DETAILED DESCRIPTION

[0086] 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 .

[0087] 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.

[0088] 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.

[0089] A. Mechanical Design

[0090] 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 120 of FIG. 1 .

[0091] 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. 2B ). 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 {fraction (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.

[0092] B. Conduit Motion

[0093] The motion of the straight sections of loops 18 and 20 are shown in three modes in FIGS. 3A, 3B and 3 C. In the drive mode shown in FIG. 3 B, 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.

[0094] The drive motion shown in FIG. 3B induces the Coriolis mode motion shown in FIG. 3 A, 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), Wis 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.

[0095] 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.

[0096] 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.

[0097] Other implementations may include different conduit designs. For example, a single loop or a straight tube section may be employed as the conduit.

[0098] C. Electronic Design

[0099] 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.

[0100] 1. Analog Control System

[0101] 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 .

[0102] 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,

[0103] 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,

[0104] where V _D and V _C are, respectively, the drive voltage and the Coriolis voltage, and ω is the drive mode angular frequency.

[0105] Voltages V _A1 and V _B1 are differenced by operational amplifier 425 to produce:

V _DRV =V _A1 −V _B1 =2 V _D sin ω t,

[0106] 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 .

[0107] 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,

[0108] where V _COR is related to the induced Coriolis motion.

[0109] 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 co so that V _M corresponds to the amplitude of V _COR at ω. This amplitude is directly proportional to the mass in the conduit.

[0110] 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.

[0111] 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 is 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.

[0112] 2. Digital Control System

[0113] 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.

[0114] 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.

[0115] 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.

[0116] 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.

[0117] 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.

[0118] 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.

[0119] 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.

[0120] FIG. 5 provides a general description of the hardware included in a digital flowmeter. A more detailed description of a specific hardware implementation is provided in the attached appendix, which is incorporated by reference.

[0121] a. Data Collection

[0122] 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.

[0123] 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.

[0124] 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.

[0125] 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.

[0126] 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 .

[0127] 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.

[0128] 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.

[0129] 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'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.

[0130] 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: 1 $\mathrm{end\_sample}\mathrm{\_SV}1\le \mathrm{start\_sample}\mathrm{\_SV}1+\frac{365}{360}*\frac{\mathrm{sample\_rate}}{\mathrm{est\_freq}*0.95}$

[0131] 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.

[0132] 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.

[0133] 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 2 $\mathrm{sv1\_in}\left(\mathrm{start\_sample}\mathrm{\_SV}1+\frac{30}{360}*\frac{\mathrm{sample\_rate}}{\mathrm{est\_freq}*1.05}\right)>0.475*\mathrm{est\_amp}\mathrm{\_SV}$

[0134] The inequalities for the other points have the same form, with the degree offset term (×/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.

[0135] Measurement processing takes place on the vectors sv 1 _in(start:end) and sv 2 _in(start:end) where:

[0136] start=min (start_sample_SV 1 , start_sample_SV 2 ), and

[0137] end=max (end_sample_SV 1 , end_sample_SV 2 ).

[0138] The difference between the start and end points for a signal is indicative of the frequency of the signal.

[0139] b. Frequency Determination

[0140] 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.

[0141] 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 +δ+σε _t ,

[0142] 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 =2 πft), 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 .

[0143] When x _t is close to 2π, sin x _t and sin ² x _t can be approximated as x _0t =x _t −2πand x _0t ² , respectively. Accordingly, for values of x _t close to 2π, a _t can be approximated as: 3 $\begin{array}{c}{a}_t^2\approx A^2x_{0t}^2+2A\left(\delta +\sigma {\varepsilon }_t\right)x_{0t}+2\delta \sigma {\varepsilon }_t+{\delta }^2+{\sigma }^2{\varepsilon }_t^2\\ \approx \left(A^2x_{0t}^2+2A\delta x_{0t}+{\delta }^2\right)+{\mathrm{\sigma \varepsilon }}_t\left(2Ax_{0t}+2\delta +\sigma {\varepsilon }_t\right).\end{array}$

[0144] 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.

[0145] 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.

[0146] 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.

[0147] 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.

[0148] 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.

[0149] 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.

[0150] 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.

[0151] 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 .

[0152] 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 ).

[0153] 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.

[0154] 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 ).

[0155] 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.

[0156] 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.

[0157] 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.

[0158] c. Zero Offset Compensation

[0159] 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.

[0160] 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.

[0161] d. Amplitude Determination

[0162] 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.

[0163] 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.

[0164] 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.

[0165] 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: 4 $\begin{array}{c}x\left(t\right)=a_0/2+a_1\cos \left(\omega t\right)+a_2\cos \left(2\omega t\right)+a_3\cos \left(3\omega t\right)+\dots +\\ b_1\sin \left(\omega t\right)+b_2\sin \left(2\omega t\right)+\dots \end{array}$

[0166] 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: 5 $\begin{array}{c}{a}_n=\frac{2}{T}{\int }_0^Tx\left(t\right)\cos n\omega t,\\ \mathrm{and}\\ b_n=\frac{2}{T}{\int }_0^Tx\left(t\right)\sin n\omega t.\end{array}$

[0167] The amplitude, A _n , of each harmonic is given by:

A _n ={square root}{square root over (a _n ² +b _n ² )}.

[0168] The integrals are calculated using Simpson's method with quadratic correction (described below). The chief computational expense of the method is calculating the pure sine and cosine functions.

[0169] e. Phase Determination

[0170] 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: 6 ${\varphi }_n={\tan }^{-1}\frac{a_n}{b_n}.$

[0171] 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 co. 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.

[0172] The amplitude and phase may be generated using a Fourier method that eliminates the 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 co of the cycle, the controller calculates: 7 $\begin{array}{c}{I}_1=\frac{2\omega }{\pi }{\int }_0^{\frac{2\pi }{\omega }}SV\left(t\right)\sin \left(\omega t\right)t,\mathrm{and}\\ I_2=\frac{2\omega }{\pi }{\int }_0^{\frac{2\pi }{\omega }}SV\left(t\right)\cos \left(\omega t\right)t,\end{array}$

[0173] 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: 8 $\begin{array}{c}\mathrm{Amp}=\sqrt{I_1^2+I_2^2},\mathrm{and}\\ \mathrm{Phase}={\tan }^{-1}\frac{I_2}{I_1}.\end{array}$

[0174] 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: 9 $\begin{array}{c}\mathrm{scaled\_phase}{\mathrm{\_SV}}_1={\mathrm{phase\_SV}}_1\frac{\mathrm{av\_freq}}{{\mathrm{freq\_SV}}_1},\\ \mathrm{scaled\_shift}{\mathrm{\_SV}}_2=\frac{\left({\mathrm{midpoint\_SV}}_2-{\mathrm{midpoint\_SV}}_1\right)h{\mathrm{freq\_SV}}_2}{360},\mathrm{and}\\ \mathrm{scaled\_phase}{\mathrm{\_SV}}_2=\left({\mathrm{phase\_SV}}_2+\mathrm{scale\_shift}{\mathrm{\_SV}}_2\right)\frac{\mathrm{av\_freq}}{{\mathrm{freq\_SV}}_2},\end{array}$

[0175] where h is the sample length and the midpoints are defined in terms of samples: 10 ${\mathrm{midpoint\_SV}}_x=\frac{\left({\mathrm{startpoint\_SV}}_x+{\mathrm{endpoint\_SV}}_x\right)}{2}$

[0176] 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.

[0177] 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 ).

[0178] 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.

[0179] 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.

[0180] 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.

[0181] The difference-amplitude method starts with: 11 ${\mathrm{SV}}_1\left(t\right)=A_1\sin \left(2\pi ft+\frac{\varphi }{2}\right)\mathrm{and}{\mathrm{SV}}_2\left(t\right)=A_2\sin \left(2\pi ft-\frac{\varphi }{2}\right),$

[0182] 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: 12 $\begin{array}{c}\mathrm{Sum}\equiv {\mathrm{SV}}_1\left(t\right)+\frac{A_1}{A_2}{\mathrm{SV}}_2\left(t\right)=2A_1\cos \frac{\varphi }{2}·\sin 2\pi ft,\mathrm{and}\\ \mathrm{Diff}\equiv {\mathrm{SV}}_1\left(t\right)-\frac{A_1}{A_2}{\mathrm{SV}}_2\left(t\right)=2A_1\sin \frac{\varphi }{2}·\cos 2\pi ft.\end{array}$

[0183] 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, φ.

[0184] As an alternative, the phase difference may be calculated using the function Prod, defined as: 13 $\mathrm{Prod}\equiv \mathrm{SumxDiff}=4A_1^2\cos \frac{\varphi }{2}·\sin \frac{\varphi }{2}·\cos 2\pi ft·\sin 2\pi ft=A_1^2\sin \varphi ·\sin 4\pi ft,$

[0185] which is a function with amplitude A ² sinφ and frequency 2f. Prod can be generated sample by sample, and φ may be calculated from the amplitude of the resulting sine wave.

[0186] 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.

[0187] f. Drive Signal Generation

[0188] 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).

[0189] 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.

[0190] 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.

[0191] 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.

[0192] 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.

[0193] Referring to FIGS. 11 A- 13 D, 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. 11 A- 11 D, 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. 11B ). The magnitudes of the drive signals vary with noise in the sensor signals ( FIGS. 11C and 11D ). However, the amplified noise is insufficient to vary the sign of the drive signals so as to induce oscillation.

[0194] FIGS. 12 A- 12 D 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