oscillating component  34  are substantially aligned with, and thus substantially cancel, the valleys of the oscillating component  36 , and vice-versa. 
     Consequently, referring to  FIG. 4C , starting at time T/2, the sum of the oscillation components  34  and  36  equals an approximately constant position x 2 =+D 1 +D 2 . That is, one may effectively increase the damping level of an under-damped system without decreasing the system&#39;s response time (e.g., the time it takes for the lens  12  to travel from position x 1  to position x 2  in  FIG. 4C ) by driving the system with a waveform similar to that of  FIG. 4A   
     Referring to FIGS.  1  and  4 A- 4 C, such a technique may reduce both the overshoot/undershoot amplitude and the settling time of the lens assembly  10 . And this technique may accomplish such a reduction without modifying the physical characteristics (e.g., the damping level or damping coefficient) of the lens assembly  10 . Therefore, this technique may reduce the time and cost of manufacturing a system such as a lens assembly  10 , because the overshoot/undershoot amplitude, the settling time, or both the overshoot/undershoot amplitude and the settling time, of the system may be adjusted by programming/modifying the drive waveform instead of physically modifying the system. 
     Still referring to FIGS.  1  and  4 A- 4 C, an embodiment for calculating the amplitudes A 1  and A 2  of the V coil  components  30  and  32  is described. 
     The position x with respect to time t of a second-order system such as the lens assembly  10  may be described according to the following equation:
 
 x ( t )= P (1 −e   −ζω0t ( A  cos ω d   t+B  sin ω d   t ))  (1)
 
where P is the step in the position x applied to the system, ζ is the damping coefficient of the system, ω 0  is the natural radial frequency of the system, and ω d  is the damped natural radial frequency of the system (ω d =ω 0 √{square root over (1−ζ 2 )})−ω d  is the actual radial frequency of the decaying oscillations of an under-damped system.
 
     Therefore, x(t) for each of the position components  34  and  36  is given by the following equations: 
     
       
         
           
             
               
                 
                   
                     
                       x 
                       34 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       D 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             ⅇ 
                             
                               
                                 - 
                                 
                                   ϛω 
                                   0 
                                 
                               
                               ⁢ 
                               t 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 A 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ω 
                                   d 
                                 
                                 ⁢ 
                                 t 
                               
                               + 
                               
                                 B 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ω 
                                   d 
                                 
                                 ⁢ 
                                 t 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       x 
                       36 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       D 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             ⅇ 
                             
                               - 
                               
                                 
                                   ϛω 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       T 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 A 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     ω 
                                     d 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       
                                         T 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 B 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     ω 
                                     d 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       
                                         T 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     For the oscillations of the components  34  and  36  to effectively cancel such that these components sum to a substantially constant position D tot =D 1 +D 2  starting at time T/2, one may set the sum of equations (2) and (3) equal to D 1 +D 2 =D tot  starting at time T/2 as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         D 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               ⅇ 
                               
                                 
                                   - 
                                   
                                     ϛω 
                                     0 
                                   
                                 
                                 ⁢ 
                                 t 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   A 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ω 
                                     d 
                                   
                                   ⁢ 
                                   t 
                                 
                                 + 
                                 
                                   B 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ω 
                                     d 
                                   
                                   ⁢ 
                                   t 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         D 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               ⅇ 
                               
                                 - 
                                 
                                   
                                     ϛω 
                                     0 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       
                                         T 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   A 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ω 
                                       d 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         
                                           T 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   B 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       ω 
                                       d 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         
                                           T 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     D 
                     tot 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Because equation (4) holds at all values of t≧T/2, one may solve for D 1  by setting t=T/2 and by setting D 2 =D tot −D 1  in equation (4). Therefore, D 1  and D 2  are given by the following equations: 
     
       
         
           
             
               
                 
                   
                     D 
                     1 
                   
                   = 
                   
                     
                       D 
                       tot 
                     
                     
                       1 
                       + 
                       
                         ⅇ 
                         
                           
                             - 
                             
                               ϛω 
                               0 
                             
                           
                           ⁢ 
                           
                             T 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     D 
                     2 
                   
                   = 
                   
                     
                       D 
                       tot 
                     
                     ⁡ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           1 
                           
                             1 
                             + 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   
                                     ϛω 
                                     0 
                                   
                                 
                                 ⁢ 
                                 
                                   T 
                                   2 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     And in an embodiment where G=V 2 −V 1  ( FIG. 4A ) is a constant multiple of D tot —one may determine the relationship between G and D tot  by testing the lens assembly  10  to develop a general relationship between the position x of the lens  12  and the coil voltage V coil —then A 1  and A 2  are given by the following equations: 
     
       
         
           
             
               
                 
                   
                     A 
                     1 
                   
                   = 
                   
                     G 
                     
                       1 
                       + 
                       
                         ⅇ 
                         
                           
                             - 
                             
                               ϛω 
                               0 
                             
                           
                           ⁢ 
                           
                             T 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     A 
                     2 
                   
                   = 
                   
                     G 
                     ⁡ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           1 
                           
                             1 
                             + 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   
                                     ϛω 
                                     0 
                                   
                                 
                                 ⁢ 
                                 
                                   T 
                                   2 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Furthermore, for ζ&lt;&lt;1 (e.g., ζ=0.05), one may approximate A 1  and A 2  from equations (7) and (8) as:
 
 A   1   =G (0.5+ζ)  (9)
 
 A   2   =G (0.5−ζ)  (10)
 
     The values of A 1  and A 2  for an expected range of G may be stored in a look-up table (omitted from  FIG. 1 ) of the system that incorporates the lens assembly  10 . 
     Still referring to  FIGS. 4A-4C , other embodiments of this technique are contemplated. For example, Vcoil may have more than two components that set up more than two position components in the lens assembly  10  such that the position components sum to substantially a constant value after a particular time period has elapsed. 
       FIG. 5  is a diagram of an embodiment of a drive circuit  40  for generating the coil drive voltage V coil  according to an embodiment of the plot of  FIG. 4A . 
     The drive circuit  40  includes an input node  42 , first and second programmable drive-signal generators  44  and  46  for respectively generating the components  30  and  32  of V coil , a programmable delay  48  for delaying the component  32 , a combiner  50  for generating V coil  from the component  30  and the delayed component  32 , and an output node  52  coupled to the coil  18  of the lens assembly  10  of  FIG. 1 . 
     In operation, the system in which the lens assembly  10  is installed generates at the input node  42  an input signal representing or having the amplitude of the voltage V 2  needed to move the lens  12  from a position x 1  to a position x 2 , where the difference V 2 −V 1  (the current drive voltage)=G. 
     The signal generator  44  generates a signal representing or having the amplitude V 1 +A 1  (A 1  is the amplitude of the component  30 ), and the signal generator  46  generates a signal representing or having the amplitude A 2  of the component  32 . For example, the signal generators  44  and  46  may obtain the amplitudes A 1  and A 2  from one or more look-up tables (not shown in  FIG. 5 ) in response to the value G. The system which incorporates the lens assembly  10  may provide the value G to the generators  44  and  46  (or to associated look-up tables), or the generators may calculate G from V 1  and V 2 . Furthermore, the generator  44  may generate the component  30  by subtracting approximately A 2  from the input signal on the node  42 . 
     The delay  48  generates a signal representing or having zero amplitude for a programmed delay time, such as T/2, and thereafter generates a signal representing or having the amplitude A 2  of the component  32 . For example, where the desired programmed delay time is T/2, then one may program the delay  48  to have a duration that approximates T/2 and that is given by the following equation: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     t 
                   
                   = 
                   
                     
                       T 
                       2 
                     
                     = 
                     
                       
                         π 
                         
                           
                             
                               ω 
                               0 
                             
                             ⁢ 
                             
                               
                                 1 
                                 - 
                                 
                                   ϛ 
                                   2 
                                 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                         
                       
                       ≈ 
                       
                         π 
                         
                           ω 
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
         
         
           
             The delay  48  may obtain the value of Δt from a look-up table (omitted from  FIG. 5 ) in response to the value of T, where the look-up table stores values of Δt for an anticipated range of T. And Δt may have a resolution suitable for the application for which the drive circuit  40  is being used. 
           
         
       
    
     The combiner generates on the output node  52  the coil drive signal V coil  having an amplitude equal to approximately V 1 +A 1  from time t=0 to time t˜T/2, and equal to approximately V 1 +A 1 +A 2 =V 2  thereafter. For example, the combiner may be an adder. 
     Still referring to  FIG. 5 , alternate embodiments of the circuit  40  are contemplated. For example, instead of generating the coil drive voltage V coil , the circuit  40  may generate the coil drive current I coil . Furthermore, the circuit  40  may be modified to drive second-order systems other than a lens assembly. Moreover, if V coil  on the output node  52  is in digital form, then the circuit  40  may include a digital-to-analog converter (DAC) to generate V coil  in analog form. In addition, the signal generated by the combiner  50  may represent V coil , and one or more other circuits may generate V coil  and apply V coil  to the coil  18 . Furthermore, any signal referred to as a voltage may be a current. 
       FIG. 5A  is a diagram of an embodiment of the programmable signal generator  44  of  FIG. 5 . Because the signal generator  44  is programmable, it may be used with a wide variety of lens assemblies or other second-order systems having a wide range of response characteristics. 
     The signal generator  44  includes adders  54  and  56  and a multiplier  58 . 
     In operation, the adder  54  subtracts G from V 2 =V 1 =G to generate a signal representing or having the amplitude V 1 . The adder  54  may receive G from the controller (omitted from  FIG. 5A ) for the system in which the signal generator  44  is installed, or from another suitable source. 
     The multiplier  58  multiplies G by 
             1     1   +     ⅇ       -     ϛω   0       ⁢     T   2                 
to generate a signal representing or having the amplitude A 1  per equation (7). The multiplier  58  may be programmed with the value
 
               1     1   +     ⅇ       -     ϛω   0       ⁢     T   2             ,         
or may receive this value from the system controller (omitted from  FIG. 5A ) or from a look-up table (omitted from  FIG. 5A ). Or, the multiplier  58  may receive only the value of T or T/2 from the controller or a look-up table, and derive the multiplier
 
             1     1   +     ⅇ       -     ϛω   0       ⁢     T   2                 
from this value. Alternatively, the multiplier  58  may multiply G by (0.5+ζ) per equation (9), and may be programmed with the value ζ, or may receive this value from the system controller or a look-up table.
 
     The adder  56  sums the signals from the adder  54  and the multiplier  58  to generate on the output node  52  the drive-signal component  30 , which represents or has the amplitude V 1 +A 1 . 
     Still referring to  FIG. 5A , alternate embodiments of the signal generator  44  are contemplated. For example, V 1  may be zero. Furthermore, the signal generator  46  of  FIG. 5  may include only a multiplier that generates A 2  from G according to equation (8) or equation (10). 
       FIG. 6  is a diagram of an embodiment of a drive circuit  60  for generating the coil drive voltage V coil  according to an embodiment of the plot of  FIG. 4A . 
     The drive circuit  60  includes an input node  62 , a counter  64  having an output node  66 , a programmable drive-signal generator  68  for generating the component  30  ( FIG. 4A ) of V coil , a multiplexer  70 , an optional DAC  72 , and an output node  74  coupled to the coil  18  of the lens assembly  10  of  FIG. 1 . 
     In operation, the system in which the lens assembly  10  is installed generates at the input node  42  an input signal representing or having an amplitude equal to the voltage V 2  needed to move the lens  12  from a position x 1  to a position x 2 . 
     In response to the input signal, the counter  64  begins counting. The counter  64  is programmed to output on the node  66  a selection signal having a first value starting from the counter&#39;s reception of the input signal until a duration approximately equal to T/2 has elapsed, and having a second value thereafter. Count values Δt (approximately equal to T/2) for the counter  64  may be calculated according to equation (11) and stored in a look-up table (omitted from  FIG. 6 ) for an expected range of T. 
     The signal generator  68  generates from the input signal a component drive signal representing or having the amplitude V 1 +A 1  of the component  30  of  FIG. 4A . For example, the generator  68  may subtract from the input signal a programmed value approximately equal to A 2 . This value may be calculated from G and equation (8) or (10), or obtained from a look-up table (omitted from  FIG. 6 ). An embodiment of the signal generator  68  may be similar to the signal generator  44  of  FIGS. 5 and 5A . 
     The multiplexer  70  outputs the component  30  from the generator  68  in response to the counter output signal having the first value (i.e., before the counter value reaches approximately T/2), and then outputs the input signal in response to the counter output signal having the second value (i.e., after the counter value reaches approximately T/2). 
     The DAC  72  converts the multiplexer output into the analog version of the drive voltage V coil  on the output node  74 . 
     Still referring to  FIG. 6 , alternate embodiments of the circuit  60  are contemplated. For example, one or more of the alternate embodiments described above in conjunction with the circuit  40  of  FIG. 5  may be applied to the circuit  60 . 
       FIG. 7  is a diagram of an embodiment of a system, such as a camera system  80 , that may incorporate the lens assembly  10  of  FIG. 1  and an embodiment of the lens drive circuit  40  of  FIG. 5  or of the lens drive circuit  60  of  FIG. 6 . But for example purposes, the system  80  is described as including an embodiment of the lens drive circuit  60  of  FIG. 6 . 
     In addition to the lens assembly  10  and the lens drive circuit  60 , the camera system  80  includes a controller  82  for generating the drive voltage (e.g., V 2 ) on the drive-circuit input node  62  ( FIG. 6 ) and a signal representing the change G in the drive voltage, and for otherwise controlling the operation of the camera system. The controller  82  may generate the drive voltage and the signal representing G in response to one or more circuits, such as an auto- or manual-focus circuit (omitted from  FIG. 7 ), or in response to an operator input (e.g., pressing a focus button). 
     The camera system  80  may also include a pixel array  84  for capturing an image, where the lens assembly  10  causes the lens  12  to focus the image onto the pixel array. 
     The circuit  60 , controller  82 , and pixel array  84 , as well as any other circuits of the camera  80 , may be disposed on a same or on different integrated circuits (ICs) and on a same or on different IC dies. 
     Although described as a camera system including a lens assembly, the system  80  may be any other type of system that incorporates a second-order subsystem other than a lens assembly, and the drive circuit  60  may be modified to drive such another second order subsystem. 
     From the foregoing it will be appreciated that, although specific embodiments have been described herein for purposes of illustration, various modifications may be made without deviating from the spirit and scope of the disclosure. Furthermore, where an alternative is disclosed for a particular embodiment, this alternative may also apply to other embodiments even if not specifically stated.