Patent Publication Number: US-2022236090-A1

Title: Signal processing

Description:
TECHNICAL FIELD 
     Aspects relate, in general, to determination of a flow rate of a fluid flowing through a conduit, and more particularly, although not exclusively to use of transducers to emit an ultrasonic signal into the fluid and receive the signal after propagating through a section of the conduit. 
     BACKGROUND 
     The extraction of flow, flow rate or fluid velocity from transit time ultrasonic signals is well known. There are a variety of existing techniques or algorithms to that can be used to extract time of flight data. For example, phase or envelope-based algorithms could be used to extract time of flight data. Similarly, peak-estimation algorithms allow efficient time-domain calculations to extract amplitude ratios and thereby flow in the absence of, or negligible, conduit-borne interference, thereby abrogating the need for pico-second measurement accuracy. 
     SUMMARY 
     According to a first aspect there is provided a method for determining a flow rate of a fluid in a conduit, the method comprising providing a first ultrasonic transit time signal under fluid flow, generated using a first transducer in a direction towards a second transducer, providing a second ultrasonic transit time signal under fluid flow, generated using the second transducer in a direction towards the first transducer, the first and second transducers being spatially separated from one another along a length of the conduit, providing a third ultrasonic transit time signal under zero fluid flow, generated using the first or second transducer in a direction towards the second or first transducer, generating respective measures of amplitude and phase of the first, second and third signals at a selected time to provide respective first, second and third flow vectors, calculating a measure of the difference in the values of amplitude and phase of the first signal and second signals at the selected time using the first and second flow vectors to provide a difference vector, generating a measure for a component of an interference vector representing a conduit-borne component signal of the first and second signals using the first and second flow vectors and the difference vector, and generating a measure for the flow rate of the fluid using the component of the interference vector, the third flow vector and the sum of the first and second vectors. 
     Generating the measure for the flow rate of the fluid may further comprise using the difference vector. The first and/or second signal may comprise the conduit-borne component signal and a pure fluid component. The third flow vector may comprise a pure fluid component and no conduit-borne component signal. The conduit-borne component signal may correspond to an ultrasonic wave that has propagated through the conduit itself and has not propagated through the fluid. The pure fluid component may correspond to an ultrasonic wave that has propagated through the fluid within the conduit. The difference vector may indicate the flow rate of fluid in the conduit affected by the conduit-borne component signal. 
     Generating a measure for the component of the interference vector may comprise using the equation: 
     
       
         
           
             
               m 
               y 
             
             = 
             
               - 
               
                 
                   | 
                   U 
                   ⁢ 
                   
                     | 
                     2 
                   
                   ⁢ 
                   
                     - 
                     
                       | 
                       D 
                       ⁢ 
                       
                         | 
                         2 
                       
                     
                   
                 
                 
                   2 
                   ⁢ 
                   Δ 
                 
               
             
           
         
       
     
     where
         m y  is the component of the interference vector,   U is the first flow vector,   D is the second flow vector, and   Δ is the difference vector.       

     Generating a measure for the flow rate of the fluid may comprise using the equation: 
     
       
         
           
             s 
             = 
             
               
                 
                   | 
                   Z 
                   ⁢ 
                   
                     | 
                     2 
                   
                   ⁢ 
                   
                     - 
                     
                       
                         m 
                         y 
                       
                       2 
                     
                   
                 
               
               - 
               
                 
                   
                     
                       ( 
                       
                         
                            
                           Σ 
                            
                         
                         2 
                       
                       ) 
                     
                     2 
                   
                   - 
                   
                     
                       m 
                       y 
                     
                     2 
                   
                 
               
             
           
         
       
         
         
           
             where 
             Z is the third flow vector, 
             m y  is the imaginary component of the interference vector, 
             Σ is a sum vector passing through the intersection of the difference vector with the real axis, and 
             s is a sagitta equal to the difference between the sum vector Σ/2 and the third vector. 
           
         
       
    
     Generating a measure for the flow rate of the fluid may comprise using the equation: 
     
       
         
           
             
               Φ 
               t 
             
             = 
             
               4 
               . 
               
                 arctan 
                 ⁡ 
                 
                   ( 
                   
                     
                       2 
                       ⁢ 
                       s 
                     
                     Δ 
                   
                   ) 
                 
               
             
           
         
       
         
         
           
             where 
             Δ is the difference vector, 
             s is a sagitta equal to the difference between a sum vector Σ/2 and the third vector, 
             wherein the sum vector Σ passes through the intersection of the difference vector with the real axis, and 
           
         
       
    
     Φ t  is the flow rate of the fluid. 
     Generating a measure for the flow rate of the fluid may comprise using the equation: 
     
       
         
           
             r 
             = 
             
               - 
               
                 
                   
                     s 
                     2 
                   
                   - 
                   
                     
                       ( 
                       
                         
                           | 
                           Δ 
                           | 
                         
                         2 
                       
                       ) 
                     
                     2 
                   
                 
                 
                   2 
                   ⁢ 
                   s 
                 
               
             
           
         
       
         
         
           
             where 
             Δ is the difference vector ( 418 ), 
             s is a sagitta ( 480 ) equal to the difference between the sum vector Σ/2 and the third flow vector ( 410 ), and 
             r is a radius ( 462 ) linked with the sagitta ( 480 ), where the radius ( 462 ) represents an amplitude of the fluid flow. 
           
         
       
    
     The method may further comprise performing synchronous demodulation on the first signal and/or second signal. 
     According to a second aspect there is provided an apparatus for determining a flow rate of a fluid in a conduit, the apparatus comprising a conduit, a first transducer configured to provide a first ultrasonic transit time signal, a second transducer configured to provide a second ultrasonic transit time signal, wherein the first a second transducers are spatially separated from one another along a length of the conduit and are further configured to provide a third ultrasonic transit time signal, a signal generator, a processor, configured to generate respective measures of amplitude and phase of the first, second and third signals at a selected time to provide respective first, second and third flow vectors, calculate a measure of the difference in the values of amplitude and phase of the first signal and second signals at the selected time using the first and second flow vectors to provide a difference vector, generate a measure for a component of an interference vector representing a conduit-borne component signal of the first and second signals using the first and second flow vectors and the difference vector, and generate a measure for the flow rate of the fluid using the component of the interference vector, the third flow vector and the sum of the first and second vectors. The processor may be configured to perform the method according to the first aspect. 
     According to a third aspect there is provided a non-transitory machine-readable storage medium encoded with instructions executable by a processor for determining a flow rate of a fluid in a conduit, the machine-readable storage medium comprising instructions to generate respective measures of an amplitude and phase of a first, second and third signals at a selected time to provide respective first, second and third flow vectors, calculate a measure of the difference in the values of amplitude and phase of the first signal and second signal at the selected time using the first and second flow vectors to provide a difference vector, generate a measure for a component of an interference vector representing a conduit-borne component signal of the first and second signals using the first and second flow vectors and the difference vector, and generate a measure for the flow rate of the fluid using a component of an interference vector, the third flow vector and a sum of the first and second vectors. 
     According to a fourth aspect, there is provided a method for determining a flow rate of a fluid in a conduit, the method comprising detecting a first ultrasonic transit time signal under fluid flow, generated using a first transducer in a direction towards a second transducer, detecting a second ultrasonic transit time signal under fluid flow, generated using the second transducer in a direction towards the first transducer, the first and second transducers being spatially separated from one another along a length of the conduit, determining a measure of a difference in phase between the first and second ultrasonic transit time signals at a selected time, and on the basis of the determined measure of difference in phase, determining the flow rate of the fluid. The method can further comprise generating a representation of phase of a signal at the selected time, and using the representation, calculating a measure of the flow rate at the selected time. The method can further comprise sampling the first and second signals at the selected time to provide first and second samples, and using the first and second samples, calculating measures for the phase of the first and second signals at the selected time. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Embodiments will now be described, by way of example only, with reference to the accompanying drawings, in which: 
         FIG. 1  shows a schematic of an apparatus for determining a flow rate of a fluid in a conduit according to an example; 
         FIG. 2A  shows a schematic of an ultrasonic ping signal detected under zero fluid flow according to an example; 
         FIG. 2B  shows a schematic of an ultrasonic ping signal detected under a flow of fluid through the conduit according to an example; 
         FIG. 3A  shows a schematic of an ultrasonic ping signal for a fluid-borne signal and conduit-borne interference detected under zero fluid flow according to an example; 
         FIG. 3B  shows a schematic of an ultrasonic ping signal detected for a fluid-borne signal and a conduit-borne interference signal under a flow of fluid through the conduit according to an example; 
         FIG. 4A  shows a schematic of an argand diagram with phase vectors representing an ultrasonic ping signal detected under a zero flow and under a flow of fluid through the conduit according to an example; 
         FIG. 4B  shows a schematic of an argand diagram with phase vectors representing an ultrasonic ping signal detected under a zero flow and under a flow of fluid through the conduit, where the detected signals have been distorted due to an additive conduit-borne interference signal according to an example; 
         FIG. 4C  shows a schematic of an argand diagram with distorted phase vector relationships having their origin translated according to conduit-borne Interference according to an example; 
         FIG. 4D  shows a schematic of an argand diagram for distorted phase vector relationships used to determine a quantity referred to as sagitta s, and an average vector according to an example; 
         FIG. 5A  shows a schematic of detected first, second and difference signals, along with demodulated plots for the detected first, second and difference signals according to an example; 
         FIG. 5B  shows a schematic for vector phase plots corresponding to detected signals and derived signals at a time of 234 μs according to an example; 
         FIG. 6A  shows demodulated plots for the zero flow vector, the radius and the interference vector according to an example; 
         FIG. 6B  shows demodulated plots for the sagitta, difference vector and phase according to an example; 
         FIG. 7A  shows vector phase plots corresponding to the detected signals and derived signals at a time of 240 μs according to an example; 
         FIG. 7B  shows vector phase plots corresponding to the detected signals and derived signals at a time of 250 μs according to an example; and 
         FIG. 7C  shows vector phase plots corresponding to the detected signals and derived signals at a time of 260 μs according to an example. 
     
    
    
     DESCRIPTION 
     Example embodiments are described below in sufficient detail to enable those of ordinary skill in the art to embody and implement the systems and processes herein described. It is important to understand that embodiments can be provided in many alternate forms and should not be construed as limited to the examples set forth herein. 
     Accordingly, while embodiments can be modified in various ways and take on various alternative forms, specific embodiments thereof are shown in the drawings and described in detail below as examples. There is no intent to limit to the particular forms disclosed. On the contrary, all modifications, equivalents, and alternatives falling within the scope of the appended claims should be included. Elements of the example embodiments are consistently denoted by the same reference numerals throughout the drawings and detailed description where appropriate. 
     The terminology used herein to describe embodiments is not intended to limit the scope. The articles “a,” “an,” and “the” are singular in that they have a single referent, however the use of the singular form in the present document should not preclude the presence of more than one referent. In other words, elements referred to in the singular can number one or more, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises,” “comprising,” “includes,” and/or “including,” when used herein, specify the presence of stated features, items, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, items, steps, operations, elements, components, and/or groups thereof. 
     Unless otherwise defined, all terms (including technical and scientific terms) used herein are to be interpreted as is customary in the art. It will be further understood that terms in common usage should also be interpreted as is customary in the relevant art and not in an idealized or overly formal sense unless expressly so defined herein. 
     There is provided an apparatus and method for determining a flow rate of a fluid in a conduit using transducers that are spatially separated from one another along the conduit. It is an object of the invention to process detected signals in order to remove interference. There is described a signal processing method to remove interference from the detected signals. 
     Interference or noise can arise originate from the conduit itself or unwanted reflections from, for example, pipe fittings. This interference is referred to as a conduit-borne component of the detected signal, where the detected signal is considered to be distorted by the conduit-borne interference. The signal processing methods described herein aim to remove pipe-borne interference and hence improve the accuracy of flow rate measurements by deriving a pure fluid component of the detected signal, where the pure fluid component is considered to be an undistorted (clean) signal. 
     In general, where reference is made to a detected signal this corresponds to an experimentally measured signal. Where reference is made to a derived signal this corresponds to a mathematically derived signal which is the signal processing of the detected signal. 
     Each detected signal and derived signal may comprise a conduit-borne component and a pure fluid component. The conduit-borne component of the signal may correspond to an ultrasonic wave that has propagated through the conduit itself (and may not have propagated through the fluid). The pure fluid component may correspond to an ultrasonic wave that has propagated through the fluid within the conduit (and may not have propagated through the conduit). 
     Fluid is determined to flow through the conduit if the fluid has a rate of flow. 
       FIG. 1  is a schematic of an apparatus for determining a flow rate of a fluid in a conduit according to an example. The apparatus comprises a conduit  105  or pipe through which a fluid may flow, for example water. There is provided at least one pair of transducers. For example, a first transducer  110  and second transducer  120  are spatially separated from one another along a length of the conduit. 
     The first transducer is configured to provide a first ultrasonic transit time signal  115  that is detected by the second transducer. The second transducer is configured to provide a second ultrasonic transit time signal  125  that is detected by the first transducer. When the first transducer is used to emit an ultrasonic signal whilst fluid is flowing through the conduit, the signal is received and detected by the second transducer, where this received signal is referred to as the first ultrasonic signal. When the second transducer is used to emit an ultrasonic signal whilst fluid is flowing through the conduit, the signal is received and detected by the first transducer, where this received signal is referred to as the second ultrasonic signal. As such, the pair of transducers are used to emit an ultrasonic signal in both respective directions along the length of conduit at which the pair of transducers are located. 
     When the first and/or second transducers are used to emit an ultrasonic signal whilst fluid is not flowing through the conduit, i.e. when the fluid has a zero flow rate, this is referred to as a third ultrasonic transit time signal which is measured under zero flow. The third ultrasonic signal may be detected with fluid present in the conduit but where there is no net flow of the fluid, i.e. the fluid is not flowing through the conduit. 
     The signal processing method described herein processes the detected first, second and third signals to derive an interference component. The derived interference component is then removed from the detected first, second and third signals to provide a more accurate measurement of flow rate having compensated for the effects of interference on the detected signals. 
     Each of the detected signals are comprised of a fluid signal component (fluid-borne signal) and an interference signal component (conduit-borne signal). As each of the ultrasonic signals traverse the conduit, the signals emitted by each transducer are modified by the flow rate of the fluid and noise from sources of interference. As such, when the signal is detected at the other transducer of the pair, the signal comprises a component corresponding to actual fluid flow and a component corresponding to interference. 
     According to an example, the component of the detected signal that corresponds to the interference signal can arise from conduit-borne noise  150 . This conduit-borne signal interferes with the pure fluid signal. For example, consider a steel conduit with a fluid (e.g. water), where the time of arrival of an ultrasonic signal is predicted by the speed of sound in the fluid. The speed of longitudinal waves in steel are two to three times faster than in the fluid but this is not true of transverse waves. For transverse waves, reflections from the end fittings and group delay in setting up oscillations in the steel are taken into consideration in order for it to be possible to calculate a correct coincident arrival time of the transverse waves. 
     There are some assumptions to be made regarding the measurement system and the conduit-borne noise:
         The system is linear: the principle of superposition applies throughout and the detected signals may be accurately modelled by the concept of a fluid signal and additive noise from the conduit.   None of the system is dispersive: there are phase shifts and amplitude changes but no changes in the frequency involved for each signal.   The first and second signals are identical at zero flow: not only assured by the principle of reciprocity but reflected in experimental observations, i.e. it is also assumed that the fluid signal components and the interference signal components of the detected signals are respectively identical under zero flow.   The fluid component of the first and second signals are identical in amplitude and form: it is noted that this is not strictly true as there are issues of attenuation and apparent path length under flow, however these are second order effects which will be ignored herein.   The conduit-borne interference signal is ‘stable’ over the timescales of the ultrasonic measurements. This is apparent from experimental observations, whereby changes associated with temperature and fluidic fluctuations are factors which take several seconds or minutes to occur and are therefore ignored herein.   The path of the interference signal is entirely outside the fluid, i.e. the conduit-borne signal has not passed through the fluid. Whilst it is possible to check that the interference signal does not pass through a fixed section of fluid (such as a reflection from the end fittings, by checking it is not affected by the end fittings), it may be less clear whether the interference signal has travelled through the fluid for any part of its history.       

     The first and second transducers are driven by a signal generator  130 . The first and second transducers are in communication with, or connected, to a processing module  140  comprising a processor for processing the detected signals according to examples described herein. 
     Examples of detected signals under zero flow will be described with reference to  FIGS. 2A and 3A . According to an example, the first and second transducers are spatially separated from one another along a length of the conduit. The first transducer is configured to emit the first ultrasonic transit time signal along the length of the conduit towards the second transducer under zero fluid flow. After the first ultrasonic signal has travelled along the section of the conduit the first signal is received at the second transducer. Similarly, the second transducer is configured to emit the second ultrasonic transit time signal along the length of the conduit towards the first transducer under zero fluid flow. After the second ultrasonic signal has travelled along the section of the conduit the second signal is received at the first transducer. 
     Examples of detected signals in the presence of a flow will be described with reference to  FIGS. 2B and 3B . According to an example, the first and second transducers are spatially separated from one another along a length of the conduit. The first transducer is configured to emit the first ultrasonic transit time signal along the length of the conduit towards the second transducer under a flow of fluid through the conduit. After the first ultrasonic signal has travelled along the section of the conduit the first signal is received at the second transducer. Similarly, the second transducer is configured to emit the second ultrasonic transit time signal along the length of the conduit towards the first transducer under a flow of fluid through the conduit. After the second ultrasonic signal has travelled along the section of the conduit the second signal is received at the first transducer. 
     According to an example, more than two transducers may be provided for redundancy. For example, further pairs of transducers may be spatially separated along different lengths or sections of the conduit. This can provide an indication of fluid flow at different sections of the conduit between the regions at which the transducers are located. 
     Referring now to  FIGS. 2A and 2B , these examples correspond to the fluid-borne signal (i.e. clean signal without any interference) and, as such, are free from additional artefacts or noise from any conduit-borne signal(s). 
       FIG. 2A  shows an example of an ultrasonic ping signal detected under zero fluid flow, as a function of amplitude and time. The detected signal  200  comprises a waveform with two traces: a first trace corresponding to the first signal, and a second trace corresponding to the second signal, where the first and second trace overlap (are coincident) due to the first and second signals being identical at zero flow. The processing module  140  can be configured to calculate a measure of the difference in the values of the amplitude and phase of the first signal and second signals at a selected time, which is shown as a difference signal  202 . The difference signal is zero under zero fluid flow. 
       FIG. 2B  shows an example of an ultrasonic ping signal detected under a flow of fluid through the conduit, as a function of amplitude and time. The detected signal comprises a waveform with two traces: a first trace  204  corresponding to the first signal, and a second trace  206  corresponding to the second signal, where the first and second trace have an amplitude and phase shift between them due to the first and second signals being measured under the flow of the fluid. As shown, there is a phase shift in both the first and second signals (in opposite directions due to the applied flow). The processing module  140  can be configured to calculate a measure of the difference in the values of the amplitude and phase of the first signal and second signals at a selected time, which is shown as a difference signal  208 . The difference signal fluctuates according to the first and second signals with an amplitude related to flow. It can be seen that the amplitude of the difference signal is non-zero under the applied flow of fluid. 
     Referring now to  FIGS. 3A and 3B , these examples correspond to the fluid-borne signal (pure or undistorted component) and the conduit-borne signal (distorted component), which includes additional artefacts or noise. 
       FIG. 3A  shows an example of an ultrasonic ping signal for a fluid-borne signal and conduit-borne interference detected under zero fluid flow, as a function of amplitude and time. The detected signal  310  comprises a waveform with two traces: a first trace corresponding to the first signal, and a second trace corresponding to the second signal, where the first and second trace overlap (are coincident) due to the first and second signals being identical at zero flow. As shown, the detected first and second signals are compromised by interference originating from the conduit, seen as a modulating envelope wave function over the first and second traces. The effect of this interference or noise on the detected signal  310  is to vary the amplitude of the ping and contribute a ping which has come through the conduit rather than purely from the fluid. Since none of the media is dispersive, there is no assumed change in frequency and the effect is that of classical one-dimensional wave interference. The processing module  140  can be configured to calculate a measure of the difference in the values of the amplitude and phase of the first signal and second signals at a selected time, which is shown as a difference signal  312 . 
     The difference signal is zero under zero fluid flow.  FIG. 3B  shows an example of an ultrasonic ping signal detected for a fluid-borne signal and a conduit-borne interference signal under a flow of fluid through the conduit, as a function of amplitude and time. The detected signal comprises a waveform with two traces: a first trace  314  corresponding to the first signal, and a second trace  316  corresponding to the second signal, where the first and second trace have an amplitude and phase shift between them due to the first and second signals being measured under the flow of the fluid. As shown, there is a phase shift in both the first and second signals (in opposite directions due to the applied flow). The processing module  140  can be configured to calculate a measure of the difference in the values of the amplitude and phase of the first signal and second signals at a selected time, which is shown as a difference signal  318 . The difference signal fluctuates according to the first and second signals with an amplitude related to flow. It can be seen that the amplitude of the difference signal is non-zero under the applied flow of fluid. The relationship of the difference signal to the first and second signals is unclear due to the first and second signals having unequal amplitudes due to being distorted or affected by the conduit-borne interference signal. 
     The processor in the processing module  140  can be configured to generate respective measures of an amplitude and a phase of the first and second ultrasonic signals measured under a fluid flow, and respective measures of an amplitude and a phase of the third signal under zero flow. The respective measures are generated at a selected time in order to provide respective first, second and third flow vectors (that respectively correspond to the first, second and third signals). As such, it is possible to represent the detected first, second and third signals diagrammatically as vector quantities drawn on an Argand diagram, where their modulus and argument indicates the amplitude and phase of each respective signal. 
     According to an example, the flow vectors are represented on an Argand vector diagram, where the magnitude of each vector indicates an amplitude of the respective signal, and the direction of each vector indicates a phase of the respective signal. 
     The method for processing the detected signal in order to obtain flow vectors for representation on an Argand diagram will now be described. 
     The processor generates the respective measures of amplitude and phase of each of the detected signals using the following methodology. A complex analytic signal (equation 1 below) is provided from a sampled signal at a known carrier frequency using synchronous demodulation, i.e. the amplitude and phase of the detected first, second and third signals can be extracted and represented as phase vectors having real and imaginary components. 
     The detected signal is represented by: 
         P ( t )·sin(2 πf   c   t +φ)
 
     The complex version of the known carrier signal (having the ping carrier frequency of f c ) is represented by: 
     
       
      
       e 
       −2πif 
       
         c 
       
       t  
      
     
     The detected signal is multiplied by the complex version of the carrier signal: 
         D ( t )= P ( t )· e   iω     c     t ·cos(ω c   t +φ)
         where   P(t) is the ping envelope,   ω is an arbitrary phase angle,   f c  is the ping carrier frequency (and ω c =2Tπf c )       

     Applying Euler&#39;s formula: 
         D ( t )= P ( t )·cos(ω c   t )·cos(ω c   t +φ)+ P ( t )· i ·sin(ω c   t )·cos(ω c   t +φ)
 
     Then applying sum identities to the phase offset φ: 
         D ( t )= P ( t )·cos(ω c   t )·cos(ω c   t )·cos(φ)− P ( t )·cos(ω c   t )·sin(ω c   t )·sin(φ)+ P ( t )· i ·sin(ω c   t )·cos(ω c   t )·cos(φ)− P ( t )· i ·sin(ω c   t )·sin(ω c   t )·sin(φ)
 
     Then applying product-to-sum identities: 
         D ( t )= P ( t )·½[1+cos(2ω c   t )]·cos( co )− P ( t )·½·sin( 2 ω c   t )·sin(φ)+ i·P ( t )·½·sin(2ω c   t )·cos(φ)− i·P ( t )·−½·[1− d  cos(2ω c   t )]·sin(φ)
 
     Then applying a low-pass filter (to effectively remove terms &gt;ω c ): 
         D ( t )= P ( t )·½·cos(φ)− i·P ( t )·½·sin(φ)  Equation 1
 
     The complex function of equation 1 has real and imaginary components dependent on the phase relationship of the detected signal to the initial reference phase (i.e. known carrier frequency) used for the synchronous demodulation. This methodology gives a means of extracting the ping envelope P(t) regardless of the value of cp. 
     Then, changing from Cartesian co-ordinates to polar co-ordinates: 
     
       
         
           
             
               
                 
                   
                     
                       | 
                       
                         D 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       | 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     P 
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   · 
                                   
                                     1 
                                     2 
                                   
                                   · 
                                   cos 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   φ 
                                   ) 
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   
                                     P 
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   · 
                                   
                                     1 
                                     2 
                                   
                                   · 
                                   sin 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   φ 
                                   ) 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         ] 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                        
                       
                         D 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                        
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               
                                 
                                   P 
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 2 
                               
                               · 
                               
                                 1 
                                 4 
                               
                               · 
                               cos 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 φ 
                                 ) 
                               
                               2 
                             
                           
                           + 
                           
                             
                               
                                 
                                   P 
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 2 
                               
                               · 
                               
                                 1 
                                 4 
                               
                               · 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 φ 
                                 ) 
                               
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                        
                       
                         D 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                        
                     
                     = 
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   P 
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 2 
                               
                               · 
                               
                                 1 
                                 4 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         ( 
                                         φ 
                                         ) 
                                       
                                       2 
                                     
                                   
                                   + 
                                   
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         ( 
                                         φ 
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         
                            
                           
                             D 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                         
                       
                       = 
                       
                         
                           1 
                           2 
                         
                         · 
                         
                           P 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     Similarly: 
     
       
         
           
             
               
                 
                   
                     
                       arg 
                       ⁡ 
                       
                         ( 
                         
                           D 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       arctan 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 P 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               · 
                               
                                 1 
                                 2 
                               
                               · 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               φ 
                               ) 
                             
                           
                           
                             
                               
                                 - 
                                 
                                   P 
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                               · 
                               
                                 1 
                                 2 
                               
                               · 
                               cos 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               φ 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       arg 
                       ⁡ 
                       
                         ( 
                         
                           D 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           arctan 
                           ⁡ 
                           
                             ( 
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   φ 
                                   ) 
                                 
                               
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   φ 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           arctan 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               tan 
                               ⁡ 
                               
                                 ( 
                                 φ 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                         
                           arg 
                           ⁡ 
                           
                             ( 
                             
                               D 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             ) 
                           
                         
                         = 
                         φ 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3 
                 
               
             
           
         
       
     
     Therefore, if the complex version of the known carrier signal is applied across the detected signals in the time domain (for discrete  8   t ) for any given phase offset, accurate values for both the phase and amplitude of the first and second signals can be obtained at time t (given by φ t ). 
     As such, the detected signals can be processed for representation on an Argand diagram. Examples of this signal processing for representation on Argand diagrams will now be described with reference to  FIGS. 4A to 4D . 
       FIG. 4A  shows an example argand diagram with phase vectors representing an ultrasonic ping signal detected under a zero flow (for example, with reference to the signal shown in  FIG. 2A ) and detected under a flow of fluid through the conduit (for example, with reference to the detected signal shown in  FIG. 2B ). The first signal (under flow) is represented by phase vector  404 . The second signal (under flow) is represented by phase vector  406 . The third signal (zero flow) is represented by phase vector  400 . The processor  104  can be configured to calculate a measure of the difference in the values of the amplitude and phase of the first signal and second signals (under flow) at the selected time using the first and second flow vectors, to provide a difference vector. Note that there is no phase difference vector for the third signal, since under zero flow the first and second signals are identical. The difference signal measured under flow is represented by the difference phase vector  408 . The difference vector  408  is represented on the Argand diagram as being perpendicular to the real axis. The difference vector  408  indicates the phase difference between the first and second signals under flow. Note that the intersection of the difference vector  408  with the real axis indicates the phase difference at zero flow (i.e. there is no phase difference at zero flow). 
     According to an example, the axis of the first and second flow vectors can be arbitrarily aligned (where the first and second flow vectors  405  coincide), such that a zero flow is parallel to the real axis, i.e. zero flow can be represented as a zero flow vector pointing along the real axis  400 . As such, the flow is indicated directly by the angle Φ t  between the first and second vectors: the greater the angle Φ t , the higher the rate of flow of the fluid through the conduit, and vice versa. 
       FIG. 4B  shows an example argand diagram with phase vectors representing an ultrasonic ping signal detected under a zero flow (for example, with reference to the signal shown in  FIG. 3A ) and under a flow of fluid through the conduit (for example, with reference to the signal shown in  FIG. 3B ), where the detected signals have been distorted due to an additive conduit-borne interference signal. It is assumed that the conduit-borne signal has the same frequency as the pure fluid signal. The distorted first, second and third signals that are detected are represented on the Argand vector diagram as flow vectors having a corresponding amplitude and phase. The first distorted signal (under flow) is represented by phase vector  414 . The second signal (under flow) is represented by phase vector  416 . The third signal (zero flow) is represented by phase vector  410 . Similarly, as before, the axis of the first and second flow vectors can be arbitrarily aligned. 
     It can then be seen in  FIG. 4B  that the detected signals, in the presence of conduit-borne interference, are affected by the conduit-borne interference signal so as to be offset or distorted by an amount ‘M’  444 . 
     For example, the detected vector at zero flow is distorted by the interference signal, such that it has moved away from the real axis. Applying some geometric constructions, the following observations of this model can be made:
         The length of the distorted first and second flow vectors ( 414 ,  416 ) represent the amplitude of the distorted first and second signals ( 314 ,  316 ) and these are detected as having differing amplitudes. This will be the case unless the interference vector (yet to be derived) is parallel to the zero flow vector  410 . This geometric construction is confirmed by observation.   The detected phase shift  413  (represented by ξ t ) is not equal to the correct phase shift  411  (represented by Φ t ) and therefore cannot be directly related to flow. This geometric construction is confirmed by observation.   The zero flow vector  410  is similarly distorted by the conduit-borne interference.   The distorted difference vector  418  is unaffected by the conduit-borne interference (in both phase and amplitude).       

     Using the above observations and geometric constructions (as will be described as follows), the offset ‘M’  444  of the interference signal can be determined, i.e. the amount of distortion created by the presence of the conduit-borne interference can be determined. 
     In order to take this analysis further, referring to  FIG. 4C , the point of reference  407  for the distorted flow vectors may be changed (for ease of visual reference to the Argand diagrams) by inverting the interference vector  444  (now no longer dashed) and applying it to the origin of the distorted signal vectors. This is shown geometrically in  FIG. 4C . Additionally, a circle  460  has been drawn around the origin  405  of the pure fluid vectors, with radius r  462  equal to the amplitude of the pure fluid signal vectors (i.e. without conduit-borne interference). This radius is effectively the locus of the endpoints of the pure fluid signals from zero flow, where the angle Φ t    411  widens as flow increases. 
     The interference vector ‘M’  444  of the interference signal comprises a real m x  and imaginary m y  component. 
     Pythagoras theorem can be applied to the distorted flow vectors to determine the first  414  (U), second  416  (D) and third  410  (Z) flow vector amplitudes: 
         Z   2 =( r+m   x ) 2   +m   y   2   Equation 4
         where Z is the third flow vector  410  (distorted zero flow),       

     
       
         
           
             
               
                 
                   
                     U 
                     2 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           r 
                           + 
                           
                             m 
                             x 
                           
                           - 
                           s 
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             m 
                             y 
                           
                           - 
                           
                             Δ 
                             2 
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   5 
                 
               
             
           
         
       
         
         
           
             where U is the distorted first flow vector  414 , and s is sagitta  480  (which, with reference to  FIG. 4D , is equal to the difference between the average vector halved (Σ/2) and the distorted third vector  410  under zero flow (Z), and where the value of sagitta s  480  is represented by equation 8 shown later), and Δ is the difference vector  418 . Furthermore: 
           
         
       
    
     
       
         
           
             
               
                 
                   
                     D 
                     2 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           r 
                           + 
                           
                             m 
                             x 
                           
                           - 
                           s 
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             m 
                             y 
                           
                           + 
                           
                             Δ 
                             2 
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   6 
                 
               
             
           
         
       
     
     where D is the distorted second flow vector  416 . 
     Combining equations 5 and 6 allows a determination of the imaginary component m y  of the interference vector  444 : 
     
       
         
           
             
               
                 U 
                 2 
               
               - 
               
                 D 
                 2 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       m 
                       y 
                     
                     - 
                     
                       Δ 
                       2 
                     
                   
                   ) 
                 
                 2 
               
               - 
               
                 
                   ( 
                   
                     
                       m 
                       y 
                     
                     + 
                     
                       Δ 
                       2 
                     
                   
                   ) 
                 
                 2 
               
             
           
         
       
     
     Such that: 
     
       
         
           
             
               
                 
                   
                     
                       | 
                       U 
                       ⁢ 
                       
                         | 
                         2 
                       
                       ⁢ 
                       
                         - 
                         
                           | 
                           D 
                           ⁢ 
                           
                             | 
                             2 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           m 
                           y 
                         
                         2 
                       
                       - 
                       
                         
                           m 
                           y 
                         
                         ⁢ 
                         Δ 
                       
                       + 
                       
                         
                           Δ 
                           2 
                         
                         4 
                       
                       - 
                       
                         
                           m 
                           y 
                         
                         2 
                       
                       - 
                       
                         
                           m 
                           y 
                         
                         ⁢ 
                         Δ 
                       
                       - 
                       
                         
                           Δ 
                           2 
                         
                         4 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                            
                           U 
                            
                         
                         2 
                       
                       - 
                       
                         
                            
                           D 
                            
                         
                         2 
                       
                     
                     = 
                     
                       
                         - 
                         2 
                       
                       ⁢ 
                       
                         m 
                         y 
                       
                       ⁢ 
                       Δ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       m 
                       y 
                     
                     = 
                     
                       - 
                       
                         
                           | 
                           U 
                           ⁢ 
                           
                             | 
                             2 
                           
                           ⁢ 
                           
                             - 
                             
                               | 
                               D 
                               ⁢ 
                               
                                 | 
                                 2 
                               
                             
                           
                         
                         
                           2 
                           ⁢ 
                           Δ 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   7 
                 
               
             
           
         
       
     
     It can be observed that the imaginary component m y  of the interference vector  444 , as represented by equation 7, is not dependent on flow (however, it is observed to be undefined at zero flow). In practical terms, this means that the value of the imaginary component m y  (for a given time, t) may be averaged over time regardless of flow. 
     As such, the processor  140  can be configured to generate a measure for a component (m y ) of an interference vector ‘M’  444  representing a conduit-borne component signal of the first and second distorted signals  314 ,  316 , using the first and second distorted flow vectors  414 ,  416  and the difference vector  418 . 
     As will now be described, the processor  140  can be configured to generate a measure for the flow rate of the fluid using the component m y  of the interference vector ‘M’  444 , the third distorted flow vector  410  and the sum of the first and second distorted vectors  414 ,  416 . 
       FIG. 4D  shows an example Argand diagram for distorted phase vector relationships used to determine a quantity referred to as sagitta s  480 , and an average vector E  470 , where the average vector Σ  470  corresponds to the sum of the first and second distorted vectors  414 ,  416 . This average vector  470  is the addition of the first and second vectors (i.e. sum of the first and second distorted signals). The average vector  470  passes through the intersection of the difference vector  418  with the real axis, where this relationship follows from a parallelogram formed by translating the origin of the distorted first and second vectors, such that the average vector  470  and the difference vector  418  bisect the area thus formed. Also shown in  FIG. 4D  is the quantity sagitta s  480 , which is equal to the difference between the average vector halved (Σ/2) and the distorted third vector  410  under zero flow (Z). The value of sagitta s  480  is represented by: 
     
       
         
           
             
               
                 
                   s 
                   = 
                   
                     
                       
                         | 
                         Z 
                         ⁢ 
                         
                           | 
                           2 
                         
                         ⁢ 
                         
                           - 
                           
                             
                               m 
                               y 
                             
                             2 
                           
                         
                       
                     
                     - 
                     
                       
                         
                           
                             ( 
                             
                               
                                 | 
                                 Σ 
                                 | 
                               
                               2 
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             m 
                             y 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   8 
                 
               
             
           
         
       
         
         
           
             where m y  is the imaginary component of the interference vector ‘M’  444 . 
           
         
       
    
     As such, the flow Φ t  (for the pure fluid signal having removed the effects of the conduit-borne interference) may be given directly by (with reference to  FIGS. 4C and 4D ): 
     
       
         
           
             
               
                 
                   
                     Φ 
                     t 
                   
                   = 
                   
                     4 
                     . 
                     
                       arctan 
                       ⁡ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             s 
                           
                           Δ 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   9 
                 
               
             
           
         
       
     
     In practice, however, these calculations are sensitive since a small error in either |Z| or |Σ| leads to a far greater error in s and thence Φ t . Moreover, as the value of s is flow dependent, it may not be refined by averaging over time. 
     A preferred (less sensitive) approach, may be found by calculating the radius r  462  from the sagitta s  480  using the formula for the radius of an arc: 
     
       
         
           
             
               
                 
                   r 
                   = 
                   
                     - 
                     
                       
                         
                           s 
                           2 
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 | 
                                 Δ 
                                 | 
                               
                               2 
                             
                             ) 
                           
                           2 
                         
                       
                       
                         2 
                         ⁢ 
                         s 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   10 
                 
               
             
           
         
       
     
     This value for r may be averaged over time. The flow Φ t  can then be computed from the difference vector Δ  418  and the radius r. 
     The derivations described herein imply that the two quantities m y  and r can be calculated from the amplitude values U, D and Z obtained via flow vectors representing the detected first, second and third signals. 
     Further, m y  and r can be averaged over time for a more accurate determination of flow since these values are independent of flow (as second order effects are ignored according to the listed assumptions above). 
     Accordingly, derived values for flow have been calculated and compared to actual flow rates that have been measured, and the linearity in these measurements reveals a sufficiently accurate, derived determination of flow. Further, the use of the Argand vector diagram representations (from the calculations) show the expected pure fluid flow behaviour. 
     Some example experimental measurements which use this signal processing method will now be described. 
       FIGS. 5-7  are examples waveforms and Argand diagrams of experimentally detected flow rates for a fluid flowing through a conduit, supporting the described derivation for determining the flow. 
     A Windows program was written in Delphi to perform the following:
         a) Process the captured data sets,   b) Show waveforms and demodulated waveforms for detected first (Up), second (Down), and difference signals,   c) Use a centre of gravity algorithm to derive a time of flight from the difference signal,   d) Calculate filters and display: the third flow vector (zero flow vector), the imaginary component m y  of the conduit-borne signal, and the radius r (i.e. pure fluid amplitude),   e) Provide vector displays for all of the above, and   f) Calculate the flow and compare this to the actual flow log.       

       FIG. 5A  shows detected first  514 , second  516  and difference  518  signals, along with magnitudes of the demodulated plots for the detected first  515 , second  517  and difference signals  519  relating to the amplitude axis on the left-hand side, according to an example. The magnitudes for the demodulated plots  515 ,  517  and  519  are found using equation (2) at a carrier frequency of around 233 kHz. The phase of derived pure signals  504 ,  506  are found using equation (3) and are shown relating to the phase axis on the right-hand side. 
       FIG. 5A  maps onto  FIG. 5B  which is has been rotated and modified (as described above) for visual ease of recognition. 
       FIG. 5B  shows vector phase plots corresponding to the detected signals and derived signals at a time of 234 μs according to an example. The distorted flow vectors are shown for: the first flow vector  524  corresponding to the detected first signal  514 ; the second flow vector  526  corresponding to the detected second signal  516 ; and difference vector  528 . The derived pure fluid signals  504 ,  506  are determined using the interference vector ‘M’  544  to compensate for the conduit-borne interference component signal, according to the derived first pure (undistorted) vector  534  and derived second pure (undistorted) vector  536 . 
       FIG. 6A  shows demodulated plots for the zero flow vector  602  (using equation 4), the radius r  604  (using equation 10) and the imaginary component of the interference vector m y    606  (using equation 7) according to an example. 
       FIG. 6B  shows demodulated plots for the sagitta s  608  (using equation 8), the radius r  604  (using equation 10), and the difference vector  610  relating to the amplitude axis on the left-hand side, and the flow Φ t    612  (using equation 9) relating to the phase axis on the right-hand side according to an example. 
       FIGS. 7A-7C  show vector phase plots implemented using equations 9 and 10 at a selected time. These vector phase plots show vector phase plots corresponding to the first (U) and second (D) detected signals (under flow/distorted) respectively  714 ,  716  and first and second derived signals respectively  704 ,  706 . The (distorted) difference vector Δ  718  for each of the detected and derived signals is shown to be overlapping. The interference vector “M”  744  is also shown. The third signal (Z) is shown by the zero flow phase vector  710 . 
       FIG. 7A  shows vector phase plots corresponding to the detected signals and derived signals at a time of 240 μs according to an example.  FIG. 7B  shows vector phase plots corresponding to the detected signals and derived signals at a time of 250 μs according to an example.  FIG. 7C  shows vector phase plots corresponding to the detected signals and derived signals at a time of 260 μs according to an example. 
     In a case in which conduit/pipe-borne noise is eliminated, for example by acoustically engineering a flow meter structure so that the effects of such noise are minimal, it is possible to determine a measure for the flow rate of a liquid in a conduit using the phase difference angle Φ t  directly. That is, according to an example, assuming the noise offset vector to be zero, Φ t  may be determined by calculating the difference between the angles of the measured up and down vectors. 
     Referring to equation 3 noted above, the phases of an up signal (ϕ u ) and a down signal (ϕ d ) can be determined. Referring to  FIG. 4B  in which the vectors are shown rotated and aligned symmetrically about the real axis, the value Φ t  may be calculated directly as: Φ r =ϕ u −ϕ d . 
     In other words, if pipe borne information is ignored, then in  FIG. 4B  vector  444  is zero and Φ t  equals ζ r . Accordingly,  FIG. 4 b    conflates to become  FIG. 4A . 
     In an example, pipe-borne interference may be removed using the vector analysis methods described above (e.g. using a real time graphical display) to set up preferred oscillation modes on a given transducer arrangement for a flow meter so as to minimise interference on the received signal. Furthermore, it is possible to change materials (e.g. axially along the length of the conduit in question) to reduce transmission of interference, and configure a flow meter to reduce internal and external reflections at critical acoustic distances to prevent “rattling”. It is also possible to apply tube damping by various means. 
     As described above, examples enable a digital representation of vector quantities to be generated, which in turn enable calculation of vector angles at any point in time on a waveform representing a signal. 
     According to an example, and by way of reference to  FIGS. 5, 6 and 7 , the vectors provide phase and magnitude information throughout a received signal window and provide a value of ϕ t  which can be used to determine flow graphically at the appropriate parts of the signal in the time domain. The vectors (or their graphical representation) thus provide direct indications of whether pipe-borne (or any other) interference exists in the system. With reference to  FIG. 5B  for example, the vectors  534  and  536  (in which interference is absent) can therefore be directly used to determine a measure of flow rate. 
     In an example, it is therefore possible to use the vector analysis and the associated display of phase angles throughout the waveform as a tool to asses and tune a system to minimise interference effects. 
     The present inventions can be embodied in other specific apparatus and/or methods. The described embodiments are to be considered in all respects as illustrative and not restrictive. In particular, the scope of the invention is indicated by the appended claims rather than by the description and figures herein. All changes that come within the meaning and range of equivalency of the claims are to be embraced within their scope.