Patent Publication Number: US-6909935-B2

Title: Methods with resolution enhancement feature for improving accuracy of conversion of required chemical mechanical polishing pressure to force to be applied by polishing head to wafer

Description:
RELATED APPLICATIONS 
     The present application is a divisional of U.S. patent application Ser. No. 09/823,151, filed on Mar. 29, 2001, now U.S. Patent No. 6,725,120, entitled “APPARATUS AND METHODS WITH RESOLUTION ENHANCEMENT FEATURE FOR IMPROVING ACCURACY OF CONVERSION OF REQUIRED CHEMICAL MECHANICAL POLISHING PRESSURE TO FORCE TO BE APPLIED BY POLISHING HEAD TO WAFER”, by Miguel A. Saldana (the “Parent Application”), priority under 35 U.S.C. 120 is hereby claimed based on the Parent Application, and such Parent Application is hereby incorporated by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates generally to high performance systems and techniques for polishing workpieces. Specifically, the present invention relates to chemical mechanical polishing (CMP) methods for improving the accuracy of conversion of data representing required CMP pressures to data representing CMP forces to be applied by a polishing (or planarization) head to a workpiece such as a semiconductor wafer, wherein quantization errors are minimized even though components having average resolution are used to provide some of the data used in the conversion operations. 
     2. Description of the Related Art 
     In the fabrication of semiconductor devices, CMP operations are performed for buffing, cleaning, planarization, and polishing of wafers. A typical semiconductor wafer may be made from silicon and may be a disk that is 200 mm or 300 mm in diameter. The term “wafer” is used below to describe and include such semiconductor wafers and other planar structures, or substrates, that are used to support electrical or electronic circuits. 
     As integrated circuit device complexity increases, there is an increased need to improve the accuracy of CMP operations for planarizing dielectric materials deposited onto wafers. Also, as more metallization line patterns are formed in the dielectric materials, there is an increased need for higher accuracy in CMP operations that remove excess metallization. 
     In a typical CMP system, a wafer is mounted on a carrier with a surface of the wafer exposed. The carrier and the wafer rotate in a direction of rotation. The CMP process may be achieved, for example, when the exposed surfaces of the rotating wafer and of a polishing pad are urged into contact with each other by a polishing force, and when the wafer and the polishing pad move laterally relative to each other. 
     Two aspects of achieving accuracy of the polishing force applied to a wafer are of interest. Once a value of a required polishing pressure is specified, that value must first be accurately converted to a corresponding required force and then to a required force signal that accurately represents the required force. The force signal is applied to a force-producing device. Secondly, the actual force applied by the force-producing device must be measured and fed back to adjust the force signal. Improvements have been made to facilitate making repeatable measurements of the actual polishing forces applied to the wafer. However, there is still a need to more accurately convert the value of the required pressure to the value of the force signal. Such need exists, for example, in CMP systems in which the value of the required CMP force must be rapidly changed in relation to rapidly changing values of the exposed area of the wafer that is in contact with the polishing pad as the lateral position of the polishing pad changes relative to the wafer. CMP systems and methods in which the value of the required polishing forces are rapidly changed according to such rapidly changing values of the contact areas are described in co-pending U.S. patent application Ser. No. 09/748708, filed Dec. 22, 2000, entitled “POLISHING APPARATUS AND METHODS HAVING HIGH PROCESSING WORKLOAD FOR CONTROLLING POLISHING PRESSURE APPLIED BY POLISHING HEAD,” by Miguel A. Saldana and Damon V. Williams (the Prior Application). Such Prior Application is hereby incorporated by reference. 
     The CMP systems and methods of the Prior Application implement a recipe (or set of instructions) for laterally moving the polishing pad relative to a wafer carrier and to a retaining ring on the carrier. The relative movement results in the rapidly changing values of the contact area between the polishing pad and the exposed surface of the wafer, and between the pad and a conditioning puck. Feedback of polishing pad position is coordinated with determinations of required values of the variable force by which such different contact areas are separately urged into contact with the polishing pad so that the pressure on each such different contact area may be controlled. The feedback is generated by an encoder that indicates the actual successive lateral positions of the polishing pad relative to the wafer, for example. The different value of each such separate contact area is determined based on the output of the encoder. For each respective pair of one such contact area and one such pressure to be applied to that contact area, a force signal is output (commanded) to represent a corresponding requested force. Each respective force signal is applied to the force-producing device (e.g., an actuator) which provides the force by which the one such contact area of the wafer, for example, is separately urged into contact with the polishing pad at the particular time at which the actual lateral position is measured. 
     Even though the invention of the Prior Application enables conversions of the value of the required pressure to the force signal, there is a need to increase the resolution of the commanded force signal when the actuator that is used displays analog controllability better than that of conventional digital control methods. For example, conventional pneumatic actuators have a low (or coarse) resolution, which provides steps or increments of 2.5 pounds of force. With such coarse resolution, the actuator may be used with the conventional digital control methods having a 10 bit resolution, for example. In detail, a range of polishing pressure may be 10 psi for a 200 mm wafer that has an area of about 50.26 square inches. The maximum force is 502.6 pounds (10 psi×50.26 sq. in.). Force increments corresponding to the 10 bits are about 0.49 pounds (the force divided by the 1024 steps of the resolution). Thus, the increments of the mechanical resolution are more coarse than the 10 bit digital increments. However, when the actuator is a high resolution actuator capable of applying force in increments substantially less than 2.5 pounds (e.g., much less than the above exemplary 0.49 pounds), the conventional digital control methods do not provide the small increments of the commanded force signal that are necessary to take advantage of the high actuator resolution. 
     Another example illustrates errors that may result from use of devices having too low a resolution. Resolution is generally defined as 2 bit, 4 bit, n bit, etc. The number of output signals (or counts or steps) is 2 to the nth power. Thus, the very low 2 bit resolution corresponds to four counts or steps. In the context of the above-described required pressure, the resolution of the above-described digital methods dictates aspects of the force computation for converting the required pressure to the required force and to the value of the required force signal, and those aspects have an effect on accuracy. For example, the very low 2 bit resolution would correspond to a very low 2 bit computational resolution. Use of the 2 bit computational resolution would provide that a 10 psi pressure range be divided into four parts, such as discrete steps at 2.5 psi intervals, i.e., pressure values of 0, 2.5, 5.0, 7.5, and 10 psi. If the CMP system performs the conversion computations with respect to a required pressure having a value of 8.25 psi, for example, the increments (or steps) of the pressure may be 0.25 psi, which may be referred to as a parameter resolution increment. Also, 7.5 psi would be the value of the available output pressure step that is closest to the required 8.25 psi pressure. An accuracy problem resulting from such low resolution is shown by an example in which the required pressure value of 8.25 psi is to be input for processing. The conversion computation must convert the value of the required pressure (e.g., from psi to counts to voltage to counts and back to psi). Ideally, after the conversions, the required pressure would be output as exactly 8.25 psi. However, if the very low 2 bit resolution is used, the value of the required pressure would not exactly match the absolute value of any of the 0, 2.5, 5.0, 7.5, or 10 psi values of the steps of the pressure range. Use of the 7.5 psi value to represent the required 8.25 psi pressure would result in an error of 0.75 psi, or an error of 9.1 percent (9.1%) of the required 8.25 psi. Such a large error in current CMP systems would be unacceptable. 
     With this example in mind, the term “quantization” is used herein to refer to a process of computation in which computational resolution is of significant importance in obtaining a computed result having an acceptable accuracy. A “quantization process” is quantization in which an initial value of a parameter is subjected to computational operations to obtain the computed result. Such exemplary 9.1% error resulting from the above exemplary quantization is referred to herein as a “quantization error”. Generally, a high value of resolution results in steps having a small absolute value. With this in mind, in a normal situation, an unacceptable quantization error may result from performing the computation using too low a value of the computational resolution. For example, the above very low resolution may be the very low computational resolution (2 bits). A high absolute value (2.5 psi) of the steps of the computational resolution in such example was determined by dividing the count value of the very low 2 bit computational resolution (i.e., 4) into the 10 psi pressure range. Such high absolute value of the computational steps results in fewer steps. On the other hand, in the example the absolute value of the pressure (or parameter) increments (0.25 psi) is much less than the absolute value 2.5 psi. As noted above, the values of the exemplary 9.1% quantization error is unacceptable. 
     If a higher computational resolution were used, such as a 3 bit resolution, then the 10 psi pressure range would be divided by 8 (2 to the third power), and each step based on the higher resolution would have a smaller absolute value (1.25 psi). Use of the 1.25 psi absolute value steps would provide a computational step of 8.0 psi closest to the exemplary required 8.25 psi, and a quantization error of 0.25 psi, or 3.03 percent (3.03%) of the required 8.25 psi. This example shows that as the computational resolution increases, the number of steps increases, the value of each step decreases, and the quantization error decreases. 
     The method of determining the quantization error in each of the above-described examples is referred to as the “normal criteria” for determining whether an acceptable quantization error will result from the use of relatively low component resolution digital devices, such a digital to analog converters and analog to digital converters. Such normal criteria is not based on the principles of the present invention. 
     Continuing to use such digital devices as one example of a component having an availability that decreases as resolution increases, such digital devices are essential in determining the values of the command signals (voltages) applied to the actuators. However, there is limited availability of such digital devices having high component resolution (e.g., in excess of about 10 or 12 bits ). Reference is made to the above-described need to increase the resolution of the commanded force signal when the actuator that is used displays analog controllability better than that of conventional digital control methods. Such need to increase component resolution is in conflict with the limited availability noted above. Therefore, as a basis for assuring availability of components, there is a need to use average resolution digital devices of 10 to 12 bits and at the same time increase the resolution of the commanded force signals. However, conventional ways of processing digital device output, and of performing the above conversions, for example, in the processing of the above-described pressure, area and force values, are in part based on use of the less available, high resolution digital devices, for example. 
     What is needed then, is a CMP method in which the accuracy of pressure and force command signals exceeds the resolution of mechanical actuating devices and which is less dependent on the use of high resolution, less available, components such as high resolution digital devices. In the required CMP method, such need is for a way to more accurately compute the value of forces to be applied to a wafer carrier, for example, as a polishing pad moves laterally relative to such wafer carrier during the CMP operation, wherein such computational accuracy does not depend on the use of high resolution digital devices. Moreover, such improved accuracy should be achieved even though the computation involves both digital and analog operations. Further, this improved computational accuracy should be achieved even though it may be necessary to convert values of required pressure or force, for example, from one set of units to a second set of units and then back to the first set of units. In such conversion, a value of a required pressure, for example, in the first set of units should have the same value after the conversion as before the conversion. In another sense, then, what is needed are methods for quantization, which are effective without the use of high resolution digital devices, and in which the resulting average computational resolution is of less importance in obtaining computed results having an acceptable accuracy, such that quantization errors are eliminated or significantly reduced. 
     SUMMARY OF THE INVENTION 
     A further aspect of the present invention relates to reducing quantization error in a computation by defining synchronization data for synchronizing computational operations of first Broadly speaking, the present invention fills these needs by providing CMP methods in which the accuracy of pressure and force computations is less dependent on the use of high resolution, less available components, such as high resolution digital devices such as digital to analog converters and analog to digital converters. The CMP methods of the present invention provide a way to more accurately compute the values of forces to be applied to a wafer carrier, for example, as a polishing pad moves laterally relative to such wafer carrier during the CMP operation. Such computational accuracy does not depend on the use of high resolution digital devices. Moreover, such improved accuracy is achieved even though the computation involves both digital and analog operations, and even though it may be necessary to convert values of required pressure or force, for example, from one set of units to a second set of units and then back to the first set of units. In such conversion, a pressure value, for example, in the first set of units may have the same value after the conversion as before the conversion. The present CMP methods enable a quantization process to be performed without the use of data from high resolution digital devices, and in which an average computational resolution is of less importance in obtaining computed results having an acceptable accuracy, such that quantization errors are eliminated or significantly reduced. 
     An aspect of the present invention relates to a method of accurately representing, for computational processing, a required value among a pressure range of values of pressure to be applied to a wafer in chemical mechanical polishing. Operations of the method include dividing the pressure range by the value of a component resolution to define scale portions of the pressure range. Another operation generates a first output signal to identify one of the scale portions that includes the required value. A final operation generates a second output signal to identify a set point that defines the requested value in the identified scale portion. 
     Yet another aspect of the present invention relates to more accurately representing, for computational processing, a required value of a variable parameter, the value being among a range of parameter values. A system component, such as a digital device, is selected and has an operational resolution defined in terms of a number of increments. A computational signal range of a computational signal is defined to represent the amount by which the required values of the parameter may vary in the parameter range. A processor is programmed to divide the computational signal range by the number of increments of the operational resolution to represent a plurality of scales within the parameter range, each of the scales having a given number of units per increment, the number of scales being about equal to the number of increments. One of the scales is selected and includes a set point that identifies the required value of the parameter, the selected scale having a scale range of units. The selected scale is represented in terms of a first output signal that is within the computational signal range, and the set point is represented in terms of a second output signal that is within the computational signal range. 
     A further aspect of the present invention relates to reducing quantization error in a computation by defining synchronization data for synchronizing computational operations of first and second digital processors. The computational operations are performed on data representing a parameter. Based on the synchronization data, first and second data converting operations are performed by the first digital processor. The first data converting operation converts an initial value of the parameter to first digital data corresponding to one scale of a plurality of scales in a scale function. The one scale identifies one range of values of the parameter within an entire set of values of the parameter. The second data converting operation converts the initial value to second digital data corresponding to a range function that identifies one set point in the one range of values corresponding to the scale. Based on the synchronization data, the second digital processor converts the first and second digital data to a data item that digitally represents the exact initial value of the parameter. 
     An additional aspect of the present invention relates to reducing quantization error in a computation of CMP pressure. The synchronization data is defined for synchronizing operations of the first and second digital processors. The synchronization data defines a computational resolution, a set of values of the pressure to be used in computations, a set of values of output pressure data for communications between the first and second digital processors, a scale data conversion function that defines a relationship between a required polishing pressure and each one of a plurality of scales into which the set of values of the pressure is divided; and a set point data conversion function that defines a relationship between a range of the pressures in a particular one of the scales and a set point that defines one value of the required pressure in the particular scale. The first processor performs a first conversion operation based on the synchronization data. The first conversion operation is performed on a required value of the pressure, and converts the required value of the pressure to first output pressure digital data representing a particular one of the scales. The first digital processor also performs a second conversion operation based on the synchronization data. The second conversion operation is performed on the required value of the pressure to convert the required value to second output pressure digital data representing the set point that defines the required pressure in the particular scale. In the second processor a third conversion operation is performed based on the synchronization data. The third conversion operation is performed to convert the first output pressure digital data to scale data representing the particular one of the scales. A fourth conversion is performed by the second digital processor based on the synchronization data. The fourth conversion operation is performed on the second output pressure digital data to convert the second output pressure digital data to digital data more accurately representing the required value of the pressure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present invention will be readily understood by the following detailed description in conjunction with the accompanying drawings, wherein like reference numerals designate like structural elements. 
         FIG. 1A  is a schematic elevational view showing a preferred embodiment of the present invention in which a polishing head contacts a contact area of a wafer mounted on a wafer carrier; 
         FIG. 1B  is a plan view of  FIG. 1A , schematically illustrating an initial position of the polishing head and by dashed lines identifying an initial contact area between the wafer and a polishing pad on the head; 
         FIG. 1C  is a plan view similar to  FIG. 1B , illustrating the initial position of the polishing head and in cross hatch lines identifying an initial contact area between a retainer ring surrounding the wafer and the polishing pad on the head, and in dashed-dot lines identifying an initial contact area between a puck carried by a pad conditioner carrier and the polishing pad on the head; 
         FIG. 1D  is a schematic view of a system of the preferred embodiment of the present invention, wherein a first processor provides first inputs to a second processor, the first inputs representing the position of the polishing pad relative to the wafer, and the second processor being shown receiving second inputs representing the pressure to be applied by the polishing pad on the wafer; 
         FIG. 2  is a schematic view of the first digital processor shown operating based on a recipe and specifying various required CMP pressures; 
         FIG. 3  depicts a flow chart illustrating operations of a method for specifying the required pressure in terms of a first scale identifier specifying a particular scale as the scale in which the required pressure is located, and a second identifier specifying a value of a set point within the specified one of the scales; 
         FIG. 4  is a schematic view illustrating the scales resulting from the method depicted in  FIG. 3 , and the set point in the specified one of the scales; 
         FIG. 5  depicts a flow chart illustrating operations of a further method performed in the first digital processor for providing scale and set point signals representing the required pressure to be applied to the wafer; 
         FIG. 6  is a schematic diagram illustrating how to join  FIGS. 6A and 6B ; 
         FIG. 6A  is a schematic diagram of one of two sections of the second digital processor that converts pressure request data to a pressure request; 
         FIG. 6B  is a schematic diagram of the second section of the second digital processor that converts pressure request data to a force request; 
         FIG. 7  depicts a flow chart illustrating operations performed by the second processor for processing a scale signal and a set point signal to define the pressure request; 
         FIG. 8  depicts a flow chart illustrating further operations performed by the second processor for defining the required force in terms of a first scale identifier specifying a particular scale as the scale in which the required force is located, and a second identifier specifying a value of a set point within the specified one of the scales; 
         FIG. 9  is a schematic diagram depicting a set of force scales and a force set point within an identified force scale to represent a required force; 
         FIG. 10  depicts a flow chart illustrating operations performed by the second digital processor for converting the force scale and force set point to define the required force in terms of force scale volts and force set point volts; 
         FIG. 11  is a schematic diagram of an analog logic preprocessor that receives data in terms of the force scale volts and force set point volts; 
         FIG. 12  depicts a flow chart illustrating operations performed by the analog logic preprocessor for converting the data in terms of force scale volts and force set point volts to define a force request; 
         FIG. 13  depicts a flow chart illustrating further operations performed by the analog logic preprocessor for defining logic and force range signals for input to an analog logic processor; 
         FIG. 14  is a schematic diagram of the analog processor which outputs the required force in terms of one analog voltage to be applied to a force actuator; and 
         FIG. 15  depicts a flow chart illustrating operations performed by the analog logic processor for defining the value of the one analog voltage to be applied to the force actuator. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     An invention is described for CMP methods that provide solutions to the above-described problems. Such CMP methods render the accuracy of CMP-related computations less dependent on the use of less available, high resolution components, such as high resolution digital devices. Such CMP methods of the present invention provide a way to more accurately compute values of required pressure and forces to be applied to a wafer carrier, for example, as a polishing pad moves laterally relative to such wafer carrier during the CMP operation. Such CMP methods enable a quantization process to be performed without the use of high resolution components, so that a resulting average computational resolution is of less importance in obtaining computed results having an acceptable accuracy. As a result, quantization errors are eliminated or significantly reduced. 
     In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention. It will be understood, however, to one skilled in the art, that the present invention may be practiced without some or all of these details. In other instances, well known process operations and structure have not been described in detail in order not to obscure the present invention. 
     Referring to  FIGS. 1A-1D , there is schematically shown a preferred embodiment of the present invention, including a system  200  having a resolution enhancement feature for improving the accuracy of conversion of required chemical mechanical polishing (CMP) pressure P to force F to be applied by a CMP head  202  to a wafer  204 , for example. Generally, the system  200  may use an encoder  208  (FIG.  1 D), to provide encoder signals  210  indicating the position of the CMP head  202  relative to the wafer  204 . The system  200  may also use a processor  212 , such as a personal computer, to process a recipe  213  that specifies the operations of the system  200  for required processing of the wafer  204 , e.g., for CMP operations. The processor  212  may be a personal computer having a rated processing capacity of a 600 MHz Pentium™ series processor, or equivalent., and running under an NT O/S and under a visual logic controller program (VLC) sold by Steeplechase, for example. The processor  212  may output separate signals  214 ,  216 , and  218  representing individual required pressures P that are required to be applied by a polishing, or planarization, pad  220 . For example, signals  214  represent values of one such pressure Pwp to be applied by the pad  220  on the wafer  204 . Signals  216  represent values of another such pressure Pcp to be applied by the pad  220  on a conditioner puck  222 . Signals  218  represent values of the other such pressure Pwp to be applied by the pad  220  on a retainer ring  224 . Use of the letter “P” refers generally to such required pressures, and is shown in FIG.  1 D. Reference to a specific one of the required pressures P is indicated by the use of Pwp, Pcp, or Prp. The term “Pressure Profiles” shown in  FIG. 1D  indicates that the recipe  213  may specify that the value of any such pressure P is to be constant, or that such value is to vary over time. 
     To illustrate the present invention, a situation is described in which the value of the pressure P is to be constant, and in which the head  202  and the pad  220  may move relative to each of the wafer  204 , the puck  222 , and the ring  224  (see arrow  226  in FIG.  1 A). Of course, the pressure P may vary in the operation of the system  200 . In the exemplary constant pressure situation, the relative motion results in changing values of areas AW (FIG.  1 B), and AC and AR (shown in  FIG. 1C ) of overlap (or contact) of the pad  220  on (or with) the respective wafer  204 , puck  222 , and ring  224 . The pressure P is in terms of the force F applied to an area A. With the values of the respective pressures Pwp, Pcp, and Prp maintained constant in this example, as the pad  220  moves in the directions of the arrow  226 , the values of the respective forces Fwp, Fcp, and Frp applied to respective areas Awp, Acp, or Arp must change in proportion to the changes in the values of the respective area AW, AC, or AR. The term “Force Profiles” shown in  FIG. 1D  indicates that in response to the recipe  213  specifying a value of any such pressure P, the corresponding value of the force F (e.g., Frp, Fcp, or Fwp) may vary over time. The encoder signals  210  and the pressure signals  214 ,  216 , and  218  are applied to a multi-axis force controller  228 , wherein one axis is for the wafer  204 , another axis is for the puck  222 , and the other axis is for the ring  224 . The force controller  228  may be a programmable signal processor (DSP) sold by Logosol, Inc. and having a per axis processing capacity of about that of a 486 series Intel ™ processor or equivalent. Such processor  228  has three axes, one corresponding to each of the three axes described above, such that the three axes may be processed at the same time. 
     The wafer axis of the controller  228  processes the encoder signal  210  in respective area processors  230 W for the area Awp,  230 C for the area Acp, and  230 R for the area Arp. Respective signals  232 W,  232 C, and  232 R represent the respective areas Awp, Acp, and Arp at a moment of time and corresponding to the particular relative position of the pad  220  and the respective wafer  204 , puck  222  and ring  224 . The respective signals  232 W,  232 C, and  232 R are applied to respective force processors  234 W,  234 C, and  234 R which convert the respective pressure signals  214 ,  216 , and  218  and the respective area signals  232 W,  232 C, and  232 R to respective signals  236 W,  236 C, and  236 R representing the respective forces Fwp, Fcp, and Frp in terms of force units such as pounds, for example. The signals  236 W,  236 C, and  236 R are applied to an analog logic processor  237  having a section corresponding to each of the signals  236 W,  236 C and  236 R. The respective sections of the analog logic processor  237  provide respective force signals  238 W,  238 C and  238 R to respective force actuators  239 W,  239 C, and  239 R ( FIG. 1A ) which urge the respective wafer  204 , puck  222 , and ring  224  toward the pad  220  to apply respective required pressure Pwp, Pcp, and Prp to the respective wafer  204 , puck  222 , and ring  224 . As described above, the force actuators  239  may be of the high resolution type, such as linear electromagnetic actuators, rather than the low resolution pneumatic actuators noted above. 
     The processor  212 , the force controller  228 , and the analog logic processor  237  are configured to minimize, if not eliminate, the above-defined quantization error. In the context of the system  200 , the above-defined term “quantization” refers to the below-described process of computation performed by the processor  212 , the force controller  228 , and the analog logic processor  237 , in which computational resolution is of significant importance in obtaining the values of the respective forces Fwp, Fcp, and Frp computed results, each of which has an acceptable accuracy. 
     In a quantization process performed with respect to one of the axes, a parameter may be the required pressure P, such as the required pressure Pwp, for example. Other quantization processes may be performed with respect to the other two axes (puck and ring), and the parameters may be the respective required pressures Pcp and Prp having appropriate initial values. Using the required pressure Pwp as an example for purposes of description of all such required pressures Pwp, Pcp, and Prp, such exemplary pressure Pwp may have an initial value of 0.005 psi, for example. Such initial value of the exemplary required pressure parameter Pwp is subjected to the below-described computational operations in the processor  212 , the force controller  228 , and the analog logic processor  237  to obtain the computed result, which is the value of the force Fwp corresponding to the initial value of the pressure Pwp. Similar operations with respect to the other required pressures Pcp and Prp result in obtaining the value of the required respective forces Fcp and Frp as the respective computed results. 
     Such quantization process may be performed with minimum, or no, quantization error, as defined above, even though the system  200  includes digital devices such as the force controller  228 , for example, having the relatively average component resolution defined below and even though the computations in the processor  212 , the force controller  228 , and the analog logic processor  237  are based on an average computational resolution. A preferred value of the selected component resolution is about six bits, and a more preferred value of the component resolution is about eight bits, and a most preferred value of the component resolution is from about ten to twelve bits. The components in the high end of this range are referred to as having a “relatively average” component resolution, which is in comparison to digital devices having high resolutions of from about fourteen bits to about 16 bits, for example. As described above with respect to component availability, digital devices having relatively average resolution are readily available, whereas as resolution increases such high resolution digital devices are less available. 
     The recipe  213  typically specifies a preferred range of required pressure P of from about zero psi to about ten psi. However, without the benefits of the present invention, the low end of the range is generally a pressure of about 1.5 psi. With the present invention, the range of the pressure P may start from about zero psi. The parameter resolution (as defined above) of the preferred pressure range is 0.001 psi, for example, which is to say that the required pressure Pwp is most preferrably specified in increments or steps of 0.001 psi. 
     With such parameter resolution and component resolution in mind, for comparative purposes the normal criteria described above may be used as follows to determine whether quantization error would normally result from the use of the selected relatively average component resolution of 10 bits (or 1024 counts), i.e., without the present invention. The exemplary absolute value of the parameter (pressure) increment is 0.001 psi. The absolute value of computational pressure range steps is determined by the pressure range of 10 psi divided by 1024 (i.e., about 0.01 psi per step). Thus, the absolute value of the parameter resolution increment is much less than the absolute value of the pressure range step. Based on the above normal criteria, one would expect significant quantization error to result because a choice of zero or 0.01 psi would be available as the steps closest to the required exemplary 0.005 psi pressure Pwp. Each of such steps of the choice would have an error of 0.005 psi, or 100%. However, as described below, such quantization error does not occur in the use of the present invention. Rather, with the ten or twelve bit component resolution of the described digital devices(e.g., the force controller  228 ) and the same computational resolution, no quantization error should result. 
     In  FIG. 1D , the processor  212 , the force controller  228 , and the analog logic processor  237  are shown as separate units. To achieve the required minimization or elimination of the quantization error, the present invention includes a method of specifying the chemical mechanical polishing pressure P (the pressure profiles in  FIG. 1D , for example). The method facilitates improvements in communicating the value of the exemplary required pressure Pwp from the processor  212  to the force controller  228 , and from the force controller  228  to the analog logic processor  237 , and to the force actuators  239 . Referring to  FIGS. 2 and 3 , the processor  212  is programmed by an instruction  240 . The method is defined by a flow chart  242  depicted in  FIG. 3 , and starts with an operation  244  implementing the instruction  240 . Operation  244  outputs the exemplary required pressure Pwp (0.005 psi) as a pressure request  246 . The method moves to an operation  248  for specifying the 10 bit computational resolution to be used in processing to obtain a computed value of the required pressure Pwp. The computed value is to have improved accuracy. In  FIG. 1D  a keyboard  250  or other input device is provided for performing operation  248 . The method moves to an operation  252  for defining the set of values representing the range of possible required pressures Pwp. The set includes the required value (0.005 psi) of the exemplary pressure Pwp. The method moves to an operation  254  to implement instruction  256 . Operation  254  divides the highest value (10 psi) of the exemplary range of possible required pressure Pwp by the value of the computational resolution (the exemplary 1024) to obtain a series of pressure scales  258 . The pressure scales  258  may be identified by 0-L 1 , L 1 -L 2 , . . . (Ln- 1 )-Ln, as shown in  FIG. 4 , for example. The pressure scales  258  represent ranges  260  of uniformly increasing possible values of the exemplary pressure Pwp, where the ranges  260  have equal amounts of the required pressure Pwp. In the exemplary situation, each of the ranges  260  equals 0.01 psi. A different first scale identifier (e.g., “I 1 ”, “I 2 ”, . . . “In”) is provided for each of the scales  258  of the exemplary range of pressure Pwp. A number (the exemplary 1024) of different first identifiers “In” is equal to the value of the computational resolution (the exemplary 1024). In the example, operation  254  results in the first scale identifier specifying scale I 1  as the scale in which the required exemplary pressure Pwp is located. 
     The instructions  256  are further implemented as the method moves to an operation  264  for specifying the required value (the exemplary 0.005 psi) of the pressure Pwp by providing a different second identifier (SP) to indicate a value of a set point  266  within any specific one of the scales  258 , e.g. scale I 1 . The set point  266  may correspond to any particular pressure value in the identified scale  258 , e.g. 0.005 psi in scale I 1 . The number of different second identifiers SP (the exemplary 1024) is equal to the value of the computational resolution (the exemplary 1024 in the exemplary situation). The set point  266  corresponding to the pressure Pwp is identified by the second identifier SP 512  in FIG.  4 . In  FIG. 3  the specifying of the exemplary CMP pressure Pwp is completed as operations  254  and  264  output the respective first and second identifiers I 1  and SP 512 . It may be understood that the computational resolution is used to obtain each of the scale identifier and the set point identifier. In other words, each of the exemplary 1024 scales  258  is divided into the exemplary 1024 possible set points. 
     Referring to  FIGS. 2 and 5 , other instructions are processed by the processor  212 , including instruction  270 ,  272 ,  274 ,  276 ,  278 , and  280 , which are implemented by operations of a flow chart  284  shown in  FIG. 5  for communicating the specific required value of the exemplary pressure Pwp to the force controller  228  for more accurate processing of the required value of the exemplary pressure Pwp. Operation  286  implements a pressure scale-to-count conversion of instruction  270 , by which the exemplary first identifier “In” is converted to a number of counts. For example, the exemplary identifier I 1  representing the first scale  258  is represented by 1 count. The identifier In representing the last scale  258  would be represented by the exemplary 1024 counts corresponding to an appropriate value of the pressure Pwp. The method moves to operation  288  which implements a pressure set point-to-count conversion of instruction  272 , by which the second identifier SP 512  is converted to a number of counts. For example, the exemplary identifier SP 512  representing the set point  266  is represented by  512  counts to correspond to the value of 0.005 psi which is one-half way between 0.00 psi and 0.01 psi. An exemplary identifier SP 1024  would identify the last set point  266  and would be represented by 1024 counts. For efficiency of operation of the force actuators  239 , the pressure scale-to-count conversion provides count values of between 0 and 1024 for the odd numbered pressure scales  258  (e.g., scales I 1 , I 3 , etc.) whereas the count values of the even numbered pressure scales  258  are between 1024 and 0. 
     The method moves to operations  290  and  292  which respectively implement instructions  274  and  276  to collectively generate one of the signals  214  when pressure Pwp is processed, or the respective signals  216  and  218  when the respective pressure Pcp or Prp are processed. Each such signal is in two parts. In the exemplary situation, one part represents the required value (0.005 psi) of the exemplary pressure Pwp in terms of a pressure scale part  214 S, and a second part represents a pressure set point part  214 SP. Operation  290  implements a pressure scale count-to-voltage conversion of instruction  274 . The implementation in operation  290  again uses the computational resolution, by which the count value of the first identifier “I 1 ” is converted to a voltage. The conversion is performed by selecting a value of a range of voltage of the output  214 , such as 10 volts. The voltage range is divided by the computational resolution to obtain a value of a pressure scale data conversion function, which in the exemplary situation is 0.01 volts per count. The one count value of the first scale identifier I 1  thus corresponds to a 0.01 volt value, which may be referred to as pressure scale volts and represents the value of the pressure scale part  214 S of the two part signal  214 . 
     The method moves to operation  292  that implements a pressure set point count-to-voltage conversion of instruction  276 . The implementation in operation  292  again uses the computational resolution, by which the count value of the second identifier “SP 512 ” is converted to a voltage. The above 10 volt value of the range of the signal  214  divided by the computational resolution provides a pressure set point data conversion function having a value of about 0.01 volts per count. The  512  count value of the second scale identifier SP 512  thus corresponds to about a 5.0 volt value, which may be referred to as pressure set point volts and represents the value of the pressure set point part  214 SP of the two part signal  214 . 
     The method moves to operation  294  to implement instructions  278  and  280 . The exemplary required pressure Pwp is defined in terms of the signal  214 S (i.e., the 0.01 volt value of the pressure scale volts) and the signal  214 SP (i.e., the 5.0 volt value of the pressure set point volts). The method is then done. As shown in  FIG. 2 , the signals  214 S and  214 SP are output from the processor  212 , and are applied to the force controller  228  shown in FIG.  6 A. The methods of flow charts  242  and  284  facilitate improved accuracy of communication of the value of the exemplary required pressure Pwp from the processor  212  to the force controller  228 , in that, as described below, the exact value of the exemplary required pressure Pwp may be obtained in the force controller  228 . 
     One aspect of the improved accuracy of communication of the value of the exemplary required pressure Pwp from the processor  212  to the force controller  228  is facilitated by defining synchronization, or pressure synchronization, data  300 . This data  300  synchronizes the computational operations of the processor  212 , which represents a first digital processor, and of the force processor  234 W of the controller  228 , which represents a second digital processor. The synchronization data  300  includes the data set forth in Table I: 
     
       
         
           
               
             
               
                 TABLE I 
               
               
                   
               
               
                 SYNCHRONIZATION DATA 300 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 The computational resolution 
               
               
                 The set of values representing the range of possible required pressures P 
               
               
                 The definition of the pressure scales 258 
               
               
                 The pressure scale data conversion function 
               
               
                 The pressure set point data conversion function 
               
               
                   
               
            
           
         
       
     
     As described above, the operations in flow charts  242  and  284  are based on one or more items of the synchronization data  300 . The processor  212  and the force controller  228  are provided with the synchronization data  300  from a hard drive  301 , for example, via a bus  296 . The data  300  is in the form of an RS232 signal applied to the force controller  228 , for example. In general, based on one or more items of the synchronization data  300 , the second digital processor (i.e., the force processor  234 W,  FIG. 1D ) converts first and second digital data (e.g., the exemplary respective 0.01 volt signal  214 S and the exemplary 5 volt signal  214 SP) to one data item  302 , which is a pressure request that ideally digitally represents the exact initial value (e.g., the exemplary 0.005 psi) of the parameter (the exemplary required pressure Pwp). 
     In more detail,  FIGS. 6A and 6B  taken together show the force processor  234 W as being provided with the synchronization data  300  (shown as the RS232 signal) from the hard drive  301  via the bus  296 . The force processor  234 W includes instructions  304 ,  306 , and  308  for processing the signal  214 S, and instructions  310 ,  312 , and  308  for processing the signal  214 SP.  FIG. 7  shows a flow chart  320  depicting operations for processing the signal  214 S. An operation  322  converts the value of the voltage of the pressure scale signal  214 S to digital data  324  representing counts and having a value corresponding to the respective exemplary specified pressure scale I 1 , i.e., 1 count. In such conversion, operation  322  uses the pressure scale data conversion function of the synchronization data  300 . The method moves to operation  326  in which instruction  306  is processed to convert the 1 count value of the digital data  324  to digital data  328  representing the one of the 1024 scales shown in FIG.  4 . In such conversion, operation  326  uses the definition of the scales  258  of the synchronization data  300 . 
     When the method moves to operation  322 , the method also moves to operation  330  for converting the value of the voltage of the pressure set point signal  214 SP to digital data  332  representing counts and having a value corresponding to the respective specified scale SP 512 , i.e., 512 counts. In such conversion, operation  330  uses the pressure set point data conversion function of the synchronization data  300 . The method moves to operation  334  in which instruction  312  is processed to convert the 512 count value of the digital data  332  to digital data  336  representing the set point in scale I 1  shown in FIG.  4 . In such conversion, operation  326  uses the definition of the scales  258 . 
     The method moves to operation  338  in which instruction  308  is processed to convert the exemplary pressure scale I 1  identity represented by the digital data  328 , and the pressure set point identity represented by the data  336 . Conversion of the pressure scale I 1  results in an identification of value of the range (zero to 0.01 psi) of the one of the 1024 scales described in  FIG. 4  that includes the exemplary pressure Pwp. Conversion of the set point SP 512  results in identifying the exact value of the exemplary required pressure Pwp, i.e., 0.005 psi. In such conversion, operation  338  uses the definition of the pressure scales  258  of the synchronization data  300 . Digital data  340  representing the value (the exemplary 0.005 psi) of the required pressure Pwp is output as the pressure request  302 . 
     Reference to  FIGS. 2 ,  6 A and  6 B indicates that the above-described use of the pressure synchronization data  300  in the force processor  234 W, the selection of the relatively average computational resolution, and the dividing of such computational resolution into both the pressure range of the exemplary required pressure Pwp and the voltage range of the output signals  214 , facilitates the improved accuracy of the communication of the value of the exemplary required pressure Pwp from the processor  212  to the force processor  234 W. 
     As described above, the encoder signals  210  and the pressure signals  214 ,  216 , and  218  are applied to the force processor  234 W of the multi-axis force controller  228 . The force controller  228  may be a programmable signal processor (DSP) sold by Logosol, Inc. and having a per axis processing capacity of about that of a 486 series Intel™ processor or equivalent. This DSP processor  228  has three axes, which means that the three axes (each of the wafer  208 , the ring  226  and the puck  222 ) may be processed at the same time.  FIGS. 6A and 6B  taken together show the details of the force processor  234 W for the one wafer axis.  FIGS. 7 and 8  show operations of methods performed by the force processor  234 W. The details for the two other axes and the method operations for such axes are similar to those shown in  FIGS. 6A ,  6 B,  7  and  8 . 
     The wafer axis of the processor  234 W shown in  FIGS. 6A and 6B  processes the encoder signal  210  in the area processor  230 W to define the area Awp at a moment of time and corresponding to the particular relative position of the pad  220  and the respective wafer  204 . It is understood that the resolution of the encoder  208  is high enough as to induce only small errors in such defining of the areas A. This processing is described in the Prior Application, and results in the signal  232 W being applied to the force processor  234 W of the force controller  228 .  FIGS. 6A and 6B  show the pressure request  302  and the area signal  232 W input to a force calculation instruction  350 . The instruction  350  is processed as also described in the Prior Application, and results in a force request  352 . The force request  352  may be in terms of digital data  354  representing the force in force units such as pounds corresponding to the exemplary required pressure Pwp to be applied to the exemplary area Awp. 
     To achieve the required minimization or elimination of the quantization error, the present invention further includes a method of specifying the CMP force F (the force profiles in  FIG. 1D , for example). The method facilitates improvements in communicating the value of the exemplary force Fwp (corresponding to the required pressure Pwp) from the force controller  228  to the analog logic preprocessor  237 PP shown in FIG.  11 . Referring to  FIGS. 6A ,  6 B,  8  and  9 , the force processor  234 W is programmed by instructions  360 . The method is defined by a flow chart  362 , and starts with an operation  364 . Operation  364  outputs an exemplary required force Fwp (7.5 pounds) representing the force request  352 . The method moves to an operation  368  for specifying a computational resolution (e.g., the average 10 or 12 bit computational resolution) to be used in processing to obtain a computed value of the force Fwp. The computational resolution (e.g., 10 bits) is read from the pressure synchronization data  300  stored in the drive  301 . The method moves to an operation  370  for defining the set of values representing the range of possible required force. The set includes the required value (exemplary 7.5 pounds) of an exemplary force Fwp. Processing of forces Fcp and Frp is performed in a manner similar to that described below with respect to the force Fwp. 
     The method moves to an operation  372  to further implement the instruction  360 . Operation  372  divides the highest value (about 1000 pounds) of the exemplary range of possible required force Fwp by the value of the computational resolution (the exemplary 1024) to obtain a series of force scales  376 . The force scales  376  may be identified by 0-M 1 , M 1 -M 2 , . . . (Mn- 1 )-Mn, as shown in  FIG. 9 , for example. The force scales  376  represent ranges  378  of uniformly increasing possible values of the exemplary force Fwp, where the ranges  378  have equal amounts of force. In the exemplary situation, each of the ranges  378  equals about 1 pound. A different first scale identifier (e.g.,“II 1 ”, “II 2 ”, −“IIn”) is provided for each of the force scales  376  of the exemplary force Fwp, and a number (the exemplary 1024) of different first identifiers “IIn” is equal to the value of the computational resolution (the exemplary 1024). In the example, operation  372  results in the first scale identifier specifying scale  118  as the force scale in which the exemplary required force Fwp is located. 
     The instructions  360  are further implemented as the method moves to an operation  380  for specifying the required value (the exemplary 7.5 pounds) of the force Fwp by providing a different second identifier (SSP) to indicate a value of a set point  382  within any specific one of the scales  376 , e.g. the exemplary force scale  118 . The force set point  382  may correspond to any particular force value in the identified force scale  376 , e.g. the exemplary 7.5 pounds in scale II 8 . The number of different second identifiers SSP (the exemplary 1024) is equal to the value of the computational resolution (1024 in the exemplary situation). The force set point  382  corresponding to the force Fwp is identified by the second identifier SSP 512  in FIG.  9 . In  FIG. 8  the specifying of the exemplary CMP force Fwp is completed as operations  372  and  380  output the respective first and second identifiers II 8  and SSP 512 . 
     Referring to  FIGS. 6A ,  6 B, and  10 , other instructions are processed by the force processor  234 W, including instructions  400 ,  402 ,  404 ,  406 ,  408 , and  410  which are implemented by operations of a flow chart  412  for communicating the specific required value of the exemplary force Fwp to the analog logic preprocessor  237 PP for more accurate processing of the required value of the exemplary force Fwp. The analog logic preprocessor  237 PP may be a programmable signal processor (DSP) sold by Logosol, Inc. and having a per axis processing capacity of about that of a 486 series Intel™ processor or equivalent, similar to that used for the force controller  228 . In  FIG. 10  operation  414  is shown for implementing a force scale identifier-to-count conversion of instruction  400 , by which the first identifier “IIn” is converted to a number of counts. For example, the exemplary identifier II 8  representing the eighth force scale  376  is represented by 8 counts; and the exemplary identifier II 1000  representing the last force scale  376  is represented by the exemplary 1024 counts. The method moves to operation  416  which implements a force set point identifier-to-count conversion of instruction  406 , by which the second identifier SSP is converted to a number of counts. For example, the exemplary identifier SSP 512  representing the set point  382  is represented by 512 counts to correspond to the value of 7.5 pounds being one-half way between 7.0 pounds and 8.0 pounds. For efficiency of operation of the force actuators  239 , the scale-to-count conversion provides count values of between 0 and 1024 for the odd numbered force scales  376  (e.g., scales I 1 , I 3 , etc.) whereas the count values of the even numbered force scales  376  are between 1024 and 0. 
     The method moves to operations  418  and  420  which respectively implement instructions  402  and  408  to collectively generate one of the signals  236 W,  236 C, and  236 R in the form of two parts. In the exemplary situation relating to signal  236 W, one part represents the required exemplary value (7.5 pounds) of the exemplary force Fwp in terms of a force scale part  236 S and a force set point part  236 SP. In more detail, the method moves to operation  418  which implements a force scale count-to-voltage conversion of instruction  402 . The implementation in operation  418  again uses the computational resolution, by which the count value of the first identifier “IIn” is converted to a voltage. The conversion is performed by selecting a value of a range of voltage of the output  236 W, such as 10 volts. The voltage range is divided by the computational resolution to obtain a value of a force scale data conversion function, which in the exemplary situation is 0.01 volts per count. The eight count value of the first scale identifier II 1  thus corresponds to a 0.08 volt value, which may be referred to as force scale volts and represents the value of the force scale part  236 S of the two part signal  236 . 
     The method moves to operation  420  that implements a force set point count-to-voltage conversion of instruction  408 . The implementation in operation  420  again uses the computational resolution, by which the count value of the second identifier “SSP 512 ” is converted to a voltage. The above exemplary 10 volt value of the range of the signal  236 SP divided by the computational resolution provides a force set point data conversion function having a value of 0.01 volts per count. The 512 count value of the second scale identifier SSP 512  thus corresponds to a 5.0 volt value, which may be referred to as force scale volts and represents the value of the force set point part  236 SP of the two part signal  236 . 
     The method moves to operation  422  in which the exemplary required force Fwp is defined in terms of the signal  236 S (i.e., the 0.08 volt value of the force scale volts) and the signal  236 SP (i.e., the 5.0 volt value of the scale volts). The method is then done. As shown in  FIGS. 6B and 11 , the signals  236 S and  236 SP are communicated from the force controller  228  to the analog logic preprocessor  237 PP. The methods of flow charts  362  and  412  facilitate improved accuracy of communication of the value of the exemplary required force Fwp from the force processor  234 W to the analog logic preprocessor  237 PP, in that, as described below, the exact value of the exemplary required force Fwp may be obtained in the analog logic preprocessor  237 PP. 
     Consistent with the use of the pressure synchronization data  300  for communications between the processor  212  and the force processor  234 W, communications between the force processor  234 W and the analog logic preprocessor  237 PP are synchronized by analog synchronization data  431  described below. This data  431  synchronizes the computational operations of the force processor  234 W, which represents a first digital processor, and of the analog logic preprocessor  237 PP, which represents a second digital processor.  FIGS. 6A and 6B  show the force processor  234 W as being provided with the analog synchronization data  431  from the hard drive  301  via the bus  296  in the form of the RS 232  signal. The analog synchronization data  431  includes the data set forth in Table II: 
     
       
         
           
               
             
               
                 TABLE II 
               
               
                   
               
               
                 ANALOG SYNCHRONIZATION DATA 431 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 The computational resolution 
               
               
                 The set of values representing the range of possible required force F 
               
               
                 The definition of the force scales 376 
               
               
                 The force scale data conversion function 
               
               
                 The force set point data conversion function 
               
               
                   
               
            
           
         
       
     
     As described above, the operations in flow charts  362  and  412  are based on one or more items of the analog synchronization data  431 . Similarly, in general, based on one or more items of the analog synchronization data  431 , if the force processor  234 W is considered a first digital processor, then a second digital processor in the form of the analog logic preprocessor  237 PP converts first and second digital data (the exemplary respective 0.08 volt signal  236 S and the exemplary 5 volt signal  236 SP) to one data item, which is a force request  450  ( FIG. 11 ) that ideally digitally represents the exact initial value (7.5 pounds) of a parameter (the exemplary required force Fwp). In more detail,  FIG. 11  shows the analog logic preprocessor  237 PP as being provided with the analog synchronization data  431  (in the form of the RS 232  signal) from a hard drive  432  via the bus  430 .  FIG. 11  also shows the analog logic preprocessor  237 PP as including instructions  452  and  454  for processing the force scale signal  236 S, and instructions  456  and  458  for processing the signal  236 ; along with instructions  460 .  FIG. 12  shows a flow chart  462  depicting operations for processing the signal  236 S. An operation  464  converts the value of the voltage of the force scale signal  236 S to digital data  466  representing counts and having a value corresponding to the respective exemplary specified force scale II 8 , i.e., 8 counts. In such conversion, operation  464  uses the force scale data conversion function of the analog synchronization data  431 . The method moves to operation  468  in which instruction  454  is processed to convert the exemplary 8 count value of the digital data  466  to digital data  470  representing the one of the 1024 scales  376  identified as the exemplary force scale II 8  in FIG.  9 . In such conversion, operation  468  uses the definition of the scales  376  of the analog synchronization data  431 . 
     When the method moves to operation  464 , the method also moves to operation  470  for converting the value of the voltage of the signal  236 SP to digital data  472  representing counts and having a value corresponding to the respective specified scale SSP 512 , i.e., 512 counts. In such conversion, operation  470  uses the force set point data conversion function of the synchronization data  431 . The method moves to operation  474  in which instruction  458  is processed to convert the 512 count value of the digital data  472  to digital data  476  representing the force set point in scale II 8  shown in FIG.  9 . In such conversion, operation  474  uses the definition of the force scale  376 . 
     The method moves to operation  478  in which instruction  460  is processed to convert the force scale II 8  identity represented by the digital data  470 , and the force set point identity represented by the data  476  to an identification of the value of the range (7.0 to 8.0 pounds) of the one force scale  376  shown in  FIG. 9  that includes the exemplary force Fwp. Conversion of the force set point SSP 512  results in identifying the exact value of the exemplary required force Fwp, i.e., 7.5 pounds. In such conversion, operation  478  uses the definition of the force scales  376  of the analog synchronization data  431 . Digital data  480  representing the exemplary required force Fwp is output as the pressure request  450 . 
     Reference to  FIGS. 6A ,  6 B, and  11  indicates that the above-described use of the analog synchronization data  431  in the analog logic preprocessor  237 PP, the selection of the relatively average computational resolution, and the dividing of such computational resolution into both the force range of the exemplary required force Fwp and the voltage range of the output signals  236 S and  236 SP, facilitates the improved accuracy of the communication of the value of the exemplary required force Fwp from the force processor  234 W to the analog logic preprocessor  237 PP. 
       FIG. 11  further shows that the analog logic preprocessor  237 PP is also provided with instructions  500  for converting the force request  450  into an analog upper range signal  502  and an analog lower range signal  504 , and to two digital logic signals  506  and  508 . The instructions  500  are implemented by a method depicted by a flow chart  510  shown in FIG.  13 . An operation  512  uses the force scale  376  and the exemplary force identifier II 8  to cause the signals  502  and  504  to define, or represent, the respective upper and lower boundaries, or range, of the one force scale  376  identified by the exemplary identifier II 8 . Thus, the signal  502  represents 8 volts and the signal  504  represents 7 volts in the exemplary situation in which the exemplary required force F is to be 7.5 pounds. The method moves to operation  514  which defines digital logic for identifying the set point  382  within the identified force scale  376 , and the method is done. The digital logic is based on the computational resolution (e.g., 10 bits in the exemplary situation). 
     For ease of description,  FIG. 14  primarily shows an example of 2 bit logic of the signals  506  and  508 , and the following description refers to how the 2 bit logic and the 10 bit logic are implemented.  FIG. 14  schematically depicts analog circuitry  511  for converting the four input signals  502 ,  504 ,  506 , and  508  to one of the analog signals  238 , in this case the exemplary analog signal  238 W shown in  FIG. 1D. A  method of operation of the circuitry  511  is shown on  FIG. 15  which depicts a flow chart  550 . In an operation  552  the range signals  502  and  504  are applied to a subtractor circuit  520  to generate an analog range-of-force signal  522  representing the difference between the values of the signals  502  and  504 . In the exemplary situation, the value of the difference is 1 volt, which is the value of the analog range-of-force signal  522 . Based on the resolution of the digital logic signals  506  and  508 , which in the example of  FIG. 14  is 2 bits, in an operation  554  a divider circuit  524  converts the value of the analog range-of-force signal  522  (i.e., the difference between the two analog force signals  502  and  504 ) to an analog force increment signal  526 , representing a value of 0.25 volts in the exemplary situation. The resolution (e.g., 2 bit) input to the divider circuit  524  may, for example, be from the drive  432  and is based on the analog synchronization data  431 . An input to the divider  524  is provided by a divider  527 . The divider  527  reduces the value of the signals  502  and  504  according to the range of the analog signals  238 . For example, in the 2 bit situation 2 bits (2×2) is divided by 1; or in the 4 bit situation, 16 is divided by 2; and in the 10 bit situation 1024 is divided by 100 (which is the exemplary value shown in FIG.  14 ). 
     Based on the logic defined by the two digital logic signals  506  and  508  via an analog logic signal  528 , a multiplier circuit  530  converts the value of the analog force increment signal  526  (the exemplary 0.25 volts) to an analog force set point signal  532 . In the exemplary situation the value of the signal  532  is 0.5 volts (0.25 times the value 2 of the analog logic signal  528 ).  FIG. 14  shows, and operation  556  describes, one of the analog force signals  502  and  504  (e. g., the lower signal  504 ) added to the analog force set point signal  532  to determine the value (in this example, 7.5 volts) of the force actuator signal  238 W. In operation  558  the force actuator signal  238 W is output and has the improved accuracy. 
     It may be understood that with the 2 bit logic shown by example in  FIG. 14 , only two logic input signals  506  and  508  are used (e.g., logic A and B). When the noted 10 bit logic is used for the logic signals, such as  506  and  508 , etc., ten such logic signals are used (e.g., logic A-J). The circuitry for the 10 bit logic will be understood by first referring to the 2 bit logic shown in FIG.  14 . An analog analysis circuit  570  receives the respective A and B logic signals  506  and  508 . The circuit  570  may be a programmable signal processor (DSP) sold by Logosol, Inc. With the 2 bit logic, two times two, or four, possible logic states  572 - 575  may be provided by the two input logic signals  506  and  508 . In the 10 bit case, 1024 logic states are achievable with 10 bit logic signals corresponding to logic A through logic J. In the 2 bit example, one of the logic states  572 - 575  outputs a logic signal  590  for any given logic input collectively defined by the signals  506  and  508 . Each logic signal  590  is accompanied by a multiplier input  592  having a one volt value. The value of the signals  590  is selected according to the required values of the analog force set point signals  532 . Generally, the values of the logic signals  590  are within the range of a 24 volt power supply. Thus, in the 10 bit example, the values of the logic signals  590  may range from 0.0 volts to about 10.0 volts (in the exemplary 0.01 volt increments shown in FIG.  14 ). In the 2 bit example, the signals  590  would be in a range of 0.0 volts to 3 volts, for example, in 1.0 volt increments, such that one exemplary signal  596  could have a 2 volt value. 
     The value of the signals  532  in turn depends on the values of the signals  526  and  528 . The corresponding multiplier input  592  and logic signal  590  are input to a respective corresponding multiplier  594 . For any given logic input to the analog logic evaluation circuit  570 , only one multiplier  594  outputs a product signal  596  having a value other than zero. The product signals  596  are added as shown by staged adders  600  to provide a series of sum signals  602 ,  604 , and  528 . The value of the last sum signal is the value of the analog logical voltage signal  528 , and depends on the logic input by the signals  506  and  508 . In the 10 bit logic example, there are 1024 multipliers  594 , and 1023 stages of the adders  600 . 
     As an example for the 2 bit logic, with the 7.5 volt value of the required force Fwp, and the value of 0.25 volts (1 volt divided by 4) of the analog force increment signal  526 , to obtain the 7.5 volt value, the sum, or analog logical voltage, signal  528  has the value of 2 volts based on the 2 volt signal  596  from one of the multipliers  594 . 2 volts times the increment 0.25 (the exemplary value of the signal  526  in the 2 bit example) gives the product 0.5 volts, which corresponds to the voltage amount above the 7 volt value of the signal  504  corresponding to the voltage value of 7.5 volts of the required force Fwp. In summary, the number of logic states in the evaluation circuit  570  equals the number of multipliers  594 , and there is one less adder  600  than the value of the computational resolution. 
     An example of the exemplary 10 bit logic is as follows when the required pressure Pwp is the exemplary 0.005 psi, and a corresponding required force Fwp is 0.25 pounds for a 200 mm wafer  208 , for example. An exemplary voltage range of the signals  236  ( FIG. 6B ) is 10 volts (which corresponds to a range of 502 pounds of the required force Fwp for a 10 psi maximum pressure P for the 200 mm wafer  208 ). The value of the inputs  592  may range from zero volts to 10.24 volts in 0.01 volt increments, and as shown in  FIG. 14 , the difference between the LR voltage signal  504  and the UR voltage signal  502  (the value of the signal  522 ) may be 9.766 millivolts. The ten logic inputs  506 ,  508 , etc. may thus cause the analog logical voltage signal  528  to change in increments of 9.537 times 10 to the minus six power. As a result, the LR voltage  504  may be increased in increments of 9.537 times 10 to the minus six power. Therefore, the double use of the relatively average 10 bit resolution results in the signals  238  (e.g., the signal  238 W in  FIG. 14 ) having a very small incremental value, which significantly improves the accuracy of the force signals  238 , and importantly may conform the increments in which the force signals  238  are valued to the increments of the high resolution electromagnetic actuators, for example. 
     In view of the foregoing description, it may be understood that in the use of the system  200  the accuracy of computations of the pressure P and the force F are less dependent on the use of high resolution, less available digital devices. The CMP system  200  and the described methods therefore provide a way to more accurately compute the values of the forces F that are to be applied to the wafer  204 , for example, as the  220  polishing pad moves laterally (arrow  226 ,  FIG. 1A ) relative to such wafer  204  during the CMP operations. Moreover, such improved accuracy is achieved even though the computation involves both the digital operations of the processor  212  and the controller  228 , for example, and the analog operations of the circuitry  511 . Importantly, such improved accuracy is achieved even though it may be necessary to convert values of the required pressure P or force F, for example, from one set of units to a second set of units and then back to the first set of units. In such conversion, it is seen that a pressure value, for example, in the first set of units may have the same value after the conversion as before the conversion. The CMP system  200  thus enable the quantization process to be performed with data from the relatively average resolution digital devices (e.g. the controller  228 ), and render such relatively average computational resolution of less importance in obtaining computed results having an acceptable accuracy, such as about one percent (1%), whereby quantization errors are eliminated or significantly reduced. 
     Although the foregoing invention has been described in some detail for purposes of clarity of understanding, it will be apparent that certain changes and modifications may be practiced within the scope of the appended claims. Accordingly, the present embodiments are to be considered as illustrative and not restrictive, and the invention is not to be limited to the details given herein, but may be modified within the scope and equivalents of the appended claims.