Figure 1. Typical double U tube CMF structure

Figure 2. The oscillation of the tube

Novel Coriolis Mass Flowmeter Signal Processing Algorithms based on DFT and Digital

Correlation

Chen Kun, Zheng Dezhi, Fan Shangchun, Li Fan, Zhao Jianhui School of Instrument Science and Opto-electronics Engineering

Beihang University Beijing, China

Chenkun_buaa@163.com Abstract—Coriolis mass flowmeter (CMF) is becoming a research hotspot in flow measurement for its high accuracy and multi-parameter measurement. The detection principle of CMF is to calculate the phase difference of two sine signals which have same frequency. Discrete forurier transform (DFT) and digital correlation are applied in the signal processing of CMF, and theoretical error in this application is analyzed. According to the true signal output from the sensors, simulation analysis is made. Results show that the algorithms’ relative error is lower than 0.2%, which is more precise than typical algorithms and can be in widely application in mass flow meter.

Keywords— CMF; phase difference measurement; DFT; digital correlation; simulation

I. INTRODUCTION Coriolis mass flowmeter (CMF) is a kind of resonant

sensor that directly measures the mass flow-rate and obtains the density of liquid through the element simultaneously. CMF is currently one of the most rapidly developed flow-meters and has a wide future in application.

CMF is designed by the principle that when mass-flow moving through a vibrating tube causes a proportional Coriolis force to act, which is detected as a phase difference between two sensors [1]. But in the industry environment there are huge ambient noise and disturbance, so SNR of the signal and the phase difference of two vibrating signal detected are very low. What’s more, influenced by the change of fluid density the vibrating frequency of the tube doesn’t equal to its natural frequency, the output signal of CMF has changeable basic frequency and complex frequency components, so phase difference of synthesized waveform is measured and calculated by the meter. The accuracy of traditional phase difference algorithm is very low with up to 1 percentage error. From the viewpoint of signal processing, frequency domain analysis and digital correlation methods are introduced and analyzed in mass flow calculation. Simulation results demonstrate the algorithms are valid and of high precision of 0.2%.

II. CMF MASS FLOW MEASURING PRINCIPLE The structure of traditional double U tube CMF is as

figure 1 shows. The vibrating of measuring tube is showed as figure 2 [2].

Lateral vibration around CC' axis is produced under the motivation when no liquid flows in the tube, and can be expressed as tsAtsx ωsin)(),( = . Where ω is basic frequency of system determined by resonant structure,

including elastic tube and elastic support. )(sA is principal mode correspond to ω . s is curvilinear coordinates along the axial direction of tube.

So angular velocity of elastic tube around CC' axis is

tsxsA

sxttsxs ωω cos

)()(

)(1

d),(d)( =⋅=Ω (1)

Where )(sx is the distance from the certain point to CC' axis.

2010 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2010), October 3-5, 2010, Penang, Malaysia

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When there is flow in the tube at velocity υ , elastic tube will vibrate upward. There will be a Coriolis force act on CBD part:

msmaF CC d)(2dd υ×Ω−=−= (2)

As the same there must be a symmetrical point on which Coriolis force is CC FF d'd −= .

The minus sign indicates that the two forces are equal and opposite. When liquid flow traverse, a moment around DD ′ axis is produced :

∫ ×= )(d2 srFM C (3)

Where )(sr is distance from the certain point to DD' axis.

ssxsrsAtQM m d

)()()(coscos2 ∫= ωαω (4)

Where α is contending angle between flow velocity and DD' axis. mQ is mass flow through the tube.

The couple caused by Coriolis effect makes the resonant tube distortion around DD' axis, which is called secondary undulation relative to principal oscillation and can be expressed as

( ) ( )ϕωω ＋tQsBtx m cos)( 11 = (5)

Where ( )sB1 is sensitivity coefficient of secondary undulation, which is related to structure and parameter of resonantor and the placement of detection point. ϕ is phase change caused by secondary undulation response.

According to the above analysis, when there is flow in the tube, the oscillation equation at B、B ′ can be described as follow. ( BL is coordinate value of B)

( ) ( )ϕωωω +−= tQLBtLAS mBBB cossin)( 1 (6)

( ) ( )ϕωωω ++= tQLBtLAS mBBB cossin)( '' 1' (7)

So mass flow rate can be calculated by means of measuring phase difference BB ′ϕ

'BBm KQ ϕω = (8)

Where K is proportional action coefficient which is related to structure, size, material of flowtube and excitation signal.

III. DFT ALGORITHM AND ERROR ANALYSIS Suppose that sine sampling signal sequence under white

Gaussian noise is

( ) ( ) 10π2sin −≤≤++= NkfkTakTx kωθ (9)

Where a f θ are amplitude frequency and initial phase of signal respectively. kω is noise signal following

Gaussian distributions (0， 2σ ). Use discrete Fourier transform, power spectrum

expression is as follow.

)()()()(1

0

π2

kjSkSenskS IR

N

n

knNj

+==∑−

=

− (10)

Real part expression of formula (10) is:

∑−

=

=1

0

π2cos)()(N

nR nk

NnskS (11)

Imaginary part expression of formula (10) is:

∑−

=

=1

0

π2sin)()(N

nR nk

NnskS (12)

Corresponding amplitude expression is:

22 )()()( kSkSkS IR += (13)

Corresponding phase expression is:

[ ])()(arctan)(argkSkSkS

R

I= (14)

Using fast Fourier transform (FFT) to sample data to find the k value corresponding to maximum power spectrum, which is recorded as mk . The frequency at mk is fundamental frequency, so the phase difference is:

)(S)(Sarctan

)(S)(Sarctan

2R

2I

1R

1I

m

m

m

m

kk

kk −=Δθ (15)

)(S1I mk )(S1R mk )(S2I mk )(S2R mk are imaginary part and real part of maximum power spectrum of two signals’ fundamental frequency after FFT.

Actually the processor can only sample and calculate finitely many elements, and Spectrum analysis with DFT can only be done just in finite interval. Because frequency fluctuating of CMF output signal, it is impossible to realize the integral period signal samples. So it’s inevitable for energy leakage caused by time domain truncation, causing in lower precision [4].

]})(1[exp{)](sin[2

)](sin[)( 00

0

0 πθππ Tfk

NNj

TfkN

TfkakS −−−×−

−×=

12/,...,2,1,0 −= Nk (16)

57

Amplitude of )(kS is:

)](sin[2

)](sin[)(0

0

TfkN

TfkakA−

−×= ππ (17)

Around main lobe of )(kS , )(kA is approximated to:

)(2)](sin[)(

0

0

TfkTfkaNkA

−−××=

ππ (18)

The k value at peak spectral line is denoted as m. )int( 0Tfm = . The amplitude at peak spectral line is

recorded as mA∧

.

)(2)](sin[

0

0

TfmTfmaNAm

−−××=

∧

ππ (19)

Tfm 0−=δ indicates the relative deviation between signal actual frequency and frequency at peak spectral line, whose variation range is (-0. 5, 0. 5). Where Δf = 1 /T is DFT’s frequency resolution.

The phase at peak spectral line is recorded as mφ ：

δπθφ )/11(0 Nm −−= (20)

It can be concluded that phase difference calculated by DFT method has an error of δπ)/11( N− , which can be up to 90° .So spectrum correction must be added to improve measuring precision. The principle of phase difference correction is described as follow. Sample two continuous data sequence from single frequency signal with the same points for each, so the two sample data have the same frequency and amplitude, different initial phase. That’s to say they have phase difference [4]. Use FFT to the two signals which have added the same window function to find the phases at peak spectral line. The phase-frequency functions after transformed have linear relation and same gradient in main lobe of window function. So correcting value can be calculated through difference of phase angle at peak spectral line to realize correction of frequency and phase, and to obtain more accurate phase difference information ulteriorly.

IV. DIGITAL CORRELATION ALGORITHM AND ERROR ANALYSIS

The tube is vibrating when CMF is working, so the signal is subjected to noise disturbances. Because the correlation between noise and vibrating signal is very tiny, correlation detection method effectively eliminate noise interference and have advantages in noise removal [5]. The two vibrating signal coupled noise can be abstracted as:

)()sin()( 111 tNtAtx ++= θω (21)

)()sin()( 222 tNtBtx ++= θω (22)

Where two signal’s amplitude are A and B respectively. )(1 tN ， )(2 tN are noise signal following Gaussian

distributions (0， 2σ ). Correlation calculation of two signals in a period is :

ttxtxT

RT

xx d)()(1)(0

2121 ∫ −= ττ (23)

If 0=τ , Correlation calculation is as follow:

ttNtBtNtAT

RT

xx d)]()[sin()]()[sin(1)0(0

221121 ∫ ++++== θωθωτ

Ideally, noise and signals have no correlation, nor between noise and noise, so after integration, it will be:

)cos(21)0( 2121 θθτ −== ABR xx (25)

So phase difference between signals can be calculated by arc-cosine function. Signal amplitude can be calculated by autocorrelation function at zero point. Discretization expression for signal after A/D sampling is:

∑−

=

==1

02121 )()(1)0(

N

kxx kxkx

NR τ (26)

Where N is sample sequence length in a signal period. So when integral period sampling is satisfied, the phase difference is:

)0(*)0()0(arccos

*)0(2arccos

21

2121

xx

xxxx

RRR

BAR ==Δϕ (27)

Foregoing discussion about phase difference calculating is on the precondition of integral period sampling, while actually the assumption can not be satisfied. When the domain of integration is not an exact period, there will be error in calculating. Farther theoretical research is done about the error’s impact on the phase difference measuring.

Take no account of noise the vibrating signals are:

)sin()( 11 θω += tAtx )sin()( 22 θω += tBtx (28)

Consider correlation calculating in the domain (0, TT Δ+ ). Where T is signal period and －T< TΔ <T.

dt)][sin()][sin(1

dt)][sin()][sin(1

d)][sin()][sin(1)0(

21

021

02121

∫

∫

∫

Δ+

Δ+

++Δ+

+

++Δ+

=

++Δ+

==

TT

T

T

TT

xx

tBtATT

tBtATT

ttBtATT

R

θωθω

θωθω

θωθωτ

58

Figure 3. Spectral analysis of CMF output signals

The former part (recorded as 1R ) is

2/)cos(

dt)][sin()][sin(1

21

0211

θθ

θωθω

−Δ+⋅⋅=

++Δ+

= ∫

TTBAT

tBtATT

RT

(30)

The latter part (recorded as 2R ) is:

dt)]2cos()(2

2/)cos(

dt)][sin()][sin(1

2121

212

∫

∫Δ+

Δ+

++Δ+

⋅−−Δ+

⋅⋅Δ=

++Δ+

=

TT

T

TT

T

tTTBA

TTBAT

tBtATT

R

θθωθθ

θωθω

The latter part (recorded as 2P ) is:

)sin()cos()(2

dt)]2cos()(2

21

212

TTTT

BA

tTTBAP

TT

T

Δ++ΔΔ+

⋅=

++Δ+

⋅= ∫Δ+

ωθθωω

θθω (32)

So the calculated result is :

)sin()cos()(2

2/)cos()0(

21

2121

TTTT

BAR xx

Δ++ΔΔ+

⋅−−==

ωθθωω

θθτ (33)

)sin()cos()(2

)( 21 TTTT

BAT Δ++ΔΔ+

⋅=Δ ωθθωω

ε (34)

Noticing that 1θ , 2θ are initial phase of two correlation vibrating signal, if correlation is calculated continuously, then two signals’ phase can be expressed as:

)0()( 11 θωθ += tt )0()( 22 θωθ += tt (35)

So the relationship between error and time is:

)sin()]0()0(2cos[)(2

)( 21 TTtTT

BAT Δ++Δ+Δ+

⋅=Δ ωθθωωω

ε (36)

When domain of correlation integration is not a full period, the result contains error that is not proportional to phase difference. The error performs as double frequency oscillating cosine function whose amplitude decided by TΔ . When domain of correlation integration is a full period, then 0=ΔT , the error is zero.

The conclusion can be expanded to multiples of full period based on periodicity of trigonometric function. Assume the domain of correlation integration is (0,

TkT Δ+ ) where k is a positive integer , T is the signal’s period and TTT <Δ<− , then the correlation result is

)(2/)cos()0( 2121 TkR xx Δ−−== εθθτ (37)

V. SIMULATION After spectral analysis is made to the output signals

from CMF sensors, we can discover that the signal contains not only power-line interference and random noise, but also frequency multiplication of tube’s first-order natural frequency and power frequency. Build output signals based on the spectral analysis to do some simulation experiments. Normalize the amplitude of basic frequency signal, the values of amplitude of multiple-frequency are listed as table.1 to make sure the same with real output signals.

The working frequency of the tube is fluctuating with the change of liquid flow, so any sampling systems can not sample in full periods exactly. Random noise has some bad effect on calculating precision. Study the two factors independently. Change the sampling frequency around the k times signal frequency, then change the noise amplitude to get the relative error of phase difference calculating results

Assume that ss ffkf Δ+= 0* , then there will be over

sampling or under sampling decided by sfΔ . Study the relative error of the phase difference with the change of

sfΔ from -100 Hz to 100 Hz. The results are showed as fig.4. When the normalized noise amplitude changes form 0 to 0.2, the relative errors of the two methods are calculated and showed as fig.5.

TABLE I. COMPONENTS OF SIMULATED SIGNAL

Signal component

Basic frequency

Hz800 =ω

Double frequency

02ω

Triple frequency

03ω

Normalized amplitude

1 0.01 0.005

Signal component

power-line interference

Hzf 500 =

Double frequency

02 f

Triple frequency

03 f

Normalized amplitude

0.02 0.005 0.0005

59

Figure 4. Relative error when sfΔ changes

Figure 5. Relative error when noise amplitude changes

We can see from the simulation results that the relative error is lower than 0.5% with sfΔ changes. But when the normalized random noise amplitude is bigger than 0.1, the relative error is much bigger with the digital correlation method. But in the real application, the noise amplitude can be controlled under 0.1 by means of filter design, to make sure the relative error is lower than 0.2%.

Because the DFT algorithm is corrected with window functions, it is much more precise and stable. Further steps should be taken in the digital correlation algorithm to enhance the precision and stability.

VI. CONCLUSION CMF is designed by the principle that when mass-flow

moving through a vibrating tube causes a proportional Coriolis force to act, which is detected as a phase difference between two sensors [6]. The traditional algorithms’ precision is very low and incline to be affected by ambient noise and disturbance. Use frequency domain analysis and digital correlation method to calculate the phase difference can get more precise results. The thesis analyzed the DFT algorithm and the digital correlation algorithm in the signal processing of CMF, and theoretical errors in this application are analyzed. According to the true signal output from the sensors, simulation analysis is made. Results show that the algorithms’ relative error is lower than 0.5%, which is more precise than typical algorithms and can be in widely

application in mass flow meter. It is very feasible to built embedded system with digital circuit for application.

ACKNOWLEDGMENT

This work was supported by the foundation item of Chinese National High-tech Research and Development Program (2008AA042207). It is also supported by the National Natural Science Foundation of China with grant No.60904094 and No.60927005.

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flowmeter. The 12th International Conference on Flow Measurement[C]. Chinese Society for Measurement Press, 2004: 455-460

[3] M. Zamora and M. P. Henry, “An FPGA Implementation of a Digital Coriolis Mass Flow Metering Drive System”, IEEE Trans. on Industrial Electronics, Vol.5,No. 7, pp. 2820-2831, Jul. 2008

[4] YU Cui-xin,XU Ke-jun and LIU Jia-jun, “An improvement of signal processing method based on DFT for Coriolis mass flowmeter” [J].Journal of Hefei university of technology,Vol.23,No.6,Dec.2000

[5] YANGJun,WUQi-sheng and SUN Hong-qi , “Study on Using Correlation to Detect Phase Difference in Coriolis Flow Meter” [J]. Chinese Journal of sensors and acruators , Vol. 20 No.1 Jan. 2007

[6] WU Junqing, “Research of Digital Measurement for Phase D ifference” [J]. Journal of basic science and engineering, Vol. 13,No. 1,March2005

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