Patent Publication Number: US-7720617-B2

Title: Flow sensor signal conversion

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is a continuation application of and claims the benefit under 35 U.s.c. §120 to pending U.S. patent application Ser. No. 11/051,209 filed Feb. 4, 2005 which is a divisional application of and claims the benefit under 35 U.S.C. §120 to U.S. patent application Ser. No. 10/400,419, filed on Mar. 26, 2003, entitled “Flow Sensor Signal Conversion,” and issued as U.S. Pat. No. 7,043,374 B3, on May 9, 2006 each of which is hereby incorporated by reference in its entirety. 
    
    
     FIELD OF THE INVENTION 
     This invention relates to the field of mass flow sensors and, more particularly, to methods and apparatus for providing flow signals that accurately indicate the actual fluid flow of arbitrary fluids through a flow path of a sensor across a wide range of flow rates. 
     BACKGROUND OF THE INVENTION 
     Mass flow sensors are used in a wide variety of applications to measure the mass flow rate of a gas or other fluid. One application in which a mass flow sensor may be used is a mass flow controller. In a conventional mass flow controller, the mass flow rate of a fluid flowing in a main flow path is regulated or controlled based upon a mass flow rate of a portion of the fluid that is diverted into a typically smaller conduit forming a part of the mass flow sensor. Assuming laminar flow in both the main flow path and the conduit of the sensor, the mass flow rate of the fluid flowing in the main flow path can be determined (and regulated or controlled) based upon the mass flow rate of the fluid flowing through the conduit of the sensor. 
     A flow sensor, in general, refers to any device or combination of devices that responds to fluid flow by providing an output indicative of the fluid flow rate. A thermal mass flow sensor refers to a flow sensor that provides an output that varies with the flow rate of a fluid in a flow path based on heat convection, conduction and/or flux resulting from the flow of the fluid. The term “fluid” is used herein to describe any type of matter in any state that is capable of flow. It is to be understood that the term fluid applies to liquids, gases, and slurries comprising any combination of matter or substance capable of flow. 
     One conventional thermal mass flow sensor is illustrated in  FIG. 1 . Thermal mass flow sensor  10  includes a conduit  12  having an upstream resistance coil  14  and a downstream resistance coil  16  wound about the conduit  12  at a predetermined spacing such that the conduit has a characteristic length related to the length l. The upstream and downstream coils are coupled to sensor electronic circuit  18 . Typically, coils  14  and  16  are of the variety wherein the resistance of the coil is a function of temperature. When no fluid is flowing through conduit  12 , the sensor is in a balanced state. For example, coils  14  and  16  may comprise one leg of respective Wheatstone bridge configurations such that the balanced state may be achieved by having electronic circuit  18  drive upstream coil  14  and downstream coil  16  to the same temperature. When both coils are at the same temperature their resistances are equal such that the voltage drop across each resistor is also equal (i.e., V 1  is equal to V 2 ). Electronic circuit  18  may be configured to detect differences between voltage V 1  and V 2  and output the difference as a voltage signal  20 , referred to as the sensor output signal or simply the sensor output. 
     As fluid flows through the conduit, the fluid transfers heat from the upstream coil  14  toward the downstream coil  16  according to properties of heat convection. As a result, the temperature of the downstream coil  16  becomes greater than that of the upstream coil  14 , thereby varying the respective resistances and unbalancing the sensor electronic circuit  18 . The voltage drop V 2  will no longer equal the voltage drop V 1  and sensor electronic circuit  18  will detect the difference in voltage drops and output the difference as the sensor output signal  20 . The amount of heat transferred by convection, and thus the sensor output, is proportional to the mass flow rate of the fluid. 
     It should be appreciated that a sensor output need not be a voltage signal but may be any of various other signals and may depend upon the design of the flow sensor. For example, a sensor output may be a voltage signal as illustrated in  FIG. 1 , a current signal, a digital or analog signal or any other signal capable of indicating fluid flow through a conduit of the sensor. 
     Many applications, for example, semiconductor fabrication processes, may require that a particular flow sensor operate with a variety of different fluids and/or combinations of fluids. In practice, providing such a flow sensor is a difficult task. For example, the sensor output of conventional thermal mass flow sensors may depend both on the mass flow rate of the fluid in the flow path and, in part, on the type of fluid. That is, the thermal and/or physical characteristics of a fluid in the flow path may affect how the sensor output voltage changes with flow rate. 
       FIG. 2  illustrates sensor response curves for eight different types of fluid. A sensor response curve refers generally to any representation based on sensor output as a function of or in association with flow rate. In  FIG. 2 , the horizontal axis represents the actual fluid flow rate of the respective fluid through the flow path. The vertical axis represents the normalized sensor output voltage resulting from the fluid flow rate. The normalized sensor output voltage is the sensor output voltage divided by the sensor output voltage resulting when a fluid is flowing at its maximum flow rate through the flow path, referred to as full scale flow. That is, the vertical axis represents the fraction of the full scale sensor output voltage. 
     Ideally, each sensor response curve would be a linear function of flow rate. However, in practice it can be seen that the sensor response curves have varying degrees of curvature and that the curvature itself may be a fluid dependent function of flow rate. Conventionally, it is not well understood how the sensor response curves vary from fluid to fluid. This unpredictability makes it difficult to develop a flow sensor that responds satisfactorily to a range of fluids over an adequate range of flow rates. More particularly, it is difficult to provide a flow sensor that operates satisfactorily with an arbitrary fluid over an adequate range of flow rates. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates a conventional thermal mass flow sensor; 
         FIG. 2  illustrates representative sensor response curves resulting from the flow of eight fluids through a conventional thermal mass flow sensor of the prior art; 
         FIG. 3  illustrates a conventional method of employing surrogate fluid sensor response curves to correct sensor output values of a process fluid using conversion factors; 
         FIG. 4  illustrates one embodiment of a sensor model describing sensor behavior according to one aspect of the present invention; 
         FIG. 5  illustrates how a sensor model may be employed to determine flow rates of an arbitrary fluid according to another aspect of the present invention; 
         FIG. 6  illustrates a method of obtaining characteristic information during calibration of a sensor with a test fluid that may be used to determine flow rates of an arbitrary fluid during operation of the sensor according to another aspect of the present invention; 
         FIG. 7  illustrates a method of employing characteristic information obtained during calibration of a sensor on a test fluid to determine flow rates of an arbitrary fluid during operation of the sensor from sensor output values resulting from fluid flow of the arbitrary fluid according to another aspect of the present invention; 
         FIG. 8  illustrates another embodiment of a sensor model describing sensor behavior according to an aspect of the present invention; 
         FIG. 9  illustrates sensor response curves of various fluids mapped to characteristic values according to the embodiment of the sensor model illustrated in  FIG. 8  using unity scale factors; 
         FIG. 10  illustrates sensor response curves of various gases mapped to characteristic values according to the embodiment of the sensor model illustrated in  FIG. 8  using fluid dependent scale factors; 
         FIG. 11  illustrates another method of obtaining characteristic information during calibration of a sensor with a test fluid that may be used to determine flow rates of an arbitrary fluid during operation of the sensor according to another aspect of the present invention; 
         FIG. 12  illustrates another method of employing characteristic information obtained during calibration of a sensor on a test fluid to determine flow rates of an arbitrary fluid during operation of the sensor from sensor output values resulting from fluid flow of the arbitrary fluid according to another aspect of the present invention; 
         FIG. 13  illustrates one embodiment of a flow sensor incorporating various aspects of the present invention; 
         FIG. 14  illustrates one embodiment of a mass flow controller including a flow sensor operating according to various aspects of the present invention; 
         FIG. 15  illustrates another embodiment of a mass flow controller including a flow sensor wherein the controller and various methods of processing a sensor output signal are implemented on a separate computer located proximate to or remote from the sensor of the mass flow controller; 
         FIG. 16  illustrates another embodiment of a mass flow controller including a flow sensor having a sensor conduit and a bypass wherein a linearization curve for an arbitrary fluid may be determined by separately correcting sensor flow and bypass flow by employing a sensor model according to the present invention; 
         FIG. 17  illustrates one method of obtaining calibration measurements of a test fluid that facilitates determining a linearization curve for an arbitrary fluid according to another aspect of the present invention; and 
         FIGS. 18A and 18B  illustrate a method of determining a linearization curve by separately correcting for sensor flow non-linearity and bypass non-linearity based on calibration measurements of a test fluid and a sensor model according to another aspect of the present invention. 
     
    
    
     SUMMARY OF INVENTION 
     Various aspects of the present invention derive from Applicant&#39;s identification and appreciation of a correlation between fluid sensor output curves and a characteristic curve of a sensor when the sensor is modeled with a transfer function operating on flow rate, at least one sensor property and at least one fluid property. The correlation facilitates accurately determining flow rates of an arbitrary fluid by employing response information obtained during calibration with a test fluid and a model of the sensor. 
     According to one aspect of the present invention, a method of generating characteristic information of a sensor with a test fluid is provided. The method comprises acts of measuring a flow rate of the test fluid through a sensor conduit of the sensor for each of a plurality of sensor output values, converting each flow rate to an operand value based on at least one sensor property and at least one first property of the test fluid, mapping each of the plurality of sensor output values to a characteristic value based on at least one second property of the test fluid and storing at least one representation associating each operand with the respective characteristic value. 
     According to one embodiment, the act of converting each flow rate and the act of mapping each of the plurality of sensor output values are performed in accordance with the relationship: 
     
       
         
           
             
               
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     wherein ρ is a density of the test fluid, Q is a volumetric fluid flow rate, L is a characteristic length of the sensor conduit, A is a cross-sectional area of the sensor conduit, c p  is a specific heat of the test fluid, κ is a thermal conductivity of the test fluid and ƒ and g are fluid dependent constants. 
     According to another aspect of the present invention, a method of generating response information of a sensor with a test fluid for operation with an arbitrary fluid is provided. The method comprises acts of measuring a flow rate of the test fluid through a sensor conduit of the sensor for each of a plurality of sensor output values, recording each flow rate in association with a respective one of the plurality of sensor output values, storing a first mapping configured to convert sensor output values resulting from flow of an arbitrary fluid through the sensor conduit to characteristic values based on at least one property of the arbitrary fluid, and storing a second mapping configured to convert the flow rates to operand values based on at least one sensor property and at least one property of the test fluid. 
     In one embodiment, the act of storing the second mapping includes an act of storing a second mapping configured to convert flow rates to operand values in accordance with the relationship: 
     
       
         
           
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     wherein Φ is the operand value, ρ is a density of the test fluid, Q is a volumetric fluid flow rate, L is a characteristic length of the sensor conduit, A is a cross-sectional area of the sensor conduit, c p  is a specific heat of the test fluid, κ is a thermal conductivity of the test fluid and g is a fluid dependent constant and the act of storing the first mapping includes an act of storing a first mapping configured to convert sensor output values to characteristic values in accordance with the relationship: 
     
       
         
           
             
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     wherein c are the characteristic values, S are the sensor output values, κ is a thermal conductivity of the arbitrary fluid and f is a fluid dependent constant. 
     According to a further aspect of the present invention, a method of predicting flow rates of a process fluid through a sensor conduit with a flow sensor including at least one representation of a characteristic function of the sensor obtained during calibration with a test fluid is provided. The method comprises acts of providing a sensor output related to a flow rate of the process fluid through the conduit, mapping the sensor output value to a characteristic value based on at least one first property of the process fluid, obtaining an operand value including a flow rate component, at least one sensor property component and at least one fluid property component by indexing the at least one representation of the characteristic function with the characteristic value, and determining the flow rate component of the operand value using at least one property of the sensor and at least one second property of the process fluid to provide an indication of the flow rate of the process fluid through the sensor conduit. 
     According to another aspect of the present invention, a sensor model is provided that may be used for predicting flow rates of an arbitrary fluid through a sensor from sensor output values resulting from flow of the arbitrary fluid through the sensor. The model comprises an operand including flow rate, at least one sensor property and at least one first fluid property of the arbitrary fluid and a transfer function that, when operating on values of the operand, produces a sensor characteristic curve related to the sensor output values by at least one second fluid property of the arbitrary fluid. 
     According to another aspect of the present invention, a computer readable medium is encoded with a representation of a characteristic function and a program for execution on at least one processor. The program, when executed on the at least one processor, performs a method of converting a sensor output signal resulting from flow of a process fluid to a flow signal. The method comprises acts of receiving as an input a sensor output value related to a sensor output signal from a flow sensor, converting the sensor output value to a characteristic value based on at least one first property of the process fluid, obtaining an operand value including at least one sensor property and at least one fluid property associated with the characteristic value from the representation of the characteristic function, and determining a flow rate value of the process fluid from the operand value using at least one sensor property value and at least one process fluid property value. 
     Another aspect of the present invention is directed to a data structure to store a representation of a characteristic function of a flow sensor obtained during calibration of the flow sensor on a test fluid. The data structure comprises a plurality of entries, each entry having a first field and a second field, wherein the first field identifies a characteristic value computed from a sensor output value of the flow sensor based on at least one property of the test fluid and the second field identifies an operand value computed from a flow rate corresponding to the sensor output value, at least one property of the sensor and at least one property of the test fluid. 
     Another embodiment according to the present invention includes a computer readable medium adapted to electronically store information comprising an encoding of at least one representation of a characteristic function of a flow sensor obtained during calibration of the flow sensor with a test fluid. 
     Another embodiment according to the present invention includes a flow sensor comprising a conduit adapted to convey fluids, a circuit coupled to the conduit and configured to produce a sensor output signal indicative of a flow rate of a fluid through the conduit and at least one storage medium storing information including at least one representation of a characteristic function of the flow sensor. 
     According to another aspect of the present invention, a mass flow controller is provided. The mass flow controller comprises a flow sensor adapted to sense fluid flow in a flow path and provide a sensor output signal indicative of the mass flow rate in the flow path, a storage medium encoded with at least one representation of a characteristic function of the flow sensor, and a processor coupled to the flow sensor to receive the sensor output signal, the processor configured to convert the sensor output signal to a flow signal based on the characteristic function. The mass flow controller further comprises a controller coupled to the flow meter and adapted to provide a drive signal based at least in part on the flow signal, a valve actuator adapted to receive the drive signal from the controller and a valve coupled to the flow path. The valve is adapted to permit flow through the flow path depending on the displacement of a controlled portion of the valve. 
     DETAILED DESCRIPTION 
     Methods have been proposed to address the difficulties that may result from variation in the behavior of a sensor when operating with different fluids. The variation may be influenced, in part, by the non-linear relationship between flow rate and sensor output. For example, restricting operation of the flow sensor to a linear range of a sensor response curve for a particular fluid may mitigate problems associated with the non-linearity of the sensor. As illustrated in  FIG. 2 , a sensor response curve for a particular fluid may have a portion that is approximately linear. By considering only flow rates within this limited range, sensor output values can be assumed to have an approximately linear, and thus predictable, relationship with the fluid flow. 
     However, a significant portion of a desired range of use of many of the sensor response curves lies outside the respective linear range and cannot be adequately approximated by a straight line. As such, limiting flow rates to the linear range of a given sensor response curve of a particular fluid may severely restrict the usefulness of a flow sensor to a very narrow range of applications and flow rates. For example, the linear range for the gas SF 6  shown in  FIG. 2  is less than 3 sccm. Outside this range, the behavior of the sensor is not directly predictable and may render the sensor unusable for applications requiring a wider range of flow rates. 
     The term “curve” refers generally to any representation of a function, distribution or relationship. For example, a curve may be a set of point pairs representing the relationship between two variables (e.g., a set of point pairs describing the relationship between sensor flow and sensor output). A curve may also be a mathematical, geometrical or other representation of a function. For example, a curve may be represented as a algebraic function (e.g., a polynomial), a trigonometric finction (e.g., a tangent curve), a piecewise approximation (e.g., a piecewise linear or conic approximation), a series sum of basis functions (e.g., splines or fourier coefficients), etc. 
     As illustrated in  FIG. 2 , the linear range of a sensor response curve may also depend on the fluid, that is, both the length and slope of a line that adequately approximates the linear portion of a sensor response curve may vary from fluid to fluid. Accordingly, the linear range for each fluid is generally established empirically by testing the fluid with a flow sensor during calibration at various known flow rates (e.g., measured by another flow measuring instrument) and measuring the sensor output. The term “calibration” refers generally to various tests and measurements made on a flow sensor to obtain information about the sensor and, more particularly, sensor response information. The information obtained during calibration may be used, for instance, to approximate the behavior of the sensor during operation. Typically, sensor calibration is carried out before the sensor is employed in the field (e.g., in an industrial application) and/or provided to a customer. 
     However, measuring the linear range for a variety of test fluids, for example, the eight test fluids shown in  FIG. 2 , is a labor intensive and time consuming procedure. Moreover, a linear range may have to be specifically computed for each of a set of fluids for which a sensor is intended to operate. In general, this type of customized calibration adds additional expense to the sensor calibration process. To complicate matters, many industrial applications operate flow sensors with process fluids having reactive, toxic and/or corrosive properties. These fluids may not be able to be handled and tested with during calibration of the flow sensor. Furthermore, which fluids a sensor will ultimately operate on in the field may not be known a priori. 
     To compensate for the inability to calibrate with a particular process fluid, either because a particular process fluid is unknown at the time of calibration or is dangerous to handle and/or harmful to the equipment, a flow sensor may be tested with a number of surrogate fluids during calibration. A surrogate fluid is generally a less corrosive and/or more generally available fluid having known properties used during calibration to obtain information about sensor behavior that may be subsequently used to approximate the response of the flow sensor operating on a process fluid known to have at least somewhat similar properties. The fluids illustrated in  FIG. 2 , for example, may be employed as surrogate fluids. 
     The term “process fluid” refers generally to any fluid on which a sensor operates or on which the sensor is intended to operate in the field for any of a variety of applications. In particular, a process fluid describes a fluid that may be used by the end user of a flow sensor. The term “test fluid” refers generally to any fluid used during calibration of a sensor, typically, in preparation for the operation of the sensor in the field. 
     It should be appreciated that test fluids and process fluids may not be different, that is, a particular test fluid used during calibration may be used in the field as a process fluid. For the purposes of this description, the terms generally describe whether a fluid is being used during calibration (i.e. a “test” fluid) or for operation in the field (a “process” fluid), and not necessarily to describe the type of fluid. As such, a surrogate fluid is at least a test fluid. 
     During calibration, a sensor may be tested with a number of test and/or surrogate fluids. In particular, for each surrogate fluid to be tested, the surrogate fluid may be introduced to a flow path coupled to a flow sensor at known flow rates and the sensor output recorded. This results in a sensor response curve for each surrogate fluid. The exemplary sensor response curves for a particular sensor for the fluids shown in  FIG. 2  were obtained in this manner. Conventionally, surrogate sensor response curves have been employed to approximate flow rates from sensor outputs resulting from flow of a process fluid by assuming that sensor response curves of a process fluid may be related to one or more surrogate sensor response curves by a conversion factor. Conventional conversion factors are often based on the ratio between the molar specific heat of the process fluid and the molar specific heat of the most similar surrogate fluid. 
       FIG. 3  illustrates how, conventionally, surrogate sensor response curves obtained during calibration may be employed to approximate flow rates during operation of the sensor on a process fluid. During calibration  305  of a sensor  310 , each of a plurality of surrogate fluids SF 1 -SFN may be introduced to sensor  310  at a predetermined set of known flow rates F. Sensor  310  responds to the flow rates by providing surrogate sensor response curves SS 1 -SSN, respectively. The surrogate sensor response curves may then be recorded in association with the known flow rates F. 
     During operation  315  of sensor  310 , it may be desirable for sensor  310  to operate on any of process fluids PF 1 -PFN. When any one of process fluids PF 1 -PFN is introduced to sensor  310  at unknown flow rates UF, the sensor responds with a corresponding process sensor response curve PS 1 -PSN, respectively. It is assumed that values of a process sensor response curve are related to values of one of the surrogate sensor response curves by the ratio of the molar specific heat of the process fluid and the molar specific heat of the surrogate fluid. This relationship is shown schematically as mapping  325 . As such, sensor output values resulting from fluid flow of a process fluid may be mapped to associated values of a surrogate sensor response curve and an associated and/or interpolated flow rate from the known flow rates in set F may be used as an approximation of the unknown flow rate of the process fluid. 
     However, the relationship based on molar specific heats may only be valid within the linear range of the process fluid. That is, there may only be an accurate mapping between process sensor output values and values on the recorded surrogate sensor response curve in the linear range of the process fluid. Accordingly, accurate approximations of process fluid flow rates may only be achieved by the method described above when a sensor output value falls within the linear range of the sensor response curve of the process fluid. As discussed in the foregoing, this is a severe limitation on the applicability of a sensor. 
     Furthermore, it should be appreciated that in order to obtain accurate approximations, even in the linear range of a process fluid, a sufficiently similar surrogate fluid is required to be tested during calibration of the sensor. As a result, not only is it difficult to ensure that a sensor will operate adequately on an arbitrary process fluid, but the accuracy of the sensor will vary depending on the availability of an appropriate surrogate fluid. In addition, calibrating a sensor that operates generally on multiple process fluids may require time-consuming effort to measure and record response information (e.g., sensor response curves) for numerous surrogate fluids. 
     Applicant has appreciated that deficiencies in conventional conversion techniques derive from the insufficiency of empirical information to adequately describe the behavior of the sensor. Conventional conversion factors have been generated largely by experimentation and may not have any physical support. In particular, Applicant has identified that conversion factors based on sensor response information as a function of flow rate are incomplete general descriptions of the response of a sensor to arbitrary fluids. 
     Accordingly, Applicant has developed a physically based general sensor model describing the behavior of a sensor from properties of the sensor and fluid that may describe the response of a sensor to arbitrary fluids. Various aspects of the present invention derive from Applicant&#39;s identification and appreciation that a transfer function of a sensor may be characterized as operating on one or more properties of the sensor and one or more properties of a fluid in addition to flow rate to produce a single characteristic curve describing the response of the sensor. As such, response information obtained from a sensor during calibration may be utilized according to a sensor model to facilitate accurate prediction of flow rates of an arbitrary fluid both within and without the linear range of the fluid&#39;s sensor response curve. 
     Following below are more detailed descriptions of various concepts related to, and embodiments of, methods and apparatus according to the present invention. It should be appreciated that various aspects of the invention, as discussed above and outlined further below, may be implemented in any of numerous ways, as the invention is not limited to any particular manner of implementation. Examples of specific implementations are provided herein for illustrative purposes only. 
     In particular, various methods according to the present invention may be employed in conjunction with a variety of flow sensors including the flow sensor described in connection with  FIG. 1  and/or the various flow sensors described in U.S. application Ser. No. 09/783,439, U.S. Application Ser. No. 60/397,139 and U.S. Application Ser. No. 60/436,207 which are herein incorporated by reference in their entirety. The present invention is not limited to any particular implementation or arrangement of a flow sensor. 
       FIG. 4  illustrates one embodiment of a sensor model according to the present invention. Sensor model  400  includes a characteristic function W that when operating on values of Φ provides characteristic curve C. The Φ operand may be any of various appropriate functions of flow rate, at least one property of the sensor, and at least one property of the fluid. By expanding the sensor model such that its transfer function  405  operates on values of the Φ operand, sensor model  400  incorporates sensor and fluid properties whereby the behavior of the sensor is accurately described by the model. The sensor model may be employed to accurately determine flow rates from the sensor output values resulting from operation on arbitrary processes fluids. 
     A transfer function of a sensor model operating on an operand including flow rate, at least one sensor property, and at least one fluid property that produces a characteristic curve that may be related to sensor output values of a sensor by at least one fluid property is referred to as a “characteristic function.” 
       FIG. 5  illustrates the relationship between a sensor  510  and one embodiment of a sensor model according to the present invention. When any of various fluids, for example, fluids f 1 , f 2 , f 3  and/or fn are introduced to a sensor  510  at various flow rates F, sensor  510  may produce a different sensor response output curve, for example, S 1 , S 2 , S 3  and Sn, for each fluid, respectively. According to sensor model  500 , each of the sensor response curves is related to a characteristic curve C by a transform  525 . Transform  525  may be any function or transform that relates the various sensor response curves to the characteristic curve C based on at least a property of the fluid associated with the respective sensor response curve. Fluid properties may include, but are not limited to, thermal conductivity, viscosity, specific heat, density, temperature, any combination or relationship thereof, etc. 
     For example, transform  525  may map each of sensor response curves S 1 , S 2 , S 3  and Sn onto characteristic curve C. That is, a sensor output value resulting from a particular flow rate of a particular fluid may be mapped to a value along the characteristic curve by applying transform  525  according to sensor model  500 . 
     The Φ operand of sensor model  500  may be any of various functions of flow rate, one or more sensor properties, and one or more fluid properties. Sensor properties may include physical dimensions of a sensor conduit, the configuration of the sensing elements of the sensor electronics, other various physical properties of a sensor, combinations of sensor characteristics, and/or relationships thereof that may influence flow. Since the Φ operand of model  500  is based on flow rate and properties of the sensor and fluid, transfer function W describes a characteristic function of sensor  510 . 
     Furthermore, according to sensor model  500 , a fluid flowing through a sensor at a particular flow rate will have an associated value of Φ that may be computed by applying transform  515 . For example, an associated value of Φ for each of the flow rates F for fluid f 1 , f 2 , f 3  and fn, etc., may be computed from properties of sensor  510  and properties of the respective fluid. 
     It should be appreciated that values along characteristic curve C can be mapped to associated values of Φ according to an inverse characteristic function W −1 , provided information on W is available and the characteristic function W is monotonic increasing or monotonic decreasing in a region of interest. That is, if a value c i  along curve C is known, a corresponding value of Φ may be computed. As such, since the model establishes that transform  525  maps sensor response curves to the same characteristic curve C and values of Φ are a function of flow rate, a sensor output value produced by an arbitrary fluid flowing through a sensor may be used to back calculate a flow rate that would result in a characteristic value of c i  according to the sensor model, provided a representation of characteristic function W is available. 
     The term “representation” refers generally to any numerical, analytical, mathematical and/or algorithmic description or approximation of a characteristic function. Exemplary representations include, but are not limited to, look-up tables (LUT), approximating functions such as splines, trigonometric or algebraic functions, series expansions, etc. In addition, a representation may include data and a set of operations and/or instructions on how to manipulate the data. For example, a representation may include a look-up table storing sensor response information (e.g., sensor output values in association with flow rate values) and one or more mapping operations adapted to convert the sensor response information to characteristic information (e.g., characteristic values in association with Φ operands). 
       FIG. 6  illustrates one embodiment of a method according to the present invention that includes obtaining a representation of a characteristic function of a sensor model during calibration of a sensor on a single test fluid. The information obtained during calibration  600  may then be employed to facilitate approximating flow rates during operation of the sensor on an arbitrary fluid. During calibration, a sensor may be coupled to various calibration equipment adapted to introduce fluids to a conduit of the sensor at adjustable and measurable flow rates. 
     In step  610 , a test fluid having known thermal and/or physical properties is introduced to a sensor at a number of desired sensor output values. For example, an operator may adjust the calibration equipment such that the flow rate of the test fluid causes the sensor to respond with a desired sensor output value. In one embodiment, the sensor output values are normalized, for example, by dividing sensor output values by the sensor output value at a nominal full scale flow through the conduit such that they represent fractions of full scale flow. 
     For example, the calibration equipment may be adjusted such that the sensor responds with a sensor output value of 0.1 (e.g., indicative of 10% of nominal full scale flow). The actual flow rate through the conduit of the sensor is then measured, for example, by a volume-displacement reference flow meter of the calibration equipment. The sensor output value and the associated actual flow rate may then be recorded. The flow rate of the test fluid may again be adjusted until the sensor responds with a second desired sensor output value, for instance, a sensor output value of 0.2 (e.g., indicative of 20% of nominal full scale flow). The actual flow rate is again measured and the pair of values recorded. This operation may be repeated until it has been determined that adequate sensor response information has been obtained for the test fluid. 
     In step  620 , the sensor output values recorded in step  610  are mapped to characteristic values according to the relationship between sensor response curves and the characteristic curve described by the sensor model being employed. The term “characteristic value” refers generally to values computed from a sensor output value produced in response to flow of a fluid by using at least one property of the fluid. A characteristic value is presumed to lie substantially along the characteristic curve of the sensor model. For example, the sensor output values recorded in step  610  may be mapped to characteristic values by multiplying the sensor output value by a property of the test fluid, for example, the thermal conductivity of the test fluid. 
     In step  630 , the actual flow rates recorded in step  610  are mapped to values of the Φ operand based on properties of the sensor and the test fluid according to the sensor model being employed. In step  640 , the values of the Φ operand computed from the recorded actual flow rates in step  630  and the characteristic values computed in step  620  from the predetermined set of sensor output values corresponding to the actual flow rates are stored in association with one another, for example, in an LUT. 
     Accordingly, during calibration, characteristic values of a test fluid are recorded as a function of Φ, that is, a representation describing the characteristic function of the sensor has been obtained. This characteristic information may be used to accurately determine the actual flow rate of an arbitrary process fluid from sensor output values obtained from the sensor operating with the process fluid. The term “characteristic information” refers generally to information describing a characteristic function of a sensor and, more particularly, to a representation of the characteristic function. For example, characteristic information may include, but is not limited to, any of various data structures such as look-up tables, approximating functions such as splines, parametric curves, trigonometric or algebraic approximations, etc. 
     It should be appreciated that test fluid flow rates and test fluid sensor output values need not be mapped to characteristic values and operand values, respectively, during calibration. For example, the values obtained in step  610  may be stored in a look-up table or other suitable data structure as sensor response information. The sensor response information may be subsequently converted to characteristic information during operation of the sensor on an arbitrary fluid. That is, steps  620  and  630  may be carried out during operation of the sensor instead of during calibration provided the appropriate mapping has been stored or otherwise been made available to the sensor during operation. Storing the sensor response information and converting it to characteristic information according to a sensor model during operation may facilitate operation of the sensor with various sensor models without requiring the sensor to be re-calibrated. 
       FIG. 7  illustrates one method of employing sensor characteristic information to determine an unknown flow rate of an arbitrary process fluid directly from sensor output values provided by the sensor. In step  710 , a process fluid is introduced to a sensor at an unknown flow rate. For example, the flow sensor may be employed to measure the flow in an industrial control application wherein knowledge of the fluid flow through the sensor is used in a feedback loop to deliver a precise amount of fluid to a process chamber. The sensor may respond to the unknown flow rate by producing a sensor output value indicative of the flow rate, for example, according to principles of heat convection as described in connection with  FIG. 1 . 
     In step  720 , the sensor output value is mapped to a characteristic value substantially along a characteristic curve of the sensor according to a mapping described by the sensor model, for example, as described in step  620  of  FIG. 6 . It should be appreciated that, typically, the same sensor model used to obtain characteristic information about the sensor during calibration is employed in predicting flow rates during operation of the flow sensor. 
     In step  730 , the characteristic value is used to reference the characteristic information in order to, ultimately, obtain a flow rate associated with the characteristic value. For example, a LUT obtained by the method described in  FIG. 6  may be indexed with the characteristic value to obtain the associated Φ operand. Since the sensor model establishes that characteristic values computed from sensor output values of different fluids lie on the same characteristic curve, a representation of which was recorded during calibration of the sensor on the test fluid, the characteristic value computed in step  720  will have an associated Φ operand that may be obtained from the representation of the characteristic function. 
     Indexing characteristic information or a representation of a characteristic function with a characteristic value refers generally to using the characteristic value to determine an operand value associated with the characteristic value according to the sensor model, that is, consistent with the representation of the characteristic function. It should be appreciated that indexing may include using the characteristic value to determine the associated operand value in a look-up table, as a parameter to an approximating and/or interpolating function representing the characteristic information, applying the characteristic value as an input variable to a function configured to determine an associated operand value, or any of various other methods designed to obtain an associated operand value from a representation of a characteristic function of the sensor with a characteristic value. 
     In step  740 , the associated Φ operand is used to determine the unknown flow rate of the process fluid through the sensor that produced the sensor output value obtained in step  710 . In particular, since the Φ operand is a function of flow rate and properties of the sensor and process fluid, the flow rate may be isolated from the Φ operand by applying known values associated with the configuration of the sensor and known process fluid properties to provide an accurate determination of the actual flow rate of the process fluid through the sensor. 
     As discussed above in connection with  FIG. 6 , in some embodiments, sensor response information is recorded during calibration along with operations for converting the sensor response information to characteristic information. Accordingly, flow rates and corresponding sensor output values stored in the sensor response information may be mapped to operand values and characteristic values, respectively, during operation of the sensor. For example, steps  620  and  630  described in connection with  FIG. 6  may be part of the operation of the sensor as described in connection with  FIG. 7 . 
     Applicant has identified that sensor output curves of various fluids may be related to a characteristic curve of a sensor by the thermal conductivity of the fluid in a sensor model having a transfer function operating on flow rate and properties of the sensor and fluid, that is, operating on a Φ operand. For example, the Φ operand may be related to the Reynold&#39;s number associated with the characteristic flow of a particular sensor, for example, by considering the physical dimensions of the sensor conduit and/or the configuration of the sensor. The Reynold&#39;s number is often used to describe momentum, heat, and mass transfer to account for dynamic similarity and/or flow characteristics. The Reynold&#39;s number may take on various forms, for example, the Reynolds number may be expressed as: 
                   Re   =       ρ   ⁢           ⁢   Q   ⁢           ⁢   L       μ   ⁢           ⁢   A               Eq   .           ⁢   1               
where, ρ is the density of the fluid, Q is the volumetric fluid flow rate, μ is the viscosity of the fluid, L is the characteristic length of the sensor, and A is the cross-sectional area of the sensor conduit. The characteristic length L may be related to the distance between sensing elements coupled to the sensor conduit to detect flow, for example, related to the length l illustrated in  FIG. 1 . Accordingly, the Φ includes a flow rate component and a sensor property component.
 
     In some embodiments, Φ may depend on the Prandtl number of a particular fluid flowing through the flow path. The Prandtl number of a fluid may be used to describe heat transfer in general and in particular heat convection. The Prandtl number of a fluid may be expressed as follows: 
                   Pr   =       μ   ·     c   p       κ             Eq   .           ⁢   2               
where, μ is the viscosity of the fluid, c p  is the specific heat of the fluid and κ is the thermal conductivity of the fluid. In one embodiment, the Φ operand includes a flow rate component, a sensor property component and a fluid property component. For example, the Φ operand may be expressed in terms of the Reynolds number associated with a sensor and the Prandtl number associated with the fluid such that the characteristic function of the sensor is related to sensor output curves of various fluids by the thermal conductivity of the fluid.
 
     According to one embodiment, the Φ operand includes both the Reynolds number associated with the flow of fluid through a sensor and the Prandtl number of the fluid flowing through the conduit, for example, by the relationship Φ=RePr. It should be appreciated that such a formulation considers the physical properties of the sensor and thermal and/or physical properties of the fluid. In addition, the above formulation is compelling in that the Φ operand is dimensionless. 
     For example, the expression Φ=RePr has the following units, 
     
       
         
           
             
               
                 
                   
                     Re 
                     · 
                     Pr 
                   
                   = 
                   
                     
                       
                         
                           ρ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             C 
                             p 
                           
                         
                         k 
                       
                       ⁡ 
                       
                         [ 
                         
                           hr 
                           
                             ft 
                             2 
                           
                         
                         ] 
                       
                     
                     · 
                     
                       Q 
                       ⁡ 
                       
                         [ 
                         
                           
                             cm 
                             3 
                           
                           min 
                         
                         ] 
                       
                     
                     · 
                     
                       
                         L 
                         A 
                       
                       ⁡ 
                       
                         [ 
                         
                           ft 
                           
                             ft 
                             2 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   3 
                 
               
             
           
         
       
     
     When the units are made consistent and by using the fact that 
                           1   ⁢           ⁢   hour     =     60   ⁢           ⁢   min   ⁢           ⁢     (       i   .   e   .     ,         60   ⁢           ⁢   min       1   ⁢           ⁢   hr       =   1       )     ⁢                 ⁢           
     ⁢           and   ⁢           ⁢     
     ⁢               1   ⁢           ⁢   ft     =     30.48   ⁢           ⁢   cm   ⁢           ⁢     (       i   .   e   .     ,         1   ⁢           ⁢   ft       30.48   ⁢           ⁢   cm       =   1       )         ,                               
the physical units in the relationship shown in Eq. 3 cancel each other out as shown below.
 
     
       
         
           
             
               
                 
                   
                     Re 
                     · 
                     Pr 
                   
                   = 
                   
                     
                       
                         
                           ρ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             C 
                             p 
                           
                         
                         k 
                       
                       ⁡ 
                       
                         [ 
                         
                           hr 
                           
                             ft 
                             2 
                           
                         
                         ] 
                       
                     
                     · 
                     
                       Q 
                       ⁡ 
                       
                         [ 
                         
                           
                             cm 
                             3 
                           
                           min 
                         
                         ] 
                       
                     
                     · 
                     
                       
                         L 
                         A 
                       
                       ⁡ 
                       
                         [ 
                         
                           ft 
                           
                             ft 
                             2 
                           
                         
                         ] 
                       
                     
                     · 
                     
                       [ 
                       
                         
                           60 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           min 
                         
                         
                           1 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           hr 
                         
                       
                       ] 
                     
                     · 
                     
                       
                         [ 
                         
                           
                             1 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ft 
                           
                           
                             30.48 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cm 
                           
                         
                         ] 
                       
                       3 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   4 
                 
               
             
           
         
       
     
     As a result, the formulation results in the dimensionless relationship, 
     
       
         
           
             
               
                 
                   
                     Re 
                     · 
                     Pr 
                   
                   = 
                   
                     
                       
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           C 
                           p 
                         
                       
                       k 
                     
                     · 
                     Q 
                     · 
                     
                       L 
                       A 
                     
                     · 
                     
                       ( 
                       0.00211888 
                       ) 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   5 
                 
               
             
           
         
       
     
     It should be appreciated that Q may be isolated from Eq. 5, such that flow rates may be computed directly from the Φ operand, that is, the flow rate component may be separated from the Φ operand provided the necessary sensor and fluid properties are available. 
       FIG. 8  illustrates one embodiment of a sensor model according to the present invention. In sensor model  800 , the Φ operand is a scaled product of the Reynolds number of a particular sensor and the Prandtl number for a particular fluid. The scale factor g may be a fluid dependent value established, for example, from empirical data obtained about the sensor operation with various fluids. A fluid dependent scale factor refers generally to a scalar value empirically or analytically derived that increases the alignment between characteristic values of a particular fluid and a characteristic curve of the sensor. 
     It should be appreciated that a scale factor need not be used, that is, scale factor g may be considered to be unity. The transfer function  805  of sensor model  800  is described as characteristic function W operating on values of Φ that produces a characteristic curve related to the sensor output values. In particular, according to sensor model  800 , sensor response curves for different fluids are related to the same characteristic curve of the sensor by their respective thermal conductivity and a fluid dependent scale factor f. As with fluid dependent scale factor g, f may be eliminated by assuming that it takes a value of one. 
       FIG. 9  illustrates sensor response curves of the eight fluids of  FIG. 2  mapped to characteristic values and plotted as a finction of the Φ operand according to the sensor model  800  illustrated in  FIG. 8  without using scale factors f and g (e.g., by assuming unity scale factors). In particular, each sensor response curve for the fluids shown in  FIG. 2  were divided by their respective thermal conductivity and plotted as a function of the product of the Reynold&#39;s number associated with the sensor and the Prandtl number associated with the fluid. It should be appreciated that the exemplary fluids shown cover a wide range of fluids having substantially different thermal physical properties. However, the characteristic values of the various sensor response curves substantially collapse to a single characteristic curve of the sensor as predicted by sensor model  800 . 
       FIG. 10  illustrates the same sensor response curves mapped to characteristic values according to the characteristic function W illustrated in  FIG. 8  using fluid dependent scale factors f and g, the values of which are listed in Table 1 below. The use of fluid dependent scale factors f and g further aligns the mapped sensor response curves with the characteristic curve of the sensor. As such, sensor model  800  provides a robust description of the behavior of the sensor that may be employed to determine flow rates for any fluid for which the appropriate thermal and/or physical properties are available. 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Fluid 
                 f 
                 g 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 N2 
                 1.05 
                 1.04 
               
               
                   
                 He 
                 1.0 
                 1.0 
               
               
                   
                 Ar 
                 1.0 
                 1.0 
               
               
                   
                 CH4 
                 1.0 
                 1.0 
               
               
                   
                 CO2 
                 1.0 
                 1.0 
               
               
                   
                 CF4 
                 1.08 
                 1.08 
               
               
                   
                 CHF3 
                 0.98 
                 0.96 
               
               
                   
                 SF6 
                 1.08 
                 1.08 
               
               
                   
                   
               
            
           
         
       
     
     It should be appreciated that a characteristic curve is a construct of the sensor model. How closely sensor output values map to this curve depends on how well the model describes the behavior of the sensor. As illustrated in  FIG. 10 , sensor output values for the various exemplary fluids align with the underlying characteristic curve even well outside the linear range of the respective fluids. As such, the response of a sensor to a fluid may be accurately determined over an extended range of flow rates, and not just over the linear range of the fluid. 
     Furthermore, since each sensor response curve may be mapped substantially to the same curve, response information from a single test fluid may be sufficient to describe the underlying characteristic curve. Accordingly, characteristic information may be computed from a single test fluid either during calibration or during operation of the sensor which may be used to determine the response of the sensor on arbitrary fluids across an extended range of flow rates. 
       FIG. 11  describes one embodiment according to the present invention of a method of calibrating a flow sensor on a single test fluid to obtain characteristic information about the sensor according to sensor model  800 . The characteristic information facilitates subsequent operation of the sensor on an arbitrary fluid by providing a predictive model of the behavior of the sensor. For example, the method described below may be employed during calibration of a flow sensor for use in an MFC configured to control the flow of a plurality of process fluids, one or more of which may not be available or known at the time of calibration. Moreover, the MFC may be employed in applications requiring flow rates substantially outside the linear range of the process fluids for which it is intended to operate. 
     In step  1100 , a test fluid  1101  having a known thermal conductivity and molar specific heat is introduced to a conduit of a sensor having a known cross-sectional area and characteristic length. As illustrated in  FIG. 1 , the characteristic length of the sensor conduit may be related to the length l formed by the two coils  14  and  16  and the spacing between them. The known properties discussed above may be, for example, stored in a database  1105  of fluid properties and a database  1107  of sensor properties. 
     As discussed in more detail below, sensor and fluid property information as well as information obtained during calibration, may be stored on any computer readable medium such as a memory included in the sensor, a component using the sensor such as an MFC, one or more computers coupled to the sensor and/or MFC, or anywhere else that information may be made available. 
     Table 2 illustrates values for eight exemplary fluids that may be stored, for example, in databases  1105  and  1107 . 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Sensor and Fluid Properties 
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
            
               
                 Symbol 
                 Property 
                 Unit 
                 N2 
                 He 
                 Ar 
                 CF4 
                 SF6 
                 CO2 
                 CH4 
                 CHF3 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
            
               
                 Mw 
                 Fluid Molecular 
                   
                 28.013 
                 4.003 
                 39.950 
                 88.000 
                 146.050 
                 44.100 
                 16.043 
                 70.000 
               
               
                   
                 Weight 
               
               
                 ρ 
                 Fluid density 
                 lbm/ft 3   
                 0.060196 
                 0.0086 
                 0.08585 
                 0.1891 
                 0.31384 
                 0.09477 
                 0.03447 
                 0.15042 
               
               
                 Cp 
                 Fluid Specific Heat 
                 Btu/ 
                 0.249169 
                 1.24176 
                 0.12439 
                 0.18309 
                 0.17749 
                 0.21514 
                 0.57888 
                 0.19398 
               
               
                   
                   
                 lbm-F. 
               
               
                 k 
                 Fluid conductivity 
                 Btu/hr- 
                 0.017157 
                 0.09691 
                 0.01182 
                 0.01251 
                 0.01075 
                 0.01218 
                 0.02497 
                 0.00981 
               
               
                   
                   
                 ft-F. 
               
               
                 μ 
                 Dynamic viscosity 
                 lbm/ft-sec 
                 1.18E−05 
                 1.31E−05 
                 1.50E−05 
                 1.15E−05 
                 9.70E−06 
                 9.92E−06 
                 7.37E−06 
                 9.72E−06 
               
               
                 Pr 
                 Prandtl No. 
                   
                 0.715 
                 0.693 
                 0.666 
                 0.706 
                 0.723 
                 0.765 
                 0.723 
                 0.848 
               
               
                 ρCp/κ 
                   
                 hr/ft 2   
                 0.874 
                 0.110 
                 0.904 
                 2.768 
                 5.184 
                 1.674 
                 0.799 
                 2.975 
               
               
                 L 
                 Sensor Length 
                 Ft 
                 0.067 
                 0.067 
                 0.067 
                 0.067 
                 0.067 
                 0.067 
                 0.067 
                 0.067 
               
               
                 D 
                 Sensor ID 
                 Ft 
                 1.17E−03 
                 1.17E−03 
                 1.17E−03 
                 1.17E−03 
                 1.17E−03 
                 1.17E−03 
                 1.17E−03 
                 1.17E−03 
               
               
                 A 
                 Cross-Sectional 
                 ft 2   
                 1.07E−06 
                 1.07E−06 
                 1.07E−06 
                 1.07E−06 
                 1.07E−06 
                 1.07E−06 
                 1.07E−06 
                 1.07E−06 
               
               
                   
                 Area 
               
               
                 Z 
                 Compressibility 
                   
                 0.9996 
                 1.0004 
                 0.9994 
                 0.9983 
                 0.9886 
                 0.9995 
                 0.9982 
                 0.9921 
               
               
                   
                 Factor 
               
               
                   
               
            
           
         
       
     
     A test fluid may be introduced to the conduit at a number of flow rates such that the sensor responds at a predetermined set S of sensor output values. For example, the sensor may be coupled to calibration equipment adapted to introduce a fluid to the sensor at adjustable flow rates, and that may accurately provide a measurement of the flow rates corresponding to each of the sensor output values in set S. 
     Initially, the flow rate of test fluid  1101  may be adjusted until the sensor responds with a value S i , for example, S i  may take on a value equal to 0.1 after any of various normalization calculations to indicate 10% of nominal full scale flow. The actual flow rate F i  of test fluid  1101  through the conduit is then measured by the calibration equipment and recorded. 
     Table 3 below illustrates data obtained during an exemplary calibration of a sensor using nitrogen (N2) as a test fluid. The first column lists a set S of desired sensor output values. The second column lists the corresponding flow rate values measured by calibration equipment resulting from introducing nitrogen to the sensor such that each of the desired sensor output values shown in the first column were produced. 
     
       
         
           
               
             
               
                 TABLE 3 
               
             
            
               
                   
               
               
                 (N2) 
               
            
           
           
               
               
               
            
               
                   
                   
                 Flow Rate 
               
               
                   
                 Sensor Output 
                 (sccm) 
               
               
                   
                   
               
            
           
           
               
               
               
            
               
                   
                 0.000 
                 0.000 
               
               
                   
                 0.100 
                 1.842 
               
               
                   
                 0.200 
                 3.709 
               
               
                   
                 0.300 
                 5.571 
               
               
                   
                 0.400 
                 7.443 
               
               
                   
                 0.500 
                 9.339 
               
               
                   
                 0.600 
                 11.281 
               
               
                   
                 0.700 
                 13.297 
               
               
                   
                 0.800 
                 15.412 
               
               
                   
                 0.900 
                 17.653 
               
               
                   
                 1.000 
                 20.007 
               
               
                   
                 1.100 
                 22.677 
               
               
                   
                 1.200 
                 25.647 
               
               
                   
                 1.300 
                 29.211 
               
               
                   
                 1.400 
                 34.011 
               
               
                   
                   
               
            
           
         
       
     
     In step  1110 , sensor output value S i  is mapped to a characteristic value substantially along the characteristic curve of the sensor according to sensor model  800 . In particular, the sensor output value is divided by the thermal conductivity of the test fluid κ t  and scaled by a scale factor of the test fluid f t  to produce characteristic value c i . 
     In step  1120 , the flow rate F i  measured in step  1100  is used in combination with sensor and fluid properties to compute the value of Φ associated with the sensor output value S i . In particular, flow rate F i  is multiplied by the product of the fluid density ρ and the sensor characteristic length L, and divided by the product of the fluid viscosity μ of the test fluid and the cross-sectional area of the sensor conduit A to form a Reynolds number associated with flow through the sensor as expressed in equation 1. The computed Reynolds number may then be multiplied by the Prandtl number of the test fluid as expressed in equation 2 and scaled by a scale factor g t  of the test fluid to produce Φ i . 
     In step  1130 , the computed values of c i  and Φ i  are stored in association, for example, as an entry in look-up table  1135 . The look-up table may be stored in a memory, for example, in the storage medium encoded with sensor and fluid properties and/or sensor response information of the test fluid. The above described steps may be repeated for each desired sensor output value in the set S such that the plurality of associated values Φ i  and c i  tabulated in look-up table  1135  form a representation of the characteristic function of the sensor. It should be appreciated that the characteristic information obtained from the single test fluid may suffice to describe the behavior of the sensor according to the model such that calibration on additional test fluids may be unnecessary. 
     Table 4 illustrates an exemplary look-up table computed during calibration of a sensor using N2 as a test fluid. The first two columns list the same values shown in Table 3. The third column lists values of each of the desired sensor output values in column one mapped to characteristic values according to the characteristic function of the sensor model demonstrated in  FIG. 8 . The fourth column lists values of each of the corresponding flow rates shown in column two mapped to an associated value of Φ, based on the Reynolds number associated with sensor flow and the Prandtl number of nitrogen. It may not be necessary to store the information in the first and second column, as the final two columns provide a description of the characteristic function of the sensor that may be used to facilitate determining flow rates through the sensor operating on an arbitrary fluid as described in more detail below. 
     Alternately, the final two columns may not need to be stored since the response information in the first two columns can be converted to characteristic information according to a sensor model at any time. Accordingly, a LUT may include response information and instructions that perform a mapping from flow rates to Φ operands and sensor output values to characteristic values. The response information may thus be mapped to characteristic information during operation of the sensor in the field. 
     
       
         
           
               
             
               
                 TABLE 4 
               
             
            
               
                   
               
               
                 (N2) 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Sensor 
                 Flow Rate 
                   
                   
               
               
                   
                 Output (S) 
                 (sccm) 
                 f * (S/k) 
                 g * (RePr) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 0.000 
                 0.000 
                 0.000 
                 0 
               
               
                   
                 0.100 
                 1.842 
                 5.829 
                 212.8 
               
               
                   
                 0.200 
                 3.709 
                 11.657 
                 428.5 
               
               
                   
                 0.300 
                 5.571 
                 17.486 
                 643.6 
               
               
                   
                 0.400 
                 7.443 
                 23.314 
                 859.9 
               
               
                   
                 0.500 
                 9.339 
                 29.143 
                 1078.8 
               
               
                   
                 0.600 
                 11.281 
                 34.971 
                 1303.2 
               
               
                   
                 0.700 
                 13.297 
                 40.800 
                 1536.1 
               
               
                   
                 0.800 
                 15.412 
                 46.628 
                 1780.4 
               
               
                   
                 0.900 
                 17.653 
                 52.457 
                 2039.4 
               
               
                   
                 1.000 
                 20.007 
                 58.285 
                 2311.3 
               
               
                   
                 1.100 
                 22.677 
                 64.114 
                 2619.7 
               
               
                   
                 1.200 
                 25.647 
                 69.943 
                 2962.8 
               
               
                   
                 1.300 
                 29.211 
                 75.771 
                 3374.6 
               
               
                   
                 1.400 
                 34.011 
                 81.600 
                 3929.1 
               
               
                   
                   
               
            
           
         
       
     
       FIG. 12  illustrates one method according to the present invention of determining flow rates of an arbitrary fluid by employing a representation of the characteristic function of the sensor obtained during sensor calibration. For example, the look-up table obtained during the calibration on a single test fluid as described in connection with  FIG. 11  may be used to determine flow rates of an arbitrary process fluid  1201 . 
     Process fluid  1201  may be introduced to a sensor  1205  which responds with a sensor output signal, for example, voltage signal V i  indicative of the flow of process fluid  1201  through the sensor. Sensor  1205  may be, for example, coupled to an MFC controlling the introduction of a process fluid into a process chamber during fabrication of a semiconductor device. 
     In step  1200 , the sensor output signal V i  may be preprocessed to place it in a desired format. For example, sensor output signal V i  may be converted into a digital signal and/or normalized, such that the signal represents a fraction of full scale flow through the sensor to provide preprocessed signal S i . 
     In step  1210 , the preprocessed sensor output signal S i  is converted to a characteristic value C i  according to a sensor model being employed. It should be appreciated that the sensor model employed to map sensor output values of the test fluid during calibration to characteristic values should also be employed to map sensor output values of the process fluid to characteristic values. 
     Alternately, as described above, calibration may include recording only response information from the test fluid. Under such circumstances, a same mapping from flow rates to Φ values and from sensor output values to characteristic values according to a desired sensor model may be employed to both map the recorded test response information to characteristic information, and to map sensor output values S i  to characteristic values during operation of the sensor. 
     For example, S i  may be multiplied by a fluid dependent scale factor f p  and divided by the thermal conductivity κ p  of the process fluid to provide characteristic value C i  similar to the step  1110  of the calibration described in connection with  FIG. 11 . Sensor output values resulting from flow of the process fluid are thus mapped to values substantially along the characteristic curve of the sensor model. 
     Since the sensor model establishes that sensor output values map to the same characteristic curve, the value C i  is expected to fall somewhere along the curve described by the representation of the characteristic function obtained during calibration, for example, look-up table  1235  (e.g., a look-up table obtained as described in connection with  FIG. 11 ). Accordingly, it may then be determined where on the curve C i  falls, by indexing look-up table  1235  to obtain the associated operand Φ i . 
     Due to the discrete nature of a look-up table, a computed characteristic value C i  is likely to fall between values stored in the LUT. Accordingly, a given characteristic value may not have a directly associated operand Φ i . However, the operand Φ i  may be approximated by interpolating between the two nearest values stored in the LUT, for example, by a piecewise linear approximation, such that a corresponding value Φ i  may be obtained. 
     In step  1230 , a flow rate predicting the actual flow of the process fluid through the sensor is computed from the obtained value Φ i . As discussed in the foregoing, the value Φ i  is a function of flow rate and known sensor and fluid properties such that the flow rate component of a particular value of Φ may be isolated from Φ i . For example, the value Φ i  may be multiplied by a ratio between the thermal conductivity and specific heat of the process fluid, and a scaled ratio of the cross sectional area of the flow path A and the characteristic length L of the sensor conduit. This operation isolates the mass flow rate component from the obtained value of the Φ operand providing flow rate F i  as an accurate determination of the actual flow rate through the sensor. 
     It should be appreciated that the same look-up table may be used regardless of the process fluid with which the sensor is operating. That is, a single representation of the characteristic function of a sensor obtained, for example, from calibration or during operation from response information of any one test fluid may be sufficient to describe the behavior of the sensor, such that accurate flow rates may be determined for any arbitrary process fluid. In addition, sensor models according to the present invention also describe the behavior of the sensor across an extended range of flow rates. As such, flow rates of an arbitrary fluid may be accurately predicted well outside the linear range of the fluid. Accordingly, the sensor may be employed effectively in a broad range of applications. 
     Tables 5-10 demonstrate actual experimental data using methods according to the present invention in comparison with conventional techniques. In particular, Tables 5, 7 and 9 illustrate determination of flow rates resulting from operating with argon (Ar), CF 4  and SF 6  using the look-up table tabulated during a calibration of the sensor with nitrogen as shown in Tables 3 and 4. 
     In Table 5, the sensor calibrated on N2 is operated with the process fluid argon. The first column lists a number of sensor output values provided by the sensor in response to flow of argon through the sensor at unknown flow rates. The second column illustrates the sensor output values mapped to characteristic values based on the thermal conductivity of argon and a scale factor f determined empirically for argon at 0.98. The third column illustrates the corresponding value of Φ obtained from the LUT table as shown in Table 4. The fourth column illustrates the determined flow rate computed from the obtained value of Φ. The fifth column lists actual flow rates as measured by, for example, a volumetric displacement flow meter. The sixth and final column illustrates the percent error obtained by using the described method. Tables 7 and 9 are arranged in the same way for process fluids CF 4  and SF 6 , respectively. 
     For contrast, Table 6 illustrates flow rates determined using conventional conversion factors. The first column lists the sensor output values as shown in column one of Table 5. The second column lists the flow rates measured during calibration with nitrogen (i.e., column two of Tables 3 and 4). The third column lists the predicted flow rates computed using a conventional conversion factor. The conversion factor used for argon is 1.428. The fourth column lists the actual measured flow rates, and the fifth column shows the percentage error obtained using conventional conversion techniques. Tables 8 and 10 are arranged in the same way listing results from conventional operation with process fluids CF 4  and SF 6 , respectively. 
     
       
         
           
               
             
               
                 TABLE 5 
               
             
            
               
                   
               
               
                 Test Fluid Nitrogen/Process Fluid Argon 
               
               
                 f = .98, g = .985 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                   
                   
                 Computed 
                 Actual 
                   
               
               
                   
                   
                   
                 Flow 
                 Flow 
                   
               
               
                   
                   
                   
                 Rate 
                 Rate 
                 Percent 
               
               
                 Sensor Output 
                 f * S/k 
                 Φ = RePr 
                 (sccm) 
                 (sccm) 
                 Error 
               
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 0.0 
                 0.000 
                 0.000 
                 0.000 
                 0.000 
                  0.0% 
               
               
                 0.1 
                 8.294 
                 308.629 
                 2.584 
                 2.603 
                 0.71% 
               
               
                 0.2 
                 16.588 
                 619.770 
                 5.190 
                 5.234 
                 0.84% 
               
               
                 0.3 
                 24.882 
                 932.778 
                 7.811 
                 7.894 
                 1.05% 
               
               
                 0.4 
                 33.176 
                 1252.886 
                 10.491 
                 10.609 
                 1.11% 
               
               
                 0.5 
                 41.471 
                 1588.059 
                 13.298 
                 13.416 
                 0.88% 
               
               
                 0.6 
                 49.765 
                 1949.001 
                 16.320 
                 16.391 
                 0.43% 
               
               
                 0.7 
                 58.059 
                 2335.807 
                 19.559 
                 19.623 
                 0.33% 
               
               
                 0.8 
                 66.353 
                 2793.436 
                 23.391 
                 23.343 
                 −0.20%   
               
               
                 0.9 
                 74.647 
                 3345.346 
                 28.012 
                 27.600 
                 −1.49%   
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 6 
               
             
            
               
                   
               
               
                 Surrogate Fluid Nitrogen/Process Fluid Argon 
               
               
                 Conversion Factor: 1.428 
               
            
           
           
               
               
               
               
               
            
               
                 Sensor 
                 Calibrated N2 
                 Computed Flow Rate 
                 Actual 
                 Percent 
               
               
                 Output 
                 Flow Rate (sccm) 
                 Conventional Method 
                 Flow 
                 Error 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 0.000 
                 0.000 
                 0.000 
                 0.000 
                   
               
               
                 0.100 
                 1.842 
                 2.630 
                 2.603 
               
               
                 0.200 
                 3.709 
                 5.296 
                 5.234 
                 −1.20% 
               
               
                 0.300 
                 5.571 
                 7.956 
                 7.894 
                 −0.78% 
               
               
                 0.400 
                 7.443 
                 10.629 
                 10.609 
                 −0.19% 
               
               
                 0.500 
                 9.339 
                 13.335 
                 13.416 
                 0.60% 
               
               
                 0.600 
                 11.281 
                 16.109 
                 16.391 
                 1.72% 
               
               
                 0.700 
                 13.297 
                 18.988 
                 19.623 
                 3.24% 
               
               
                 0.800 
                 15.412 
                 22.008 
                 23.343 
                 5.72% 
               
               
                 0.900 
                 17.653 
                 25.209 
                 27.600 
                 8.66% 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 7 
               
             
            
               
                   
               
               
                 Test Fluid Nitrogen/Process Fluid CF4 
               
               
                 f = 1.012, g = 1.013 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                   
                   
                 Computed Flow 
                   
                   
               
               
                 Sensor 
                   
                   
                 Rate 
                 Actual Flow 
                 Percent 
               
               
                 Output 
                 f * S/k 
                 Φ = RePr 
                 (sccm) 
                 Rate (sccm) 
                 Error 
               
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 0.0 
                 0.000 
                 0.000 
                 0.000 
                 0.000 
                   
               
               
                 0.1 
                 8.092 
                 292.723 
                 0.800 
                 0.771 
               
               
                 0.2 
                 16.184 
                 587.925 
                 1.607 
                 1.578 
               
               
                 0.3 
                 24.277 
                 884.535 
                 2.418 
                 2.394 
                 −0.98% 
               
               
                 0.4 
                 32.369 
                 1187.568 
                 3.246 
                 3.225 
                 −0.67% 
               
               
                 0.5 
                 40.461 
                 1503.049 
                 4.109 
                 4.089 
                 −0.47% 
               
               
                 0.6 
                 48.553 
                 1841.992 
                 5.035 
                 5.010 
                 −0.49% 
               
               
                 0.7 
                 56.645 
                 2206.149 
                 6.030 
                 6.025 
                 −0.09% 
               
               
                 0.8 
                 64.738 
                 2622.363 
                 7.168 
                 7.217 
                 0.68% 
               
               
                 0.9 
                 72.830 
                 3126.168 
                 8.545 
                 8.624 
                 0.91% 
               
               
                 1.0 
                 80.922 
                 3815.034 
                 10.428 
                 10.497 
                 0.65% 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 8 
               
             
            
               
                   
               
               
                 Surrogate Fluid Nitrogen/Process Fluid CF4 
               
               
                 Conversion Factor: .437 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Calibrated N2 
                   
                   
                   
               
               
                 Sensor 
                 Flow Rate 
                 Computed Flow Rate 
                   
                 Percent 
               
               
                 Output 
                 (sccm) 
                 Conventional Method 
                 Actual Flow 
                 Error 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 0.000 
                 0.000 
                 0.000 
                 0.000 
                   
               
               
                 0.100 
                 1.842 
                 0.805 
                 0.771 
               
               
                 0.200 
                 3.709 
                 1.621 
                 1.578 
               
               
                 0.300 
                 5.571 
                 2.435 
                 2.394 
                 −1.67% 
               
               
                 0.400 
                 7.443 
                 3.253 
                 3.225 
                 −0.87% 
               
               
                 0.500 
                 9.339 
                 4.081 
                 4.089 
                 0.21% 
               
               
                 0.600 
                 11.281 
                 4.930 
                 5.010 
                 1.61% 
               
               
                 0.700 
                 13.297 
                 5.811 
                 6.025 
                 3.56% 
               
               
                 0.800 
                 15.412 
                 6.735 
                 7.217 
                 6.69% 
               
               
                 0.900 
                 17.653 
                 7.715 
                 8.624 
                 10.55% 
               
               
                 1.000 
                 20.007 
                 8.743 
                 10.497 
                 16.71% 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 9 
               
             
            
               
                   
               
               
                 Test Fluid Nitrogen/Process Fluid SF6 
               
               
                 f = 1.0, g = 1.0 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                   
                   
                 Computed Flow 
                   
                   
               
               
                 Sensor 
                   
                   
                 Rate 
                 Actual Flow 
                 Percent 
               
               
                 Output 
                 f * S/k 
                 Φ = RePr 
                 (sccm) 
                 Rate (sccm) 
                 Error 
               
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 0.0 
                 0.000 
                 0.000 
                 0.000 
                   
                   
               
               
                 0.1 
                 9.306 
                 341.462 
                 0.498 
               
               
                 0.2 
                 18.613 
                 685.428 
                 1.001 
               
               
                 0.3 
                 27.919 
                 1032.870 
                 1.508 
               
               
                 0.4 
                 37.226 
                 1393.283 
                 2.034 
                 2.018 
                 −0.78% 
               
               
                 0.5 
                 46.532 
                 1776.372 
                 2.593 
                 2.586 
                 −0.28% 
               
               
                 0.6 
                 55.839 
                 2197.182 
                 3.207 
                 3.210 
                 0.09% 
               
               
                 0.7 
                 65.145 
                 2680.433 
                 3.913 
                 3.936 
                 0.60% 
               
               
                 0.8 
                 74.451 
                 3281.356 
                 4.790 
                 4.833 
                 0.89% 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 10 
               
             
            
               
                   
               
               
                 Surrogate Fluid Nitrogen/Process Fluid SF6 
               
               
                 Conversion Factor: .437 
               
            
           
           
               
               
               
               
               
            
               
                   
                   
                 Computed Flow 
                   
                   
               
               
                 Sensor 
                 Calibrated N2 
                 Rate Conventional 
                   
                 Percent 
               
               
                 Output 
                 Flow Rate (sccm) 
                 Method 
                 Actual Flow 
                 Error 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 0.000 
                 0.000 
                 0.000 
                   
                   
               
               
                 0.100 
                 1.842 
                 0.510 
               
               
                 0.200 
                 3.709 
                 1.027 
               
               
                 0.300 
                 5.571 
                 1.543 
               
               
                 0.400 
                 7.443 
                 2.062 
                 2.018 
                 −2.17% 
               
               
                 0.500 
                 9.339 
                 2.587 
                 2.586 
                 −0.03% 
               
               
                 0.600 
                 11.281 
                 3.125 
                 3.210 
                 2.66% 
               
               
                 0.700 
                 13.297 
                 3.683 
                 3.936 
                 6.43% 
               
               
                 0.800 
                 15.412 
                 4.269 
                 4.833 
                 11.67% 
               
               
                   
               
            
           
         
       
     
     As shown, methods according to the present invention provide accurate flow rate predictions across a wide range of flow rates. By contrast, conventional conversion techniques provide accurate results only within a limited range of flow values. In particular, as shown in Tables 6, 8 and 10, flow rates errors become unsatisfactorily high outside the linear range of the sensor response curve for the particular fluid. 
     Applicant has appreciated that various aspects of the present invention may be used to provide a general purpose sensor capable of operating accurately on arbitrary process fluids across a wide range of flow rates.  FIG. 13  illustrates one embodiment of a sensor according to the present invention. Sensor  1310  includes a conduit  1320 , sensor electronics  1330 , analog-to-digital (A/D) converter  1335  and computer  1340  having a processor  1345  and a storage medium  1347 . Sensor electronics are coupled to the conduit  1320  and respond to flow in the conduit by providing a sensor output signal  1333  indicative of the fluid flow through the conduit. Sensor electronics  1330  may be any circuit or combination of circuits responsive to fluid flow, for example, the thermal mass flow sensor described in connection with  FIG. 1 . It should be appreciated that various aspects of the invention may be employed in a variety of sensors such as those described in previously incorporated applications, and is not limited to any particular sensor configuration. 
     Sensor output signal  1333  may be, for instance, an electronic analog signal such as a voltage or current level. A/D converter  1335  receives sensor output signal  1333  and converts it to a digital signal  1336  that is provided to processor  1345  of computer  1340 . Processor  1345  may be any device or combination of devices capable of arithmetic and/or logic computation. Processor  1345  may preprocess the digital signal  1336 , for instance, to normalize the signal, such that it is expressed as a fraction of the full scale flow through conduit  1320 . 
     Processor  1345  may be implemented in a number of ways including using dedicated hardware (e.g., various circuitry, programmable logic arrays, etc.) designed and/or configured to perform any one of a variety of functions described herein, and/or using one or more computers or processors (e.g., microprocessors) that are programmed using microcode (i.e., software) to perform any one or more of the variety of functions described herein. 
     Processor  1345  may be coupled to storage medium  1347 . Storage medium  1347  may store various sensor and fluid information that facilitates operating the sensor with any number of process fluids. For example, storage medium  1347  may be encoded with physical and/or thermal properties of various fluids for which the sensor is intended to accurately measure flow, fluid dependent properties of the fluid, physical properties and/or design parameters of the sensor, etc. Storage medium  1347  may be any of various storage devices and computer readable media, and can be implemented in numerous ways, such as, but not limited to, RAM, ROM, PROM, EPROM, EEPROM, CD, DVD, optical disks, floppy disks, magnetic tape, and the like. 
     In the embodiment illustrated in  FIG. 13 , storage medium  1347  stores a representation  1360  of the characteristic function W of sensor  1310 . In addition, storage medium  1347  stores properties of various fluids  1362 , including thermal conductivity, viscosity, and specific heat, and fluid dependent properties  1364  for each of a set of process fluids. Storage medium  1347  may also store sensor properties  1366 , including the cross-sectional area and the characteristic length of conduit  1320 . 
     Processor  1345  may convert digital signal  1336  into flow signal  1355  based on data stored on storage medium  1347 . In particular, processor  1345  may convert digital signal  1336  to flow signal  1355  according to the representation of characteristic finction  1360  and properties of the sensor and process fluid fΦ or which operation of the sensor is desired. For example, storage medium  1347  may be encoded with instructions in the form of one or more programs (e.g., program  1370 ) that, when executed by processor  1345  performs one more conversion methods according to the present invention, for instance, the methods described in connection with  FIGS. 7 and 11 . 
     In particular, program  1370  may include instructions that when executed by processor  1345  multiply digital signal  1336  by a ratio of the thermal conductivity of the process fluid flowing through the conduit and a scale factor for the process fluid to provide signal  1346 . Program  1370  may further include instruction programmed to use signal  1346  to index into the representation  1360  of the characteristic function W stored in storage medium  1347  to obtain an associated value of Φ according to instructions in program  1370 . Further instructions included in program  1370  when executed by processor  1345  may multiply the obtained value of Φ by properties of the sensor and the process fluid to provide flow signal  1355 . Flow signal  1355  may then be provided to other devices, components and/or control circuits as an accurate indication of the actual fluid flowing through the sensor conduit. 
     It should be appreciated that computer  1340  may be resident on sensor  1310 , for example, as an embedded microcontroller. Alternatively, computer  1340  may be part of a device operating in conjunction with sensor  1310 . For example, computer  1340  may be part of a mass flow controller or mass flow meter that includes sensor  1310  as a component part or may be a part of a separate controller located proximate to or remote from the sensor  1320 . As discussed briefly in the foregoing, mass flow controllers often include a processor and a memory or may be coupled to, for example, a general purpose computer having a processor and memory which may be used to implement the present invention and which may be employed to control the flow of fluids. 
     Conventional mass flow controllers generally include four main portions: a flow meter, a control valve, a valve actuator, and a controller. The flow meter measures the mass flow rate of a fluid in a flow path and provides a signal indicative of that flow rate. The flow meter may include a mass flow sensor and a bypass. The mass flow sensor measures the mass flow rate of fluid in a sensor conduit that is fluidly coupled to the bypass. The mass flow rate of fluid in the sensor conduit is approximately proportional to the mass flow rate of fluid flowing in the bypass, with the sum of the two being the total flow rate through the flow path controlled by the mass flow controller. However, it should be appreciated that some mass flow controllers may not employ a bypass, as such, all of the fluid may flow through the sensor conduit. 
     In many mass flow controllers, a thermal mass flow sensor is used that includes a pair of resistors that are wound about the sensor conduit at spaced apart positions, each having a resistance that varies with temperature similar to that described in connection with  FIG. 1 . A control valve may be positioned in the main fluid flow path (typically downstream of the bypass and mass flow sensor) and can be controlled (e.g., opened or closed) to vary the mass flow rate of fluid flowing through the main fluid flow path and provided by the mass flow controller. The valve is typically controlled by a valve actuator, examples of which include solenoid actuators, piezoelectric actuators, stepper actuators, etc. Some mass flow controllers may not include a valve or valve actuator and function, rather, as general purpose mass flow meters. 
     Control electronics control the position of the control valve based upon a set point indicative of the mass flow rate of fluid that is desired to be provided by the mass flow controller, and a flow signal from the mass flow sensor indicative of the actual mass flow rate of the fluid flowing in the sensor conduit. Traditional feedback control methods such as proportional control, integral control, proportional-integral (PI) control, derivative control, proportional-derivative (PD) control, integral-derivative (ID) control, and proportional-integral-derivative (PID) control are then used to control the flow of fluid in the mass flow controller. In each of the aforementioned feedback control methods, a control signal (e.g., a control valve drive signal) is generated based upon an error signal based on the difference between a set point signal indicative of the desired mass flow rate of the fluid and a feedback signal that is related to the actual mass flow rate sensed by the mass flow sensor. One example of a mass flow controller is described in U.S. patent application Ser. No. 10/131,603, which is hereby incorporated by reference in its entirety. 
       FIG. 14  illustrates a schematic block diagram of a mass flow controller according to one embodiment of the present invention. The mass flow controller illustrated in  FIG. 14  includes a flow meter  1410 , a controller  1450 , a valve actuator  1460 , and a valve  1470 . The flow meter  1410  is coupled to a flow path  1403 . The flow meter  1410  senses the flow rate of a fluid in the flow path, or portion of the flow path, (e.g., sensor conduit  1420 ) and provides a flow signal FS 2  indicative of the sensed flow rate. The flow signal FS 2  is provided to a first input of a Gain Lead-Lag (GLL) controller  1450 . 
     Controller  1450  also includes a second input to receive a set point signal SI 2 . A set point refers to an indication of the desired fluid flow to be provided by the mass flow controller  1400 . Any of a variety of signals capable of providing indication of the desired fluid flow is considered a suitable set point signal. The term set point, without reference to a particular signal, describes a value that represents a desired fluid flow. 
     Based in part on the flow signal FS 2  and the set point signal SI 2 , the GLL controller  1450  provides a drive signal DS to the valve actuator  1460  that controls the valve  1470 . The valve  1470  may be positioned downstream from the flow meter  1410 , although the present invention is not limited to such an implementation, as the valve may be alternatively disposed upstream of the flow meter. The valve permits a certain mass flow rate depending, at least in part, upon the displacement of a controlled portion of the valve. The controlled portion of the valve may be, for example, a moveable plunger placed across a cross-section of the flow path. The valve controls the flow rate in the fluid path by increasing or decreasing the area of an opening in the cross section where fluid is permitted to flow. Typically, mass flow rate is controlled by mechanically displacing the controlled portion of the valve by a desired amount. The term displacement is used generally to describe the variable of a valve on which mass flow rate is, at least in part, dependent. 
     The displacement of the valve is often controlled by a valve actuator, such as a solenoid actuator, a piezoelectric actuator, a stepper actuator, etc. In  FIG. 14 , valve actuator  1460  may be a solenoid type actuator, however, the present invention is not so limited, as other alternative types of valve actuators may be used. The valve actuator  1460  receives drive signal DS from the controller and converts the signal DS into a mechanical displacement of the controlled portion of the valve. 
     Controller  1450  displaces valve  1470  in such a way as to reduce the error between the actual fluid flow as indicated by flow signal FS 2  and the desired flow as indicated by SI 2 . Stated differently, controller  1450  attempts to bring the actual flow and the desired flow into agreement based upon measurements made by the flow meter  1410 . Accordingly, the precision with which the mass flow controller operates depends on how accurately flow signal FS 2  approximates the actual flow through flow path  1403 . 
     As discussed in the foregoing, the accuracy of conventional flow sensors may depend on the type of fluid flowing through flow path, the desired flow rate, and the similarity between a process fluid introduced to the sensor and the available surrogate fluids. Furthermore, when flow through the flow path is outside the linear range of a sensor response curve for a particular process fluid, the accuracy of the flow signal may suffer to the extent that it cannot be used for certain applications and/or adversely effect the process for which it is employed. For example, when the accuracy of the flow signal is outside the tolerances of a semiconductor fabrication process, the mass flow controller may be unsatisfactory to control the process or may have adverse effects on the fabrication yield of, for instance, integrated circuits, computer chips, etc. 
     As such, in one embodiment according to the present invention, MFC  1400  includes a computer  1440  having a processor  1445  and a memory  1447 . Computer  1440  may be similar to that described in connection with  FIG. 13 . In particular, memory  1447  may be encoded with a representation of the characteristic function of the sensor and, for example, a database of sensor and fluid properties. Accordingly, computer  1440  may be adapted to convert sensor output signal SO to flow signal FS 2  based on information included in the representation of the characteristic function. 
     For example, computer  1440  may include a program stored in memory  1447  that when executed by processor  1445  performs conversion methods according to the present invention, for instance, the methods described in connection with  FIGS. 7 and 11 . Accordingly, accurate flow indications may be obtained for arbitrary process fluids across an extended range of flow rates substantially beyond the linear range of the particular process fluid directly from sensor output values resulting from flow of an arbitrary fluid. 
     It should be appreciated that the computer  1440  need not be a separate processor distinct from the controller  1450 , but may correspond to a program that is implemented by the controller  1450  or vice versa. Moreover, computer  1440  and controller  1450  need not be integral with or proximate to other components of the MFC but may be located remotely. 
       FIG. 15  illustrates a schematic block diagram of a mass flow controller  1500  having a remote computer  1580 . Portion  1505  of MFC  1500  generally includes the components that are coupled to the flow path  1503  of the MFC, for example, a flow meter  1510 , a valve actuator  1560 , and a valve  1570  coupled to flow path  1503 . The flow meter  1510  may be any suitable device for providing a sensor output signal (e.g. sensor output SO) indicative of the flow rate of fluid through the flow path  1503 . The sensor output SO may be provided to a first input of a remote computer  1580 . 
     Remote computer  1580  may be any processor based device or combination of devices, for example, any of various general-purpose computers such as those based on a Intel PENTIUM-type processor, a Motorola PowerPC processor, Sun UltraSPARC processor, Hewlett-Packard PA-RISC processors, or any other type of processor. 
     For example, remote computer  1580  may include a processor  1545  and a memory  1547 . Memory  1547  may be encoded with similar information as the memory described in connection with  FIG. 13 . In particular, memory  1547  may include a representation  1549  of the characteristic finction of the sensor and, for example, a database of sensor and fluid properties. Memory  1547  may also include one or more programs  1548  including instructions that, when executed on processor  1545 , may determine flow rates of an arbitrary process fluid from sensor output signal SO based on information included in the representation of the characteristic function. 
     In addition, memory  1547  may include one or more programs depicted schematically in memory as controller  1550  that, when executed on processor  1545 , determines a drive signal based on flow rates determined by program  1548  and a set point signal S 12  provided as a second input to remote computer  1580 . In particular, controller  1550  may include instructions that apply various feedback control methods such as proportional control, integral control, PI control, derivative control, PD control, ID control, PID control, GLL control, etc., that attempt to reduce the difference between the flow rate provided by program  1548  and the set point signal S 12  by providing a drive signal DS to the valve actuator  1560  that controls valve  1570 . 
     In many devices that employ flow sensors, a flow path adapted to convey fluids is comprised of a sensor conduit and a bypass.  FIG. 16  illustrates a schematic block diagram of an MFC  1600  including a flow sensor  1610  having sensor electronics  1630  (i.e., circuitry responsive to fluid flow) coupled to conduit  1620  which diverts some portion of the fluid flowing through the flow path  1603 . The remainder and often majority of the fluid flowing through flow path  1603  flows through bypass  1640 . The bypass  1640  may be formed in any manner, for example, from a plurality of tubes or conduits similar in nature to the sensor conduit packaged together to form a path by which fluid may bypass the sensor conduit. The amount of fluid flowing through the conduit  1620  is generally proportional to the amount of fluid flowing in the bypass  1640 . 
     However, across the range of flow rates that may be introduced to the flow path  1603 , the relationship between the flow rate in the conduit  1620  and the flow rate in the bypass  1640  may not be linear. That is, the ratio between fluid flowing through the sensor conduit, referred to as sensor flow, and fluid flowing through the bypass, referred to as bypass flow, is not constant for all flow rates. 
     Flow rates in flow path  1603  may vary between zero flow and some full scale flow value. Often one or more flow signals provided by the sensor (e.g., sensor output SO) will be normalized such that they indicate a fraction of the full scale flow value. For example, SO may be normalized by full scale flow such that, ideally, a value of 0.0 indicates that there is no fluid flowing through the fluid path, a value of 0.3 indicates a flow of 30% of full scale flow, a value of 0.5 indicates a flow of 50% of full scale flow, etc. However, in practice, this linear relationship may not result for the following two different reasons: 1) SO may not be linear with respect to sensor flow; and 2) the ratio of sensor flow to bypass flow may not be constant across a desired range of flow rates. 
     In conventional flow sensors, a linearization curve is often generated that maps sensor output values (e.g., sensor output signal SO) to indicated flow values having a substantially more linear relationship with respect to the total flow through the flow path than the sensor output values. The term “indicated flow” refers to any signal or indication of flow based on a sensor output signal that has been corrected and/or adjusted to compensate for error and/or nonlinearity in the sensor (i.e., the sensor conduit and sensor electronics, the bypass or both). 
     The term “total flow” refers generally to the complete flow rate of fluid through the flow path. As such, when the flow path consists of a sensor conduit the total flow is equal to the sensor flow. When the flow path comprises a sensor conduit and a bypass, the total flow is equal to the sum of the sensor flow and the bypass flow. 
     During a conventional calibration procedure, one or more surrogate fluids may be introduced to the flow path at a variety of flow rates. The sensor may respond with a sensor output signal, values of which are recorded along with the associated total flow rate through the sensor. The recorded values may indicate the relationship between total flow and sensor output for a particular surrogate fluid. Since the desired relationship between total flow and sensor output may be linear, a linearization curve may be generated that maps each sensor output value to an indicated flow value that has a substantially more linear relationship with total flow. 
     However, the relationship between total flow and sensor output will change from fluid to fluid such that a linearization curve for a particular surrogate fluid may not be valid for an arbitrary process fluid. To compensate, conventional methods have generated multiple linearization curves for a number of surrogate fluids. During operation on a process fluid, the linearization curve generated for the most similar surrogate fluid is then used and a constant conversion factor is applied. However, such constant conversion factors are largely empirical and highly inaccurate outside the linear range of the fluid. In addition to inaccuracies, obtaining calibration measurements for use in generating linearization curves for numerous surrogate fluids is a time consuming process. 
     Applicant has appreciated that conventional conversion factors may be deficient in part because the relationship between total flow and sensor output varies from fluid to fluid according to multiple influences. As noted by Applicant, this relationship may have components including the nonlinearity between sensor flow and sensor output and between sensor flow and bypass flow. These two components may vary differently from fluid to fluid and are often insufficiently compensated or corrected for by conventional conversion factors. 
     In particular, a change from a first fluid to a second fluid may cause the relationship between sensor flow and sensor output to change in a different manner than the relationship between total flow and sensor flow. For example, respective components may be effected by different properties of the fluid or operating conditions of the sensor. As such, conventional conversion factors which are largely empirical do not address the independent variation of the different component influences on the sensor and are therefore often inadequate. 
     Applicant has identified and appreciated that by employing a sensor model according to the present invention, the relationship between sensor output and sensor flow can be characterized and accurately described from fluid to fluid. An accurate description of this relationship facilitates decoupling the effects due to the non-linearity between sensor flow and sensor output and sensor flow and bypass flow such that each can be independently corrected for when operating on an arbitrary process fluid. 
     Accordingly, one aspect of the invention includes obtaining calibration measurements with a test fluid that facilitate separating flow meter behavior into component parts such that the behavior of each component can be addressed and/or independently corrected when the sensor is operating on an arbitrary process fluid. 
       FIG. 17  illustrates one embodiment of the present invention including calibrating a sensor on a test fluid to obtain calibration measurements that facilitate correcting separately sensor non-linearity and bypass non-linearity by employing a sensor model when operating the sensor on an arbitrary process fluid. 
     In step  1710 , a test fluid  1701 , for example nitrogen, having known thermal/physical properties is introduced to a sensor before it has been coupled to a bypass. That is, the flow path consists of the sensor conduit. The test fluid  1701  is introduced to the sensor conduit at various flow rates such that the sensor responds with a set S of desired sensor output values using calibration equipment as described in the foregoing. The flow rates giving rise to the desired sensor output values are measured by, for example, a volumetric flow meter, and recorded such that the relationship between sensor flow and sensor output is obtained. This sensor response information may then be stored as part of calibration measurements  1705 . 
     In step  1720 , the sensor may be coupled to a bypass such that the flow path of the sensor includes both the sensor conduit and the bypass. Similar to step  1710 , test fluid  1701  may be introduced to the sensor flow path to provide the various desired sensor output values in set S and the corresponding flow rates are recorded such that the relationship between total flow and sensor output is obtained. This total response information may be further stored as part of calibration measurements  1705 . 
     In step  1730 , a linearization curve may be generated that, when applied to the sensor output values resulting from flow of the test fluid, maps sensor output values to indicated flow values having a substantially linear relationship with total flow. This linearization curve may also be stored as part of calibration measurements  1705 . 
     For example, a curve may be fit to the set of points indicating total flow rate as a function of sensor output by a piecewise linear approximation of the set of points. A linearization curve may then be determined that maps values along the total flow curve to values along a straight line such that the relationship between sensor output and total flow is linear for the test fluid. For example, a linearization curve may be generated by performing a least squares fit of the total flow curve to a line connecting the sensor output value at zero flow and the sensor output values at full scale flow. Various methods of generating a linearization curve or linearization information will occur to those skilled in the art. However, any method of mapping a total flow curve substantially to a line is considered to be within the scope of the invention. 
     The various calibration measurements obtained as described in the foregoing may be stored, for example, as calibration measurements  1687  in the memory of computer  1680  of  FIG. 16 , and subsequently used to facilitate an independent correction of sensor and bypass response to operating with an arbitrary process fluid. 
     It should be appreciated that the set of points indicating the relationship between sensor flow and sensor output is a sensor response curve (e.g., the set of points obtained in step  1710  and stored in calibration measurements  1705 ). As such, in conjunction with a sensor model according to the present invention, a sensor response curve obtained with a test fluid includes characteristic information about the sensor. That is, a representation of the characteristic function may be obtained by applying the sensor model to the sensor response curve of the test fluid. This representation of the characteristic function may facilitate computing a sensor response curve for an arbitrary process fluid. By determining a process sensor response curve, the sensor behavior can be corrected for independently of the bypass behavior. 
     Referring again to  FIG. 16 , sensor  1610  may be coupled to computer  1680  having a processor  1685  and memory  1686 . Memory  1686  may store various calibration measurements  1687  obtained using a test fluid during calibration of the sensor as described above in connection with  FIG. 17 . In particular, memory  1686  may include a sensor response curve of the test fluid, a total flow response curve of the test fluid and a linearization curve corresponding to the total flow response curve. In addition, memory  1686  may include at least one program containing instructions that, when executed by processor  1685 , maps sensor output values to characteristic values and corresponding flow rates to operand values of the characteristic function of the sensor. In addition, one or more programs stored in memory  1686  may include instructions configured to access a representation of a characteristic function to obtain an associated operand value and to compute a flow rate from the obtained operand value as discussed in connection with the various embodiments described in the foregoing. 
       FIG. 18  illustrates one method according to the present invention that may be performed by a processor (e.g., processor  1685 ) executing one or more programs stored in memory (e.g., memory  1686 ) to separate and independently correct sensor and bypass effects of a sensor operating on an arbitrary fluid. 
     When it is desired to operate a sensor (e.g., sensor  1610  incorporated in MFC  1600 ) on a process fluid different than the test fluid, a linearization curve generated during calibration may need to be corrected to compensate for the behavior differences the sensor may exhibit when operating on different fluids. In step  1810 , the total flow response curve of the test fluid obtained during calibration of the sensor may be divided by the sensor response curve of the test fluid also obtained during calibration of the sensor. 
     It should be appreciated that since the sensor response curve indicates the relationship between sensor flow and sensor output, and the total flow curve indicates the relationship between the total flow and sensor output with the test fluid, the result is an indication of the relationship between sensor flow and total flow, that is, it represents a bypass flow curve. As a result, sensor effects and bypass effects on the test fluid have been separated, that is, two curves have been separated from the total flow response curve: 1) a test sensor response curve; and 2) a test bypass flow curve. The test sensor response curve may be corrected in sensor correction  1815  and the bypass flow curve may be corrected in bypass correction  1850 . 
     In sensor correction  1815 , the test sensor response curve may be converted into a process sensor response curve according to various methods described in the foregoing. In particular, in step  1820 , each of the sensor output values of the test sensor response curve may be mapped to characteristic values using at least one property of the test fluid. For example, the sensor output values may be multiplied by a scale factor of the test fluid and divided by the thermal conductivity of the test fluid. Each of the flow rate values of the test sensor response curve may be mapped to operand values using at least one property of the sensor and at least one property of the test fluid. For example, the operand values may be computed using a product of the Reynold&#39;s number associated with flow through the sensor and the Prandtl number of the test fluid. Accordingly, a representation of the characteristic function of the sensor may be obtained. 
     In step  1830 , each of the sensor output values of the test sensor response curve is mapped to characteristic values using the same mapping as that used in step  1820  except that properties of the process fluid are used instead of the test fluid. For example, each of the sensor output values may be multiplied by a scale factor of the process fluid and divided by the thermal conductivity of the process fluid. Since the sensor model establishes that the characteristic values mapped using process fluid properties will lie along the same curve as characteristic values mapped using test fluid properties, this information can be used to obtain an operand value associated with flow of the process fluid. 
     In step  1840 , the characteristic values computed according to the process fluid are used to index the representation of the characteristic function obtained in step  1820  to obtain an associated operand value. A flow rate associated with each of the obtained operand values may be calculated by reversing the operation of step  1820  using properties of the sensor and the process fluid. Accordingly, a sensor response curve for the process fluid may be obtained by employing the representation of the characteristic function of the sensor. It should be appreciated that the process sensor response curve indicates the relationship between sensor flow and sensor output for the process fluid, that is, the sensor response has been corrected independently of the bypass response. 
     In bypass correction  1850 , the test bypass curve as separated from the test total response flow curve in step  1810  may be corrected to compensate for the behavior of the bypass when operating on the process fluid. In one embodiment, the bypass curve is assumed to remain substantially constant from fluid to fluid such that the test bypass curve is considered equal to the process bypass curve. In other embodiments, a bypass curve may be assumed to depend on a variety of factors including physical properties of the bypass and sensor, for example, the difference in length between the sensor conduit and the bypass, one or more physical properties of the process fluid such as viscosity, temperature of the process fluid, etc. Accordingly, a test bypass curve may be corrected and/or converted to a process bypass curve according to any of various models that describe how the bypass behavior changes when operating on different fluids. 
     In step  1860 , the process sensor response curve is multiplied by the process bypass curve to obtain a process total flow response curve of the sensor with the process fluid. Accordingly, in step  1870 , a process linearization curve may be computed that maps values of the process total flow response curve to a substantially straight line. In this manner, a linearization curve for the flow sensor operating on an arbitrary process fluid may be obtained from calibration measurements obtained during calibration of the sensor with a single test fluid. 
     Having described several embodiments of the invention in detail, various modifications and improvements will readily occur to those skilled in the art. Such modifications and improvements are intended to be within the scope of the invention. While some examples presented herein involve specific combinations of functions or structural elements, it should be understood that those functions and elements may be combined in other ways according to the present invention to accomplish the same or different objectives. In particular, acts, elements and features discussed in connection with one embodiment are not intended to be excluded from a similar role in other embodiments. Accordingly, the foregoing description is by way of example only, and is not intended as limiting. The invention is limited only as defined by the following claims and the equivalents thereto.