Patent Publication Number: US-6981424-B2

Title: Correcting for two-phase flow in a digital flowmeter

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. application Ser. No. 10/339,623, filed Jan. 10, 2003, now U.S. Pat. No. 6,758,102 and titled “CORRECTING FOR TWO-PHASE FLOW IN A DIGITAL FLOWMETER,” which is a continuation of U.S. application Ser. No. 09/815,273, filed Mar. 23, 2001, and titled “CORRECTING FOR TWO-PHASE FLOW IN A DIGITAL FLOWMETER,” now U.S. Pat. No. 6,505,519. U.S. application Ser. No. 09/815,273 claims priority to U.S. Provisional Application No. 60/191,465, filed Mar. 23, 2000, titled “A NEURAL NETWORK TO CORRECT MASS FLOW ERRORS CAUSED BY TWO PHASE FLOW IN A DIGITAL CORIOLIS MASS FLOWMETER,” and is a continuation of U.S. application Ser. No. 09/716,644, filed Nov. 21, 2000, and titled “DIGITAL FLOWMETER,” now abandoned. The subject matter from all of the above applications is incorporated by reference. 
    
    
     TECHNICAL FIELD 
     The invention relates to flowmeters. 
     BACKGROUND 
     Flowmeters provide information about materials being transferred through a conduit. For example, mass flowmeters provide a direct indication of the mass of material being transferred through a conduit. Similarly, density flowmeters, or densitometers, provide an indication of the density of material flowing through a conduit. Mass flowmeters also may provide an indication of the density of the material. 
     Coriolis-type mass flowmeters are based on the well-known Coriolis effect, in which material flowing through a rotating conduit becomes a radially traveling mass that is affected by a Coriolis force and therefore experiences an acceleration. Many Coriolis-type mass flowmeters induce a Coriolis force by sinusoidally oscillating a conduit about a pivot axis orthogonal to the length of the conduit. In such mass flowmeters, the Coriolis reaction force experienced by the traveling fluid mass is transferred to the conduit itself and is manifested as a deflection or offset of the conduit in the direction of the Coriolis force vector in the plane of rotation. 
     Energy is supplied to the conduit by a driving mechanism that applies a periodic force to oscillate the conduit. One type of driving mechanism is an electromechanical driver that imparts a force proportional to an applied voltage. In an oscillating flowmeter, the applied voltage is periodic, and is generally sinusoidal. The period of the input voltage is chosen so that the motion of the conduit matches a resonant mode of vibration of the conduit. This reduces the energy needed to sustain oscillation. An oscillating flowmeter may use a feedback loop in which a sensor signal that carries instantaneous frequency and phase information related to oscillation of the conduit is amplified and fed back to the conduit using the electromechanical driver. 
     SUMMARY 
     In one general aspect, the flowmeter includes a vibratable conduit, a driver connected to the conduit and operable to impart motion to the conduit, and a sensor connected to the conduit and operable to sense the motion of the conduit and to generate a sensor signal. A controller is connected to receive the sensor signal, and is operable to generate a raw mass-flow measurement from the sensor signal, detect a single-phase flow condition and process the raw mass-flow measurement using a first process during the single-phase flow condition to generate a first mass-flow measurement, and detect a two-phase flow condition and correct the raw mass-flow measurement using a second process during the two-phase flow condition to generate a second mass-flow measurement. 
     The second process may include a neural network processor to predict a mass-flow error and to calculate an error correction factor used to generate the second mass-flow measurement. The neural network processor may receive at least one input parameter and apply a set of predetermined coefficients to the input parameter. 
     The neural network processor may be a multi-layer perceptron neural network processor including an input layer for receiving input parameters, a hidden layer having processing nodes for applying the set of predetermined coefficients to the input parameters, and an output layer that generates an output parameter The input parameters may include a temperature parameter, a damping parameter, a density parameter, and an apparent flow rate parameter. The flowmeter may further include a training module connected to the neural network processor to calculate an updated set of coefficients when supplied with training data. In other implementations of the flowmeter, the neural network processor may be a radial basis function network. 
     In other instances, the second process includes a processor to analyze a density drop parameter, to predict a mass-flow error, and to calculate an error correction factor used to generate the second mass-flow measurement. The processor may execute a bubble model routine to analyze the density drop parameter. 
     The first mass-flow measurement may be a validated mass-flow measurement comprising the raw mass-flow measurement and an uncertainty parameter calculated by the controller. The second mass-flow measurement may be a validated mass-flow measurement comprising a corrected mass-flow measurement generated from the raw mass-flow measurement and an uncertainty parameter calculated by the controller. The controller may generate a measurement status parameter associated with the first mass-flow measurement. The controller also may generate a measurement status parameter associated with the second mass-flow measurement. 
     The flowmeter may include a memory for storing sensor signal data generated from the sensor signal for use by the controller. The controller associated with the flowmeter also may include a sensor parameter processing module to analyze the sensor signal data and generate sensor signal parameters. The flowmeter also may include a state machine that uses the raw mass-flow measurement and the sensor signal parameters to detect the single-phase flow condition and the two-phase flow condition. 
     The controller may include an output signal generator that is operable to receive the sensor signal parameters from the sensor parameter processing module and to generate a drive signal for the driver based the sensor signal parameters. The controller may also include circuitry to generate a drive signal based on the sensor signal. The drive signal may be a digital drive signal for operating the driver. Alternatively, the drive signal may be an analog drive signal for operating the driver. 
     The flowmeter may include a second sensor connected to the conduit for sensing the motion of the conduit and generating a second sensor signal, with the controller being connected to receive the second sensor signal and generate the drive signal based on the first sensor signal and the second sensor signal using digital signal processing. The flowmeter may generate a measurement of a property of material flowing through the conduit based on the first and second sensor signals. The flowmeter may include a second driver, and the controller may generate different drive signals for the two drivers. 
     The controller may generate the measurement of the property by estimating a frequency of the first sensor signal, calculating a phase difference using the first sensor signal, and generating the measurement using the calculated phase difference. The controller may compensate for amplitude differences in the sensor signals by adjusting the amplitude of one of the sensor signals. The controller may determine a frequency, amplitude and phase offsets for each sensor signal, and scale the phase offsets to an average of the frequencies of the sensor signals. The controller may calculate the phase difference using multiple approaches and selects a result of one of the approaches as the calculated based difference. The controller may combine the sensor signals to produce a combined signal and to generate the drive signal based on the combined signal. The controller may generate the drive signal by applying a gain to the combined signal. 
     The controller may generate the drive signal by applying a large gain to the combined signal to initiate motion of the conduit and generate a periodic signal having a phase and frequency based on a phase and frequency of a sensor signal as the drive signal after motion has been initiated. 
     The flowmeter may include a second sensor connected to the conduit and operable sense the motion of the conduit, and the controller includes a first signal processor to generate the measurement, a first analog to digital converter connected between the first sensor and the first signal processor to provide a first digital sensor signal to the first signal processor, and a second analog to digital converter connected between the second sensor and the first signal processor to provide a second digital sensor signal to the controller. The first signal processor may combine the digital sensor signals to produce a combined signal and generate a gain signal based on the first and second digital sensor signals. The controller may further include a multiplying digital to analog converter connected to receive the combined signal and the gain signal from the first signal processor to generate the drive signal as a product of the combined signal and the gain signal. 
     The flowmeter also may include circuitry for measuring current supplied to the driver. The controller may determine a frequency of the sensor signal by detecting zero-crossings of the sensor signal and counting samples between zero crossings. The flowmeter also may include a connection to a control system, wherein the controller transmits the measurement and results of a uncertainty analysis to the control system. 
     In another aspect, the flowmeter includes a vibratable conduit, a driver connected to the conduit and operable to impart motion to the conduit, and a sensor connected to the conduit and operable to sense the motion of the conduit and generate a sensor signal. A controller is connected to receive the sensor signal, and is operable to detect a single-phase flow condition and process the sensor signal using a first process during the single-phase flow condition to generate a validated mass-flow measurement, and detect a two-phase flow condition and process the sensor signal using a second process during the two-phase flow condition to generate the validated mass-flow measurement. 
     In another aspect, the digital flowmeter includes a vibratable conduit, a driver connected to the conduit and operable to impart motion to the conduit, and a sensor connected to the conduit and operable to sense the motion of the conduit and generate a sensor signal. A controller is connected to receive the sensor signal and detect a single-phase flow condition and a two-phase flow condition. The controller includes a first processing module to analyze the sensor signal during the single-phase flow condition and to generate a first mass-flow measurement, and includes a second processing module to analyze the sensor signal during the two-phase flow condition and to generate a second mass-flow measurement. The second processing module includes a neural network processor to predict a mass-flow error and to calculate an error correction factor used to generate the second mass-flow measurement. The controller also includes output circuitry to generate a drive signal to operate the driver. 
     The neural network processor receives at least one input parameter and applies a set of predetermined coefficients to the input parameter. The input parameter may include a set of input parameters including a temperature parameter, a damping parameter, a density parameter, and an apparent flow rate parameter. 
     The controller may further include a memory for storing sensor signal data generated from the sensor signal, wherein the controller uses the stored sensor signal data. 
     The controller may include a sensor parameter processing module to analyze the sensor signal data and generate sensor signal parameters. 
     The digital flowmeter may include a state machine that receives the sensor signal parameters and detects the signal-phase flow condition and the two-phase flow condition. The output circuitry is operable to receive the sensor signal parameters from the sensor parameter processing module and to generate the drive signal based on the sensor signal parameters. 
     The controller may include a memory for storing driver signal data generated from the driver, wherein the controller uses the driver signal data. The controller may include a memory for storing sensor signal data generated from the sensor signal and driver signal data generated from the driver, wherein the controller uses the sensor signal data and the driver signal data. 
     The controller may generate an uncertainty parameter associated with one of the first mass-flow measurement and the second mass-flow measurement. The controller may also generate a measurement status parameter associated with one of the first mass-flow measurement and the second mass-flow measurement. 
     In another aspect, a method of generating a measurement of a property of material flowing through a conduit is implemented. The method includes sensing motion of the conduit, generating a measurement of a property of material flowing through the conduit based on the sensed motion, and detecting a flow condition based on the measurement of the property. If the flow condition is a single-phase flow condition, a first analysis process is applied to the measurement of the property using digital signal processing to generate a first measurement signal. If the flow condition is a two-phase flow condition, a second analysis process is applied to the measurement of the property using a neural network processor. The neural network processor operates to predict a mass-flow error based on the measurement of the property, generates an error correction factor, and applies the error correction factor to the measurement of the property to generate a second measurement signal. In addition, a drive signal is generated and used to impart motion in the conduit based on the measurement of the property. 
     Generating the measurement of the property may further include storing sensor signal data in a memory. The sensor signal data may be retrieved from the memory, and sensor variables, and sensor variable statistics may be calculated from the sensor signal data. 
     The sensor variable statistics may include mean, standard deviation, and slope for each of the sensor variables. 
     The flowmeter may perform an uncertainty analysis on the first measurement signal and generate an uncertainty parameter associated with the first measurement signal to produce a validated mass-flow measurement signal. A measurement status parameter associated with the validated measurement signal may also be-generated. The flowmeter may perform an uncertainty analysis on the second measurement signal and generate an uncertainty parameter associated with the second measurement signal to produce a validated mass-flow measurement signal. A measurement status parameter associated with the validated measurement signal may also be generated. 
     The techniques may be implemented in a digital flowmeter, such as a digital mass flowmeter, that uses a control and measurement system to control oscillation of the conduit and to generate mass flow and density measurements. Sensors connected to the conduit supply signals to the control and measurement system. The control and measurement system processes the signals to produce a measurement of mass flow and uses digital signal processing to generate a signal for driving the conduit. The drive signal then is converted to a force that induces oscillation of the conduit. 
     The digital mass flowmeter provides a number of advantages over traditional, analog approaches. For example, a processor of the flowmeter allows for the detection of a single-phase flow condition and a two-phase flow condition. The mass-flow measurement can be corrected when the flowmeter detects two-phase flow. From a control perspective, use of digital processing techniques permits the application of precise, sophisticated control algorithms that, relative to traditional analog approaches, provide greater responsiveness, accuracy and adaptability. 
     The digital control system also permits the use of negative gain in controlling oscillation of the conduit. Thus, drive signals that are 180° out of phase with conduit oscillation may be used to reduce the amplitude of oscillation. The practical implications of this are important, particularly in high and variable damping situations where a sudden drop in damping can cause an undesirable increase in the amplitude of oscillation. One example of a variable damping situation is when aeration occurs in the material flowing through the conduit. 
     The ability to provide negative feedback also is important when the amplitude of oscillation is controlled to a fixed setpoint that can be changed under user control. With negative feedback, reductions in the oscillation setpoint can be implemented as quickly as increases in the setpoint. By contrast, an analog meter that relies solely on positive feedback must set the gain to zero and wait for system damping to reduce the amplitude to the reduced setpoint. 
     From a measurement perspective, the digital mass flowmeter can provide high information bandwidth. For example, a digital measurement system may use analog-to-digital converters operating at eighteen bits of precision and sampling rates of 55 kHz. The digital measurement system also may use sophisticated algorithms to filter and process the data, and may do so starting with the raw data from the sensors and continuing to the final measurement data. This permits extremely high precision, such as, for example, phase precision to five nanoseconds per cycle. Digital processing starting with the raw sensor data also allows for extensions to existing measurement techniques to improve performance in non-ideal situations, such as by detecting and compensating for time-varying amplitude, frequency, and zero offset. 
     The control and measurement improvements interact to provide further improvements. For example, control of oscillation amplitude is dependent upon the quality of amplitude measurement. Under normal conditions, the digital mass flowmeter may maintain oscillation to within twenty parts per million of the desired setpoint. Similarly, improved control has a positive benefit on measurement. Increasing the stability of oscillation will improve measurement quality even for meters that do not require a fixed amplitude of oscillation (i.e., a fixed setpoint). For example, with improved stability, assumptions used for the measurement calculations are likely to be valid over a wider range of conditions. 
     The digital mass flowmeter also permits the integration of entirely new functionality (e.g., diagnostics) with the measurement and control processes. For example, algorithms for detecting the presence of process aeration can be implemented with compensatory action occurring for both measurement and control if aeration is detected. 
     In another aspect a flowmeter controller includes an input module operable to receive sensor signals from a sensor connected to a vibratable flowtube. The sensor signals relate to a mass flow through the flowtube, wherein the mass flow includes a first material having a first phase and a second material having a second phase. A signal processing system is operable to receive the sensor signal, determine sensor signal characteristics, and output drive signal characteristics for a drive signal applied to the flowtube, wherein the drive signal characteristics include a drive gain necessary to maintain oscillation of the flowtube during the mass flow. A measurement system is operable to determine an actual flow condition of the mass flow, based on an apparent flow condition derived from the sensor signal characteristics, and an output module is operable to output the drive signal. 
     Other advantages of the digital mass flowmeter result from the limited amount of hardware employed, which makes the meter simple to construct, debug, and repair in production and in the field. Quick repairs in the field for improved performance and to compensate for wear of the mechanical components (e.g, loops, flanges, sensors and drivers) are possible because the meter uses standardized hardware components that may be replaced with little difficulty, and because software modifications may be made with relative ease. In addition, integration of diagnostics, measurement, and control is simplified by the simplicity of the hardware and the level of functionality implemented in software. New functionality, such as low power components or components with improved performance, can be integrated without a major redesign of the overall control system. 
     Other features and advantages of the invention will be apparent from the following description, including the drawings, and from the claims. 
    
    
     
       DESCRIPTION OF DRAWINGS 
         FIG. 1  is a block diagram of a digital mass flowmeter. 
         FIGS. 2A and 2B  are perspective and side views of mechanical components of a mass flowmeter. 
         FIGS. 3A-3C  are schematic representations of three modes of motion of the flowmeter of FIG.  1 . 
         FIG. 4  is a block diagram of an analog control and measurement circuit. 
         FIG. 5  is a block diagram of a digital mass flowmeter. 
         FIG. 6  is a flow chart showing operation of the meter of FIG.  5 . 
         FIGS. 7A and 7B  are graphs of sensor data. 
         FIGS. 8A and 8B  are graphs of sensor voltage relative to time. 
         FIG. 9  is a flow chart of a curve fitting procedure. 
         FIG. 10  is a flow chart of a procedure for generating phase differences. 
         FIGS. 11A-11D ,  12 A- 12 D, and  13 A- 13 D illustrate drive and sensor voltages at system startup. 
         FIG. 14  is a flow chart of a procedure for measuring frequency, amplitude, and phase of sensor data using a synchronous modulation technique. 
         FIGS. 15A and 15B  are block diagrams of a mass flowmeter. 
         FIG. 16  is a flow chart of a procedure implemented by the meter of  FIGS. 15A and 15B . 
         FIG. 17  illustrates log-amplitude control of a transfer function. 
         FIG. 18  is a root locus diagram. 
         FIGS. 19A-19D  are graphs of analog-to-digital converter performance relative to temperature. 
         FIGS. 20A-20C  are graphs of phase measurements. 
         FIGS. 21A and 21B  are graphs of phase measurements. 
         FIG. 22  is a flow chart of a zero offset compensation procedure. 
         FIGS. 23A-23C ,  24 A, and  24 B are graphs of phase measurements. 
         FIG. 25  is a graph of sensor voltage. 
         FIG. 26  is a flow chart of a procedure for compensating for dynamic effects. 
         FIGS. 27A-35E  are graphs illustrating application of the procedure of FIG.  29 . 
         FIGS. 36A-36L  are graphs illustrating phase measurement. 
         FIG. 37A  is a graph of sensor voltages. 
         FIGS. 37B and 37C  are graphs of phase and frequency measurements corresponding to the sensor voltages of FIG.  37 A. 
         FIGS. 37D and 37E  are graphs of correction parameters for the phase and frequency measurements of  FIGS. 37B and 37C . 
         FIGS. 38A-38H  are graphs of raw measurements. 
         FIGS. 39A-39H  are graphs of corrected measurements. 
         FIGS. 40A-40H  are graphs illustrating correction for aeration. 
         FIG. 41  is a block diagram illustrating the effect of aeration in a conduit. 
         FIG. 42  is a flow chart of a setpoint control procedure. 
         FIGS. 43A-43C  are graphs illustrating application of the procedure of FIG.  41 . 
         FIG. 44  is a graph comparing the performance of digital and analog flowmeters. 
         FIG. 45  is a flow chart showing operation of a self-validating meter. 
         FIG. 46  is a block diagram of a two-wire digital mass flowmeter. 
         FIGS. 47A-47C  are graphs showing response of the digital mass flowmeter under wet and empty conditions. 
         FIG. 48A  is a chart showing results for batching from empty trials. 
         FIG. 48B  is a diagram showing an experimental flow rig. 
         FIG. 49  is a graph showing mass-flow errors against drop in apparent density. 
         FIG. 50  is a graph showing residual mass-flow errors after applying corrections. 
         FIG. 51  is a graph showing on-line response of the self-validating digital mass flowmeter to the onset of two-phase flow. 
         FIG. 52  is a block diagram of a digital controller implementing a neural network processor that may be used with the digital mass flowmeter. 
         FIG. 53  is a flow diagram showing the technique for implementing the neural network to predict the mass-flow error and generate an error correction factor to correct the mass-flow measurement signal when two-phase flow is detected. 
         FIG. 54  is a 3Dgraph showing damping changes under two-phase flow conditions. 
         FIG. 55  is a flow diagram illustrating the experimental flow rig. 
         FIG. 56  is a 3Dgraph showing true mass flow error under two-phase flow conditions. 
         FIG. 57  is a 3Dgraph showing corrected mass flow error under two-phase flow conditions. 
         FIG. 58  is a graph comparing the uncorrected mass flow measurement signal with the neural network corrected mass flow measurement signal. 
     
    
    
     DETAILED DESCRIPTION 
     Techniques are provided for accounting for the effects of two-phase flow-in, for example, a digital flowmeter. Before the techniques are described starting with reference to  FIG. 40 , digital flowmeters are discussed with reference to  FIGS. 1-39 . 
     Referring to  FIG. 1 , a digital mass flowmeter  100  includes a digital controller  105 , one or more motion sensors  110 , one or more drivers  115 , a conduit  120  (also referred to as a flowtube), and a temperature sensor  125 . The digital controller  105  may be implemented using one or more of, for example, a processor, a field-programmable gate array, an ASIC, other programmable logic or gate arrays, or programmable logic with a processor core. The digital controller generates a measurement of mass flow through the conduit  120  based at least on signals received from the motion sensors  110 . The digital controller also controls the drivers  115  to induce motion in the conduit  120 . This motion is sensed by the motion sensors  110 . 
     Mass flow through the conduit  120  is related to the motion induced in the conduit in response to a driving force supplied by the drivers  115 . In particular, mass flow is related to the phase and frequency of the motion, as well as to the temperature of the conduit. The digital mass flowmeter also may provide a measurement of the density of material flowing through the conduit. The density is related to the frequency of the motion and the temperature of the conduit. Many of the described techniques are applicable to a densitometer that provides a measure of density rather than a measure of mass flow. 
     The temperature in the conduit, which is measured using the temperature sensor  125 , affects certain properties of the conduit, such as its stiffness and dimensions. The digital controller compensates for these temperature effects. The temperature of the digital controller  105  affects, for example, the operating frequency of the digital controller. In general, the effects of controller temperature are sufficiently small to be considered negligible. However, in some instances, the digital controller may measure the controller temperature using a solid state device and may compensate for effects of the controller temperature. 
     A. Mechanical Design 
     In one implementation, as illustrated in  FIGS. 2A and 2B , the conduit  120  is designed to be inserted in a pipeline (not shown) having a small section removed or reserved to make room for the conduit. The conduit  120  includes mounting flanges  12  for connection to the pipeline, and a central manifold block  16  supporting two parallel planar loops  18  and  20  that are oriented perpendicularly to the pipeline. An electromagnetic driver  46  and a sensor  48  are attached between each end of loops  18  and  20 . Each of the two drivers  46  corresponds to a driver  115  of  FIG. 1 , while each of the two sensors  48  corresponds to a sensor  110  of FIG.  1 . 
     The drivers  46  on opposite ends of the loops are energized with current of equal magnitude but opposite sign (i.e., currents that are 180° out-of-phase) to cause straight sections  26  of the loops  18 ,  20  to rotate about their co-planar perpendicular bisector  56 , which intersects the tube at point P (FIG.  2 B). Repeatedly reversing (e.g., controlling sinusoidally) the energizing current supplied to the drivers causes each straight section  26  to undergo oscillatory motion that sweeps out a bow tie shape in the horizontal plane about line  56 - 56 , the axis of symmetry of the loop. The entire lateral excursion of the loops at the lower. rounded turns  38  and  40  is small, on the order of 1/16 of an inch for a two foot long straight section  26  of a pipe having a one inch diameter. The frequency of oscillation is typically about 80 to 90 Hertz. 
     B. Conduit Motion 
     The motion of the straight sections of loops  18  and  20  are shown in three modes in  FIGS. 3A ,  3 B and  3 C. In the drive mode shown in  FIG. 3B , the loops are driven 180° out-of-phase about their respective points P so that the two loops rotate synchronously but in the opposite sense. Consequently, respective ends such as A and C periodically come together and go apart. 
     The drive motion shown in  FIG. 3B  induces the Coriolis mode motion shown in  FIG. 3A , which is in opposite directions between the loops and moves the straight sections  26  slightly toward (or away) from each other. The Coriolis effect is directly related to mvW, where m is the mass of material in a cross section of a loop, v is the velocity at which the mass is moving (the volumetric flow rate), W is the angular velocity of the loop (W=W o  sin ωt), and mv is the mass flow rate. The Coriolis effect is greatest when the two straight sections are driven sinusoidally and have a sinusoidally varying angular velocity. Under these conditions, the Coriolis effect is 90° out-of-phase with the drive signal. 
       FIG. 3C  shows an undesirable common mode motion that deflects the loops in the same direction. This type of motion might be produced by an axial vibration in the pipeline in the embodiment of  FIGS. 2A and 2B  because the loops are perpendicular to the pipeline. 
     The type of oscillation shown in  FIG. 3B  is called the antisymmetrical mode, and the Coriolis mode of  FIG. 3A  is called the symmetrical mode. The natural frequency of oscillation in the antisymmetrical mode is a function of the torsional resilience of the legs. Ordinarily the resonant frequency of the antisymmetrical mode for conduits of the shape shown in  FIGS. 2A and 2B  is higher than the resonant frequency of the symmetrical mode. To reduce the noise sensitivity of the mass flow measurement, it is desirable to maximize the Coriolis force for a given mass flow rate. As noted above, the loops are driven at their resonant frequency, and the Coriolis force is directly related to the frequency at which the loops are oscillating (i.e., the angular velocity of the loops). Accordingly, the loops are driven in the antisymmetrical mode, which tends to have the higher resonant frequency. 
     Other implementations may include different conduit designs. For example, a single loop or a straight tube section may be employed as the conduit. 
     C. Electronic Design 
     The digital controller  105  determines the mass flow rate by processing signals produced by the sensors  48  (i.e., the motion sensors  110 ) located at opposite ends of the loops. The signal produced by each sensor includes a component corresponding to the relative velocity at which the loops are driven by a driver positioned next to the sensor and a component corresponding to the relative velocity of the loops due to Coriolis forces induced in the loops. The loops are driven in the antisymmetrical mode, so that the components of the sensor signals corresponding to drive velocity are equal in magnitude but opposite in sign. The resulting Coriolis force is in the symmetrical mode so that the components of the sensor signals corresponding to Coriolis velocity are equal in magnitude and sign. Thus, differencing the signals cancels out the Coriolis velocity components and results in a difference that is proportional to the drive velocity. Similarly, summing the signals cancels out the drive velocity components and results in a sum that is proportional to the Coriolis velocity, which, in turn, is proportional to the Coriolis force. This sum then may be used to determine the mass flow rate. 
     1. Analog Control System 
     The digital mass flowmeter  100  provides considerable advantages over traditional, analog mass flowmeters. For use in later discussion,  FIG. 4  illustrates an analog control system  400  of a traditional mass flowmeter. The sensors  48  each produce a voltage signal, with signal V A0  being produced by sensor  48   a  and signal V B0  being produced by sensor  48   b . V A0  and V B0  correspond to the velocity of the loops relative to each other at the positions of the sensors. Prior to processing, signals V A0  and V B0  are amplified at respective input amplifiers  405  and  410  to produce signals V A1  and V B1 . To correct for imbalances in the amplifiers and the sensors, input amplifier  410  has a variable gain that is controlled by a balance signal coming from a feedback loop that contains a synchronous demodulator  415  and an integrator  420 . 
     At the output of amplifier  405 , signal V A1  is of the form:
 
 V   A1   =V   D  sin ω t+V   c  cos ω t, 
 
and, at the output of amplifier  410 , signal V B1  is of the form:
 
 V   B1   =−V   D  sin ω t+V   c  cos ω t, 
 
where V D  and V C  are, respectively, the drive voltage and the Coriolis voltage, and ω is the drive mode angular frequency.
 
     Voltages V A1  and V B1  are differenced by operational amplifier  425  to produce:
 
 V   DRV   =V   A1   −V   B1 =2 VD  sin ω t, 
 
where V DRV  corresponds to the drive motion and is used to power the drivers. In addition to powering the drivers, V DRV  is supplied to a positive going zero crossing detector  430  that produces an output square wave F DRV  having a frequency corresponding to that of V DRV  (ω=2πF DRV ). F DRV  is used as the input to a digital phase locked loop circuit  435 . F DRV  also is supplied to a processor  440 .
 
     Voltages V A1  and V B1  are summed by operational amplifier  445  to produce:
 
 V   COR   =V   A1   +V   B1 =2 V   C  cos ω t, 
 
where V COR  is related to the induced Coriolis motion.
 
     V COR  is supplied to a synchronous demodulator  450  that produces an output voltage V M  that is directly proportional to mass by rejecting the components of V COR  that do not have the same frequency as, and are not in phase with, a gating signal Q. The phase locked loop circuit  435  produces Q, which is a quadrature reference signal that has the same frequency (ω) as V DRV  and is 90° out of phase with V DRV  (i.e., in phase with V COR ). Accordingly, synchronous demodulator  450  rejects frequencies other than ω so that V M  corresponds to the amplitude of V COR  at ω. This amplitude is directly proportional to the mass in the conduit. 
     V M  is supplied to a voltage-to-frequency converter  455  that produces a square wave signal F M  having a frequency that corresponds to the amplitude of V M . The processor  440  then divides F M  by F DRV  to produce a measurement of the mass flow rate. 
     Digital phase locked loop circuit  435  also produces a reference signal I that is in phase with V DRV  and is used to gate the synchronous demodulator  415  in the feedback loop controlling amplifier  410 . When the gains of the input amplifiers  405  and  410  multiplied by the drive components of the corresponding input signals are equal, the summing operation at operational amplifier  445  produces zero drive component (i.e., no signal in phase with V DRV ) in the signal V COR . When the gains of the input amplifiers  405  and  410  are not equal, a drive component exists in V COR . This drive component is extracted by synchronous demodulator  415  and integrated by integrator  420  to generate an error voltage that corrects the gain of input amplifier  410 . When the gain is too high or too low, the synchronous demodulator  415  produces an output voltage that causes the integrator to change the error voltage that modifies the gain. When the gain reaches the desired value, the output of the synchronous modulator goes to zero and the error voltage stops changing to maintain the gain at the desired value. 
     2. Digital Control System 
       FIG. 5  provides a block diagram of an implementation  500  of the digital mass flowmeter  100  that includes the conduit  120 , drivers  46 , and sensors  48  of  FIGS. 2A and 2B , along with a digital controller  505 . Analog signals from the sensors  48  are converted to digital signals by analog-to-digital (“A/D”) converters  510  and supplied to the controller  505 . The A/D converters may be implemented as separate converters, or as separate channels of a single converter. 
     Digital-to-analog (“D/A”) converters  515  convert digital control signals from the controller  505  to analog signals for driving the drivers  46 . The use of a separate drive signal for each driver has a number of advantages. For example, the system may easily switch between symmetrical and antisymmetrical drive modes for diagnostic purposes. In other implementations, the signals produced by converters  515  may be amplified by amplifiers prior to being supplied to the drivers  46 . In still other implementations, a single D/A converter may be used to produce a drive signal applied to both drivers, with the drive signal being inverted prior to being provided to one of the drivers to drive the conduit  120  in the antisymmetrical mode. 
     High precision resistors  520  and amplifiers  525  are used to measure the current supplied to each driver  46 . A/D converters  530  convert the measured current to digital signals and supply the digital signals to controller  505 . The controller  505  uses the measured currents in generating the driving signals. 
     Temperature sensors  535  and pressure sensors  540  measure, respectively, the temperature and the pressure at the inlet  545  and the outlet  550  of the conduit. A/D converters  555  convert the measured values to digital signals and supply the digital signals to the controller  505 . The controller  505  uses the measured values in a number of ways. For example, the difference between the pressure measurements may be used to determine a back pressure in the conduit. Since the stiffness of the conduit varies with the back pressure, the controller may account for conduit stiffness based on the determined back pressure. 
     An additional temperature sensor  560  measures the temperature of the crystal oscillator  565  used by the A/D converters. An A/D converter  570  converts this temperature measurement to a digital signal for use by the controller  505 . The input/output relationship of the A/D converters varies with the operating frequency of the converters, and the operating frequency varies with the temperature of the crystal oscillator. Accordingly, the controller uses the temperature measurement to adjust the data provided by the A/D converters, or in system calibration. 
     In the implementation of  FIG. 5 , the digital controller  505  processes the digitized sensor signals produced by the A/D converters  510  according to the procedure  600  illustrated in  FIG. 6  to generate the mass flow measurement and the drive signal supplied to the drivers  46 . Initially, the controller collects data from the sensors (step  605 ). Using this data, the controller determines the frequency of the sensor signals (step  610 ), eliminates zero offset from the sensor signals (step  615 ), and determines the amplitude (step  620 ) and phase (step  625 ) of the sensor signals. The controller uses these calculated values to generate the drive signal (step  630 ) and to generate the mass flow and other measurements (step  635 ). After generating the drive signals and measurements, the controller collects a new set of data and repeats the procedure. The steps of the procedure  600  may be performed serially or in parallel, and may be performed in varying order. 
     Because of the relationships between frequency, zero offset, amplitude, and phase, an estimate of one may be used in calculating another. This leads to repeated calculations to improve accuracy. For example, an initial frequency determination used in determining the zero offset in the sensor signals may be revised using offset-eliminated sensor signals. In addition, where appropriate, values generated for a cycle may be used as starting estimates for a following cycle. 
     a. Data Collection 
     For ease of discussion, the digitized signals from the two sensors will be referred to as signals SV 1  and SV 2 , with signal SV 1  coming from sensor  48   a  and signal SV 2  coming from sensor  48   b . Although new data is generated constantly, it is assumed that calculations are based upon data corresponding to one complete cycle of both sensors. With sufficient data buffering, this condition will be true so long as the average time to process data is less than the time taken to collect the data. Tasks to be carried out for a cycle include deciding that the cycle has been completed, calculating the frequency of the cycle (or the frequencies of SV 1  and SV 2 ), calculating the amplitudes of SV 1  and SV 2 , and calculating the phase difference between SV 1  and SV 2 . In some implementations, these calculations are repeated for each cycle using the end point of the previous cycle as the start for the next. In other implementations, the cycles overlap by 180° or other amounts (e.g., 90°) so that a cycle is subsumed within the cycles that precede and follow it. 
       FIGS. 7A and 7B  illustrate two vectors of sampled data from signals SV 1  and SV 2 , which are named, respectively, sv 1 _in and sv 2 _in. The first sampling point of each vector is known, and corresponds to a zero crossing of the sine wave represented by the vector. For sv 1 _in, the first sampling point is the zero crossing from a negative value to a positive value, while for sv 2 _in the first sampling point is the zero crossing from a positive value to a negative value. 
     An actual starting point for a cycle (i.e., the actual zero crossing) will rarely coincide exactly with a sampling point. For this reason, the initial sampling points (start_sample_SV 1  and start_sample_SV 2 ) are the sampling points occurring just before the start of the cycle. To account for the difference between the first sampling point and the actual start of the cycle, the approach also uses the position (start_offset_SV 1  or start_offset_SV 2 ) between the starting sample and the next sample at which the cycle actually begins. 
     Since there is a phase offset between signals SV 1  and SV 2 , sv 1 _in and sv 2 _in may start at different sampling points. If both the sample rate and the phase difference are high, there may be a difference of several samples between the start of sv 1 _in and the start of sv 2 _in. This difference provides a crude estimate of the phase offset, and may be used as a check on the calculated phase offset, which is discussed below. For example, when sampling at 55 kHz, one sample corresponds to approximately 0.5 degrees of phase shift, and one cycle corresponds to about 800 sample points. 
     When the controller employs functions such as the sum (A+B) and difference (A−B), with B weighted to have the same amplitude as A, additional variables (e.g., start_sample_sum and start_offset_sum) track the start of the period for each function. The sum and difference functions have a phase offset halfway between SV 1  and SV 2 . 
     In one implementation, the data structure employed to store the data from the sensors is a circular list for each sensor, with a capacity of at least twice the maximum number of samples in a cycle. With this data structure, processing may be carried out on data for a current cycle while interrupts or other techniques are used to add data for a following cycle to the lists. 
     Processing is performed on data corresponding to a full cycle to avoid errors when employing approaches based on sine-waves. Accordingly, the first task in assembling data for a cycle is to determine where the cycle begins and ends. When nonoverlapping cycles are employed, the beginning of the cycle may be identified as the end of the previous cycle. When overlapping cycles are employed, and the cycles overlap by 180°, the beginning of the cycle may be identified as the midpoint of the previous cycle, or as the endpoint of the cycle preceding the previous cycle. 
     The end of the cycle may be first estimated based on the parameters of the previous cycle and under the assumption that the parameters will not change by more than a predetermined amount from cycle to cycle. For example, five percent may be used as the maximum permitted change from the last cycle&#39;s value, which is reasonable since, at sampling rates of 55 kHz, repeated increases or decreases of five percent in amplitude or frequency over consecutive cycles would result in changes of close to 5,000 percent in one second. 
     By designating five percent as the maximum permissible increase in amplitude and frequency, and allowing for a maximum phase change of 5° in consecutive cycles, a conservative estimate for the upper limit on the end of the cycle for signal SV 1  may be determined as: 
         end_sample   ⁢   _SV1     ≤       start_sample   ⁢   _SV1     +       365   360     *     sample_rate     est_freq   *   0.95               
 
where start_sample_SV 1  is the first sample of sv 1 _in, sample_rate is the sampling rate, and est_freq is the frequency from the previous cycle. The upper limit on the end of the cycle for signal SV 2  (end_sample_SV 2 ) may be determined similarly.
 
     After the end of a cycle is identified, simple checks may be made as to whether the cycle is worth processing. A cycle may not be worth processing when, for example, the conduit has stalled or the sensor waveforms are severely distorted. Processing only suitable cycles provides considerable reductions in computation. 
     One way to determine cycle suitability is to examine certain points of a cycle to confirm expected behavior. As noted above, the amplitudes and frequency of the last cycle give useful starting estimates of the corresponding values for the current cycle. Using these values, the points corresponding to 30°, 150°, 210° and 330° of the cycle may be examined. If the amplitude and frequency were to match exactly the amplitude and frequency for the previous cycle, these points should have values corresponding to est_amp/2, est_amp/2, −est_amp/2, and −est_amp/2, respectively, where est_amp is the estimated amplitude of a signal (i.e., the amplitude from the previous cycle). Allowing for a five percent change in both amplitude and frequency, inequalities may be generated for each quarter cycle. For the 30° point, the inequality is 
         sv1_in   ⁢           ⁢     (       start_sample   ⁢   _SV1     +       30   360     *     sample_rate     est_freq   *   1.05           )       &gt;     0.475   *   est_amp   ⁢   _SV         
 
The inequalities for the other points have the same form, with the degree offset term (x/360) and the sign of the est_amp_SV 1  term having appropriate values. These inequalities can be used to check that the conduit has vibrated in a reasonable manner.
 
     Measurement processing takes place on the vectors sv 1 _in(start:end) and sv 2 _in(start:end) where:
 
start=min(start_sample_SV 1 , start_sample_SV 2 ), and
 
end=max(end sample_SV 1  , end_sample_SV 2 ).
 
The difference between the start and end points for a signal is indicative of the frequency of the signal.
 
     b. Frequency Determination 
     The frequency of a discretely-sampled pure sine wave may be calculated by detecting the transition between periods (i.e., by detecting positive or negative zero-crossings) and counting the number of samples in each period. Using this method, sampling, for example, an 82.2 Hz sine wave at 55 kHz will provide an estimate of frequency with a maximum error of 0.15 percent. Greater accuracy may be achieved by estimating the fractional part of a sample at which the zero-crossing actually occurred using, for example, start_offset_SV 1  and start_offset_SV 2 . Random noise and zero offset may reduce the accuracy of this approach. 
     As illustrated in  FIGS. 8A and 8B , a more refined method of frequency determination uses quadratic interpolation of the square of the sine wave. With this method, the square of the sine wave is calculated, a quadratic function is fitted to match the minimum point of the squared sine wave, and the zeros of the quadratic function are used to determine the frequency. If
 
 sv   t   =A  sin  x   t +δσε t ,
 
where sv t  is the sensor voltage at time t, A is the amplitude of oscillation, x t  is the radian angle at time t (i.e., x t =2πft), δ is the zero offset, ε t  is a random variable with distribution N(0,1), and σ is the variance of the noise, then the squared function is given by:
 
 sv   t   2   =A   2  sin 2    x   t +2 A (δ+σε t )sin  x   t +2δσε t +δ 2 +σ 2 ε t   2 .
 
When x t  is close to 2π, sin x t  and sin 2 x t  can be approximated as x 0t =x t −2π and x 0t   2 , respectively. Accordingly, for values of x t  close to 2π, a t  can be approximated as:
 
 a   t   2   ≈A   2   x   0t   2 +2 A (δ+σε t ) x   0t +2δσε t +δ 2 +π 2 ε t   2 .
 
≈( A   2   x   0t   2 +2 Aδx   0t +δ 2 )+σε t (2 Ax   0t +2δ+σε t )
 
This is a pure quadratic (with a non-zero minimum, assuming δ=0) plus noise, with the amplitude of the noise being dependent upon both σ and δ. Linear interpolation also could be used.
 
     Error sources associated with this curve fitting technique are random noise, zero offset, and deviation from a true quadratic. Curve fitting is highly sensitive to the level of random noise. Zero offset in the sensor voltage increases the amplitude of noise in the sine-squared function, and illustrates the importance of zero offset elimination (discussed below). Moving away from the minimum, the square of even a pure sine wave is not entirely quadratic. The most significant extra term is of fourth order. By contrast, the most significant extra term for linear interpolation is of third order. 
     Degrees of freedom associated with this curve fitting technique are related to how many, and which, data points are used. The minimum is three, but more may be used (at greater computational expense) by using least-squares fitting. Such a fit is less susceptible to random noise.  FIG. 8A  illustrates that a quadratic approximation is good up to some 20° away from the minimum point. Using data points further away from the minimum will reduce the influence of random noise, but will increase the errors due to the non-quadratic terms (i.e., fourth order and higher) in the sine-squared function. 
       FIG. 9  illustrates a procedure  900  for performing the curve fitting technique. As a first step, the controller initializes variables (step  905 ). These variables include end_point, the best estimate of the zero crossing point; ep_int, the integer value nearest to end_point; s[0 . . . i], the set of all sample points; z[k], the square of the sample point closest to end_point; z[0 . . . n−1], a set of squared sample points used to calculate end_point; n, the number of sample points used to calculate end_point (n=2k+1); step_length, the number of samples in s between consecutive values in z; and iteration_count, a count of the iterations that the controller has performed. 
     The controller then generates a first estimate of end_point (step  910 ). The controller generates this estimate by calculating an estimated zero-crossing point based on the estimated frequency from the previous cycle and searching around the estimated crossing point (forwards and backwards) to find the nearest true crossing point (i.e., the occurrence of consecutive samples with different signs). The controller then sets end_point equal to the sample point having the smaller magnitude of the samples surrounding the true crossing point. 
     Next, the controller sets n, the number of points for curve fitting (step  915 ). The controller sets n equal to 5 for a sample rate of 11 kHz, and to 21 for a sample rate of 44 kHz. The controller then sets iteration_count to 0 (step  920 ) and increments iteration_count (step  925 ) to begin the iterative portion of the procedure. 
     As a first step in the iterative portion of the procedure, the controller selects step_length (step  930 ) based on the value of iteration_count. The controller sets step_length equal to 6, 3, or 1 depending on whether iteration_count equals, respectively, 1, 2 or 3. 
     Next, the controller determines ep_int as the integer portion of the sum of end_point and 0.5 (step  935 ) and fills the z array (step  940 ). For example, when n equals 5, z[0]=s[ep_int−2*step_length] 2, z[ 1]=s[ep_int−step_length] 2 , z[ 2 ]=s[ep_int] 2 , z[3]=s[ep_int+step_length] 2 , and z[4]=s[ep_int+2*step_length] 2 . 
     Next, the controller uses a filter, such as a Savitzky-Golay filter, to calculate smoothed values of z[k−1], z[k] and z[k+1] (step  945 ). Savitzky-Golay smoothing filters are discussed by Press et al. in  Numerical Recipes in C, pp.  650-655 (2nd ed., Cambridge University Press, 1995), which is incorporated by reference. The controller then fits a quadratic to z[k−1], z[k] and z[k+1] (step  950 ), and calculates the minimum value of the quadratic (z*) and the corresponding position (x*) (step  955 ). 
     If x* is between the points corresponding to k−1 and k+1 (step  960 ), then the controller sets end_point equal to x* (step  965 ). Thereafter, if iteration_count is less than 3 (step  970 ), the controller increments iteration_count (step  925 ) and repeats the iterative portion of the procedure. 
     If x* is not between the points corresponding to k−1 and k+1 (step  960 ), or if iteration_count equals 3 (step  970 ), the controller exits the iterative portion of the procedure. The controller then calculates the frequency based on the difference between end_point and the starting point for the cycle, which is known (step  975 ). 
     In essence, the procedure  900  causes the controller to make three attempts to home in on end_point, using smaller step_lengths in each attempt. If the resulting minimum for any attempt falls outside of the points used to fit the curve (i.e., there has been extrapolation rather than interpolation), this indicates that either the previous or new estimate is poor, and that a reduction in step size is unwarranted. 
     The procedure  900  may be applied to at least three different sine waves produced by the sensors. These include signals SV 1  and SV 2  and the weighted sum of the two. Moreover, assuming that zero offset is eliminated, the frequency estimates produced for these signals are independent. This is clearly true for signals SV 1  and SV 2 , as the errors on each are independent. It is also true, however, for the weighted sum, as long as the mass flow and the corresponding phase difference between signals SV 1  and SV 2  are large enough for the calculation of frequency to be based on different samples in each case. When this is true, the random errors in the frequency estimates also should be independent. 
     The three independent estimates of frequency can be combined to provide an improved estimate. This combined estimate is simply the mean of the three frequency estimates. 
     c. Zero Offset Compensation 
     An important error source in a Coriolis transmitter is zero offset in each of the sensor voltages. Zero offset is introduced into a sensor voltage signal by drift in the pre-amplification circuitry and the analog-to-digital converter. The zero offset effect may be worsened by slight differences in the pre-amplification gains for positive and negative voltages due to the use of differential circuitry. Each error source varies between transmitters, and will vary with transmitter temperature and more generally over time with component wear. 
     An example of the zero offset compensation technique employed by the controller is discussed in detail below. In general, the controller uses the frequency estimate and an integration technique to determine the zero offset in each of the sensor signals. The controller then eliminates the zero offset from those signals. After eliminating zero offset from signals SV 1  and SV 2 , the controller may recalculate the frequency of those signals to provide an improved estimate of the frequency. 
     d. Amplitude Determination 
     The amplitude of oscillation has a variety of potential uses. These include regulating conduit oscillation via feedback, balancing contributions of sensor voltages when synthesizing driver waveforms, calculating sums and differences for phase measurement, and calculating an amplitude rate of change for measurement correction purposes. 
     In one implementation, the controller uses the estimated amplitudes of signals SV 1  and SV 2  to calculate the sum and difference of signals SV 1  and SV 2 , and the product of the sum and difference. Prior to determining the sum and difference, the controller compensates one of the signals to account for differences between the gains of the two sensors. For example, the controller may compensate the data for signal SV 2  based on the ratio of the amplitude of signal SV 1  to the amplitude of signal SV 2  so that both signals have the same amplitude. 
     The controller may produce an additional estimate of the frequency based on the calculated sum. This estimate may be averaged with previous frequency estimates to produce a refined estimate of the frequency of the signals, or may replace the previous estimates. 
     The controller may calculate the amplitude according to a Fourier-based technique to eliminate the effects of higher harmonics. A sensor voltage x(t) over a period T (as identified using zero crossing techniques) can be represented by an offset and a series of harmonic terms as:
 
 x ( t )= a   0 /2 +a   1  cos(ω t )+ a   2  cos(2 ωt )+ a   3  cos(3 ωt )+ . . . + b   1  sin(ω t )+ b   2  sin(2 ωt )+ . . . 
 
     With this representation, a non-zero offset a 0  will result in non-zero cosine terms a n . Though the amplitude of interest is the amplitude of the fundamental component (i.e., the amplitude at frequency ω), monitoring the amplitudes of higher harmonic components (i.e., at frequencies kω, where k is greater than 1) may be of value for diagnostic purposes. The values of a n  and b n  may be calculated as: 
           a   n     =       2   T     ⁢       ∫   0   T     ⁢       x   ⁡     (   t   )       ⁢           ⁢   cos   ⁢           ⁢   n   ⁢           ⁢   ω   ⁢           ⁢     ⅆ   t             ,       
 
and 
         b   n     =       2   T     ⁢       ∫   0   T     ⁢       x   ⁡     (   t   )       ⁢           ⁢   sin   ⁢           ⁢   n   ⁢           ⁢   ω   ⁢           ⁢       ⅆ   t     .               
 
The amplitude, A n , of each harmonic is given by:
 
 A   n   =√{square root over (a     n           2     +b     n           2     )}. 
 
The integrals are calculated using Simpson&#39;s method with quadratic correction (described below). The chief computational expense of the method is calculating the pure sine and cosine functions.
 
     e. Phase Determination 
     The controller may use a number of approaches to calculate the phase difference between signals SV 1  and SV 2 . For example, the controller may determine the phase offset of each harmonic, relative to the starting time at t=0, as: 
         φ   n     =       tan     -   1       ⁢         a   n       b   n       .           
 
The phase offset is interpreted in the context of a single waveform as being the difference between the start of the cycle (i.e., the zero-crossing point) and the point of zero phase for the component of SV(t) of frequency ω. Since the phase offset is an average over the entire waveform, it may be used as the phase offset from the midpoint of the cycle. Ideally, with no zero offset and constant amplitude of oscillation, the phase offset should be zero every cycle. The controller may determine the phase difference by comparing the phase offset of each sensor voltage over the same time period.
 
     The amplitude and phase may be generated using a Fourier method that eliminates he effects of higher harmonics. This method has the advantage that it does not assume that both ends of the conduits are oscillating at the same frequency. As a first step in the method, a frequency estimate is produced using the zero crossings to measure the time between the start and end of the cycle. If linear variation in frequency is assumed, this estimate equals the time-averaged frequency over the period. Using the estimated, and assumed time-invariant, frequency ω of the cycle, the controller calculates: 
           I   1     =         2   ⁢           ⁢   ω     π     ⁢       ∫   0       2   ⁢           ⁢   π     ω       ⁢       SV   ⁡     (   t   )       ⁢           ⁢   sin   ⁢           ⁢     (     ω   ⁢           ⁢   t     )     ⁢     ⅆ   t             ,   and       
           I   2     =         2   ⁢           ⁢   ω     π     ⁢       ∫   0       2   ⁢           ⁢   π     ω       ⁢       SV   ⁡     (   t   )       ⁢           ⁢     cos   ⁡     (     ω   ⁢           ⁢   t     )       ⁢     ⅆ   t             ,       
 
where SV(t) is the sensor voltage waveform (i.e., SV 1 (t) or SV 2 (t)). The controller then determines the estimates of the amplitude and phase: 
         Amp   =         I   1   2     +     I   2   2           ,           ⁢   and       
       Phase   =       tan     -   1       ⁢         I   2       I   1       .           
 
     The controller then calculates a phase difference, assuming that the average phase and frequency of each sensor signal is representative of the entire waveform. Since these frequencies are different for SV 1  and SV 2 , the corresponding phases are scaled to the average frequency. In addition, the phases are shifted to the same starting point (i.e., the midpoint of the cycle on SV 1 ). After scaling, they are subtracted to provide the phase difference: 
                 scaled_phase   ⁢     _SV   1       =       phase_SV   1     ⁢     av_freq     freq_SV   1           ,                   scaled_shift   ⁢     _SV   2       =         (       midpoint_SV   2     -     midpoint_SV   1       )     ⁢   h   ⁢           ⁢     freq_SV   2       360       ,             and                 scaled_phase   ⁢     _SV   2       =       (       phase_SV   2     +     scale_shift   ⁢     _SV   2         )     ⁢     av_freq     freq_SV   2           ,             
 
where h is the sample length and the midpoints are defined in terms of samples: 
         midpoint_SV   k     =       (       startpoint_SV   k     +     endpoint_SV   x       )     2         
 
     In general, phase and amplitude are not calculated over the same time-frame for the two sensors. When the flow rate is zero, the two cycle mid-points are coincident. However, they diverge at high flow rates so that the calculations are based on sample sets that are not coincident in time. This leads to increased phase noise in conditions of changing mass flow. At full flow rate, a phase shift of 4° (out of 360°) means that only 99% of the samples in the SV 1  and SV 2  data sets are coincident. Far greater phase shifts may be observed under aerated conditions, which may lead to even lower rates of overlap. 
       FIG. 10  illustrates a modified approach 1000 that addresses this issue. First, the controller finds the frequencies (f 1 , f 2 ) and the mid-points (m 1 , m 2 ) of the SV 1  and SV 2  data sets (d 1 , d 2 ) (step  1005 ). Assuming linear shift in frequency from the last cycle, the controller calculates the frequency of SV 2  at the midpoint of SV 1  (f 2m1 ) and the frequency of SV 1  at the midpoint of SV 2  (f 1m2 ) (step  1010 ). 
     The controller then calculates the starting and ending points of new data sets (d 1m2  and d 2m1 ) with mid-points m 2  and m 1  respectively, and assuming frequencies of f 1m2  and f 2m1  (step  1015 ). These end points do not necessarily coincide with zero crossing points. However, this is not a requirement for Fourier-based calculations. 
     The controller then carries out the Fourier calculations of phase and amplitude on the sets d 1  and d 2m1 , and the phase difference calculations outlined above (step  1020 ). Since the mid-points of d 1  and d 2m1  are identical, scale-shift_SV 2  is always zero and can be ignored. The controller repeats these calculations for the data sets d 2  and d 1m2  (step  1025 ). The controller then generates averages of the calculated amplitude and phase difference for use in measurement generation (step  1030 ). When there is sufficient separation between the mid points m 1  and m 2 , the controller also may use the two sets of results to provide local estimates of the rates of change of phase and amplitude. 
     The controller also may use a difference-amplitude method that involves calculating a difference between SV 1  and SV 2 , squaring the calculated difference, and integrating the result. According to another approach, the controller synthesizes a sine wave, multiplies the sine wave by the difference between signals SV 1  and SV 2 , and integrates the result. The controller also may integrate the product of signals SV 1  and SV 2 , which is a sine wave having a frequency 2f (where f is the average frequency of signals SV 1  and SV 2 ), or may square the product and integrate the result. The controller also may synthesize a cosine wave comparable to the product sine wave and multiply the synthesized cosine wave by the product sine wave to produce a sine wave of frequency 4f that the controller then integrates. The controller also may use multiple ones of these approaches to produce separate phase measurements, and then may calculate a mean value of the separate measurements as the final phase measurement. 
     The difference-amplitude method starts with: 
             SV   1     ⁡     (   t   )       =         A   1     ⁢     sin   ⁡     (       2   ⁢   π   ⁢           ⁢   ft     +     φ   2       )       ⁢           ⁢   and   ⁢           ⁢       SV   2     ⁡     (   t   )         =       A   2     ⁢     sin   ⁡     (       2   ⁢   π   ⁢           ⁢   ft     -     φ   2       )             ,       
 
where φ is the phase difference between the sensors. Basic trigonometric identities may be used to define the sum (Sum) and difference (Diff) between the signals as: 
           Sum   ≡         SV   1     ⁡     (   t   )       +         A   1       A   2       ⁢       SV   2     ⁡     (   t   )             =     2   ⁢     A   1     ⁢   cos   ⁢       φ   2     ·   sin     ⁢           ⁢   2   ⁢   π   ⁢           ⁢   ft       ,           ⁢   and       
         Diff   ≡         SV   1     ⁡     (   t   )       -         A   1       A   2       ⁢       SV   2     ⁡     (   t   )             =     2   ⁢     A   1     ⁢   sin   ⁢       φ   2     ·   cos     ⁢           ⁢   2   ⁢   π   ⁢           ⁢     ft   .           
 
These functions have amplitudes of 2A 1  cos(φ2) and 2A 1  sin(φ2), respectively. The controller calculates data sets for Sum and Diff from the data for SV 1  and SV 2 , and then uses one or more of the methods described above to calculate the amplitude of the signals represented by those data sets. The controller then uses the calculated amplitudes to calculate the phase difference, p.
 
     As an alternative, the phase difference may be calculated using the function Prod, defined as: 
               Prod   ≡     Sum   ×   Diff       =       ⁢     4   ⁢     A   1   2     ⁢   cos   ⁢       φ   2     ·   sin     ⁢       φ   2     ·   cos     ⁢           ⁢   2   ⁢       π   ⁢   ft     ·   sin     ⁢           ⁢   2   ⁢     π   ⁢   ft                     =       ⁢       A   1   2     ⁢   sin   ⁢           ⁢     φ   ·   sin     ⁢           ⁢   4   ⁢     π   ⁢   ft         ,             
 
which is a function with amplitude A 2  sin φ and frequency 2f. Prod can be generated sample by sample, and φ may be calculated from the amplitude of the resulting sine wave.
 
     The calculation of phase is particularly dependent upon the accuracy of previous calculations (i.e., the calculation of the frequencies and amplitudes of SV 1  and SV 2 ). The controller may use multiple methods to provide separate (if not entirely independent) estimates of the phase, which may be combined to give an improved estimate. 
     f. Drive Signal Generation 
     The controller generates the drive signal by applying a gain to the difference between signals SV 1  and SV 2 . The controller may apply either a positive gain (resulting in positive feedback) or a negative gain (resulting in negative feedback). 
     In general, the Q of the conduit is high enough that the conduit will resonate only at certain discrete frequencies. For example, the lowest resonant frequency for some conduits is between 65 Hz and 95 Hz, depending on the density of the process fluid, and irrespective of the drive frequency. As such, it is desirable to drive the conduit at the resonant frequency to minimize cycle-to-cycle energy loss. Feeding back the sensor voltage to the drivers permits the drive frequency to migrate to the resonant frequency. 
     As an alternative to using feedback to generate the drive signal, pure sine waves having phases and frequencies determined as described above may be synthesized and sent to the drivers. This approach offers the advantage of eliminating undesirable high frequency components, such as harmonics of the resonant frequency. This approach also permits compensation for time delays introduced by the analog-to-digital converters, processing, and digital-to-analog converters to ensure that the phase of the drive signal corresponds to the mid-point of the phases of the sensor signals. This compensation may be provided by determining the time delay of the system components and introducing a phase shift corresponding to the time delay. 
     Another approach to driving the conduit is to use square wave pulses. This is another synthesis method, with fixed (positive and negative) direct current sources being switched on and off at timed intervals to provide the required energy. The switching is synchronized with the sensor voltage phase. Advantageously, this approach does not require digital-to-analog converters. 
     In general, the amplitude of vibration of the conduit should rapidly achieve a desired value at startup, so as to quickly provide the measurement function, but should do so without significant overshoot, which may damage the meter. The desired rapid startup may be achieved by setting a very high gain so that the presence of random noise and the high Q of the conduit are sufficient to initiate motion of the conduit. In one implementation, high gain and positive feedback are used to initiate motion of the conduit. Once stable operation is attained, the system switches to a synthesis approach for generating the drive signals. 
     Referring to  FIGS. 11A-13D , synthesis methods also may be used to initiate conduit motion when high gain is unable to do so. For example, if the DC voltage offset of the sensor voltages is significantly larger than random noise, the application of a high gain will not induce oscillatory motion. This condition is shown in  FIGS. 11A-11D , in which a high gain is applied at approximately 0.3 seconds. As shown in  FIGS. 11A and 11B , application of the high gain causes one of the drive signals to assume a large positive value ( FIG. 11A ) and the other to assume a large negative value (FIG.  11 B). The magnitudes of the drive signals vary with noise in the sensor signals (FIGS.  11 C and  11 D). However, the amplified noise is insufficient to vary the sign of the drive signals so as to induce oscillation. 
       FIGS. 12A-12D  illustrate that imposition of a square wave over several cycles can reliably cause a rapid startup of oscillation. Oscillation of a conduit having a two inch diameter may be established in approximately two seconds. The establishment of conduit oscillation is indicated by the reduction in the amplitude of the drive signals, as shown in  FIGS. 12A and 12B .  FIGS. 13A-13D  illustrate that oscillation of a one inch conduit may be established in approximately half a second. 
     A square wave also may be used during operation to correct conduit oscillation problems. For example, in some circumstances, flow meter conduits have been known to begin oscillating at harmonics of the resonant frequency of the conduit, such as frequencies on the order of 1.5 kHz. When such high frequency oscillations are detected, a square wave having a more desirable frequency may be used to return the conduit oscillation to the resonant frequency. 
     g. Measurement Generation 
     The controller digitally generates the mass flow measurement in a manner similar to the approach used by the analog controller. The controller also may generate other measurements, such as density. 
     In one implementation, the controller calculates the mass flow based on the phase difference in degrees between the two sensor signals (phase_diff), the frequency of oscillation of the conduit (freq), and the process temperature (temp):
 
 T   2 =temp− T   c ,
 
noneu_mf=tan (π*phase_diff/180), and
 
massflow=16( MF   1   *T   z   2   +MF   2   *T   z   +MF   3 )*noneu_mf/freq,
 
where T c  is a calibration temperature, MF 1 -MF 3  are calibration constants calculated during a calibration procedure, and noneu_mf is the mass flow in non-engineering units.
 
     The controller calculates the density based on the frequency of oscillation of the conduit and the process temperature:
 
 T   z =temp− T   c ,
 
 c   2 =freq 2 , and
 
density=( D   1   *T   z   2   +D   2   *T   z   +D   3 )/ c   2   +D   4   *T   z   2 ,
 
where D 1 -D 4  are calibration constants generated during a calibration procedure.
 
D. Integration Techniques
 
     Many integration techniques are available, with different techniques requiring different levels of computational effort and providing different levels of accuracy. In the described implementation, variants of Simpson&#39;s method are used. The basic technique may be expressed as: 
             ∫     t   n       t     n   +   1         ⁢     y   ⁢           ⁢     ⅆ   t         ≈       h   3     ⁢     (       y   n     +     4   ⁢     y     n   +   1         +     y     n   +   2         )         ,       
 
where t k  is the time at sample k, y k  is the corresponding function value, and h is the step length. This rule can be applied repeatedly to any data vector with an odd number of data points (i.e., three or more points), and is equivalent to fitting and integrating a cubic spline to the data points. If the number of data points happens to be even, then the so-called ⅜ths rule can be applied at one end of the interval: 
           ∫     t   n       t     n   +   1         ⁢     y   ⁢           ⁢     ⅆ   t         ≈         3   ⁢   h     8     ⁢       (       y   n     +     3   ⁢     y     n   +   1         +     3   ⁢     y     n   +   2         +     y     n   +   3         )     .           
 
     As stated earlier, each cycle begins and ends at some offset (e.g., start_offset_SV 1 ) from a sampling point. The accuracy of the integration techniques are improved considerably by taking these offsets into account. For example, in an integration of a half cycle sine wave, the areas corresponding to partial samples must be included in the calculations to avoid a consistent underestimate in the result. 
     Two types of function are integrated in the described calculations: either sine or sine-squared functions. Both are easily approximated close to zero where the end points occur. At the end points, the sine wave is approximately linear and the sine-squared function is approximately quadratic. 
     In view of these two types of functions, three different integration methods have been evaluated. These are Simpson&#39;s method with no end correction, Simpson&#39;s method with linear end correction, and Simpson&#39;s method with quadratic correction. 
     The integration methods were tested by generating and sampling pure sine and sine-squared functions, without simulating any analog-to-digital truncation error. Integrals were calculated and the results were compared to the true amplitudes of the signals. The only source of error in these calculations was due to the integration techniques. The results obtained are illustrated in tables A and B. 
     
       
         
           
               
             
               
                 TABLE A 
               
             
            
               
                   
               
               
                 Integration of a sine function 
               
            
           
           
               
               
               
               
            
               
                 % error (based 
                   
                   
                   
               
               
                 on 1000 simulations) 
                 Av. Errors (%) 
                 S.D. Error (%) 
                 Max. Error (%) 
               
               
                   
               
               
                 Simpson Only 
                 −3.73e−3  
                 1.33e−3 
                 6.17e−3 
               
               
                 Simpson + 
                 3.16e−8 
                 4.89e−8 
                 1.56e−7 
               
               
                 linear correction 
               
               
                 Simpson + 
                 2.00e−4 
                 2.19e−2 
                 5.18e−1 
               
               
                 quadratic correction 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE B 
               
             
            
               
                   
               
               
                 Integration of a sine-squared function 
               
            
           
           
               
               
               
               
            
               
                 % error (based 
                   
                   
                   
               
               
                 on 1000 simulations 
                 Av. Errors (%) 
                 S.D. Error (%) 
                 Max. Error (%) 
               
               
                   
               
               
                 Simpson Only 
                 −2.21e−6  
                 1.10e−6 
                 4.39e−3 
               
               
                 Simpson + 
                 2.21e−6 
                 6.93e−7 
                 2.52e−6 
               
               
                 linear correction 
               
               
                 Simpson + 
                  2.15e−11 
                  6.83e−11 
                  1.88e−10 
               
               
                 quadratic correction 
               
               
                   
               
            
           
         
       
     
     For sine functions, Simpson&#39;s method with linear correction was unbiased with the smallest standard deviation, while Simpson&#39;s method without correction was biased to a negative error and Simpson&#39;s method with quadratic correction had a relatively high standard deviation. For sine-squared functions, the errors were generally reduced, with the quadratic correction providing the best result. Based on these evaluations, linear correction is used when integrating sine functions and quadratic correction is used when integrating sine-squared functions. 
     E. Synchronous Modulation Technique 
       FIG. 14  illustrates an alternative procedure  1400  for processing the sensor signals. Procedure  1400  is based on synchronous modulation, such as is described by Denys et al., in “Measurement of Voltage Phase for the French Future Defence Plan Against Losses of Synchronism”, IEEE Transactions on Power Delivery, 7(1), 62-69, 1992 and by Begovic et al. in “Frequency Tracking in Power Networks in the Presence of Harmonics”, IEEE Transactions on Power Delivery, 8(2), 480-486, 1993, both of which are incorporated by reference. 
     First, the controller generates an initial estimate of the nominal operating frequency of the system (step  1405 ). The controller then attempts to measure the deviation of the frequency of a signal x[k] (e.g., SV 1 ) from this nominal frequency:
 
 x[k ]A sin [(ω 0 +Δω) kh +Φ]+ε( k ),
 
where A is the amplitude of the sine wave portion of the signal, ω 0  is the nominal frequency (e.g., 88 Hz), Δω is the deviation from the nominal frequency, h is the sampling interval, Φ is the phase shift, and ε(k) corresponds to the added noise and harmonics.
 
     To generate this measurement, the controller synthesizes two signals that oscillate at the nominal frequency (step  1410 ). The signals are phase shifted by 0 and π/2 and have amplitude of unity. The controller multiplies each of these signals by the original signal to produce signals y 1  and y 2  (step  1415 ): 
           y   1     =         x   ⁡     [   k   ]       ⁢     cos   ⁡     (       ω   0     ⁢   kh     )         =         A   2     ⁢     sin   ⁡     [         (       2   ⁢     ω   0       +   Δω     )     ⁢   kh     +   Φ     ]         +       A   2     ⁢     sin   ⁡     (       Δω   ⁢           ⁢   kh     +   Φ     )               ,           ⁢   and       
           y   2     =         x   ⁡     [   k   ]       ⁢     sin   ⁡     (       ω   0     ⁢   kh     )         =         A   2     ⁢     cos   ⁡     [         (       2   ⁢     ω   0       +   Δω     )     ⁢   kh     +   Φ     ]         +       A   2     ⁢     cos   ⁡     (       Δω   ⁢           ⁢   kh     +   Φ     )               ,       
 
     where the first terms of y 1  and y 2  are high frequency (e.g., 176 Hz) components and the second terms are low frequency (e.g., 0 Hz) components. The controller then eliminates the high frequency components using a low pass filter (step  1420 ): 
           y   1   ′     =         A   2     ⁢     sin   ⁡     (       Δω   ⁢           ⁢   kh     +   Φ     )         +       ɛ   1     ⁡     [   k   ]           ,           ⁢   and       
           y   1   ′     =         A   2     ⁢     cos   ⁡     (       Δω   ⁢           ⁢   kh     +   Φ     )         +       ɛ   2     ⁡     [   k   ]           ,       
 
where ε 1 [k] and ε 2 [k] represent the filtered noise from the original signals. The controller combines these signals to produce u[k] (step  1425 ): 
               u   ⁡     [   k   ]       =       ⁢       (         y   1   ′     ⁡     [   k   ]       +       jy   2   ′     ⁡     [   k   ]         )     ⁢     (         y   1   ′     ⁡     [     k   -   1     ]       +       jy   2   ′     ⁡     [     k   -   1     ]         )                   =       ⁢         u   1     ⁡     [   k   ]       +       ju   2     ⁡     [   k   ]                       =       ⁢           A   2     4     ⁢     cos   ⁡     (     Δω   ⁢           ⁢   h     )         +     j   ⁢       A   2     4     ⁢     sin   ⁡     (     Δω   ⁢           ⁢   h     )             ,             
 
which carries the essential information about the frequency deviation. As shown, u 1 [k] represents the real component of u[k], while u 2 [k] represents the imaginary component.
 
     The controller uses the real and imaginary components of u[k] to calculate the frequency deviation, Δf (step  1430 ): 
         Δ   ⁢           ⁢   f     =       1   h     ⁢   arctan   ⁢           u   2     ⁡     [   k   ]           u   1     ⁡     [   k   ]         .           
 
The controller then adds the frequency deviation to the nominal frequency (step  1435 ) to give the actual frequency:
 
 f=Δf+f   0 .
 
The controller also uses the real and imaginary components of u[k] to determine the amplitude of the original signal. In particular, the controller determines the amplitude as (step  1440 ):
 
 A   2 =4 √{square root over (u     1           2     [k]+u     2           2     [k])}. 
 
     Next, the controller determines the phase difference between the two sensor signals (step  1445 ). Assuming that any noise (ε 1 [k] and ε 2 [k]) remaining after application of the low pass filter described below will be negligible, noise free versions of y 1 ′[k] and y 2 ′[k] (y 1 *[k] and y 2 *[k]) may be expressed as: 
             y   1   *     ⁡     [   k   ]       =       A   2     ⁢     sin   ⁡     (       Δω   ⁢           ⁢   kh     +   Φ     )           ,           ⁢   and       
           y   2   *     ⁡     [   k   ]       =       A   2     ⁢       cos   ⁡     (       Δω   ⁢           ⁢   kh     -   Φ     )       .           
 
Multiplying these signals together gives: 
       v   =         y   1   *     ⁢     y   2   *       =           A   2     8     ⁡     [       sin   ⁡     (     2   ⁢   Φ     )       +     sin   ⁡     (     2   ⁢   Δω   ⁢           ⁢   kh     )         ]       .           
 
Filtering this signal by a low pass filter having a cutoff frequency near 0 Hz removes the unwanted component and leaves: 
           v   ′     =         A   2     8     ⁢     sin   ⁡     (     2   ⁢   Φ     )           ,       
 
from which the phase difference can be calculated as: 
       Φ   =       1   2     ⁢   arcsin   ⁢         8   ⁢     v   ′         A   2       .           
 
     This procedure relies on the accuracy with which the operating frequency is initially estimated, as the procedure measures only the deviation from this frequency. If a good estimate is given, a very narrow filter can be used, which makes the procedure very accurate. For typical flowmeters, the operating frequencies are around 95 Hz (empty) and 82 Hz (full). A first approximation of half range (88 Hz) is used, which allows a low-pass filter cut-off of 13 Hz. Care must be taken in selecting the cut-off frequency as a very small cut-off frequency can attenuate the amplitude of the sine wave. 
     The accuracy of measurement also depends on the filtering characteristics employed. The attenuation of the filter in the dead-band determines the amount of harmonics rejection, while a smaller cutoff frequency improves the noise rejection. 
     F. Meter with PI Control 
       FIGS. 15A and 15B  illustrate a meter  1500  having a controller  1505  that uses another technique to generate the signals supplied to the drivers. Analog-to-digital converters  1510  digitize signals from the sensors  48  and provide the digitized signals to the controller  1505 . The controller  1505  uses the digitized signals to calculate gains for each driver, with the gains being suitable for generating desired oscillations in the conduit. The gains may be either positive or negative. The controller  1505  then supplies the gains to multiplying digital-to-analog converters  1515 . In other implementations, two or more multiplying digital-to-analog converters arranged in series may be used to implement a single, more sensitive multiplying digital-to-analog converter. 
     The controller  1505  also generates drive signals using the digitized sensor signals. The controller  1505  provides these drive signals to digital-to-analog converters  1520  that convert the signals to analog signals that are supplied to the multiplying digital-to-analog converters  1515 . 
     The multiplying digital-to-analog converters  1515  multiply the analog signals by the gains from the controller  1505  to produce signals for driving the conduit. Amplifiers  1525  then amplify these signals and supply them to the drivers  46 . Similar results could be obtained by having the controller  1505  perform the multiplication performed by the multiplying digital-to-analog converter, at which point the multiplying digital-to-analog converter could be replaced by a standard digital-to-analog converter. 
       FIG. 15B  illustrates the control approach in more detail. Within the controller  1505 , the digitized sensor signals are provided to an amplitude detector  1550 , which determines a measure, a(t), of the amplitude of motion of the conduit using, for example, the technique described above. A summer  1555  then uses the amplitude a(t) and a desired amplitude a 0  to calculate an error e(t) as:
   e ( t )= a   0   −a ( t ). 
The error e(t) is used by a proportional-integral (“PI”) control block  1560  to generate a gain K 0 (t). This gain is multiplied by the difference of the sensor signals to generate the drive signal. The PI control block permits high speed response to changing conditions. The amplitude detector  1550 , summer  1555 , and PI control block  1560  may be implemented as software processed by the controller  1505 , or as separate circuitry.
 
     1. Control Procedure 
     The meter  1500  operates according to the procedure  1600  illustrated in FIG.  16 . Initially, the controller receives digitized data from the sensors (step  1605 ). Thereafter, the procedure  1600  includes three parallel branches: a measurement branch  1610 , a drive signal generation branch  1615 , and a gain generation branch  1620 . 
     In the measurement branch  1610 , the digitized sensor data is used to generate measurements of amplitude, frequency, and phase, as described above (step  1625 ). These measurements then are used to calculate the mass flow rate (step  1630 ) and other process variables. In general, the controller  1505  implements the measurement branch  1610 . 
     In the drive signal generation branch  1615 , the digitized signals from the two sensors are differenced to generate the signal (step  1635 ) that is multiplied by the gain to produce the drive signal. As described above, this differencing operation is performed by the controller  1505 . In general, the differencing operation produces a weighted difference that accounts for amplitude differences between the sensor signals. 
     In the gain generation branch  1620 , the gain is calculated using the proportional-integral control block. As noted above, the amplitude, a(t), of motion of the conduit is determined (step  1640 ) and subtracted from the desired amplitude a 0  (step  1645 ) to calculate the error e(t). Though illustrated as a separate step, generation of the amplitude, a(t), may correspond to generation of the amplitude in the measurement generation step  1625 . Finally, the PI control block uses the error e(t) to calculate the gain (step  1650 ). 
     The calculated gain is multiplied by the difference signal to generate the drive signal supplied to the drivers (step  1655 ). As described above, this multiplication operation is performed by the multiplying D/A converter or may be performed by the controller. 
     2. PI Control Block 
     The objective of the PI control block is to sustain in the conduit pure sinusoidal oscillations having an amplitude a 0 . The behavior of the conduit may be modeled as a simple mass-spring system that may be expressed as:
 
 {umlaut over (x)}+ 2ζω n   {dot over (x)}+ω   n   2   x= 0,
 
where x is a function of time and the displacement of the mass from equilibrium, ω n  is the natural frequency, and ζ is a damping factor, which is assumed to be small (e.g., 0.001). The solution to this force equation as a function of an output y(t) and an input i(t) is analogous to an electrical network in which the transfer function between a supplied current, i(s), and a sensed output voltage, y(s), is: 
           y   ⁡     (   s   )         i   ⁡     (   s   )         =       ks       s   2     +     2   ⁢     ζω   n     ⁢   s     +     ω   n   2         .         
 
     To achieve the desired oscillation in the conduit, a positive-feedback loop having the gain K 0 (t) is automatically adjusted by a ‘slow’ outer loop to give:
 
 {umlaut over (x)} +(2ζω n   −kK   0 ( t )) {dot over (x)}+ω   n   2   x =0.
 
The system is assumed to have a “two-time-scales” property, which means that variations in K 0 (t) are slow enough that solutions to the equation for x provided above can be obtained by assuming constant damping.
 
     A two-term PI control block that gives zero steady-state error may be expressed as:
 
 K   0 ( t )= K   p   e ( t )+ K   i ∫ 0   t   e ( t ) dt, 
 
where the error, e(t) (i.e., a 0 −a(t)), is the input to the PI control block, and K p  and K i  are constants. In one implementation, with a 0 =10, controller constants of K p =0.02 and K i =0.0005 provide a response in which oscillations build up quickly. However, this PI control block is nonlinear, which may result in design and operational difficulties.
 
     A linear model of the behavior of the oscillation amplitude may be derived by assuming that x(t) equals Aε jωt , which results in:
 
 {dot over (x)}={dot over (A)}e   jωt   +jωe   jωt , and
 
 {umlaut over (x)}=[Ä−ω   2   A]e   jωt +2 jω{dot over (A)}e   jωt .
 
Substituting these expressions into the expression for oscillation of the loop, and separating into real and imaginary terms, gives:
 
 jω {2 {dot over (A)} +(2ζω n   −kK   0 ) A }=0, and
 
 Ä +(2ζω n   −kK   0 ( A +(ω n   2 −ω 2 ) A =0.
 
A(t) also may be expressed as: 
           A   .     A     =       -     ζω   n       +         kK   0     2     ⁢     t   .             
 
A solution of this equation is: 
         log   ⁢           ⁢     A   ⁡     (   t   )         =       (       -     ζω   n       +       kK   0     2       )     ⁢     t   .           
 
Transforming variables by defining a(t) as being equal to log A(t), the equation for A(t) can be written as: 
             ⅆ   a       ⅆ   t       =       -     ζω   n       +         kK   0     ⁡     (   t   )       2         ,       
 
where K 0  is now explicitly dependent on time. Taking Laplace transforms results in: 
           a   ⁡     (   s   )       =         -     ζω   n       -         kK   0     ⁡     (   s   )       /   2       s       ,       
 
which can be interpreted in terms of transfer-functions as in FIG.  17 . This figure is of particular significance for the design of controllers, as it is linear for all K o  and a, with the only assumption being the two-time-scales property. The performance of the closed-loop is robust with respect to this assumption so that fast responses which are attainable in practice can be readily designed.
 
     From  FIG. 17 , the term ζω n , is a “load disturbance” that needs to be eliminated by the controller (i.e., kK o /2 must equal ζω n  for a(t) to be constant). For zero steady-state error this implies that the outer-loop controller must have an integrator (or very large gain). As such, an appropriate PI controller, C(s), may be assumed to be K p (1+1/sT i ), where T i  is a constant. The proportional term is required for stability. The term ζω n , however, does not affect stability or controller design, which is based instead on the open-loop transfer function: 
           C   ⁡     (   s   )       ⁢     G   ⁡     (   s   )         =         a   ⁡     (   s   )         e   ⁡     (   s   )         =           kK   p     ⁡     (     1   +     sT   i       )         2   ⁢     s   2     ⁢     T   i         =             kK   p     /   2     ⁢     (     s   +     1   /     T   i         )         s   2       .             
 
     The root locus for varying K p  is shown in FIG.  18 . For small K p , there are slow underdamped roots. As K p  increases, the roots become real at the point P for which the controller gain is K p =8/(kT i ). Note in particular that the theory does not place any restriction upon the choice of T i . Hence, the response can, in principle, be made critically damped and as fast as desired by appropriate choices of K p  and T i . 
     Although the poles are purely real at point P, this does not mean there is no overshoot in the closed-loop step response. This is most easily seen by inspecting the transfer function between the desired value, a 0 , and the error e: 
             e   ⁡     (   s   )           a   0     ⁡     (   s   )         =         s   2         s   2     +     0.5   ⁢       kK   p     ⁡     (     s   +     1   /     T   i         )             =       s   2         p   2     ⁡     (   s   )             ,       
 
where P 2  is a second-order polynomial. With a step input, a o (s)=α/s, the response can be written as αp′(t), where p(t) is the inverse transform of 1/p 2 (s) and equals a 1  exp(−λ 1 t)+a 2  exp(−λ 2 t). The signal p(t) increases and then decays to zero so that e(t), which is proportional to p′, must change sign, implying overshoot in a(t). The set-point a o  may be prefiltered to give a pseudo set-point a o *: 
             a   0   *     ⁡     (   s   )       =       1     1   +     sT   i         ⁢       a   0     ⁡     (   s   )           ,       
 
where T i  is the known controller parameter. With this prefilter, real controller poles should provide overshoot-free step responses. This feature is useful as there may be physical constraints on overshoot (e.g., mechanical interference or overstressing of components).
 
     The root locus of  FIG. 18  assumes that the only dynamics are from the inner-loop&#39;s gain/log-amplitude transfer function ( FIG. 16 ) and the outer-loop&#39;s PI controller C(s) (i.e., that the log-amplitude a=log A is measured instantaneously). However, A is the amplitude of an oscillation which might be growing or decaying and hence cannot in general be measured without taking into account the underlying sinusoid. There are several possible methods for measuring A, in addition to those discussed above. Some are more suitable for use in quasi-steady conditions. For example, a phase-locked loop in which a sinusoidal signal s(t)=sin(ω n t+Φ 0 ) locks onto the measured waveform y(t)=A(t)sin(ω n t+Φ 1 ) may be employed. Thus, a measure of the amplitude a=log A is given by dividing these signals (with appropriate safeguards and filters). This method is perhaps satisfactory near the steady-state but not for start-up conditions before there is a lock. 
     Another approach uses a peak-follower that includes a zero-crossing detector together with a peak-following algorithm implemented in the controller. Zero-crossing methods, however, can be susceptible to noise. In addition, results from a peak-follower are available only every half-cycle and thereby dictate the sample interval for controller updates. 
     Finally, an AM detector may be employed. Given a sine wave y(t)=A sinω n t, an estimate of A may be obtained from Â 1   0.5π{abs(y)}, where F{ } is a suitable low-pass filter with unity DC gain. The AM detector is the simplest approach. Moreover, it does not presume that there are oscillations of any particular frequency, and hence is usable during startup conditions. It suffers from a disadvantage that there is a leakage of harmonics into the inner loop which will affect the spectrum of the resultant oscillations. In addition, the filter adds extra dynamics into the outer loop such that compromises need to be made between speed of response and spectral purity. In particular, an effect of the filter is to constrain the choice of the best T   i . 
     The Fourier series for abs(y) is known to be: 
         A   ⁢           ⁢     abs   ⁡     (     sin   ⁢           ⁢     ω     n   ⁢               ⁢   t     )         =           2   ⁢   A     π     ⁡     [     1   +       2   3     ⁢   cos   ⁢           ⁢   2   ⁢     ω   n     ⁢   t     -       2   15     ⁢   cos   ⁢           ⁢   4   ⁢     ω   n     ⁢   t     +       2   35     ⁢   cos   ⁢           ⁢   6   ⁢     ω   n     ⁢   t     +   …     ]       .         
 
As such, the output has to be scaled by π/2 to give the correct DC output A, and the (even) harmonic terms a k  cos 2kω n t have to be filtered out. As all the filter needs to do is to pass the DC component through and reduce all other frequencies, a “brick-wall” filter with a cut-off below 2ω n  is sufficient. However, the dynamics of the filter will affect the behavior of the closed-loop. A common choice of filter is in the Butterworth form. For example, the third-order low-pass filter with a design break-point frequency ω b  is: 
         F   ⁡     (   s   )       =       1     1   +     2   ⁢     s   /     ω   b         +     2   ⁢       s   2     /     ω   b   2         +       s   3     /     ω   b   3           .         
 
At the design frequency the response is 3 dB down; at 2ω b  it is −18 dB (0.12), and at 4ω b  it is −36 dB (0.015) down. Higher-order Butterworth filters have a steeper roll-off, but most of their poles are complex and may affect negatively the control-loop&#39;s root locus.
 
G. Zero Offset Compensation
 
     As noted above, zero offset may be introduced into a sensor voltage signal by drift in the pre-amplification circuitry and by the analog-to-digital converter. Slight differences in the pre-amplification gains for positive and negative voltages due to the use of differential circuitry may worsen the zero offset effect. The errors vary between transmitters, and with transmitter temperature and component wear. 
     Audio quality (i.e., relatively low cost) analog-to-digital converters may be employed for economic reasons. These devices are not designed with DC offset and amplitude stability as high priorities.  FIGS. 19A-19D  show how offset and positive and negative gains vary with chip operating temperature for one such converter (the AD1879 converter). The repeatability of the illustrated trends is poor, and even allowing for temperature compensation based on the trends, residual zero offset and positive/negative gain mismatch remain. 
     If phase is calculated using the time difference between zero crossing points on the two sensor voltages, DC offset may lead to phase errors. This effect is illustrated by  FIGS. 20A-20C . Each graph shows the calculated phase offset as measured by the digital transmitter when the true phase offset is zero (i.e., at zero flow). 
       FIG. 20A  shows phase calculated based on whole cycles starting with positive zero-crossings. The mean value is 0.00627 degrees. 
       FIG. 20B  shows phase calculated starting with negative zero-crossings. The mean value is 0.0109 degrees. 
       FIG. 20C  shows phase calculated every half-cycle.  FIG. 20C  interleaves the data from  FIGS. 20A and 20B . The average phase (−0.00234) is closer to zero than in  FIGS. 20A and 20B , but the standard deviation of the signal is about six times higher. 
     More sophisticated phase measurement techniques, such as those based on Fourier methods, are immune to DC offset. However, it is desirable to eliminate zero offset even when those techniques are used, since data is processed in whole-cycle packets delineated by zero crossing points. This allows simpler analysis of the effects of, for example, amplitude modulation on apparent phase and frequency. In addition, gain mismatch between positive and negative voltages will introduce errors into any measurement technique. 
     The zero-crossing technique of phase detection may be used to demonstrate the impact of zero offset and gain mismatch error, and their consequent removal.  FIGS. 21A and 21B  illustrate the long term drift in phase with zero flow. Each point represents an average over one minute of live data.  FIG. 21A  shows the average phase, and  FIG. 21B  shows the standard deviation in phase. Over several hours, the drift is significant. Thus, even if the meter were zeroed every day, which in many applications would be considered an excessive maintenance requirement, there would still be considerable phase drift. 
     1. Compensation Technique 
     A technique for dealing with voltage offset and gain mismatch uses the computational capabilities of the digital transmitter and does not require a zero flow condition. The technique uses a set of calculations each cycle which, when averaged over a reasonable period (e.g., 10,000 cycles), and excluding regions of major change (e.g., set point change, onset of aeration), converge on the desired zero offset and gain mismatch compensations. 
     Assuming the presence of up to three higher harmonics, the desired waveform for a sensor voltage SV(t) is of the form:
 
 SV ( t )= A   1  sin(ω t )+ A   2  sin(2ω t )+ A   3  sin(3 ωt )+ A   4  sin(4 ωt )
 
where A 1  designates the amplitude of the fundamental frequency component and A 2 -A 4  designate the amplitudes of the three harmonic components. However, in practice, the actual waveform is adulterated with zero offset Z O  (which has a value close to zero) and mismatch between the negative and positive gains G n  and G p . Without any loss of generality, it can be assumed that G p  equals one and that G n  is given by:
 
 G   n =1+ε G ,
 
where ε G  represents the gain mismatch.
 
     The technique assumes that the amplitudes A i  and the frequency ω are constant. This is justified because estimates of Z o  and ε G  are based on averages taken over many cycles (e.g., 10,000 interleaved cycles occurring in about 1 minute of operation). When implementing the technique, the controller tests for the presence of significant changes in frequency and amplitude to ensure the validity of the analysis. The presence of the higher harmonics leads to the use of Fourier techniques for extracting phase and amplitude information for specific harmonics. This entails integrating SV(t) and multiplying by a modulating sine or cosine function. 
     The zero offset impacts the integral limits, as well as the functional form. Because there is a zero offset, the starting point for calculation of amplitude and phase will not be at the zero phase point of the periodic waveform SV(t). For zero offset Z o , the corresponding phase offset is, approximately, 
         φ     Z   0       =     -     sin   ⁡     (       Z   0       A   1       )             
 
For small phase 
           φ     Z   0       =     -       Z   0       A   1           ,       
 
with corresponding time delay 
         t     Z   0       =         φ     Z   0       ω     .         
 
     The integrals are scaled so that the limiting value (i.e., as Z o  and ω G  approach zero) equals the amplitude of the relevant harmonic. The first two integrals of interest are: 
           I     1   ⁢   Ps       =         2   ⁢   ω     π     ⁢       ∫     t     z   0           π   ω     +     t     z   0           ⁢         (       SV   ⁡     (   t   )       +     Z   0       )     ·     sin   ⁡     [     ω   ⁡     (     t   -     t     Z   0         )       ]         ⁢           ⁢     ⅆ   t             ,           ⁢       and   ⁢     
     ⁢     I     1   ⁢   Ns         =         2   ⁢   ω     π     ⁢     (     1   +     ɛ   G       )     ⁢       ∫       π   ω     ⁢     t     z   0             2   ⁢     π   ω       +     t     z   0           ⁢         (       SV   ⁡     (   t   )       +     Z   0       )     ·     sin   ⁡     [     ω   ⁡     (     t   -     t     Z   0         )       ]         ⁢           ⁢       ⅆ   t     .                 
 
These integrals represent what in practice is calculated during a normal Fourier analysis of the sensor voltage data. The subscript 1 indicates the first harmonic, N and P indicate, respectively, the negative or positive half cycle, and s and c indicate, respectively, whether a sine or a cosine modulating function has been used.
 
     Strictly speaking, the mid-zero crossing point, and hence the corresponding integral limits, should be given by π/ω−t Zo , rather than π/tω+t Zo . However, the use of the exact mid-point rather than the exact zero crossing point leads to an easier analysis, and better numerical behavior (due principally to errors in the location of the zero crossing point). The only error introduced by using the exact mid-point is that a small section of each of the above integrals is multiplied by the wrong gain (1 instead of 1+ε G  and vice versa). However, these errors are of order Z o   2 ε G  and are considered negligible. 
     Using computer algebra and assuming small Z o  and ε G , first order estimates for the integrals may be derived as: 
           I     1   ⁢   Ps_est       =       A   1     +       4   π     ⁢       Z   0     ⁡     [     1   +       2   3     ⁢       A   2       A   1         +       4   15     ⁢       A   4       A   1           ]             ,           ⁢   and       
         I     1   ⁢   Ns_est       =         (     1   +     ɛ   G       )     ⁡     [       A   1     -       4   π     ⁢       Z   0     ⁡     [     1   +       2   3     ⁢       A   2       A   1         +       4   15     ⁢       A   4       A   1           ]           ]       .         
 
     Useful related functions including the sum, difference, and ratio of the integrals and their estimates may be determined. The sum of the integrals may be expressed as:
 
Sum 1s =( I   1Ps   +I   1Ns ),
 
while the sum of the estimates equals: 
         Sum     1   ⁢   s_est       =         A   1     ⁡     (     2   +     ɛ   G       )       -       4   π     ⁢     Z   0     ⁢         ɛ   G     ⁡     [     1   +       2   3     ⁢       A   2       A   1         +       4   15     ⁢       A   4       A   1           ]       .             
 
Similarly, the difference of the integrals may be expressed as:
 
Diff 1s   =I   1Ps   −I   1Ns ,
 
while the difference of the estimates is: 
         Diff     1   ⁢   s_est       =         A   1     ⁢     ɛ   G       +       4   π     ⁢           Z   0     ⁡     (     2   +     ɛ   G       )       ⁡     [     1   +       2   3     ⁢       A   2       A   1         +       4   15     ⁢       A   4       A   1           ]       .             
 
Finally, the ratio of the integrals is: 
           Ratio     1   ⁢   s       =       I     1   ⁢   Ps         I     1   ⁢   Ns           ,       
 
while the ratio of the estimates is: 
         Ratio     1   ⁢   s_est       =         1     1   +     ɛ   G         ⁡     [     1   +       Z   0     ⁡     [       8   15     ⁢         15   ⁢     A   1       +     10   ⁢     A   2       +     4   ⁢     A   4           π   ⁢           ⁢     A   1   2           ]         ]       .         
 
     Corresponding cosine integrals are defined as: 
           I     1   ⁢   Pc       =         2   ⁢   ω     π     ⁢       ∫     t     Z   0           π   ω     +     t     Z   0           ⁢       (       SV   ⁡     (   t   )       +     Z   0       )     ⁢     cos   ⁡     [     ω   ⁡     (     t   -     t     z   0         )       ]       ⁢           ⁢     ⅆ   t             ,           ⁢   and       
           I     1   ⁢   Nc       =         2   ⁢   ω     π     ⁢     (     1   +     ɛ   G       )     ⁢       ∫       π   ω     +     t     Z   0               2   ⁢   π     ω     +     t     Z   0           ⁢       (       SV   ⁡     (   t   )       +     Z   0       )     ⁢     cos   ⁡     [     ω   ⁡     (     t   -     t     z   0         )       ]       ⁢           ⁢     ⅆ   t             ,       
 
with estimates: 
           I     1   ⁢   Pc_est       =       -     Z   0       +         40   ⁢     A   2       +     16   ⁢     A   4           15   ⁢   π           ,           ⁢   and       
           I     1   ⁢   Nc_est       =       (     1   +     ɛ   G       )     ⁡     [       Z   0     +         40   ⁢     A   2       +     16   ⁢     A   4           15   ⁢   π         ]         ,       
 
and sums: 
           Sum     1   ⁢   C       =       I     1   ⁢   Pc       +     I     1   ⁢   Nc           ,           ⁢   and       
         Sum     1   ⁢   C_est       =         ɛ   G     ⁡     [       Z   0     +         40   ⁢     A   2       +     16   ⁢     A   4           15   ⁢   π         ]       .         
 
     Second harmonic integrals are: 
           I     2   ⁢   Ps       =         2   ⁢   ω     π     ⁢       ∫     t     z   0           π   ω     +     t     z   0           ⁢       (       SV   ⁡     (   t   )       +     Z   0       )     ⁢     sin   ⁡     [     2   ⁢     ω   ⁡     (     t   -     t     z   0         )         ]       ⁢           ⁢     ⅆ   t             ,           ⁢   and       
         I     2   ⁢   Ns       =         2   ⁢   ω     π     ⁢     (     1   +     ɛ   G       )     ⁢       ∫       π   ω     +     t     z   0               2   ⁢   π     ω     +     t     z   0           ⁢     (         SV   ⁡     (   t   )       +       Z   0     ⁢     sin   ⁡     [     2   ⁢     ω   ⁡     (     t   -     t     Z   0         )         ]       ⁢           ⁢     ⅆ   t         ,               
 
with estimates: 
           I     2   ⁢   Ps_est       =       A   2     +       8     15   ⁢   π       ⁢       Z   0     ⁡     [       -   5     +     9   ⁢       A   3       A   1           ]             ,           ⁢   and       
           I     2   ⁢   Ps_est       =       (     1   +     ɛ   G       )     ⁡     [       A   2     -       8     15   ⁢   π       ⁢       Z   0     ⁡     [       -   5     +     9   ⁢       A   3       A   1           ]           ]         ,       
 
and sums: 
           Sum     2   s       =       I     2   ⁢   Ps       +     I     2   ⁢   Ns           ,           ⁢   and       
         Sum     2   ⁢   Ps_est       =         A   2     ⁡     (     2   +     ɛ   G       )       -       8     15   ⁢   π       ⁢     ɛ   G     ⁢         Z   0     ⁡     [       -   5     +     9   ⁢       A   3       A   1           ]       .             
 
     The integrals can be calculated numerically every cycle. As discussed below, the equations estimating the values of the integrals in terms of various amplitudes and the zero offset and gain values are rearranged to give estimates of the zero offset and gain terms based on the calculated integrals. 
     2. Example 
     The accuracy of the estimation equations may be illustrated with an example. For each basic integral, three values are provided: the “true” value of the integral (calculated within Mathcad using Romberg integration), the value using the estimation equation, and the value calculated by the digital transmitter operating in simulation mode, using Simpson&#39;s method with end correction. 
     Thus, for example, the value for lips calculated according to: 
         I     1   ⁢   Ps       =         2   ⁢   ω     π     ⁢       ∫     t     z   0           π   ω     +     t     z   0           ⁢       (       SV   ⁡     (   t   )       +     Z   0       )     ⁢     sin   ⁡     [     ω   ⁡     (     t   -     t     z   0         )       ]       ⁢     ⅆ   t               
 
is 0.101353, while the estimated value (I 1Ps     —     est ) calculated as: 
         I     1   ⁢   Ps_est       =       A   1     +       4   π     ⁢       Z   0     ⁡     [     1   +       2   3     ⁢       A   2       A   1         +       4   15     ⁢       A   4       A   1           ]               
 
is 0.101358. The value calculated using the digital transmitter in simulation mode is 0.101340. These calculations use the parameter values illustrated in Table C.
 
                             TABLE C               Parameter   Value   Comment                  ω   160π   This corresponds to frequency = 80 Hz, a typical               value.       A 1     0.1   This more typically is 0.3, but it could be smaller               with aeration.       A 2     0.01   This more typically is 0.005, but it could be larger               with aeration.       A 3  and A 4     0.0   The digital Coriolis simulation mode only offers two               harmonics, so these higher harmonics are ignored.               However, they are small (&lt;0.002).       Z o     0.001   Experience suggests that this is a large value for               zero offset.       ε G     0.001   Experience suggests this is a large value for gain               mismatch.                    
The exact, estimate and simulation results from using these parameter values are illustrated in Table D.
 
     
       
         
           
               
               
               
               
             
               
                 TABLE D 
               
               
                   
               
               
                 Integral 
                 ‘Exact’ Value 
                 Estimate 
                 Digital Coriolis Simulation 
               
               
                   
               
             
            
               
                 I 1Ps   
                 0.101353 
                 0.101358 
                 0.101340 
               
               
                 I 1Ns   
                 0.098735 
                 0.098740 
                 0.098751 
               
               
                 I 1Pc   
                 0.007487 
                 0.007488 
                 0.007500 
               
               
                 I 1Nc   
                 −0.009496  
                 −0.009498  
                 −0.009531  
               
               
                 I 2Ps   
                 0.009149 
                 0.009151 
                 0.009118 
               
               
                 I 2Ns   
                 0.010857 
                 0.010859 
                 0.010885 
               
               
                   
               
            
           
         
       
     
     Thus, at least for the particular values selected, the estimates given by the first order equations are extremely accurate. As Z o  and ε G  approach zero, the errors in both the estimate and the simulation approach zero. 
     3. Implementation 
     The first order estimates for the integrals define a series of non-linear equations in terms of the amplitudes of the harmonics, the zero offset, and the gain mismatch. As the equations are non-linear, an exact solution is not readily available. However, an approximation followed by corrective iterations provides reasonable convergence with limited computational overhead. 
     Conduit-specific ratios may be assumed for A 1 -A 4 . As such, no attempt is made to calculate all of the amplitudes A 1 -A 4 . Instead, only A 1  and A 2  are estimated using the integral equations defined above. Based on experience of the relative amplitudes, A 3  may be approximated as A 2 /2, and A 4  may be approximated as A 2 /10. 
     The zero offset compensation technique may be implemented according to the procedure  2200  illustrated in FIG.  22 . During each cycle, the controller calculates the integrals I 1Ps , I 1Ns , I 1Pc , I 1Nc , I 2Ps , I 2Ns  and related functions sum 1s , ratio 1s , sum 1c  and sum 2s , (step  2205 ). This requires minimal additional calculation beyond the conventional Fourier calculations used to determine frequency, amplitude and phase. 
     Every 10,000 cycles, the controller checks on the slope of the sensor voltage amplitude A 1 , using a conventional rate-of-change estimation technique (step  2210 ). If the amplitude is constant (step  2215 ), then the controller proceeds with calculations for zero offset and gain mismatch. This check may be extended to test for frequency stability. 
     To perform the calculations, the controller generates average values for the functions (e.g., sum 1s ) over the last 10,000 cycles. The controller then makes a first estimation of zero offset and gain mismatch (step  2225 ):
 
 Z   0 =−Sum 1c /2, and
 
ε G =1/Ratio 1s −1
 
     Using these values, the controller calculates an inverse gain factor (k) and amplitude factor (amp_factor) (step  2230 ):
 
 k =1.0/(1.0+0.5*ε G ), and
 
amp_factor=1+50/75*Sum 2s /Sum 1s 
 
The controller uses the inverse gain factor and amplitude factor to make a first estimation of the amplitudes (step  2235 ):
 
 A   1   =k *[Sum 1s /2+2/ π*Z   o *ε G *amp_factor], and
 
 A   2   =k *[Sum 2s /2−4/(3*π)*Z o *ε G 
 
     The controller then improves the estimate by the following calculations, iterating as required (step  2240 ):
 
x 1 =Z o ,
 
x 2 =ε G ,
 
ε G =[1+8/π* x   1   /A   1 *amp_factor]/Ratio 1s −1.0,
 
 Z   o =−Sum 1c /2+ x   2 *( x   1 +2.773/π *A   2 )/2,
 
 A   1   =k *[Sum 1s /2+2/π* x   1   *x   2 *amp_factor],
 
 A   2   =k *[Sum 2s /2−4/(15*π)* x   1   *x   2 *(5−4.5 *A   2 )].
 
The controller uses standard techniques to test for convergence of the values of Z o  and ε G . In practice the corrections are small after the first iteration, and experience suggests that three iterations are adequate.
 
     Finally, the controller adjusts the raw data to eliminate Z o  and ε G  (step  2245 ). The controller then repeats the procedure. Once zero offset and gain mismatch have been eliminated from the raw data, the functions (i.e., sum 1s ) used in generating subsequent values for Z o  and ε G  are based on corrected data. Accordingly, these subsequent values for Z o  and ε G  reflect residual zero offset and gain mismatch, and are summed with previously generated values to produce the actual zero offset and gain mismatch. In one approach to adjusting the raw data, the controller generates adjustment parameters (e.g., S 1 _off and S 2 _off) that are used in converting the analog signals from the sensors to digital data. 
       FIGS. 23A-23C ,  24 A and  24 B show results obtained using the procedure  2200 . The short-term behavior is illustrated in  FIGS. 23A-23C . This shows consecutive phase estimates obtained five minutes after startup to allow time for the procedure to begin affecting the output. Phase is shown based on positive zero-crossings, negative zero-crossings, and both. 
     The difference between the positive and negative mean values has been reduced by a factor of  20 , with a corresponding reduction in mean zero offset in the interleaved data set. The corresponding standard deviation has been reduced by a factor of approximately 6. 
     Longer term behavior is shown in  FIGS. 24A and 24B . The initial large zero offset is rapidly corrected, and then the phase offset is kept close to zero over many hours. The average phase offset, excluding the first few values, is 6.14e − 6, which strongly suggests that the procedure is successful in compensating for changes in voltage offset and gain imbalance. 
     Typical values for Z o  and ε G  for the digital Coriolis meter are Z o =−7.923e −4  and ε G =−1.754e −5  for signal SV 1 , and Z o =−8.038e −4  and ε G =+6.93e −4  for signal SV 2 . 
     H. Dynamic Analysis 
     In general, conventional measurement calculations for Coriolis meters assume that the frequency and amplitude of oscillation on each side of the conduit are constant, and that the frequency on each side of the conduit is identical and equal to the so-called resonant frequency. Phases generally are not measured separately for each side of the conduit, and the phase difference between the two sides is assumed to be constant for the duration of the measurement process. Precise measurements of frequency, phase and amplitude every half-cycle using the digital meter demonstrate that these assumptions are only valid when parameter values are averaged over a time period on the order of seconds. Viewed at 100 Hz or higher frequencies, these parameters exhibit considerable variation. For example, during normal operation, the frequency and amplitude values of SV 1  may exhibit strong negative correlation with the corresponding SV 2  values. Accordingly, conventional measurement algorithms are subject to noise attributable to these dynamic variations. The noise becomes more significant as the measurement calculation rate increases. Other noise terms may be introduced by physical factors, such as flowtube dynamics, dynamic non-linearities (e.g. flowtube stiffness varying with amplitude), or the dynamic consequences of the sensor voltages providing velocity data rather than absolute position data. 
     The described techniques exploit the high precision of the digital meter to monitor and compensate for dynamic conduit behavior to reduce noise so as to provide more precise measurements of process variables such as mass flow and density. This is achieved by monitoring and compensating for such effects as the rates of change of frequency, phase and amplitude, flowtube dynamics, and dynamic physical non-idealities. A phase difference calculation which does not assume the same frequency on each side has already been described above. Other compensation techniques are described below. 
     Monitoring and compensation for dynamic effects may take place at the individual sensor level to provide corrected estimates of phase, frequency, amplitude or other parameters. Further compensation may also take place at the conduit level, where data from both sensors are combined, for example in the calculation of phase difference and average frequency. These two levels may be used together to provide comprehensive compensation. 
     Thus, instantaneous mass flow and density measurements by the flowmeter may be improved by modelling and accounting for dynamic effects of flowmeter operation. In general, 80% or more of phase noise in a Coriolis flowmeter may be attributed to flowtube dynamics (sometimes referred to as “ringing”), rather than to process conditions being measured. The application of a dynamic model can reduce phase noise by a factor of 4 to 10, leading to significantly improved flow measurement performance. A single model is effective for all flow rates and amplitudes of oscillation. Generally, computational requirements are negligible. 
     The dynamic analysis may be performed on each of the sensor signals in isolation from the other. This avoids, or at least delays, modelling the dynamic interaction between the two sides of the conduit, which is likely to be far more complex than the dynamics at each sensor. Also, analyzing the individual sensor signals is more likely to be successful in circumstances such as batch startup and aeration where the two sides of the conduit are subject to different forces from the process fluid. 
     In general, the dynamic analysis considers the impact of time-varying amplitude, frequency and phase on the calculated values for these parameters. While the frequency and amplitude are easily defined for the individual sensor voltages, phase is conventionally defined in terms of the difference between the sensor voltages. However, when a Fourier analysis is used, phase for the individual sensor may be defined in terms of the difference between the midpoint of the cycle and the average 180° phase point. 
     Three types of dynamic effects are measurement error and the so-called “feedback” and “velocity” effects. Measurement error results because the algorithms for calculating amplitude and phase assume that frequency, amplitude, and phase are constant over the time interval of interest. Performance of the measurement algorithms may be improved by correcting for variations in these parameters. 
     The feedback effect results from supplying energy to the conduit to make up for energy loss from the conduit so as to maintain a constant amplitude of oscillation. The need to add energy to the conduit is only recognized after the amplitude of oscillation begins to deviate from a desired setpoint. As a result, the damping term in the equation of motion for the oscillating conduit is not zero, and, instead, constantly dithers around zero. Although the natural frequency of the conduit does not change, it is obscured by shifts in the zero-crossings (i.e., phase variations) associated with these small changes in amplitude. 
     The velocity effect results because the sensor voltages observe conduit velocity, but are analyzed as being representative of conduit position. A consequence of this is that the rate of change of amplitude has an impact on the apparent frequency and phase, even if the true values of these parameters are constant. 
     1. Sensor-Level Compensation for Amplitude Modulation 
     One approach to correcting for dynamic effects monitors the amplitudes of the sensor signals and makes adjustments based on variations in the amplitudes. For purposes of analyzing dynamic effects, it is assumed that estimates of phase, frequency and amplitude may be determined for each sensor voltage during each cycle. As shown in  FIG. 25 , calculations are based on complete but overlapping cycles. Each cycle starts at a zero crossing point, halfway through the previous cycle. Positive cycles begin with positive voltages immediately after the initial zero-crossing, while negative cycles begin with negative voltages. Thus cycle n is positive, while cycles n−1 and n+1 are negative. It is assumed that zero offset correction has been performed so that zero offset is negligible. It also is assumed that higher harmonics may be present. 
     Linear variation in amplitude, frequency, and phase are assumed. Under this assumption, the average value of each parameter during a cycle equals the instantaneous value of the parameter at the mid-point of the cycle. Since the cycles overlap by 180 degrees, the average value for a cycle equals the starting value for the next cycle. 
     For example, cycle n is from time 0 to π/ω. The average values of amplitude, frequency and phase equal the instantaneous values at the mid-point, π/ω, which is also the starting point for cycle n+1, which is from time π/ω to 3π/ω. Of course, these timings are approximate, since co also varies with time. 
     a. Dynamic Effect Compensation Procedure 
     The controller accounts for dynamic effects according to the procedure  2600  illustrated in FIG.  26 . First, the controller produces a frequency estimate (step  2605 ) by using the zero crossings to measure the time between the start and end of the cycle, as described above. Assuming that frequency varies linearly, this estimate equals the time-averaged frequency over the period. 
     The controller then uses the estimated frequency to generate a first estimate of amplitude and phase using the Fourier method described above (step  2610 ). As noted above, this method eliminates the effects of higher harmonics. 
     Phase is interpreted in the context of a single waveform as being the difference between the start of the cycle (i.e., the zero-crossing point) and the point of zero phase for the component of SV(t) of frequency ω, expressed as a phase offset. Since the phase offset is an average over the entire waveform, it may be used as the phase offset from the midpoint of the cycle. Ideally, with no zero offset and constant amplitude of oscillation, the phase offset should be zero every cycle. In practice, however, it shows a high level of variation and provides an excellent basis for correcting mass flow to account for dynamic changes in amplitude. 
     The controller then calculates a phase difference (step  2615 ). Though a number of definitions of phase difference are possible, the analysis assumes that the average phase and frequency of each sensor signal is representative of the entire waveform. Since these frequencies are different for SV 1  and SV 2 , the corresponding phases are scaled to the average frequency. In addition, the phases are shifted to the same starting point (i.e., the midpoint of the cycle on SV 1 ). After scaling, they are subtracted to provide the phase difference. 
     The controller next determines the rate of change of the amplitude for the cycle n 
                     roc_amp   n     ≈       ⁢         amp   ⁡     (     end   ⁢           ⁢   of   ⁢           ⁢   cycle     )       -     amp   ⁡     (     start   ⁢           ⁢   of   ⁢           ⁢   cycle     )           period   ⁢           ⁢   of   ⁢           ⁢   cycle                   =       ⁢       (       amp     n   +   1       -     amp     n   -   1         )     ⁢       freq   n     .                   
 
This calculation assumes that the amplitude from cycle n+1 is available when calculating the rate of change of cycle n. This is possible if the corrections are made one cycle after the raw amplitude calculations have been made. The advantage of having an accurate estimate of the rate of change, and hence good measurement correction, outweighs the delay in the provision of the corrected measurements, which, in one implementation, is on the order of 5 milliseconds. The most recently generated information is always used for control of the conduit (i.e., for generation of the drive signal).
 
     If desired, a revised estimate of the rate of change can be calculated after amplitude correction has been applied (as described below). This results in iteration to convergence for the best values of amplitude and rate of change. 
     b. Frequency Compensation for Feedback and Velocity Effects 
     As noted above, the dynamic aspects of the feedback loop introduce time varying shifts in the phase due to the small deviations in amplitude about the set-point. This results in the measured frequency, which is based on zero-crossings, differing from the natural frequency of the conduit. If velocity sensors are used, an additional shift in phase occurs. This additional shift is also associated with changes in the positional amplitude of the conduit. A dynamic analysis can monitor and compensate for these effects. Accordingly, the controller uses the calculated rate of amplitude change to correct the frequency estimate (step  2625 ). 
     The position of an oscillating conduit in a feedback loop that is employed to maintain the amplitude of oscillation of the conduit constant may be expressed as: 
       X=A ( t )sin(ω 0   t −θ( t )), 
     where θ(t) is the phase delay caused by the feedback effect. The mechanical Q of the oscillating conduit is typically on the order of 1000, which implies small deviations in amplitude and phase. Under these conditions, θ(t) is given by: 
         θ   ⁡     (   t   )       ≈     -           A   .     ⁡     (   t   )         2   ⁢     ω   0     ⁢     A   ⁡     (   t   )           .           
 
Since each sensor measures velocity: 
               SV   ⁡     (   t   )       =       ⁢       X   .     ⁡     (   t   )                   =       ⁢           A   .     ⁡     (   t   )       ⁢     sin   ⁡     [         ω   0     ⁢   t     -     θ   ⁡     (   t   )         ]         +       [       ω   0     -       θ   .     ⁡     (   t   )         ]     ⁢     A   ⁡     (   t   )       ⁢     cos   ⁡     [         ω   0     ⁢   t     -     θ   ⁡     (   t   )         ]                         =       ⁢       ω   0     ⁢         A   ⁡     (   t   )       ⁡     [         (     1   -     θ     ω   0         )     2     +       (         A   .     ⁡     (   t   )           ω   0     ⁢     A   ⁡     (   t   )           )     2       ]         1   /   2       ⁢     cos   ⁡     (         ω   0     ⁢   t     -     θ   ⁡     (   t   )       -     γ   ⁡     (   t   )         )           ,             
 
where γ(t) is the phase delay caused by the velocity effect: 
         γ   ⁡     (   t   )       =         tan     -   1       (         A   .     ⁡     (   t   )           ω   0     ⁢     A   ⁡     (   t   )       ⁢     (     1   -       θ   .       ω   0         )         )     .         
 
Since the mechanical Q of the conduit is typically on the order of 1000, and, hence, variations in amplitude and phase are small, it is reasonable to assume: 
           θ   .       ω   0       ⁢     &lt;&lt;   1     ⁢           ⁢   and   ⁢           ⁢         A   .     ⁡     (   t   )           ω   0     ⁢     A   ⁡     (   t   )           ⁢     &lt;&lt;   1.         
 
This means that the expression for SV(t) may be simplified to:
 
 SV ( t )≈ω 0   A ( t )cos(ω 0   t −θ( t )−γ( t )),
 
and for the same reasons, the expression for the velocity offset phase delay may be simplified 
         γ   ⁡     (   t   )       ≈           A   .     ⁡     (   t   )           ω   0     ⁢     A   ⁡     (   t   )           .         
 
Summing the feedback and velocity effect phase delays gives the total phase delay: 
           φ   ⁡     (   t   )       =           θ   ⁡     (   t   )       +     γ   ⁡     (   t   )         ≈       -         A   .     ⁡     (   t   )         2   ⁢     ω   0     ⁢     A   ⁡     (   t   )             +         A   .     ⁡     (   t   )           w   0     ⁢     A   ⁡     (   t   )               =         A   .     ⁡     (   t   )         2   ⁢     ω   0     ⁢     A   ⁡     (   t   )               ,       
 
and the following expression for SV(t):
 
  SV ( t )≈ω 0   A ( t )cos[ω o   t −φ( t )].
 
     From this, the actual frequency of oscillation may be distinguished from the natural frequency of oscillation. Though the former is observed, the latter is useful for density calculations. Over any reasonable length of time, and assuming adequate amplitude control, the averages of these two frequencies are the same (because the average rate of change of amplitude must be zero). However, for improved instantaneous density measurement, it is desirable to compensate the actual frequency of oscillation for dynamic effects to obtain the natural frequency. This is particularly useful in dealing with aerated fluids for which the instantaneous density can vary rapidly with time. 
     The apparent frequency observed for cycle n is delineated by zero crossings occurring at the midpoints of cycles n−1 and n+1. The phase delay due to velocity change will have an impact on the apparent start and end of the cycle: 
               obs_freq   n     =       ⁢       obs_freq     n   -   1       +         true_freq   n       2   ⁢   π       ⁢     (       φ     n   +   1       -     φ     n   -   1         )                     =       ⁢       obs_freq     n   -   1       +         true_freq     n   -   1         2   ⁢   π       ⁢     (           A   .       n   +   1         4   ⁢   π   ⁢           ⁢     true_freq   n     ⁢     A     n   +   1           -                           ⁢         A   .       n   -   1         4   ⁢   π   ⁢           ⁢     true_freq   n     ⁢     A     n   -   1           )               =       ⁢       obs_freq     n   -   1       +       1     8   ⁢     π   2         ⁢       (           A   .       n   +   1         A     n   +   1         -         A   .       n   -   1         A     n   -   1           )     .                   
 
Based on this analysis, a correction can be applied using an integrated error term: 
           error_sum   n     =       error_sum     n   -   1       -       1     8   ⁢     π   2         ⁢     (           A   .       n   +   1         A     n   +   1         -         A   .       n   -   1         A     n   -   1           )           ,           ⁢   and       
           est_freq   n     =       obs_freq   n     -     error_sum   n         ,       
 
where the value of error_sum at startup (i.e., the value at cycle zero) is: 
         error_sum   0     =       -     1     8   ⁢     π   2           ⁢       (           A   .     0       A   0       +         A   .     1       A   1         )     .           
 
Though these equations include a constant term having a value of 1/87π 2 , actual data has indicated that a constant term of ⅛π is more appropriate. This discrepancy may be due to unmodelled dynamics that may be resolved through further analysis.
 
     The calculations discussed above assume that the true amplitude of oscillation, A, is available. However, in practice, only the sensor voltage SV is observed. This sensor voltage may be expressed as: 
       SV ( t )≈ω 0   {dot over (A)} ( t )cos(ω 0   t −ω( t )) 
     The amplitude, amp_SV(t), of this expression is:
 
amp —   SV ( t )≈ω 0   {dot over (A)} ( t ).
 
The rate of change of this amplitude is:
 
roc_amp —   SV ( t )≈ω 0   {dot over (A)} ( t )
 
so that the following estimation can be used: 
             A   .     ⁡     (   t   )         A   ⁡     (   t   )         ≈         roc_amp   ⁢   _SV   ⁢     (   t   )         amp_SV   ⁢     (   t   )         .         
 
     c. Application of Feedback and Velocity Effect Frequency Compensation 
       FIGS. 27A-32B  illustrate how application of the procedure  2600  improves the estimate of the natural frequency, and hence the process density, for real data from a meter having a one inch diameter conduit. Each of the figures shows 10,000 samples, which are collected in just over 1 minute. 
       FIGS. 27A and 27B  show amplitude and frequency data from SV 1 , taken when random changes to the amplitude set-point have been applied. Since the conduit is full of water and there is no flow, the natural frequency is constant. However, the observed frequency varies considerably in response to changes in amplitude. The mean frequency value is 81.41 Hz, with a standard deviation of 0.057 Hz. 
       FIGS. 28A and 28B  show, respectively, the variation in frequency from the mean value, and the correction term generated using the procedure  2600 . The gross deviations are extremely well matched. However, there is additional variance in frequency which is not attributable to amplitude variation. Another important feature illustrated by  FIG. 28B  is that the average is close to zero as a result of the proper initialization of the error term, as described above. 
       FIGS. 29A and 92B  compare the raw frequency data ( FIG. 29A ) with the results of applying the correction function (FIG.  29 B). There has been a negligible shift in the mean frequency, while the standard deviation has been reduced by a factor of 4.4. From  FIG. 29B , it is apparent that there is residual structure in the corrected frequency data. It is expected that further analysis, based on the change in phase across a cycle and its impact on the observed frequency, will yield further noise reductions. 
       FIGS. 30A and 30B  show the corresponding effect on the average frequency, which is the mean of the instantaneous sensor voltage frequencies. Since the mean frequency is used to calculate the density of the process fluid, the noise reduction (here by a factor of 5.2) will be propagated into the calculation of density. 
       FIGS. 31A and 31B  illustrate the raw and corrected average frequency for a 2″ diameter conduit subject to a random amplitude set-point. The 2″ flowtube exhibits less frequency variation that the 1″, for both raw and corrected data. The noise reduction factor is 4.0. 
       FIGS. 32A and 32B  show more typical results with real flow data for the one inch flowtube. The random setpoint algorithm has been replaced by the normal constant setpoint. As a result, there is less amplitude variation than in the previous examples, which results in a smaller noise reduction factor of 1.5. 
     d. Compensation of Phase Measurement for Amplitude Modulation 
     Referring again to  FIG. 26 , the controller next compensates the phase measurement to account for amplitude modulation assuming the phase calculation provided above (step  2630 ). The Fourier calculations of phase described above assume that the amplitude of oscillation is constant throughout the cycle of data on which the calculations take place. This section describes a correction which assumes a linear variation in amplitude over the cycle of data. 
     Ignoring higher harmonics, and assuming that any zero offset has been eliminated, the expression for the sensor voltage is given by:
 
 SV ( t )≈ A   1  (1+λ A   t )sin(ω t )
 
where λ A  is a constant corresponding to the relative change in amplitude with time. As discussed above, the integrals I 1  and I 2  may be expressed as: 
           I   1     =         2   ⁢   ω     π     ⁢       ∫   0       2   ⁢   π     ω       ⁢       SV   ⁡     (   t   )       ⁢     sin   ⁡     (     ω   ⁢           ⁢   t     )       ⁢           ⁢     ⅆ   t             ,           ⁢   and       
         I   2     =         2   ⁢   ω     π     ⁢       ∫   0       2   ⁢   π     ω       ⁢       SV   ⁡     (   t   )       ⁢     cos   ⁡     (     ω   ⁢           ⁢   t     )       ⁢           ⁢       ⅆ   t     .               
 
Evaluating these integrals results in: 
           I   1     =       A   1     ⁡     (     1   +       π   ω     ⁢     λ   A         )         ,           ⁢   and       
         I   2     =       A   1     ⁢     1     2   ⁢   ω       ⁢       λ   A     .           
 
Substituting these expressions into the calculation for amplitude and expanding as a series in λ A  results in: 
       Amp   =         A   1     ⁡     (     1   +       π   ω     ⁢     λ   A       +       1     8   ⁢     ω   2         ⁢     λ   A   2       +   …     )       .         
 
     Assuming λ A  is small, and ignoring all terms after the first order term, this may be simplified to: 
       Amp   =         A   1     ⁡     (     1   +       π   ω     ⁢     λ   A         )       .         
 
This equals the amplitude of SV(t) at the midpoint of the cycle (t=π/ω). Accordingly, the amplitude calculation provides the required result without correction.
 
     For the phase calculation, it is assumed that the true phase difference and frequency are constant, and that there is no voltage offset, which means that the phase value should be zero. However, as a result of amplitude modulation, the correction to be applied to the raw phase data to compensate for amplitude modulation is: 
       Phase   =         tan     -   1       ⁡     (       λ   A       2   ⁢     (       πλ   A     +   ω     )         )       .         
 
Assuming that the expression in brackets is small, the inverse tangent function can be ignored.
 
     A more elaborate analysis considers the effects of higher harmonics. Assuming that the sensor voltage may be expressed as:
 
 SV ( t )=(1+λ A   t )[ A   1  sin(ω t )+A 2  sin(2 ωt )+ A   3  sin(ω t )+ A   4  sin(4 ωt )]
 
such that all harmonic amplitudes increase at the same relative rate over the cycle, then the resulting integrals may be expressed as: 
           I   1     =       A   1     ⁡     (     1   +       π   ω     ⁢     λ   A         )         ,       
 
and 
         I   2     ⁢       -   1       60   ⁢   ω       ⁢       λ   A     ⁡     (       30   ⁢     A   1       +     80   ⁢     A   2       +     45   ⁢     A   3       +     32   ⁢     A   4         )           
 
for positive cycles, and 
         I   2     ⁢       -   1       60   ⁢   ω       ⁢       λ   A     ⁡     (       30   ⁢     A   1       -     80   ⁢     A   2       +     45   ⁢     A   3       -     32   ⁢     A   4         )           
 
for negative cycles.
 
     For amplitude, substituting these expressions into the calculations establishes that the amplitude calculation is only affected in the second order and higher terms, so that no correction is necessary to a first order approximation of the amplitude. For phase, the correction term becomes: 
           -   1     60     ⁢       λ   A     ⁡     (         30   ⁢     A   1       +     80   ⁢     A   2       +     45   ⁢     A   3       +     32   ⁢     A   4             A   1     ⁡     (       πλ   A     +   ω     )         )           
 
for positive cycles, and 
           -   1     60     ⁢       λ   A     ⁡     (         30   ⁢     A   1       -     80   ⁢     A   2       +     45   ⁢     A   3       -     32   ⁢     A   4             A   1     ⁡     (       πλ   A     +   ω     )         )           
 
for negative cycles. These correction terms assume the availability of the amplitudes of the higher harmonics. While these can be calculated using the usual Fourier technique, it is also possible to approximate some or all them using assumed ratios between the harmonics. For example, for one implementation of a one inch diameter conduit, typical amplitude ratios are A 1 =1.0, A 2 =0.01, A 3 =0.005, and A 4 =0.001.
 
     e. Application of Amplitude Modulation Compensation to Phase 
     Simulations have been carried out using the digital transmitter, including the simulation of higher harmonics and amplitude modulation. One example uses f=80 Hz, A 1 (t=0)=0.3, A 2 =0, A 3  =0, A 4 =0, λ A = 1e −5 *48 KHz (sampling rate)=0.47622, which corresponds to a high rate of change of amplitude, but with no higher harmonics. Theory suggests a phase offset of −0.02706 degrees. In simulation over 1000 cycles, the average offset is −0.02714 degrees, with a standard deviation of only 2.17e − 6. The difference between simulation and theory (approx 0.3% of the simulation error) is attributable to the model&#39;s assumption of a linear variation in amplitude over each cycle, while the simulation generates an exponential change in amplitude. 
     A second example includes a second harmonic, and has the parameters f=80 Hz, A 1 (t=0)=0.3, A 2 (t=0)=0.003, A 3 =0, A 4 =0, λ A =−1e −6 *48 KHz (sampling rate)=−0.047622. For this example, theory predicts the phase offset to be +2.706e −3 , +/−2.66% for positive or negative cycles. In simulation, the results are 2.714e −3+/−2.66 %, which again matches well. 
       FIGS. 33A-34B  give examples of how this correction improves real flowmeter data.  FIG. 33A  shows raw phase data from SV 1 , collected from a 1″ diameter conduit, with low flow assumed to be reasonably constant.  FIG. 33B  shows the correction factor calculated using the formula described above, while  FIG. 33C  shows the resulting corrected phase. The most apparent feature is that the correction has increased the variance of the phase signal, while still producing an overall reduction in the phase difference (i.e., SV 2 −SV 1 ) standard deviation by a factor of 1.26, as shown in  FIGS. 34A and 34B . The improved performance results because this correction improves the correlation between the two phases, leading to reduced variation in the phase difference. The technique works equally well in other flow conditions and on other conduit sizes. 
     f. Compensation to Phase Measurement for Velocity Effect 
     The phase measurement calculation is also affected by the velocity effect. A highly effective and simple correction factor, in radians, is of the form 
             c   v     ⁡     (     t   k     )       =       1   π     ⁢   Δ   ⁢           ⁢     SV   ⁡     (     t   k     )           ,       
 
where ΔSV(t k ) is the relative rate of change of amplitude and may be expressed as: 
           Δ   ⁢           ⁢     SV   ⁡     (     t   k     )         =           SV   ⁡     (     t     k   +   1       )       -     SV   ⁡     (     t     k   -   1       )             t     k   +   1       -     t     k   -   1           ·     1     SV   ⁡     (     t   k     )             ,       
 
where t k  is the completion time for the cycle for which ΔSV(t k ) is being determined, t k+1  is the completion time for the next cycle, and t k−l  is the completion time of the previous cycle. ΔSV is an estimate of the rate of change of SV, scaled by its absolute value, and is also referred to as the proportional rate of change of SV.
 
       FIGS. 35A-35E  illustrate this technique.  FIG. 35A  shows the raw phase data from a single sensor (SV 1 ), after having applied the amplitude modulation corrections described above.  FIG. 35B  shows the correction factor in degrees calculated using the equation above, while  FIG. 35C  shows the resulting corrected phase. It should be noted that the standard deviation of the corrected phase has actually increased relative to the raw data. However, when the corresponding calculations take place on the other sensor (SV 2 ), there is an increase in the negative correlation (from −0.8 to −0.9) between the phases on the two signals. As a consequence, the phase difference calculations based on the raw phase measurements ( FIG. 35D ) have significantly more noise than the corrected phase measurements (FIG.  35 E). 
     Comparison of  FIGS. 35D and 35E  shows the benefits of this noise reduction technique. It is immediately apparent from visual inspection of  FIG. 35E  that the process variable is decreasing, and that there are significant cycles in the measurement, with the cycles being attributable, perhaps, to a poorly conditioned pump. None of this is discernable from the uncorrected phase difference data of FIG.  35 D. 
     g. Application of Sensor Level Noise Reduction 
     The combination of phase noise reduction techniques described above results in substantial improvements in instantaneous phase difference measurement in a variety of flow conditions, as illustrated in  FIGS. 36A-36L . Each graph shows three phase difference measurements calculated simultaneously in real time by the digital Coriolis transmitter operating on a one inch conduit. The middle band  3600  shows phase data calculated using the simple time-difference technique. The outermost band  3605  shows phase data calculated using the Fourier-based technique described above. 
     It is perhaps surprising that the Fourier technique, which uses far more data, a more sophisticated analysis, and much more computational effort, results in a noisier calculation. This can be attributed to the sensitivity of the Fourier technique to the dynamic effects described above. The innermost band of data  3610  shows the same Fourier data after the application of the sensor-level noise reduction techniques. As can be seen, substantial noise reduction occurs in each case, as indicated by the standard deviation values presented on each graph. 
       FIG. 36A  illustrates measurements with no flow, a full conduit, and no pump noise.  FIG. 36B  illustrates measurements with no flow, a full conduit, and the pumps on.  FIG. 36C  illustrates measurements with an empty, wet conduit.  FIG. 36D  illustrates measurements at a low flow rate.  FIG. 36E  illustrates measurements at a high flow rate.  FIG. 36F  illustrates measurements at a high flow rate and an amplitude of oscillation of 0.03V.  FIG. 36G  illustrates measurements at a low flow rate with low aeration.  FIG. 36H  illustrates measurements at a low flow rate with high aeration.  FIG. 361  illustrates measurements at a high flow rate with low aeration.  FIG. 36J  illustrates measurements at a high flow rate with high aeration.  FIG. 36K  illustrates measurements for an empty to high flow rate transition.  FIG. 36L  illustrates measurements for a high flow rate to empty transition. 
     2. Flowtube Level Dynamic Modelling 
     A dynamic model may be incorporated in two basic stages. In the first stage, the model is created using the techniques of system identification. The flowtube is “stimulated” to manifest its dynamics, while the true mass flow and density values are kept constant. The response of the flowtube is measured and used in generating the dynamic model. In the second stage, the model is applied to normal flow data. Predictions of the effects of flowtube dynamics are made for both phase and frequency. The predictions then are subtracted from the observed data to leave the residual phase and frequency, which should be due to the process alone. Each stage is described in more detail below. 
     a. System Identification 
     System identification begins with a flowtube full of water, with no flow. The amplitude of oscillation, which normally is kept constant, is allowed to vary by assigning a random setpoint between 0.05 V and 0.3 V, where 0.3 V is the usual value. The resulting sensor voltages are shown in  FIG. 37A , while  FIGS. 37B and 37C  show, respectively, the corresponding calculated phase and frequency values. These values are calculated once per cycle. Both phase and frequency show a high degree of “structure.” Since the phase and frequency corresponding to mass flow are constant, this structure is likely to be related to flowtube dynamics. Observable variables that will predict this structure when the true phase and frequency are not known to be constant may be expressed as set forth below. 
     First, as noted above, ΔSV(t k ) may be expressed as: 
         Δ   ⁢           ⁢   S   ⁢           ⁢     V   ⁡     (     t   k     )         =           S   ⁢           ⁢     V   ⁡     (     t     k   +   1       )         -     S   ⁢           ⁢     V   ⁡     (     t     k   -   1       )               t     k   +   1       -     t     k   -   1           ·       1     S   ⁢           ⁢     V   ⁡     (     t   k     )           .           
 
This expression may be used to determine ΔSV 1  and ΔSV 2 .
 
     The phase of the flowtube is related to Δ, which is defined as ΔSV 1 −ΔSV 2 , while the frequency is related to Δ + , which is defined as ΔSV 1 +ΔSV 2 . These parameters are illustrated in  FIGS. 37D and 37E . Comparing  FIG. 37B  to FIG.  37 D and  FIG. 37C  to  FIG. 37E  shows the striking relationship between Δ −  and phase and between Δ +  and frequency. 
     Some correction for flowtube dynamics may be obtained by subtracting a multiple of the appropriate prediction function from the phase and/or the frequency. Improved results may be obtained using a model of the form:
 
 y ( k )+a 1   y ( k− 1)+ . . . + a   n   y ( k−n )= b   0   u ( k )+ b   1   u ( k− 1)+ . . . + b   m   u ( k−m ),
 
where y(k) is the output (i.e., phase or frequency) and u is the prediction function (i.e., Δ − or Δ   + ). The technique of system identification suggests values for the orders n and m, and the coefficients a i  and b j , of what are in effect polynomials in time. The value of y(k) can be calculated every cycle and subtracted from the observed phase or frequency to get the residual process value.
 
     It is important to appreciate that, even in the absence of dynamic corrections, the digital flowmeter offers very good precision over a long period of time. For example, when totalizing a batch of 200 kg, the device readily achieves a repeatability of less that 0.03%. The purpose of the dynamic modelling is to improve the dynamic precision. Thus, raw and compensated values should have similar mean values, but reductions in “variance” or “standard deviation.” 
       FIGS. 38A and 39A  show raw and corrected frequency values. The mean values are similar, but the standard deviation has been reduced by a factor of 3.25. Though the gross deviations in frequency have been eliminated, significant “structure” remains in the residual noise. This structure appears to be unrelated to the Δ +  function. The model used is a simple first order model, where m=n=1. 
       FIGS. 38B and 39B  show the corresponding phase correction. The mean value is minimally affected, while the standard deviation is reduced by a factor of 7.9. The model orders are n=2 and m=10. Some structure appears to remain in the residual noise. It is expected that this structure is due to insufficient excitation of the phase dynamics by setpoint changes. 
     More effective phase identification has been achieved through further simulation of flowtube dynamics by continuous striking of the flowtube during data collection (set point changes are still carried out).  FIGS. 38C and 39C  show the effects of correction under these conditions. As shown, the standard deviation is reduced by a factor of 31. This more effective model is used in the following discussions. 
     b. Application to Flow Data 
     The real test of an identified model is the improvements it provides for new data. At the outset, it is useful to note a number of observations. First, the mean phase, averaged over, for example, ten seconds or more, is already quite precise. In the examples shown, phase values are plotted at 82 Hz or thereabouts. The reported standard deviation would be roughly ⅓ of the values shown when averaged to 10 Hz, and 1/9 when averages to 1 Hz. For reference, on a one inch flow tube, one degree of phase difference corresponds to about 1 kg/s flow rate. 
     The expected benefit of the technique is that of providing a much better dynamic response to true process changes, rather than improving average precision. Consequently, in the following examples, where the flow is non-zero, small flow step changes are introduced every ten seconds or so, with the expectation that the corrected phase will show up the step changes more clearly. 
       FIGS. 38D and 39D  show the correction applied to a full flowtube with zero flow, just after startup. The ring-down effect characteristic of startup is clearly evident in the raw data (FIG.  38 D), but this is eliminated by the correction (FIG.  39 D), leading to a standard deviation reduction of a factor of 23 over the whole data set. Note that the corrected measurement closely resembles white noise, suggesting most flowtube dynamics have been captured. 
       FIGS. 38E and 39E  show the resulting correction for a “drained” flowtube. Noise is reduced by a factor of 6.5 or so. Note, however, that there appears to be some residual structure in the noise. 
     The effects of the technique on low (FIGS.  38 F and  39 F), medium (FIGS.  38 G and  39 G), and high ( FIGS. 38H and 39H ) flow rates are also illustrated, each with step changes in flow every ten seconds. In each case, the pattern is the same: the corrected average flows ( FIGS. 39F-39H ) are identical to the raw average flows (FIGS.  38 F- 38 H), but the dynamic noise is reduced considerably. In  FIG. 39H , this leads to the emergence of the step changes which previously had been submerged in noise (FIG.  38 H). 
     3. Extensions of Dynamic Monitoring and Compensation Techniques 
     The previous sections have described a variety of techniques (physical modelling, system identification, heuristics) used to monitor and compensate for different aspects of dynamic behavior (frequency and phase noise caused by amplitude modulation, velocity effect, flowtube dynamics at both the sensor and the flowtube level). By natural extension, similar techniques well-known to practitioners of control and/or instrumentation, including those of artificial intelligence, neural networks, fuzzy logic, and genetic algorithms, as well as classical modelling and identification methods, may be applied to these and other aspects of the dynamic performance of the meter. Specifically, these might include monitoring and compensation for frequency, amplitude and/or phase variation at the sensor level, as well as average frequency and phase difference at the flowtube level, as these variations occur within each measurement interval, as well as the time between measurement intervals (where measurement intervals do not overlap). 
     This technique is unusual in providing both reduced noise and improved dynamic response to process measurement changes. As such, the technique promises to be highly valuable in the context of flow measurement. 
     I. Aeration (Two-Phase Flow) 
     The digital flowmeter provides improved performance in the presence of aeration (also known as two-phase flow) in the conduit. Aeration causes energy losses in the conduit that can have a substantial negative impact on the measurements produced by a mass flowmeter and can result in stalling of the conduit. Experiments have shown that the digital flowmeter has substantially improved performance in the presence of aeration relative to traditional, analog flowmeters. This performance improvement may stem from the meter&#39;s ability to provide a very wide gain range, to employ negative feedback, to calculate measurements precisely at very low amplitude levels, and to compensate for dynamic effects such as rate of change of amplitude and flowtube dynamics. The performance improvement also may stem from the meter&#39;s use of a precise digital amplitude control algorithm. 
     The digital flowmeter detects the onset of aeration when the required driver gain rises simultaneously with a drop in apparent fluid density. The digital flowmeter then may directly respond to detected aeration. In general, the meter monitors the presence of aeration by comparing the observed density of the material flowing through the conduit (i.e., the density measurement obtained through normal measurement techniques) to the known, non-aerated density of the material. The controller determines the level of aeration based on any difference between the observed and actual densities. The controller then corrects the mass flow measurement accordingly. 
     The controller determines the non-aerated density of the material by monitoring the density over time periods in which aeration is not present (i.e., periods in which the density has a stable value). Alternatively, a control system to which the controller is connected may provide the non-aerated density as an initialization parameter. 
     In one implementation, the controller uses three corrections to account for the effects of aeration: bubble effect correction, damping effect correction, and sensor imbalance correction.  FIGS. 40A-40H  illustrate the effects of the correction procedure. 
       FIG. 40A  illustrates the error in the phase measurement as the measured density decreases (i.e., as aeration increases) for different mass flow rates, absent aeration correction. As shown, the phase error is negative and has a magnitude that increases with increasing aeration.  FIG. 40B  illustrates that the resulting mass flow error also is negative. It also is significant to note that the digital flowmeter operates at high levels of aeration. By contrast, as indicated by the vertical bar  4000 , traditional analog meters tend to stall in the presence of low levels of aeration. 
     A stall occurs when the flowmeter is unable to provide a sufficiently large driver gain to allow high drive current at low amplitudes of oscillation. If the level of damping requires a higher driver gain than can be delivered by the flowtube in order to maintain oscillation at a certain amplitude, then insufficient drive energy is supplied to the conduit. This results in a drop in amplitude of oscillation, which in turn leads to even less drive energy supplied due to the maximum gain limit. Catastrophic collapse results, and flowtube oscillation is not possible until the damping reduces to a level at which the corresponding driver gain requirement can be supplied by the flowmeter. 
     The bubble effect correction is based on the assumption that the mass flow decreases as the level of aeration, also referred to as the void fraction, increases. Without attempting to predict the actual relationship between void fraction and the bubble effect, this correction assumes, with good theoretical justification, that the effect on the observed mass flow will be the same as the effect on the observed density. Since the true fluid density is known, the bubble effect correction corrects the mass flow rate by the same proportion. This correction is a linear adjustment that is the same for all flow rates.  FIGS. 40C and 40D  illustrate, respectively, the residual phase and mass flow errors after correction for the bubble effect. As shown, the residual errors are now positive and substantially smaller in magnitude than the original errors. 
     The damping factor correction accounts for damping of the conduit motion due to aeration. In general, the damping factor correction is based on the following relationship between the observed phase, Φ obs , and the actual phase, Φ true : 
           φ   obs     =       φ   true     ⁡     (     1   +       k   ⁢           ⁢     λ   2     ⁢     φ   true         f   2         )         ,       
 
where λ is a damping coefficient and k is a constant.  FIG. 40E  illustrates the damping correction for different mass flow rates and different levels of aeration.  FIG. 40F  illustrates the residual phase error after correction for damping. As shown, the phase error is reduced substantially relative to the phase error remaining after bubble effect correction.
 
     The sensor balance correction is based on density differences between different ends of the conduit. As shown in  FIG. 41 , a pressure drop between the inlet and the outlet of the conduit results in increasing bubble sizes from the inlet to the outlet. Since material flows serially through the two loops of the conduit, the bubbles at the inlet side of the conduit (i.e., the side adjacent to the first sensor/driver pair) will be smaller than the bubbles at the outlet side of the conduit (i.e., the side adjacent to the second sensor/driver pair). This difference in bubble size results in a difference in mass and density between the two ends of the conduit. This difference is reflected in the sensor signals (SV 1  and SV 2 ). Accordingly, the sensor balance correction is based on a ratio of the two sensor signals. 
       FIG. 40G  illustrates the sensor balance correction for different mass flow rates and different levels of aeration.  FIG. 40H  illustrates the residual phase error after applying the sensor balance correction. At low flow rates and low levels of aeration, the phase error is improved relative to the phase error remaining after damping correction. 
     Other correction factors also may be used. For example, the phase angle of each sensor signal may be monitored. In general, the average phase angle for a signal should be zero. However, the average phase angle tends to increase with increasing aeration. Accordingly, a correction factor could be generated based on the value of the average phase angle. Another correction factor could be based on the temperature of the conduit. 
     In general, application of the correction factors tends to keep the mass flow errors at one percent or less. Moreover, these correction factors appear to be applicable over a wide range of flows and aeration levels. 
     J. Setpoint Adjustment 
     The digital flowmeter provides improved control of the setpoint for the amplitude of oscillation of the conduit. In an analog meter, feedback control is used to maintain the amplitude of oscillation of the conduit at a fixed level corresponding to a desired peak sensor voltage (e.g., 0.3 V). A stable amplitude of oscillation leads to reduced variance in the frequency and phase measurements. 
     In general, a large amplitude of oscillation is desirable, since such a large amplitude provides a large Coriolis signal for measurement purposes. A large amplitude of oscillation also results in storage of a higher level of energy in the conduit, which provides greater robustness to external vibrations. 
     Circumstances may arise in which it is not possible to maintain the large amplitude of oscillation due to limitations in the current that can be supplied to the drivers. For example, in one implementation of an analog transmitter, the current is limited to 100 mA for safety purposes. This is typically 5-10 times the current needed to maintain the desired amplitude of oscillation. However, if the process fluid provides significant additional damping (e.g., via two-phase flow), then the optimal amplitude may no longer be sustainable. 
     Similarly, a low-power flowmeter, such as the two-wire meter described below, may have much less power available to drive the conduit. In addition, the power level may vary when the conduit is driven by capacitive discharge. 
     Referring to  FIG. 42 , a control procedure  4200  implemented by the controller of the digital flowmeter may be used to select the highest sustainable setpoint given a maximum available current level. In general, the procedure is performed each time that a desired drive current output is selected, which typically is once per cycle, or once every half-cycle if interleaved cycles are used. 
     The controller starts by setting the setpoint to a default value (e.g., 0.3 V) and initializing filtered representations of the sensor voltage (filtered_SV) and the drive current (filtered_DC) (step  4205 ). Each time that the procedure is performed, the controller updates the filtered values based on current values for the sensor voltage (SV) and drive current (DC) (step  4210 ). For example, the controller may generate a new value for filtered_SV as the sum of ninety nine percent of filtered_SV and one percent of SV. 
     Next, the controller determines whether the procedure has been paused to provide time for prior setpoint adjustments to take effect (step  4215 ). Pausing of the procedure is indicated by a pause cycle count having a value greater than zero. If the procedure is paused, the controller performs no further actions for the cycle and decrements the pause cycle count (step  4220 ). 
     If the procedure has not been paused, the controller determines whether the filtered drive current exceeds a threshold level (step  4225 ). In one implementation, the threshold level is ninety five percent of the maximum available current. If the current exceeds the threshold, the controller reduces the setpoint (step  4230 ). To allow time for the meter to settle after the setpoint change, the controller then implements a pause of the procedure by setting the pause cycle count equal to an appropriate value (e.g.,  100 ) (step  4235 ). 
     If the procedure has not been paused, the controller determines whether the filtered drive current is less than a threshold level (step  4240 ) and the setpoint is less than a maximum permitted setpoint (step  4245 ). In one implementation, the threshold level equals seventy percent of the maximum available current. If both conditions are met, the controller determines a possible new setpoint (step  4250 ). In one implementation, the controller determines the new setpoint as eighty percent of the maximum available current multiplied by the ratio of filtered_SV to filtered_DC. To avoid small changes in the setpoint (i.e., chattering), the controller then determines whether the possible new setpoint exceeds the current setpoint by a sufficient amount (step  4255 ). In one implementation, the possible new setpoint must exceed the current setpoint by 0.02 V and by ten percent. 
     If the possible new setpoint is sufficiently large, the controller determines if it is greater than the maximum permitted setpoint (step  4260 ). If so, the controller sets the setpoint equal to the maximum permitted setpoint (step  4265 ). Otherwise, the controller sets the setpoint equal to the possible new setpoint (step  4270 ). The controller then implements a pause of the procedure by setting the pause cycle count equal to an appropriate value (step  4235 ). 
       FIGS. 43A-43C  illustrate operation of the setpoint adjustment procedure. As shown in  FIG. 43C , the system starts with a setpoint of 0.3 V. At about eight seconds of operation, aeration results in a drop in the apparent density of the material in the conduit (FIG.  43 A). Increased damping accompanying the aeration results in an increase in the drive current ( FIG. 43B ) and increased noise in the sensor voltage (FIG.  43 C). No changes arc made at this time, since the meter is able to maintain the desired setpoint. 
     At about fifteen seconds of operation, aeration increases and the apparent density decreases further (FIG.  43 A). At this level of aeration, the driver current ( FIG. 43B ) reaches a maximum value that is insufficient to maintain the 0.3 V setpoint. Accordingly, the sensor voltage drops to 0.26 V (FIG.  43 C), the voltage level that the maximum driver current is able to maintain. In response to this condition, the controller adjusts the setpoint (at about 28 seconds of operation) to a level (0.23 V) that does not require generation of the maximum driver current. 
     At about  38  seconds of operation, the level of aeration decreases and the apparent density increases (FIG.  43 A). This results in a decrease in the drive current (FIG.  43 B). At 40 seconds of operation, the controller responds to this condition by increasing the setpoint (FIG.  43 C). The level of aeration decreases and the apparent density increases again at about  48  seconds of operation, and the controller responds by increasing the setpoint to 0.3 V. 
     K. Performance Results 
     The digital flowmeter has shown remarkable performance improvements relative to traditional analog flowmeters. In one experiment, the ability of the two types of meters to accurately measure a batch of material was examined. In each case, the batch was fed through the appropriate flowmeter and into a tank, where the batch was weighed. For 1200 and 2400 pound batches, the analog meter provided an average offset of 500 pounds, with a repeatability of 200 pounds. By contrast, the digital meter provided an average offset of 40 pounds, with a repeatability of two pounds, which clearly is a substantial improvement. 
     In each case, the conduit and surrounding pipework were empty at the start of the batch. This is important in many batching applications where it is not practical to start the batch with the conduit full. The batches were finished with the flowtube full. Some positive offset is expected because the flowmeter is measuring the material needed to fill the pipe before the weighing tank starts to be filled. Delays in starting up, or offsets caused by aerated flow or low amplitudes of oscillation, are likely to introduce negative offsets. For real batching applications, the most important issue is the repeatability of the measurement. 
     The results show that with the analog flowmeter there are large negative offsets and repeatability of only 200 pounds. This is attributable to the length of time taken to startup after the onset of flow (during which no flow is metered), and measurement errors until full amplitude of oscillation is achieved. By comparison, the digital flowmeter achieves a positive offset, which is attributable to filling up of the empty pipe, and a repeatability of two pounds. 
     Another experiment compared the general measurement accuracy of the two types of meters.  FIG. 44  illustrates the accuracy and corresponding uncertainty of the measurements produced by the two types of meters at different percentages of the meters&#39;maximum recommended flow rate. At high flow rates (i.e., at rates of 25% or more of the maximum rate), the analog meter produces measurements that correspond to actual values to within 0.15% or less, compared to 0.005% or less for the digital meter. At lower rates, the offset of the analog meter is on the order of 1.5%, compared to 0.25% for the digital meter. 
     L. Self-Validating Meter 
     The digital flowmeter may used in a control system that includes self-validating sensors. To this end, the digital flowmeter may be implemented as a self-validating meter. Self-validating meters and other sensors are described in U.S. Pat. No. 5,570,300, titled “SELF-VALIDATING SENSORS”, which is incorporated by reference. 
     In general, a self-validating meter provides, based on all information available to the meter, a best estimate of the value of a parameter (e.g., mass flow) being monitored. Because the best estimate is based, in part, on nonmeasurement data, the best estimate does not always conform to the value indicated by the current, possibly faulty, measurement data. A self-validating meter also provides information about the uncertainty and reliability of the best estimate, as well as information about the operational status of the sensor. Uncertainty information is derived from known uncertainty analyses and is provided even in the absence of faults. 
     In general, a self-validating meter provides four basic parameters: a validated measurement value (VMV), a validated uncertainty (VU), an indication (MV status) of the status under which the measurement was generated, and a device status. The VMV is the meter&#39;s best estimate of the value of a measured parameter. The VU and the MV status are associated with the VMV. The meter produces a separate VMV, VU and MV status for each measurement. The device status indicates the operational status of the meter. 
     The meter also may provide other information. For example, upon a request from a control system, the meter may provide detailed diagnostic information about the status of the meter. Also, when a measurement has exceeded, or is about to exceed, a predetermined limit, the meter can send an alarm signal to the control system. Different alarm levels can be used to indicate the severity with which the measurement has deviated from the predetermined value. 
     VMV and VU are numeric values. For example, VMV could be a temperature measurement valued at 200 degrees and VU, the uncertainty of VMV, could be 9 degrees. In this case, there is a high probability (typically 95%) that the actual temperature being measured falls within an envelope around VMV and designated by VU (i.e., from 191 degrees to 209 degrees). 
     The controller generates VMV based on underlying data from the sensors. First, the controller derives a raw measurement value (RMV) that is based on the signals from the sensors. In general, when the controller detects no abnormalities, the controller has nominal confidence in the RMV and sets the VMV equal to the RMV. When the controller detects an abnormality in the sensor, the controller does not set the VMV equal to the RMV. Instead, the controller sets the VMV equal to a value that the controller considers to be a better estimate than the RMV of the actual parameter. 
     The controller generates the VU based on a raw uncertainty signal (RU) that is the result of a dynamic uncertainty analysis of the RMV. The controller performs this uncertainty analysis during each sampling period. Uncertainty analysis, originally described in “Describing Uncertainties in Single Sample Experiments,” S. J. Kline &amp; F. A. McClintock,  Mech. Eng.,  75, 3-8 (1953), has been widely applied and has achieved the status of an international standard for calibration. Essentially, an uncertainty analysis provides an indication of the “quality” of a measurement. Every measurement has an associated error, which, of course, is unknown. However, a reasonable limit on that error can often be expressed by a single uncertainty number (ANSI/ASME PTC 19.1-1985 Part 1, Measurement Uncertainty: Instruments and Apparatus). 
     As described by Kline &amp; McClintock, for any observed measurement M, the uncertainty in M, w M , can be defined as follows:
 
 M   true   ε[M−w   M   , M+w   M ]
 
where M is true (M true ) with a certain level of confidence (typically 95%). This uncertainty is readily expressed in a relative form as a proportion of the measurement (i.e. w M /M).
 
     In general, the VU has a non-zero value even under ideal conditions (i.e., a faultless sensor operating in a controlled, laboratory environment). This is because the measurement produced by a sensor is never completely certain and there is always some potential for error. As with the VMV, when the controller detects no abnormalities, the controller sets the VU equal to the RU. When the controller detects a fault that only partially affects the reliability of the RMV, the controller typically performs a new uncertainty analysis that accounts for effects of the fault and sets the VU equal to the results of this analysis. The controller sets the VU to a value based on past performance when the controller determines that the RMV bears no relation to the actual measured value. 
     To ensure that the control system uses the VMV and the VU properly, the MV status provides information about how they were calculated. The controller produces the VMV and the VU under all conditions—even when the sensors are inoperative. The control system needs to know whether VMV and VU are based on “live” or historical data. For example, if the control system were using VMV and VU in feedback control and the sensors were inoperative, the control system would need to know that VMV and VU were based on past performance. 
     The MV status is based on the expected persistence of any abnormal condition and on the confidence of the controller in the RMV. The four primary states for MV status are generated according to Table 1. 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Expected 
                 Confidence 
                   
               
               
                   
                 Persistence 
                 in RMV 
                 MV Status 
               
               
                   
                   
               
             
            
               
                   
                 not applicable 
                 nominal 
                 CLEAR 
               
               
                   
                 not applicable 
                 reduced 
                 BLURRED 
               
               
                   
                 short 
                 zero 
                 DAZZLED 
               
               
                   
                 long 
                 zero 
                 BLIND 
               
               
                   
                   
               
            
           
         
       
     
     A CLEAR MV status occurs when RMV is within a normal range for given process conditions. A DAZZLED MV status indicates that RMV is quite abnormal, but the abnormality is expected to be of short duration. Typically, the controller sets the MV status to DAZZLED when there is a sudden change in the signal from one of the sensors and the controller is unable to clearly establish whether this change is due to an as yet undiagnosed sensor fault or to an abrupt change in the variable being measured. A BLURRED MV status indicates that the RMV is abnormal but reasonably related to the parameter being measured. For example, the controller may set the MV status to BLURRED when the RMV is a noisy signal. A BLIND MV status indicates that the RMV is completely unreliable and that the fault is expected to persist. 
     Two additional states for the MV status are UNVALIDATED and SECURE. The MV status is UNVALIDATED when the controller is not performing validation of VMV. MV status is SECURE when VMV is generated from redundant measurements in which the controller has nominal confidence. 
     The device status is a generic, discrete value summarizing the health of the meter. It is used primarily by fault detection and maintenance routines of the control system. Typically, the device status  32  is in one of six states, each of which indicates a different operational status for the meter. These states are: GOOD, TESTING, SUSPECT, IMPAIRED, BAD, or CRITICAL. A GOOD device status means that the meter is in nominal condition. A TESTING device status means that the meter is performing a self check, and that this self check may be responsible for any temporary reduction in measurement quality. A SUSPECT device status means that the meter has produced an abnormal response, but the controller has no detailed fault diagnosis. An IMPAIRED device status means that the meter is suffering from a diagnosed fault that has a minor impact on performance. A BAD device status means that the meter has seriously malfunctioned and maintenance is required. Finally, a CRITICAL device status means that the meter has malfunctioned to the extent that the meter may cause (or have caused) a hazard such as a leak, fire, or explosion. 
       FIG. 45  illustrates a procedure  4500  by which the controller of a self-validating meter processes digitized sensor signals to generate drive signals and a validated mass flow measurement with an accompanying uncertainty and measurement status. Initially, the controller collects data from the sensors (step  4505 ). Using this data, the controller determines the frequency of the sensor signals (step  45   10 ). If the frequency falls within an expected range (step  4515 ), the controller eliminates zero offset from the sensor signals (step  4520 ), and determines the amplitude (step  4525 ) and phase (step  4530 ) of the sensor signals. The controller uses these calculated values to generate the drive signal (step  4535 ) and to generate a raw mass flow measurement and other measurements (step  4540 ). 
     If the frequency does not fall within an expected range (step  4515 ), then the controller implements a stall procedure (step  4545 ) to determine whether the conduit has stalled and to respond accordingly. In the stall procedure, the controller maximizes the driver gain and performs a broader search for zero crossings to determine whether the conduit is oscillating at all. 
     If the conduit is not oscillating correctly (i.e., if it is not oscillating, or if it is oscillating at an unacceptably high frequency (e.g., at a high harmonic of the resonant frequency)) (step  4550 ), the controller attempts to restart normal oscillation (step  4555 ) of the conduit by, for example, injecting a square wave at the drivers. After attempting to restart oscillation, the controller sets the MV status to DAZZLED (step  4560 ) and generates null raw measurement values (step  4565 ). If the conduit is oscillating correctly (step  4550 ), the controller eliminates zero offset (step  4520 ) and proceeds as discussed above. 
     After generating raw measurement values (steps  4540  or  4565 ), the controller performs diagnostics (step  4570 ) to determine whether the meter is operating correctly (step  4575 ). (Note that the controller does not necessarily perform these diagnostics during every cycle.) 
     Next, the controller performs an uncertainty analysis (step  4580 ) to generate a raw uncertainty value. Using the raw measurements, the results of the diagnostics, and other information, the controller generates the VMV, the VU, the MV status, and the device status (step  4585 ). Thereafter, the controller collects a new set of data and repeats the procedure. The steps of the procedure  4500  may be performed serially or in parallel, and may be performed in varying order. 
     In another example, when aeration is detected, the mass flow corrections are applied as described above, the MV status becomes blurred, and the uncertainty is increased to reflect the probable error of the correction technique. For example, for a flowtube operating at 50% flowrate, under normal operating conditions, the uncertainty might be of the order of 0.1−0.2% of flowrate. If aeration occurs and is corrected for using the techniques described above, the uncertainty might be increased to perhaps 2% of reading. Uncertainty values should decrease as understanding of the effects of aeration improves and the ability to compensate for aeration gets better. In batch situations, where flow rate uncertainty is variable (e.g. high at start/end if batching from/to empty, or during temporary incidents of aeration or cavitation), the uncertainty of the batch total will reflect the weighted significance of the periods of high uncertainty against the rest of the batch with nominal low uncertainty. This is a highly useful quality metric in fiscal and other metering applications. 
     M. Two Wire Flowmeter 
     As shown in  FIG. 46 , the techniques described above may be used to implement a “two-wire” Coriolis flowmeter  4600  that performs bidirectional communications on a pair of wires  4605 . A power circuit  4610  receives power for operating a digital controller  4615  and for powering the driver(s)  4620  to vibrate the conduit  4625 . For example, the power circuit may include a constant output circuit  4630  that provides operating power to the controller and a drive capacitor  4635  that is charged using excess power. The power circuit may receive power from the wires  4605  or from a second pair of wires. The digital controller receives signals from one or more sensors  4640 . 
     When the drive capacitor is suitably charged, the controller  4615  discharges the capacitor  4635  to drive the conduit  4625 . For example, the controller may drive the conduit once during every 10 cycles. The controller  4615  receives and analyzes signals from the sensors  4640  to produce a mass flow measurement that the controller then transmits on the wires  4605 . 
     N. Batching From Empty 
     The digital mass flowmeter  100  provides improved performance in dealing with a challenging application condition that is referred to as batching from empty. There are many processes, particularly in the food and petrochemical industries, where the high accuracy and direct mass-flow measurement provided by Coriolis technology is beneficial in the metering of batches of material. In many cases, however, ensuring that the flowmeter remains full of fluid from the start to the end of the batch is not practical, and is highly inefficient. For example, in filling or emptying a tanker, air entrainment is difficult to avoid. In food processing, hygiene regulations may require pipes to be cleaned out between batches. 
     In conventional Coriolis meters, batching from empty may result in large errors. For example, hydraulic shock and a high gain requirement may be caused by the onset of flow in an empty flowtube, leading to large measurement errors and stalling. 
     The digital mass flowmeter  100  is robust to the conditions experienced when batching from empty. More specifically, the amplitude controller has a rapid response; the high gain range prevents flowtube stalling; measurement data can be calculated down to 0.1% of the normal amplitude of oscillation; and there is compensation for the rate of change of amplitude. 
     These characteristics are illustrated in  FIGS. 47A-47C , which show the response of the digital mass flowmeter  100  driving a wet and empty 25 mm flowtube during the first seconds of the onset of full flow. As shown in  FIG. 47A , the drive gain required to drive the wet and empty tube prior to the onset of flow (at about 4.0 seconds) has a value of approximately 0.1, which is higher than the value of approximately 0.034 needed for a full flowtube. The onset of flow is characterized by a substantial increase in gain and a corresponding drop in amplitude of oscillation. With reference to  FIG. 47B , at about 1.0 second after initiation, the selection of a reduced setpoint assists in the stabilization of amplitude while the full flow regime is established. After about 2.75 seconds, the last of the entrained air is purged, the conventional setpoint is restored, and the drive gain assumes the nominal value of 0.034. The raw and corrected phase difference behavior is shown in FIG.  47 C. 
     As shown in  FIGS. 47A-47C , phase data is given continuously throughout the transition. In similar circumstances, the analog control system stalls, and is unable to provide measurement data until the required drive gain returns to a near-nominal value and the lengthy start-up procedure is completed. As also shown, the correction for the rate of change of amplitude is clearly beneficial, particularly after 1.0 seconds. Oscillations in amplitude result in substantial swings in the Fourier-based and time-based calculations of phase, but these are substantially reduced in the corrected phase measurement. Even in the most difficult part of the transition, from 0.4-1.0 seconds, the correction provides some noise reduction. 
     Of course there are still erroneous data in this interval. For example, flow generating a phase difference in excess of about 5 degrees is physically not possible. However, from the perspective of a self validating sensor, such as is discussed above, this phase measurement still constitutes raw data that may be corrected. In some implementations, a higher level validation process may identify the data from 0.4-1.0 seconds as unrepresentative of the true process value (based on the gain, amplitude and other internal parameters), and may generate a DAZZLED mass-flow to suppress extreme measurement values. 
     Referring to  FIG. 48A , the response of the digital mass flowmeter  100  to the onset of flow results in improved accuracy and repeatability. An exemplary flow rig  4800  is shown in FIG.  48 B. In producing the results shown in  FIG. 48A , as in producing the results shown in  FIG. 44 , fluid was pumped through a magnetic flowmeter  4810  and a Coriolis flowmeter  4820  into a weighing tank  4830 , with the Coriolis flowmeter being either a digital flowmeter or a traditional analog flowmeter. Valves  4840  and  4860  were used to ensure that the magnetic flowmeter  4810  was always full, while the flowtube of the Coriolis flowmeter  4820  began each batch empty. At the start of the batch, the totalizers in the magnetic flowmeter  4810  and in the Coriolis flowmeter  4820  were reset and flow commenced. At the end of the batch, the shutoff valve  4850  was closed and the totals were frozen (hence the Coriolis flowmeter  4820  was full at the end of the batch). Three totals were recorded, with one each from the magnetic flowmeter  4810  and the Coriolis flowmeter  4820 , and one from the weigh scale associated with the weighing tank  4830 . These totals are not expected to agree, as there is a finite time delay before the Coriolis flowmeter  4820  and then finally the weighing tank  4830  observes fluid flow. Thus, it would be expected that the magnetic flowmeter  4810  would record the highest total flow, the Coriolis flowmeter  4820  would record the second highest total, and the weighing tank  4830  would record the lowest total. 
       FIG. 48A  shows the results obtained from a series of trial runs using the flow rig  4800  of  FIG. 48B , with each trial run transporting about  550  kg of material through the flow rig. The monitored value shown is the offset observed between the weigh scale and either the magnetic flowmeter  4810  or the Coriolis flowmeter  4820 . As explained above, positive offsets are expected from both instruments. The magnetic flowmeter  4810  (always full) delivers a consistently positive offset, with a repeatability (defined here as the maximum difference in reported value for identical experiments) of 4.0 kg. The analog control system associated with the magnetic flowmeter  4810  generates large negative offsets, with a mean of −164.2 kg and a repeatability of 87.7 kg. This poor performance is attributable to the analog control system&#39;s inability to deal with the onset of flow and the varying time taken to restart the flowtube. By contrast, the digital Coriolis mass flowmeter  4820  shows a positive offset averaging 25.6 kg and a repeatability of 0.6 kg. 
     It would be difficult to assess the true mass-flow through the flowtube, given its initially empty state. The reported total mass falls between that of the magnetic flowmeter  4810  and the weigh scale, as expected. In an industrial application, the issue of repeatability is often of greater importance, as batch recipes are often adjusted to accommodate offsets. Of course, the repeatability of the filling process is a lower bound on the repeatability of the Coriolis flowmeter total. Similar repeatability could be achieved in an arbitrary industrial batch process. Moreover, as shown, the digital mass flowmeter  100  provides a substantial performance improvement over its analog equivalent (magnetic flowmeter  4810 ) under the same conditions. Again, the conclusion drawn is that the digital mass flowmeter  100  in these conditions is not a significant source of measurement error. 
     O. Two-phase Flow 
     As discussed above with reference to  FIG. 40A , two-phase flow, which may result from aeration, is another flow condition which presents difficulties for analog control systems and analog mass flowmeters. Two-phase flow may be sporadic or continuous and results when the material in the flowmeter includes a gas component and a fluid component traveling through the flowtube. The underlying mechanisms are very similar to the case of batching from empty, in that the dynamics of two-phase gas-liquid flow cause high damping. To maintain oscillation, a high drive gain is required. However, the maximum drive gain of the analog control system typically is reached at low levels of gas fraction in the two-phase material, and, as a result, the flowtube stalls. 
     The digital mass flowmeter  100  is able to maintain oscillation in the presence of two-phase flow. In summary, laboratory experiments conducted thus far have been unable to stall a tube of any size with any level of gas phase when controlled by the digital controller  105 . By contrast, a typical analog control system stalls with about 2% gas phase. 
     Maintaining oscillation is only the first step in obtaining a satisfactory measurement performance from the flowmeter. As briefly discussed above, a simple model, referred to as the “bubble” model, has been developed as one technique to predict the mass-flow error. 
     In the “bubble” or “effective mass” model, a sphere or bubble of low density gas is surrounded by fluid of higher density. If both are subject to acceleration (for example in a vibrating tube), then the bubble moves within the fluid, causing a drop in the observed inertia of the whole system. Defining the void fraction a as the proportion of gas by volume, then the effective mass drops by a proportion R, with 
       R   =         2   ⁢           ⁢   α       1   -   α       .         
 
     Applied to a Coriolis flowmeter, the model predicts that the apparent mass-flow will be less than the true mass-flow by the factor R as will, by extension, the observed density.  FIG. 49  shows the observed mass-flow errors for a series of runs at different flow rates, all using a 25 mm flowtube in horizontal alignment, and a mixture of water and air at ambient temperature. The x-axis shows the apparent drop in density, rather than the void fraction. It is possible in the laboratory to calculate the void fraction, for example by measuring the gas pressure and flow rate prior to mixing with the fluid, together with the pressure of the two-phase mixture. However, in a plant, only the observed drop in density is available, and the true void fraction is not available. Note that with the analog flowmeter, air/water mixtures with a density drop of over 5% cause flowtube stalling so that no data can be collected. 
     The dashed line  4910  shows the relationship between mass-flow error and density drop as predicted by the bubble model. The experimental data follow a similar set of curves, although the model almost always predicts a more negative mass-flow error. As discussed above with respect to  FIG. 40A , it is possible to develop empirical corrections to the mass-flow rate based upon the apparent density as well as several other internally observed variables, such as the drive gain and the ratio of sensor voltages. It is reasonable to assume that the density of the pure fluid is known or can be learned. For example, in many applications the fluid density is relatively constant (particularly if a temperature coefficient is accommodated within the controller software). 
       FIG. 50  shows the corrected mass-flow measurements. The correction is based on a least squares fit of several internal variables, as well as the bubble model itself The correction process has only limited applicability, and it is less accurate for lower flow rates (the biggest errors are for 1.5-1.6 kg/second). In horizontal orientation, the gas and liquid phase begin to separate for lower flow rates, and much larger mass-flow errors are observed. In these circumstances, the assumptions of the bubble model are no longer valid. However, the correction is reasonable for higher flow rates. During on-line experimental trials, a similar correction algorithm has restricted mass flow errors to within about 2.5% of the mass flow reading. 
       FIG. 51  shows how a self validating digital mass flowmeter  100  responds to the onset of two-phase flow in its reporting of the mass-flow rate. The lower waveform  5110  shows an uncorrected mass-flow measurement under a two-phase flow condition, and the upper waveform  5120  shows a corrected mass-flow measurement and uncertainty bound under the same two-phase flow condition. With single phase flow (up to t=7 sec.), the mass-flow measurement is CLEAR and has a small uncertainty of about 0.2% of the mass flow reading. With the onset of two-phase flow, a number of processes become active. First, the two-phase flow is detected on the basis of the behavior of internally observed parameters. Second, a measurement correction process is applied, and the measurement status output along with the corrected measurement is set to BLURRED. Third, the uncertainty of the mass flow increases with the level of void fraction to a maximum of about 2.3% of the mass flow reading. For comparison, the uncorrected mass-flow measurement  5110  is shown directly below the corrected mass-flow measurement  5120 . The user thus has the option of continuing operation with the reduced quality of the corrected mass-flow rate, switching to an alternative measurement if available, or shutting down the process. 
     P. Application of Neural Networks 
     Another technique for improving the accuracy of the mass flow measurement during two-phase flow conditions is through the use of a neural network to predict the mass-flow error and to generate an error correction factor for correcting any error in the mass-flow measurement resulting from two-phase flow effects. The correction factor is generated using internally observed parameters as inputs to the digital signal processor and the neural network, and has been observed to keep errors to within 2%. The internally observed parameters may include temperature, pressure, gain, drop in density, and apparent flow rate. 
       FIG. 52  shows a digital controller  5200  that can be substituted for digital controller  105  or  505  of the digital mass flowmeters  100 ,  500  of  FIGS. 1 and 5 . In this implementation of digital controller  5200 , process sensors  5204  connected to the flowtube generate process signals including one or more sensor signals, a temperature signal, and one or more pressure signals (as described above). The analog process signals are converted to digital signal data by A/D converters  5206  and stored in sensor and driver signal data memory buffers  5208  for use by the digital controller  5200 . The drivers  5245  connected to the flowtube generate a drive current signal and may communicate this signal to the A/D converters  5206 . The drive current signal then is converted to digital data and stored in the sensor and driver signal data memory buffers  5208 . Alternatively, a digital drive gain signal and a digital drive current signal may be generated by the amplitude control module  5235  and communicated to the sensor and driver signal data memory buffers  5208  for storage and use by the digital controller  5200 . 
     The digital process sensor and driver signal data are further analyzed and processed by a sensor and driver parameters processing module  5210  that generates physical parameters including frequency, phase, current, damping and amplitude of oscillation. A raw mass-flow measurement calculation module  5212  generates a raw mass-flow measurement signal using the techniques discussed above with respect to the flowmeter  500 . 
     A flow condition state machine  5215  receives as input the physical parameters from the sensor and driver parameters processing module  5210 , the raw mass-flow measurement signal, and a density measurement  5214  that is calculated as described above. The flow condition state machine  5215  then detects a flow condition of material traveling through the digital mass flowmeter  100 . In particular, the flow condition state machine  5215  determines whether the material is in a single-phase flow condition or a two-phase flow condition. The flow condition state machine  5215  also inputs the raw mass-flow measurement signal to a mass-flow measurement output block  5230 . 
     When a single-phase flow condition is detected, the output block  5230  validates the raw mass-flow measurement signal and may perform an uncertainty analysis to generate an uncertainty parameter associated with the validated mass-flow measurement. In particular, when the state machine  5215  detects that a single-phase flow condition exists, no correction factor is applied to the raw mass-flow measurement, and the output block  5230  validates the mass-flow measurement, If the controller  5200  does not detect errors in producing the measurement, the output block  5230  may assign to the measurement the conventional uncertainty parameter associated with the fault free measurement, and may set the status flag associated with the measurement to CLEAR. If errors are detected by the controller  5200  in producing the measurement, the output block  5230  may modify the uncertainty parameter to a greater uncertainty value, and may set the status flag to another value such as BLURRED. 
     When the flow condition state machine  5215  detects that a two-phase flow condition exists, a two-phase flow error correction module  5220  receives the raw mass-flow measurement signal. The two-phase flow error correction module  5220  includes a neural network processor for predicting a mass-flow error and for calculating an error correction factor. The neural network processor can be implemented in a software routine, or alternatively may be implemented as a separate programmed hardware processor. Operation of the neural network processor is described in greater detail below. 
     A neural network coefficients and training module  5225  stores a predetermined set of neural network coefficients that are used by the neural network processor. The neural network coefficients and training module  5225  also may perform an online training function using training data so that an updated set of coefficients can be calculated for use by the neural network. While the predetermined set of neural network coefficients are generated through extensive laboratory testing and experiments based upon known two-phase mass-flow rates, the online training function performed by module  5225  may occur at the initial commissioning stage of the flowmeter, or may occur each time the flowmeter is initialized. 
     The error correction factor generated by the error correction module  5220  is input to the mass-flow measurement output block  5230 . Using the raw mass-flow measurement and the error correction factor (if received from the error correction module  5220  indicating two-phase flow), the mass-flow measurement output block  5230  applies the error correction factor to the raw mass-flow measurement to generate a corrected mass-flow measurement. The measurement output block  5230  then validates the corrected mass-flow measurement, and may perform an uncertainty analysis to generate an uncertainty parameter associated with the validated mass-flow measurement. The measurement output block  5230  thus generates a validated mass-flow measurement signal that may include an uncertainty and status associated with each validated mass-flow measurement, and a device status. 
     The sensor parameters processing module  5210  also inputs a damping parameter and an amplitude of oscillation parameter (previously described) to an amplitude control module  5235 . The amplitude control module  5235  further processes the damping parameter and the amplitude of oscillation parameter and generates digital drive signals. The digital drive signals are converted to analog drive signals by D/A converters  5240  for operating the drivers  5245  connected to the flowtube of the digital flowmeter. In an alternate implementation, the amplitude control module  5235  may process the damping parameter and the amplitude of oscillation parameter and generate analog drive signals for operating the drivers  5245  directly. 
       FIG. 53  shows a procedure  5250  performed by the digital controller  5200 . After processing begins (step  5251 ), measurement signals generated by the process sensors  5204  and drivers  5245  are quantified through an analog to digital conversion process (as described above), and the memory buffers  5208  are filled with the digital sensor and driver data (step  5252 ). For every new processing cycle, the sensor and driver parameters processing module  5210  retrieves the sensor and driver data from the buffers  5208  and calculates sensor and driver variables from the sensor data (step  5254 ). In particular, the sensor and driver parameters processing module  5210  calculates sensor voltages, sensor frequencies, drive current, and drive gain. 
     The sensor and driver parameters processing module  5210  then executes a diagnose_flow_condition processing routine (step  5256 ) to calculate statistical values including the mean, standard deviation, and slope for each of the sensor and driver variables. Based upon the statistics calculated for each of the sensor and driver variables, the flow condition state machine  5215  detects transitions between one of three valid flow-condition states: FLOW_CONDITION_SHOCK, FLOW_CONDITION_HOMOGENEOUS, AND FLOW_CONDITION_MIXED. 
     If the state FLOW_CONDITION_SHOCK is detected (step  5258 ), the mass-flow measurement analysis process is not performed due to irregular sensor inputs. On exit from this condition, the processing routine starts a new cycle (step  5251 ). The processing routine then searches for a new sinusoidal signal to track within the sensor signal data and resumes processing. As part of this tracking process, the processing routine must find the beginning and end of the sine wave using the zero crossing technique described above. If the state FLOW_CONDITION_SHOCK is not detected, the processing routine calculates the raw mass-flow measurement of the material flowing through the flowmeter  100  (step  5260 ). 
     If two-phase flow is not detected (i.e., the state FLOW_CONDITION_HOMOGENOUS is detected) (step  5270 ), the material flowing through the flowmeter  100  is assumed to be a single-phase material. If so, the validated mass-flow rate is generated from the raw mass-flow measurement (step  5272 ) by the mass-flow measurement output block  5230 . At this point, the validated mass-flow rate along with its uncertainty parameter and status flag can be transmitted to another process controller. Processing then begins a new cycle (step  5251 ). 
     If two-phase flow is detected (i.e., the state FLOW_CONDITION_MIXED is detected) (step  5270 ), the material flowing through the flowmeter  100  is assumed to be a two-phase material. In this case, the two-phase flow error correction module  5220  predicts the mass-flow error and generates an error correction factor using the neural network processor (step  5274 ). The corrected mass-flow rate is generated by the mass-flow measurement output block  5230  using the error correction factor (step  5276 ). A validated mass-flow rate then may be generated from the corrected mass-flow rate. At this point, the validated mass-flow rate along with its uncertainty parameter and status flag can be transmitted to another process controller. Processing then begins a new cycle (step  5251 ). 
     Referring again to  FIG. 52 , the neural network processor forming part of the two-phase flow error correction module  5220  is a feed-forward neural network that provides a non-parametric framework for representing a non-linear functional mapping between an input and an output space. The neural network is applied to predict mass flow errors during two-phase flow conditions in the digital mass flowmeter. Once the error is predicted by the neural network, an error correction factor is applied to the two-phase mass flow measurement to correct the errors. Thus, the system allows the errors to be predicted on-line by way of the neural network using only internally observable parameters derived from the sensor signal, sensor variables, and sensor statistics. 
     Of the various neural network models available, the multi-layer perceptron (MLP) and the radial basis function (RBF) networks have been used for implementations of the digital flowmeter. The MLP with one hidden layer (each unit having a sigmoidal activation function) can approximate arbitrarily well any continuous mapping. Therefore, this type of neural network is suitable to model the non-linear relationship between the mass flow error of the flowmeter under two-phase flow and some of the flowmeter&#39;s internal parameters. 
     The network weights necessary for accomplishing the desired mapping are determined during a training or optimization process. During supervised training, the neural network is repeatedly presented with the training set (a collection of input examples x i  and their corresponding desired outputs d i ), and the weights are updated such that an error function is minimized. For the interpolation problem associated with the present technique, a suitable error function is the sum-of-squares error, which for an MLP with one output may be represented as: 
       I   =         ∑     i   =   1     P     ⁢           ⁢         e   i     ⁡     (   t   )       2       =       ∑     i   =   1     P     ⁢       (       d   i     -     y   i       )     2             
 
where d i  is the target corresponding to input x i ; y i  is the actual neural network output to x i ; and P is the number of examples in the training set.
 
     An alternative neural network architecture that has been used is the RBF network. The RBF methods have their origins in techniques for performing exact interpolations of a set of data points in a multi-dimensional space. A RBF network generally has a simple architecture of two layers of weights, in which the first layer contains the parameters of the basis functions, and the second layer forms linear combinations of the activation of the basis functions to generate the outputs. This is achieved by representing the output of the network as a linear superposition of basis functions, one for each data point in the training set. In this form, training is faster than for a MLP network. 
     The internal sensor parameters of interest include observed density, damping, apparent flow rate, and temperature. Each of these parameters is discussed below. 
     1. Observed Density 
     The most widely used metric of two-phase flow is the void (or gas) fraction defined as the proportion of gas by volume. The equation 
       R   =       2   ⁢           ⁢   α       1   -   α           
 
models the mass flow error given the void fraction. For a Coriolis mass flowmeter, the reported process fluid density provides an indirect measure of void fraction assuming the “true” liquid density in known. This reported process density is subject to errors similar to those in the mass-flow measurement in the presence of two-phase flow. These errors are highly repeatable and a drop in density is a suitable monotonic but non-linear indicator of void fraction, that can be monitored on-line within the flowmeter. It should be noted that outside of a laboratory environment, the true void fraction cannot be independently assessed, but rather, must be modeled as described above.
 
     Knowledge of the “true” single-phase liquid density can be obtained on-line or can be configured by the user (possibly including a temperature coefficient). Both approaches have been implemented and appear satisfactory. 
     For the purposes of these descriptions, the drop in density will be used as the x-axis parameter in graphs showing two-phase flow behavior. It should be noted that in the 3D plots of FIGS.  54  and  56 - 57 , which capture the results from  134  on-line experiments, a full range of density drop points is not possible at high flow rates due to air pressure limitations in the flow system rig. Also, the effects of temperature, though not shown in the graphs, have been determined experimentally. 
     2. Damping 
     Most Coriolis flowmeters use positive feedback to maintain flowtube oscillation. The sensor signals provide the frequency and phase of the flowtube oscillation, which are multiplied by a drive gain, K 0  to provide the currents supplied to the drivers  5245 : 
         K   0     =         drive   ⁢           ⁢   signal   ⁢           ⁢   out   ⁢           ⁢     (   Amps   )         sensor   ⁢           ⁢   signal   ⁢           ⁢   in   ⁢           ⁢     (   Volts   )         =         I   D       V   A       .           
 
     Normally, the drive gain is modified to ensure a constant amplitude of oscillation, and is roughly proportional to the damping factor of the flowtube. 
     One of the most characteristic features of two-phase flow is a rapid rise in damping. For example, a normally operating 25 mm flowtube has V A =0.3V, I D =10 mA, and hence K 0 =0.033. With two-phase flow, values might be as extreme as V A =0.03V, I D =100 mA, and K 0 =3.3, a hundred-fold increase.  FIG. 54  shows how damping varies with two-phase flow. 
     3. Apparent Flow Rate and Temperature 
     As  FIG. 49  shows, the mass flow error varies with the true flow rate. Temperature variation also has been observed. However, the true mass-flow rate is not available within the transmitter (or digital controller) itself when the flowmeter is subject to two-phase flow. However, observed (faulty) flow rate, as well as the temperature, are candidates as input parameters to the neural network processor. 
     Q. Network Training and On-Line Correction of Mass Flow Errors 
     Implementing the neural network analysis for the mass flow measurement error prediction includes training the neural network processor to recognize the mass flow error pattern in training experimental data, testing the performance of the neural network processor on a new set of experimental data, and on-line implementation of the neural network processor for measurement error prediction and correction. 
     The prediction quality of the neural network processor depends on the wealth of the training data. To collect the neural network data, a series of two-phase air/water experiments were performed using an experimental flow rig  5500  shown in FIG.  55 . The flow loop includes a master meter  5510 , the self validating SEVA® Coriolis flowmeter  100 , and a diverter  5520  to transfer material from the flowtube to a weigh scale  5530 . The Coriolis flowmeter  100  has a totalization function which can be triggered by external signals. The flow rig control is arranged so that both the flow diverter  5520  (supplying the weigh scale) and Coriolis totalization are triggered by the master meter  5510  at the beginning of the experiment, and again after the master meter  5510  has observed 100 kg of flow. The weigh scale total is used as the reference to calculate mass flow error by comparison with the totalized flow from the digital flowmeter  100 , with the master meter  5510  acting as an additional check. The uncertainty of the experimental rig is estimated at about 0.1% on typical batch sizes of 100 kg. For single-phase experiments, the digital flowmeter  100  delivers mass flow totals within 0.2% of the weigh scale total. For two-phase flow experiments, air is injected into the flow after the master meter  5510  and before the Coriolis flowmeter  100 . At low flow rates, density drops of up to 30% are achieved. At higher flow rates, at least 15% density drops are achieved. 
     At the end of each batch, the Coriolis flowmeter  100  reports the batch average of each of the following parameters: temperature, damping, density, flow rate, and the total (uncorrected) flow. These parameters are thus available for use as the input data of the neural network processor. 
     The output or the target of the neural network is the mass-flow error in percentage: 
         mass_error   ⁢           ⁢   %     =         coriolis   -   weighscale     weighscale     ×   100         
 
       FIG. 56  shows how the mass-flow error varies with flow rate and density drop. Although the general trend follows the bubble model, there are additional features of interest. For example, for high flow rates and low density drops, the mass-flow error becomes slightly positive (by as much as 1%), while the bubble model only predicts a negative error. As is apparent from  FIG. 56 , in this region of the experimental space, and for this flowtube design, some other physical process is taking place to overcome the missing mass effects of two-phase flow. 
     The best results were obtained using only four input parameters to the neural network: temperature, damping, density drop, and apparent flow rate. Less satisfactory perhaps is the result that the best fit was achieved using the neural network on its own, rather than as a correction to the bubble model or to a simplified curve fit. 
     As part of the implementation, a MLP neural network was used for on-line implementation. Comparisons between RBF and MLP networks with the same data sets and inputs have shown broadly similar performance on the test set. It is thus reasonable to assume that the input set delivering the best RBF design also will deliver a good (if not the best) MLP design. The MLP neural nets were trained using a scaled conjugate gradient algorithm. The facilities of Neural Network Toolbox of the MATLAB® software package were used for neural network training. After further design options were explored, the best performance came from an 4-9-1 MLP having as inputs the temperature, damping, drop in density, and flow rate, along with the mass flow error as output. 
     Against a validation set, the best neural network provided mass flow rate predictions to within 2% of the true value. Routines for the detection and correction for two-phase flow have been coded and incorporated into the digital Coriolis transmitter.  FIG. 57  shows the residual mass-flow error when corrected on-line over 134 new experiments. All errors are with 2%, and most are considerably less. The random scatter is due primarily to residual error in the neural network correction algorithm (as stated earlier, the uncertainty of the flow rig is 0.1%). Any obvious trend in the data would imply scope for further correction. These errors are of course for the average corrected mass flow rate (i.e. over a batch). 
       FIG. 58  shows how the on-line detection and correction for two-phase flow is reflected in the self-validating interface generated for the mass flow measurement. In the graph, the lower, continuous line  5810  is the raw mass flow rate. The upper line  5820  is a measurement surrounded by the uncertainty band, and represents the corrected or validated mass-flow rate. The dashed line  5830  is the mass flow rate from the master meter, which is positioned prior to the air injection point (FIG.  55 ). 
     With single-phase flow (up to t=5 sec.), the mass flow measurement has a measurement value status of CLEAR and a small uncertainty of about 0.2% of reading. Once two-phase flow is detected, the neural network correction is applied every interleaved cycle (i.e. at 180 Hz), based on the values of the internal parameters averaged (using a moving window) over the last second. During two-phase flow, the measurement value status is set to BLURRED, and the uncertainty increases to reflect the reduced accuracy of the corrected measurement. The uncorrected measurement (lower dark line) shows a large offset error of about 30%. 
     The master meter reading agrees with a first approximation with the corrected mass flow measurement. The apparent delay in its response to the incidence of two-phase flow is attributable to communication delays in the rig control system, and its squarewave-like response is due to the control system update rate of only once per second. Both the raw and corrected measurements from the digital transmitter show a higher degree of variation than with single phase flow. The master meter reading gives a useful measure of the water phase entering the two-phase zone, and there is a clear similarity between the master meter reading and the ‘average’ corrected reading. However, plug flow and air compressibility within the complex 3-D geometry of the flowtube may result not only in flow rate variations, but with the instantaneous mass-flow into the system being different from the mass-flow out of the system. 
     Using the self-validating sensor processing approach, the measurement is not simply labeled good or bad by the sensor. Instead, if a fault occurs, a correction is applied as far as possible and the quality of the resulting measurement is indicated through the BLURRED status and increased uncertainty. The user thus assesses application-specific requirements and options in order to decide whether to continue operation with the reduced quality of the corrected mass-flow rate, to switch to an alternative measurement if available, or to shut down the process. If two-phase flow is present only during part of a batch run (e.g. at the beginning or end), then there will be a proportionate weighting given to the uncertainty of the total mass of the batch. 
     The above description provides an overview of the digital Coriolis mass flowmeter, describing its background, implementation, examples of its operation, and a comparison to prior analog controllers and transmitters. A number of improvements over analog controller performance have been achieved, including: high precision control of flowtube operation, even with operation at very low amplitudes; the maintenance of flowtube operation even in highly damped conditions; high precision and high speed measurement; compensation for dynamic changes in amplitude; compensation for two-phase flow; and batching to/from empty. This combination of benefits suggests that the digital mass flowmeter represents a significant step forward, not simply a gradual evolution from analog technology. The ability to deal with two-phase flow and external vibration means that the digital mass flowmeter  100  gives improved performance in conventional Coriolis applications while expanding the range of applications to which the flow technology can be applied. The digital platform also is a useful and flexible vehicle for carrying out research into Coriolis metering in that it offers high precision, computing power, and data rates. 
     R. Source Code Listing 
     The following source code, which is hereby incorporated into this application, is used to implement the mass-flow rate processing routine in accordance with one implementation of the flowmeter. It will be appreciated that it is possible to implement the mass-flow rate processing routine using different computer code without departing from the scope of the invention. This listing is provided merely to illustrate the best mode currently known to the inventors to practice the invention. Thus, neither the foregoing description nor the following source code listing is intended to limit the invention which is defined in the appended claims. 
                             Source code listing                                    void calculate_massflow(meas_data_type *p,                  meas_data_type *op,                  int validating)       {                       double   Tz, z1, z2, z3, z4, z5, z6, z7, t, x, dd, m, g,           flow_error;         double   noneu_mass_flow, phase_bias_unc, phase_prec_unc,           this_density;         int   reset, freeze;                   /* calculate non- engineering units mass flow */         if (amp_sv1 &lt; 1e−6)           noneu_mass_flow = 0.0;         else           noneu_mass_flow = tan(my_pi * p-&gt;phase_diff/180);         /* convert to engineering units */         Tz = p-&gt;temperature_value − 20;         p-&gt;massflow_value = flow_factor * 16.0 * (FC1 * Tz + FC3 *           Tz*Tz + FC2) * noneu_mass_flow /p-&gt;v_freq;         /* apply two-phase flow correction if necessary */         if (validating &amp;&amp; do_two_phase_correction) {         /* call neural net for calculation of mass flow correction */                             t   = VMV_temp_stats.mean;   // mean VMV temperature           x   = RMV_dens_stats.mean;   // mean RMV density                         dd   = (TX_true_density − x) / TX_true_density * 100.0;                             m   = RMV_mass_stats.mean;   // mean RMV mass flow;           g   = gain_stats.mean;   // mean gain;                     nn_predict(t, dd, m, g, &amp;flow_error);           p-&gt;massflow_value = 100.0*m/(100.0 + flow_error);           }                    
S. Notice of Copyright
 
     A portion of the disclosure of this patent document contains material which is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by any one of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright rights whatsoever. 
     Other embodiments are within the scope of the following claims.