Abstract:
An ion implantation system for implanting ions into a workpiece is provided, having a process chamber and an energy source configured to produce a plasma of ions within the process chamber. A workpiece support having a support surface configured to position the workpiece within an interior region of the process chamber is configured to expose an implantation surface of the workpiece to the plasma of ions. A pulse generator is in electrical communication with the workpiece support, wherein the pulse generator is configured to apply an electrical pulse to the support, therein attracting ions to the implantation surface of the workpiece and implanting ions into the workpiece. A calorimeter is further associated with the workpiece support, wherein a controller is configured to monitor a signal from the calorimeter and to control the implantation of ions into the workpiece based, at least in part, on the signal from the calorimeter.

Description:
TECHNICAL FIELD 
       [0001]    The present invention relates generally to ion implantation dose measurement systems and methods, and more specifically to an in-situ dose measurement system comprising a calorimeter. 
       BACKGROUND 
       [0002]    In the semiconductor industry, ions are implanted into a workpiece, such as a semiconductor wafer, in order to provide specific characteristics in the workpiece. Various different systems and methodologies are available for implanting the ions; one of which is a plasma immersion ion implantation (PIII) system. In a PIII system, the workpiece is maintained at a predetermined potential, and the implantation is performed in distinct pulses, wherein a large volume of plasma is pulsed for a very short duration. During the pulse, the ions in the plasma are attracted to the workpiece, therein depleting all the ions in the plasma. The plasma is then switched off, allowed to recharge, and then pulsed again. This process is repetitively performed until a desired amount of ions are implanted into the workpiece. 
         [0003]    One of the ongoing problems with a PIII system is the measurement of the implant dose during the implantation, and the associated determination of when the implant should end. When the plasma is pulsed at a relatively high voltage (e.g., 6500V) for a very short duration (e.g., 60 microseconds), the ions in the plasma are accelerated onto the workpiece. In the past, a Faraday cup has been used to measure the dose, however, various shortcomings have been experienced using a Faraday cup to measure the total dose. Another method for measuring the total implant dose is to measure a temperature of a given thermal mass at the beginning of the implant, and measure its temperature at the end of the implant, and then back-calculate the dose using the change in potential energy of the thermal mass. Such a methodology, however, is often adversely affected by various environmental factors, such as radiation loss and conductive loss from electrodes used to make the measurement (e.g., thermocouples, etc.). On low energy implants (e.g., an implant depositing energy on the order of 5 Joules), a relatively low thermal mass is necessitated for such a methodology, thus demanding the thermal resistance to surroundings to be high. Such a scenario is often difficult to achieve. Accordingly, a need exists for a new and more robust measurement system and methodology for measuring dosage of an implantation during implantation. 
       SUMMARY 
       [0004]    The present invention overcomes the limitations of the prior art by providing a system and method for measuring implant dosage in a plasma emersion implant system utilizing a calorimeter. Accordingly, the following presents a simplified summary of the disclosure in order to provide a basic understanding of some aspects of the invention. This summary is not an extensive overview of the invention. It is intended to neither identify key or critical elements of the invention nor delineate the scope of the invention. Its purpose is to present some concepts of the invention in a simplified form as a prelude to the more detailed description that is presented later. 
         [0005]    In accordance with the present disclosure, an ion implantation system for implanting ions into a workpiece is provided. A process chamber is provided having an energy source configured to produce a plasma of ions within the process chamber. A workpiece support having a support surface configured to position the workpiece within an interior region of the process chamber is configured to expose an implantation surface of the workpiece to the plasma of ions. A pulse generator is in electrical communication with the workpiece support, wherein the pulse generator is configured to apply an electrical pulse to the support, therein attracting ions to the implantation surface of the workpiece and implanting ions into the workpiece. A calorimeter is further associated with the workpiece support, wherein a controller is configured to monitor a signal from the calorimeter and to control the implantation of ions into the workpiece based, at least in part, on the signal from the calorimeter. 
         [0006]    The calorimeter, in one exemplary aspect, comprises a micro-calorimeter, wherein ion implantation deposition energy is measured directly. The micro-calorimeter, for example, measures the deposition energy of ions transmitted through a known aperture area. In one example, the micro-calorimeter comprises a low mass absorption calorimeter, wherein the calorimeter is designed to dissipate approximately a small amount of energy at a controlled temperature greater than an internal temperature of the process chamber. The electronics, for example, are battery powered and communicate to ground through fiber optic links. The batteries, for example, are recharged during workpiece exchange and vacuum recovery periods. 
         [0007]    The above summary is merely intended to give a brief overview of some features of some embodiments of the present invention, and other embodiments may comprise additional and/or different features than the ones mentioned above. In particular, this summary is not to be construed to be limiting the scope of the present application. Thus, to the accomplishment of the foregoing and related ends, the invention comprises the features hereinafter described and particularly pointed out in the claims. The following description and the annexed drawings set forth in detail certain illustrative embodiments of the invention. These embodiments are indicative, however, of a few of the various ways in which the principles of the invention may be employed. Other objects, advantages and novel features of the invention will become apparent from the following detailed description of the invention when considered in conjunction with the drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0008]      FIG. 1  is a block diagram of an ion implantation system according to several aspects of the present disclosure. 
           [0009]      FIG. 2  illustrates a schematic diagram of an ion implantation dose measuring system in accordance with one example of the disclosure. 
           [0010]      FIG. 3  illustrates a graph of a modeled control loop of an ion implantation, according to another exemplary aspect. 
           [0011]      FIG. 4  illustrates a graph of a measured dosage and calorimeter power versus an input dosage, according to another exemplary aspect. 
           [0012]      FIG. 5  illustrates a graph of measurement error versus time from a start of an ion implantation, according to yet another exemplary aspect. 
           [0013]      FIG. 6  illustrates a methodology for controlling a dosage of an ion implantation according to still another aspect. 
       
    
    
     DETAILED DESCRIPTION 
       [0014]    The present disclosure is directed generally toward a system, apparatus, and method for measuring a dosage of an ion implantation on a workpiece via a utilization of a calorimeter. Accordingly, the present invention will now be described with reference to the drawings, wherein like reference numerals may be used to refer to like elements throughout. It is to be understood that the description of these aspects are merely illustrative and that they should not be interpreted in a limiting sense. In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the present invention. It will be evident to one skilled in the art, however, that the present invention may be practiced without these specific details. Further, the scope of the invention is not intended to be limited by the embodiments or examples described hereinafter with reference to the accompanying drawings, but is intended to be only limited by the appended claims and equivalents thereof. 
         [0015]    It is also noted that the drawings are provided to give an illustration of some aspects of embodiments of the present disclosure and therefore are to be regarded as schematic only. In particular, the elements shown in the drawings are not necessary to scale with each other, and the placement of various elements in the drawings is chosen to provide a clear understanding of the respective embodiment and is not to be construed as necessarily being a representation of the actual relative locations of the various components in implementations according to an embodiment of the invention. Furthermore, the features of the various embodiments and examples described herein may be combined with each other unless specifically noted otherwise. 
         [0016]    It is also to be understood that in the following description, any direct connection or coupling between functional blocks, devices, components, circuit elements or other physical or functional units shown in the drawings or described herein could also be implemented by an indirect connection or coupling. Furthermore, it is to be appreciated that functional blocks or units shown in the drawings may be implemented as separate features or circuits in one embodiment, and may also or alternatively be fully or partially implemented in a common feature or circuit in another embodiment. For example, several functional blocks may be implemented as software running on a common processor, such as a signal processor. It is further to be understood that any connection which is described as being wire-based in the following specification may also be implemented as a wireless communication, unless noted to the contrary. 
         [0017]    Referring now to the figures,  FIG. 1  illustrates an exemplary ion implantation system  100 . In particular, the present disclosure is directed toward a plasma immersion ion implantation (Pill) system  102 , however, the present invention has utility in various other ion implantation systems  100 , such as ion beam-based systems (not shown). As illustrated, the ion implantation system  100  comprises a process chamber  104 , wherein a workpiece support  106  is generally positioned within process chamber. The workpiece support  106 , for example, is configured to provide a surface for holding a workpiece  108 , such as a semiconductor wafer (e.g., a silicon wafer). The workpiece support  106 , for example, may comprise an electrostatic chuck or a mechanical clamping apparatus (not shown) configured to clamp the workpiece  108  about at its periphery to a support surface  110  of the workpiece support. The workpiece support  106 , for example, is at least partially electrically conductive. The workpiece support  106  thus supports the workpiece  108 , while further providing an electrical connection to the workpiece. It should be noted that while the workpiece support  106  is described in the present example as supporting one workpiece  108 , various other configurations are also contemplated, such as a configuration of the workpiece support to concurrently support a plurality of workpieces. 
         [0018]    A load lock  112  is operably coupled to the process chamber  104 , wherein the load lock generally permits an internal environment  114  of the process chamber to be maintained at a predetermined pressure with respect to an external environment  116  (e.g., atmospheric pressure). The load lock  112  thus comprises a valve  118  configured to selectively permit a workpiece  108  to move into and out of the process chamber  104  while maintaining the predetermined pressure within the process chamber. A vacuum pump  120 , for example, is further selectively fluidly coupled to the process chamber  104  via a vacuum valve  122 , wherein the vacuum pump is configured to maintain the internal environment  114  at a reduced pressure. A gas source  124  is further selectively fluidly coupled to the process chamber  104  via a gas source valve  126 , wherein the gas source is configured to supply an ionizable gas to the internal environment  114  of the process chamber. 
         [0019]    In accordance with one example, an energy source  128  is provided above the workpiece support  106 , wherein the energy source is configured to inject energy into the process chamber in order to ionize the gas from the gas source  124 , therein producing a plasma of ions  130  in a plasma region  132  within the process chamber between the energy source and the workpiece support. The energy source  128 , for example, is positioned within the process chamber  104 , or alternatively, is provided along a wall  134  of the process chamber (e.g., a quartz plate, not shown), wherein an RF coil (not shown) operating at a predetermined frequency (e.g., between 2 MHz and 15 MHz) that transmits energy toward the workpiece  108  positioned on the workpiece support  106 . 
         [0020]    RF energy from the energy source  128  thus produces the plasma of ions  130  (also called an ion plasma) from gas molecules that are pumped into the process chamber  104  from the gas source  124 . The pressure within the process chamber  104 , for example, is maintained in the range of 0.2 to 5.0 millitorr. As one example, the gas source  124  provides nitrogen gas into the process chamber  104 , wherein the nitrogen gas is ionized by the RF energy entering the process chamber via the energy source  128 . Accordingly, the RF energy ionizes the gas molecules, therein producing the plasma of ions  130 . It is noted that various other gases, techniques, and/or apparatus known for producing a plasma of ions  130  can be utilized, as all such gases, techniques, and/or apparatus are contemplated as falling within the scope of the present invention. 
         [0021]    In accordance with the present disclosure, once the plasma of ions  130  is set up in the plasma region  132 , the ions are accelerated into contact with the workpiece  108  positioned on the workpiece support  106 . The workpiece support  106 , for example, is at least partially electrically conductive. The plasma of ions  130 , for example, are positively charged, such that an application of an electric field of suitable magnitude and direction in the plasma region  132  will generally cause the ions in the plasma to accelerate toward and impact a surface  136  of the workpiece  108 . In accordance with one example, a pulse generator  138  (also called a modulator) supplies voltage pulses  140  (e.g., less than 10 kV) to the workpiece support  106 , therein biasing workpiece support with respect to conductive inner walls  142  of the process chamber  104 , thus inducing an electric field in the plasma region  132  and accelerating the plasma of ions  130  into the workpiece. The pulse generator  138 , in one example, provides pulses in a range of 100 to 7000 volts, in 1 to 60 microseconds in duration and a pulse repetition rate up to 10 KHz. A controller  144  is further provided to control overall operation of the ion implantation system  100 . For example, the controller  144  is configured to control the pulse generator  138 , supply of gas from the gas source  124 , movement of the workpiece  108  through the load lock  112 , as well as other conditions associated with the ion implantation system  100 . 
         [0022]    It will be appreciated that while specific parameters for the pulse generator  138  and modulation of the voltage pulses  140  are provided as one example, other values and parameters may be utilized, and all such values and parameters are contemplated as falling within the scope of the present invention. The pulse voltage, for example, is selected to implant the positive ions to a desired depth in the workpiece  108 . The number and duration of the pulses are further selected to provide a desired dose of impurity material into the workpiece  108 . The current per pulse is also a function of pulse voltage, gas pressure and species, as well as any variable position of the electrodes. For example, the spacing between the energy source  128  and the workpiece support can be adjusted for various voltages. 
         [0023]    Once the workpiece  108  is implanted with ions, the workpiece is removed from the process chamber  104  via the load lock  112 , wherein further processing or fabrication of the workpiece can be performed. It is highly desirable, however, to tightly control the total energy implanted or deposited on the workpiece  108  during implantation, as resultant devices formed on the workpiece  108  are commonly dependent on proper doping during ion implantation. Accordingly, measurement of the total deposition energy during ion implantation is desirable in order to maintain proper manufacturing yields. 
         [0024]    One method for determining total deposition energy comprises measuring a temperature of a predetermined thermal mass within the process chamber at the beginning of the ion implantation, followed by measuring the temperature of the thermal mass at the end of ion implantation, and then calculating the total energy that is deposited based on the temperature difference of the thermal mass. Such a methodology is moderately effective; however, environmental factors such as radiation losses from the thermal mass and conductive losses from electrodes (e.g., thermocouples, wiring, etc.) used for the temperature measurement can have deleterious effects on the resultant calculation. In low energy implants (e.g., deposits of energy of 5 Joules or less), a relatively low thermal mass is needed, and thermal resistance to surroundings needs to be substantially high. 
         [0025]    Rather than simply measuring temperature differences, however, the present disclosure utilizes calorimetry, therein integrating an amount of power needed to maintain a constant temperature into the determination of the total deposition energy of the ion implantation being performed. Thus, in accordance with the present disclosure, a dosimetry system  146  is provided, where a calorimeter  148  is provided within the process chamber  104 , wherein the calorimeter is generally exposed to the plasma of ions  130  during the implantation. The dosimetry system  146  is illustrated as a schematic  150  in  FIG. 2 , wherein the calorimeter  148  comprises of a resistor  152  (e.g., a thick film resistor) formed or positioned over a ceramic substrate  154  (e.g., a 0.5 mm thick alumina substrate). The ceramic substrate  154  thus provides a thermal mass for absorbing energy from the plasma of ions  130  during the implantation. The ceramic substrate  154 , for example, is comprised of alumina (aluminum oxide) or another suitable ceramic material. The calorimeter  148 , for example, further comprises a ring  156  generally encircling the ceramic substrate  154 , wherein one or more wires  158  (e.g., four wires radiating from the ceramic substrate and generally equidistantly spaced about the ceramic substrate) thermally couple the ceramic substrate to the ring. The one or more wires  158 , for example, are comprised of copper or tungsten. The ring  156 , for example, is operably coupled to a thermal cooling apparatus  160 , wherein the thermal cooling apparatus is configured to generally remove heat from the ring. The thermal cooling apparatus  160 , for example, comprises a fluid circulation system (e.g., chilled water) configured to remove heat from the ring  156 . 
         [0026]    Accordingly, the ceramic substrate  154  has a fixed conductive loss through the one or more wires  158  connecting the substrate to the ring  156  that surrounds the ceramic substrate. In accordance with one example, the calorimeter  148  comprises an aperture  162  positioned along the support surface  110  of the workpiece support  106 , wherein the aperture defines an area  164  of the aperture of the calorimeter that is exposed to the plasma of ions  130 . 
         [0027]    The resistor  152  is thus configured to be heated with a predetermined power (e.g., approximately 1 watt) in order to maintain a predetermined constant temperature (e.g., 50 degrees C.) of the calorimeter  148  above ambient temperature. By heating the calorimeter  148  to a constant temperature differential above the ambient temperature of the internal environment  114  of  FIG. 1 , a thermal loss is provided to the internal environment, thus providing a constant power loss or “calorimeter constant”. If the power going into the calorimeter is measured during the implantation of ions, the integral of the calorimeter constant over that period of time minus the integral of the power going into the calorimeter  148  will provide the change in energy attributed to the ion implantation, itself. 
         [0028]    In one example, the controller  144  further comprises a PID controller  166  configured to maintain the temperature of the calorimeter  148  at the predetermined constant. Thus, the power delivered to the calorimeter  148  is generally continuously monitored, and a calorimeter constant Kc is updated during periods between implants, thus correcting for variations in ambient temperatures. The calorimeter  148 , for example, is powered via one or more batteries  168  and configured to communicate to the controller  144  via a non-electrically conductive signal transmitter  170  associated with therewith. Thus, the calorimeter  148  is controlled while generally preventing stray capacitance associated with the communication of the signal. 
         [0029]    In one example, the non electrically-conductive signal transmitter  170  comprises a fiber optic signal transmitter  172 , wherein the signal is communicated to the controller via a fiber optic cable  174 . Alternatively, the non electrically-conductive signal transmitter  170  comprises a wireless transmitter (not shown), wherein the signal is communicated to the controller via the wireless transmitter to a wireless receiver (not shown) associated with the controller  144 . The one or more batteries  168 , for example, are configured to be recharged during one or more of a transfer or exchange of workpieces  108  and vacuum recovery periods, wherein the internal environment  114  is stabilized. 
         [0030]    In accordance with another aspect of the present disclosure, the energy or Power P provided to the calorimeter  148  can be stated as: 
         [0000]    
       
         
           
             
               
                 
                   P 
                   = 
                   
                     
                       V 
                       2 
                     
                     R 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where V=voltage provided to the calorimeter to maintain the constant predetermined temperature and R=resistance of the resistor  152 . The measured energy into the calorimeter E c  during an implant from time t 0  to t 1  can be written as: 
         [0000]        E   c   =K   C ( t   1   −t   0 )−∫ t     0     t     1     Pdt   (2)
 
         [0000]    where K C =the calorimeter constant in watts. 
         [0031]    The dosage of the implant Dose (e.g., expressed in ions/cm 2 ) can be written as: 
         [0000]    
       
         
           
             
               
                 
                   Dose 
                   = 
                   
                     
                       E 
                       c 
                     
                     
                       
                         E 
                         b 
                       
                        
                       Aq 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where E b  is the ion beam or plasma energy (e.g., expressed in eV), A=the area of the aperture  164  of the calorimeter  148  (e.g., expressed in cm 2 ), and q=the electron charge (e.g., 1.602×10 −19  coulombs). 
         [0032]    Thus, the Dose of the implantation of ions into the workpiece  108  can be finally calculated as: 
         [0000]    
       
         
           
             
               
                 
                   Dose 
                   = 
                   
                     
                       
                         
                           
                             K 
                             c 
                           
                            
                           
                             ( 
                             
                               
                                 t 
                                 1 
                               
                               - 
                               
                                 t 
                                 0 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             ∫ 
                             
                               t 
                               0 
                             
                             
                               t 
                               1 
                             
                           
                            
                           
                             P 
                              
                             
                                 
                             
                              
                             
                                
                               t 
                             
                           
                         
                       
                       
                         
                           E 
                           b 
                         
                          
                         Aq 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0033]    In accordance with one example, the temperature of the calorimeter  148  is controlled in a tight range (e.g., +/−0.1 degrees C.). In one example, since the PID controller  166  is employed to maintain a predetermined constant (e.g., 50 degrees C.) difference between the calorimeter  148  and its surroundings, environmental factors are automatically compensated for, such as day to day temperature changes. The temperature control equation for the PID controller is: 
         [0000]                    P   n     =       P     n   -   1       +     A        [     1   -       T   n       T   s         ]       -     B        [     1   -       T     n   -   1         T   s         ]       +     C        [       (     1   -       T   n       T   s         )     -     (     1   -       T     n   -   1         T   s         )       ]                 (   5   )               where: 
         [0000]        A=k   i   +k   p   (6)
 
         [0000]        B=k   p   (7)
 
         [0000]        C=k   d   (8)
 
         [0000]    and n=a loop counter indexed at a constant frequency. 
         [0034]    A model of the functionality of the dosimetry system  146  will now be described, wherein the thermal response characteristics of the calorimeter  148  are provided for an exemplary implantation of ions. For example,  FIG. 3  illustrates a graph  176  of the temperature time response of the dosimetry control system  146  of  FIG. 1  from the warm up of the ion implantation system  100  to a stabilization  178  of the PID control and a commencement  180  of the ion implantation. In the present example, the ion implantation was simulated using impulses of 1×10 14  dose, with the impulses spaced 100 msec apart. The dose impulses thus create a disturbance in the control loop, causing the temperature to rise momentarily. In turn, the power supplied to the calorimeter  148  decreases proportionately. As shown in graph  182  of  FIG. 4 , the integrator of the PID control measures a drop in heater power  184  (e.g., also called power excursions) and converts it to an implant dose which can be seen in the staircase-like response  186  of accumulated implant dose shown in the graph. Accordingly, the accumulated implant dose D n  is used for end-point measurement to control the implantation of ions. 
         [0035]      FIG. 5  is a graph  188  illustrating an error envelope  190  versus implant time, wherein a measurement error  192  is illustrated well within the desired operating range of the system. Each impulse of deposition energy to the calorimeter  148  of  FIG. 1 , for example, is reflected as a momentary drop in the applied heater power  184  shown in the graph  182  of  FIG. 4 . The PID controller  166  of  FIG. 1 , for example, responds relatively slowly to the impulse, thus allowing a momentary rise in calorimeter temperature and causing the input power to drop momentarily. The equation for the power excursions Q n  in heater power  184  shown in  FIG. 4  is: 
         [0000]        Q   n =( Kc−P   n )( t   n   −t   n-1 )  (9).
 
         [0000]    The equation for the staircase ramp  186  in accumulated implant dose D n  is: 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                     n 
                   
                   = 
                   
                     
                       
                         Q 
                         n 
                       
                       
                         
                           E 
                           b 
                         
                          
                         Aq 
                       
                     
                     + 
                     
                       
                         D 
                         
                           n 
                           - 
                           1 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Equations 9 and 10 thus represent the quantization of implant dose as a function of the calorimeter power difference. 
         [0036]    In accordance with another exemplary aspect of the invention,  FIG. 6  illustrates an exemplary method  200  for measuring dosage during a plasma emersion ion implantation using a calorimeter. It should be noted that while exemplary methods are illustrated and described herein as a series of acts or events, it will be appreciated that the present invention is not limited by the illustrated ordering of such acts or events, as some steps may occur in different orders and/or concurrently with other steps apart from that shown and described herein, in accordance with the invention. In addition, not all illustrated steps may be required to implement a methodology in accordance with the present invention. Moreover, it will be appreciated that the methods may be implemented in association with the systems illustrated and described herein as well as in association with other systems not illustrated. 
         [0037]    The method  200  of  FIG. 6  begins at act  202 , wherein a workpiece is provided on a workpiece support in a process chamber. The workpiece support, for example, comprises a calorimeter, such as the calorimeter  148  of the dosimetry system  146  of  FIGS. 1 and 2 . In act  204  of  FIG. 6 , a dosage D n  of implanted ions (also called a dose counter) is initially set to zero (D n =D 0 =0). A plasma of ions is provided in act  206 , wherein an amount of ions are implanted into the workpiece for a period of time. In act  208 , the dosage D n  (e.g., the accumulated amount of ions implanted into the workpiece) is determined via the calorimeter associated with the workpiece support and dosimetry system. For example, the dose D n  is updated in act  208  at a rate n that is equal to a clock frequency of the PID controller  166  of  FIG. 1 . In act  210 , a determination is made regarding whether the dosage D n  has reached a predetermined preset dosage D preset  (also called a final implant dose). If the determination in act  210  is such that the preset dosage D preset  is achieved (e.g., D n &gt;=D preset ), the implantation is halted and the workpiece is removed from the process chamber in act  212 . If the determination in act  210  is such that the preset dosage D preset  has not been achieved, the implantation continues by continuing to provide ions to the workpiece in act  206 . It is noted that a residual error in the dosage D n  measurement in act  208  may be seen due to a time delay of the PID controller; however, the residual error is acceptably small, as evidenced in  FIG. 5 . 
         [0038]    Although the invention has been shown and described with respect to a certain embodiment or embodiments, it should be noted that the above-described embodiments serve only as examples for implementations of some embodiments of the present invention, and the application of the present invention is not restricted to these embodiments. In particular regard to the various functions performed by the above described components (assemblies, devices, circuits, etc.), the terms (including a reference to a “means”) used to describe such components are intended to correspond, unless otherwise indicated, to any component which performs the specified function of the described component (i.e., that is functionally equivalent), even though not structurally equivalent to the disclosed structure which performs the function in the herein illustrated exemplary embodiments of the invention. In addition, while a particular feature of the invention may have been disclosed with respect to only one of several embodiments, such feature may be combined with one or more other features of the other embodiments as may be desired and advantageous for any given or particular application. Accordingly, the present invention is not to be limited to the above-described embodiments, but is intended to be limited only by the appended claims and equivalents thereof.