A Novel Phase Difference Measurement Method for Coriolis Mass Flowmeter Based on Correlation Theory

Abstract

Aiming at the poor precision problem in phase difference measurements with unknown frequencies in engineering practice, a new phase difference measurement method is proposed for Coriolis mass flowmeter based on correlation theories. Firstly, the signal frequency was estimated by using an adaptive notch filter, which was applied to filter the waves and determined the integer period of the sampling signals, and the non-integer period sampling signals needed to be extended. Then, the Hilbert transformation was conducted relative to the extended signals, and the correlation functions of these extended signals with the transformed signals can be computed. Finally, the formula of phase difference can be obtained by utilizing the sinusoidal function. Compared to traditional methods, such as the correlation method, the Hilbert transformation method, and sliding Goertzel algorithm, the proposed method is suitable for both integer period and non-integer period sampling signals, and its accuracy, real-time, and dynamic performance is superior. Simulation and experiment results validate the superiority and effectiveness of the proposed method.

1. Introduction

A Coriolis mass flowmeter (CMF) can directly measure the mass flowrate with high precision, and it is widely used in many industries. Compared with other types of flow meters, a Coriolis mass flowmeter has many advantages, such as direct high precision measurement of fluid mass flow; multi-parameter measurement; wide measurement range of fluid; ease in installation, maintenance, and repair. From the measurement principle of CMF, the mass flowrate is calculated by measuring the phase difference or time interval between two signals detected by electromagnetic sensors [1,2,3,4,5,6,7]. The phase difference estimation between two additive white Gaussian noise signals in the same frequency band has attracted considerable attention in acoustics, electricity, petroleum, chemistry, and other fields [8,9,10,11].

Phase difference measurement methods have been studied and developed in the past several decades [12,13,14]. According to the method of signal processing, the measurement methods for phase difference of sinusoidal signal can be divided into two kinds. One is the time domain method, and the other is frequency domain method. The commonly used time domain methods have the advantages of clear physical meaning, simple principle, convenient calculation, and it is easy to realize in hardware, such as zero-crossing detection method, correlation analysis method, Hilbert Transformation method, and so on. The frequency domain methods have the advantages of high measurement accuracy and strong anti-noise performance. Currently, common frequency domain methods include the spectrum analysis method, cross-spectrum estimation method, higher-order spectrum estimation method, and so on.

Quadrature delay estimator (QDE) in Ref. [15] has been proposed based on cross-correlation, which makes use of the received in-phase and quadrature-phase components, and it obtains a high-resolution phase shift estimate. QDE improves the phase difference measurement precision under high signal-to-noise ratio (SNR), but it causes a bias when calculating the non-integer number of signal periods. A modified method is presented in Ref. [16], called the unbiased QDE (UQDE), which uses all in-phase and quadrature-phase components. Compared with QDE method, UQDE has no bias, even if the correlation length is the number of non-integer periods. However, if the introduced phase delay is not exactly π/2, both two methods show bias and even invalid results when the signal frequency is unknown. Ref. [17] proposed a new method based on multiple correlations, which improved the phase difference estimation precision at low SNR. However, if the frequency of generated signal is not equal to the frequency of original signals, it shows significant bias. Ref. [18] proposed a new method based on Ref. [17], and the similarity lies in the use of multiple cross-correlations; the difference in the correlation operation signal source is different.The non-integer period problemsof phase difference in Refs. [17,18] are not fundamentally solved, because Hilbert transformationsare used in these methods, while the phase difference measurement accuracy of Hilbert transformation is also affected by the length of the non-integer period.The Hilbert transformation (HT) method in Ref. [19] can be used independently to calculate the phase difference of unknown signal frequency. The phase difference is estimated by calculating the phase of two analytic signals. Although this method can calculate the dynamic phase, the performance of anti-interference is poor. Hilbert transformation of the non-integer period sampling signal produces end point effects; this is the fundamental cause of the phase difference measurement error, which leads to deviation.

In addition to the above-mentioned time domain methods, there are many frequency domain methods also used in the phase difference measurement for Coriolis mass flowmeter. The spectrum analysis method uses Fourier transform to obtain a discrete spectrum, and it calculates the phase difference of the two signals at the maximum spectral line. The spectrum analysis method transforms the finite length signal from time domain to frequency domain and then processes it in the condition of non-whole period sampling. The energy leakage caused by the truncation of time domain signal reduces the precision of spectrum analysis [20]. The discrete spectrum is obtained by fast Fourier transform (FFT), and the frequency value and the phase value will cause the big error; therefore, a correction must be carried out. Therefore, there are a lot of methods to perform spectrum correction. The spectrum analysis method can restrain the interference of harmonic and noise to a certain extent and improve the measurement precision of phase difference, but the algorithm requires an entire sampling period. In order to improve the measurement precision of phase difference, windowing or interpolation is usually used to reduce the impact of spectrum leakage. One of the unavoidable disadvantages of windowing method is that it can increase the width of the main lobe and reduce the spectrum’s resolution while suppressing the spectrum’s leakage, there are some problems, such as complex calculation, a large amount of calculation, and poor real-time performance [21]. Although these methods are used to correct the signal or its spectrum and have achieved some results, it has not really solved the fundamental problem of spectrum leakage. The fundamental reason for the spectrum leakage is that the number of sampling points in the discrete Fourier transform (DFT) calculation window is not an integer period.

Currently, the phase difference estimation methods for CMF mainly include sliding Goertzel algorithm (SGA) [22] and discrete time Fourier transform (DTFT) [23,24]. SGA decreases the spectrum leakage, which is caused by non-integer periods sampling of discrete Fourier transform, but there is a numerical overflow problem and a slow convergence rate in practice, which leads to measurement bias. DTFT considers negative frequency contribution, which obtains a higher precision and anti-interference ability than traditional DTFT methods, while the entire iterative process produces a lot of computation. Both methods require a prior knowledge of the frequency to compute the phase difference.

On the basis of the above research studies, in order to further improve the measurement accuracy of phase difference and remove the bias for unknown signal frequencies, a novel method is proposed in this paper for the non-integer period sampling signal. Adaptive notch filter is used to filter out interference signals and to estimate the frequency signal. The idea of data extension is used to make the length of sampling data equal to or close to integer periods. Then, Hilbert transformation can be conducted on extended signals, and the phase difference can be computed by using correlation functions. The entire process of the new method is presented in Section 2, including error analysis and the derivation of the proposed method. In Section 3, the performances of the proposed method are evaluated by simulation and experiments. Finally, Section 4 presents the conclusions.

2. Development of the Proposed Method

2.1. Error Analysis of the Cross-Correlation Method

Two real sinusoids signals are explained as follows:

$$\begin{array}{l}x(n)=A\cos (\omega \cdot n+{\theta }_1)\\ \mathrm{y}(n)={\rm B}\cos (\omega \cdot n+{\theta }_2)\end{array},n=0,1,\cdots ,N-1.$$

(1)

where $\omega$ represents the frequency, A and B represent amplitudes, and ${\theta }_1$ and ${\theta }_2$ represent initial phases. The cross-correlation operation between signals is calculated as follows.

$$R_{xy}\left(0\right)=\frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}x}(n)y(n)=\frac{AB}{2}\cos ({\theta }_2-{\theta }_1)+\underset{\_}{\frac{AB}{2N}{\displaystyle \sum _{n=0}^{N-1}\cos (2\omega n+}{\theta }_1+{\theta }_2)}$$

(2)

$$R_{xx}\left(0\right)=\frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}{x}^2}(n)=\frac{A^2}{2}+\underset{\_}{\frac{A^2}{2N}{\displaystyle \sum _{n=0}^{N-1}\cos (2\omega n+2}{\theta }_1)}$$

(3)

$$R_{yy}\left(0\right)=\frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}{y}^2}(n)=\frac{B^2}{2}+\underset{\_}{\frac{B^2}{2N}{\displaystyle \sum _{n=0}^{N-1}\cos (2\omega n+2}{\theta }_2)}$$

(4)

The phase difference $\Delta \theta$ can be provided as follows.

$$\Delta \theta ={\theta }_2-{\theta }_1=\mathrm{arc}\cos \left(\frac{2R_{xy}\left(0\right)}{AB}\right)=\mathrm{arc}\cos \left(\frac{R_{xy}\left(0\right)}{\sqrt{R_{xx}\left(0\right)}\sqrt{R_{yy}\left(0\right)}}\right)$$

(5)

According to the above derivation process, when the correlation length is a non-integer period number, the underlined part is not equal to 0, and there will be deviation in the calculation with cross-correlation method, which has two times frequency of the original signal. The fundamental reason for the error of cross-correlation method is that the influence of correlation length includes non-integer periods.

2.2. Error Analysis of the Hilbert Transformation Method

X(k) is gained by discrete Fourier transform (DFT) of x(n).

$$X(k)=DFT\left[x(n)\right]={\displaystyle \sum _{n=0}^{N-1}x(n)\cdot e^{-j\frac{2\pi }{N}nk}},k=0,1,\cdots N-1$$

(6)

According to X(k), Z(k) can be computed as follows.

$$Z(k)=\{\begin{array}{ll}X(k)& k=0\\ 2X(k)& k=1,2,\cdots ,\frac{N}{2}-1\\ 0& k=\frac{N}{2},\cdots ,N-1\end{array}$$

(7)

By making the DFT of Z(k) inverse, the analytic signal z(n) can be calculated as follows.

$$z(n)=IDFT\left[Z(k)\right]=\frac{1}{N}{\displaystyle \sum _{k=0}^{\frac{N}{2}-1}Z(k)\cdot e^{j\frac{2\pi }{N}nk}}$$

(8)

Then, Equation (8) can be deduced as follows.

$$\begin{array}{ll}z(n)\stackrel{\omega =\frac{2\pi k_1}{N}}{=}& \frac{1}{N}{\displaystyle \sum _{n_1=0}^{N-1}x(n_1)}\\ & +\frac{A}{N}{\displaystyle \sum _{k=1}^{\frac{N}{2}-1}\left\{\left[{\displaystyle \sum _{n_1=0}^{N-1}{e}^{j\theta }\cdot e^{j\frac{2\pi }{N}[k_1-k]n_1}}\right]e^{j\frac{2\pi }{N}nk}\right\}}\\ & +\frac{A}{N}{\displaystyle \sum _{k=1}^{\frac{N}{2}-1}\left\{\left[{\displaystyle \sum _{n_1=0}^{N-1}{e}^{-j\theta }\cdot e^{-j\frac{2\pi }{N}[k_1+k]n_1}}\right]e^{j\frac{2\pi }{N}nk}\right\}}\end{array}$$

(9)

If k₁equals k, Equation (9) can be simplified as follows:

$$z(n)=x(n)+j\hat{x}(n)$$

(10)

where $\hat{x}(n)$ represents the Hilbert transformation result of $x(n)$. It is observed that, from Equation (8), if the sampling signals are of the integer signal periods, there will be no end point effects for Hilbert transformation. Although the method can calculate the dynamic phase, the performance of anti-interference is poor, which limited the extension and application of the method. Hilbert transformation of the non-integer period sampling signal produces end-point effects; this is the fundamental cause of the phase difference measurement error, which leads to the deviation.

2.3. The Derivation of the Proposed Method

Frequency estimation is another important parameter in CMF signal processing. Adaptive notch filter (ANF) has been widely used for simple algorithm, fast convergence speed, and high frequency tracking accuracy [25,26]. In this paper, frequency estimation methods are not the focus of our research. The proposed phase difference measurement method adopts the adaptive notch filter elaborated in Ref. [22] for it has firstly been used to estimate frequency and filter signals. Then, the signal data length matches integer period sampling or it is not judged, and the non-integer period sampling sequence is extended to an integer period sequence, which makes the signal data length close to or equal to an integer period sequence. Thus, there are no end-point effects for Hilbert transformation. At last, the phase difference can be calculated by the correlation method.

Assume that the filtered sampling signals as follows.

$$x=[x_1,x_2,\cdots ,x_{N-1},x_N];y=[y_1,y_2,\cdots ,y_{N-1},y_N]$$

(11)

Assume that the signal period is estimated as P, and $P\in Z^{+}$ denotes the number of sampling data points in a period. Assume N/P = q, m represents the remainder of the expression. If remainder m equals 0, the data length matches integer period sampling. Otherwise, the non-integer period sampling sequence needs to be extended to an integer period sequence.

Find out sequences $[x_{(q-1)P+m+1},\cdots ,x_{qP}]$,$[y_{(q-1)P+m+1},\cdots ,y_{qP}]$ from sampling data x and y, based on the periodicity of the signals, and then add it to the sampling data sequence to so that it approaches integer periods of sampling signals. Data extension signals x_e and y_e are obtained by means of data extension of two sampling signals.

$$x_e=[x_1,x_2,\cdots ,x_N,x_{(q-1)P+m+1},\cdots ,x_{qP}];y_e=[y_1,y_2,\cdots ,y_N,y_{(q-1)P+m+1},\cdots ,y_{qP}]$$

(12)

Analytic signals can be obtained by Hilbert transformation. Moreover, cross-correlation functions are computed as follows

$$\begin{array}{l}{R}_{x_e{y}_e}\left(0\right)=\frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}{x}_e}(n)y_e(n),R_{x_e{y}_e^{\prime }}\left(0\right)=\frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}{x}_e}(n)y_e^{\prime }(n)\\ R_{x_e^{\prime }{y}_e}\left(0\right)=\frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}{x}_e^{\prime }}(n)y_e(n),R_{x_e^{\prime }{y}_e^{\prime }}\left(0\right)=\frac{1}{N}{\displaystyle \sum _{n=0}^{N-1}{x}_e^{\prime }}(n)y_e^{\prime }(n)\end{array}$$

(13)

The expectation values of $R_{x_e{y}_e}\left(0\right),R_{x_e{y}_e^{\prime }}\left(0\right),R_{x_e^{\prime }{y}_e}\left(0\right),R_{x_e^{\prime }{y}_e^{\prime }}\left(0\right)$ are as follows.

$$E\{R_{x_e{y}_e}\left(0\right)\}=\frac{AB}{2}\cos ({\theta }_2-{\theta }_1)+\frac{AB}{2N}{\displaystyle \sum _{n=0}^{N-1}\cos (2\omega n+}{\theta }_1+{\theta }_2)$$

(14)

$$E\{R_{x_e{y}_e^{\prime }}\left(0\right)\}=\frac{AB}{2}\sin ({\theta }_2-{\theta }_1)+\frac{AB}{2N}{\displaystyle \sum _{n=0}^{N-1}\sin (2\omega n+}{\theta }_1+{\theta }_2)$$

(15)

$$E\{R_{x_e^{\prime }{y}_e}\left(0\right)\}=-\frac{AB}{2}\sin ({\theta }_2-{\theta }_1)+\frac{AB}{2N}{\displaystyle \sum _{n=0}^{N-1}\sin (2\omega n+}{\theta }_1+{\theta }_2)$$

(16)

$$E\{R_{x_e^{\prime }{y}_e^{\prime }}\left(0\right)\}=\frac{AB}{2}\cos ({\theta }_2-{\theta }_1)-\frac{AB}{2N}{\displaystyle \sum _{n=0}^{N-1}\cos (2\omega n+}{\theta }_1+{\theta }_2)$$

(17)

$$E\{R_{x_e{y}_e^{\prime }}\left(0\right)-R_{x_e^{\prime }{y}_e}\left(0\right)\}=AB\sin ({\theta }_2-{\theta }_1)$$

(18)

$$E\{R_{x_e{y}_e}\left(0\right)+R_{x_e^{\prime }{y}_e^{\prime }}\left(0\right)\}=AB\cos ({\theta }_2-{\theta }_1)$$

(19)

The phase difference can be obtained as follows.

$$\Delta \theta ={\tan }^{-1}(\frac{R_{x_e{y}_e^{\prime }}\left(0\right)-R_{x_e^{\prime }{y}_e}\left(0\right)}{R_{x_e{y}_e}\left(0\right)+R_{x_e^{\prime }{y}_e^{\prime }}\left(0\right)})$$

(20)

It can be seen from Equation (20) that the proposed method can obtain high phase difference precision, whether the signal frequency is known or not. In addition, even if the signal amplitude is unknown and the number of sampling data points is anon-integer period number, the proposed method shows no bias. This is the advantage of the proposed method in this paper.

Based on the above analysis, the entire process of the proposed method is implemented as follows.

(1)

Frequency is estimated by the adaptive notch filter.

(2)

Data extension signals x_e and y_e are obtained by Equation (12).

(3)

Analytic signals $x_e^{\prime }$ and $y_e^{\prime }$ are gained by Hilbert transformation.

(4)

Cross-correlation functions are computed by utilizing equations from Equation (14) to Equation (17).

(5)

Phase difference is calculated by Equation (20).

3. Simulation and Experiment Results

3.1. Simulation Results

To validate the advantage of the proposed method, simulations for ANF, HT, the correlation, SGA, DTFT and the proposed method are tested, respectively. Simulation parameters are set as A = B = 1, f = 100 Hz, f_s = 2000 Hz, and $\Delta \theta$ = 30°, and the additive white Gaussian noise is added into the signals. The performance of these methods is evaluated by Root Mean Square Error (RMSE), and each experiment is independently tested for 100 runs.

To evaluate the frequency tracking accuracy and filtering effect of the modified ANF, simulations are provided in Figure 1 and Figure 2. The parameters of specific signal models are given in Ref. [22]. It is observed that the modified ANF not only has good frequency estimating ability but also has good filtering effect. The presented method overcomes the problem that the original method cannot keep track of the frequency for a long time, and it retains high precision in terms of frequency estimation. It can be seen from Figure 2 that the notch filter can obtain better filtering effects even if SNR is low. In order to further highlight the advantages of the method, the original signal is processed instead by the filtered signal for extension processing.

To verify the phase difference estimation performances of the three methods under different SNRs, the SNR of non-integer periodic sampling signal (N = 17) is simulated and compared in Figure 3. As shown in Figure 3, the precision of the three phase difference estimation methods improved with an increase in SNR. The phase difference estimation accuracy of HT and correlation method is low for these methods are affected by non-integer number of periods. While the accuracy of the proposed method is still high, for the proposed method presented in this paper, it has already solved this problem, which also validated the fact that the presented method is better than the other two methods.

To validate the phase difference estimation performance of these methods under different correlation lengths, simulations are conducted at SNR = 20 dB, and the results are illustrated in Figure 4. It was observed that the estimation performance of HT and correlation methods varies periodically due to the influence of non-integer number of periods. Only when correlation length N is close to the integer number of sampling signal, HT and correlation methods have better precision of phase difference estimation. As shown in Figure 3 and Figure 4, it can be concluded that the proposed method overcomes the influence of non-integer period sampling, which also reflects the superiority of the method in this paper.

The SGA and DTFT methods described above are the main frequency-domain phase difference estimation methods for CMF. In order to validate the performance of phase difference estimation under the condition of phase difference variation, simulations are conducted at SNR = 20 dB and the results are illustrated in Figure 5. As shown in Figure 5, the proposed method is closer to the actual value, which shows that it has better estimation performance. SGA lags behind because it has a slow convergence rate. The accuracy of the proposed method is similar to DTFT, but it is better than DTFT as the minimum computation length of the proposed method can be set as N = 2, while the DTFT method is N = 8. The computational complexity of the proposed method is obviously less than that of the DTFT method. Compared with SGA and DTFT methods, the proposed method has the advantages of less computation, higher precision, and better dynamic performance.

3.2. Experiment Results

To demonstrate that the proposed method is effective in practical application, the performances of the SGA, DTFT, and the proposed method are tested in a RHEONIK CMF with a RHE08 transmitter. The frequency of the sensor is approximately 146 Hz and the sampling frequency equals 10 KHz. The mass flowrate varies from 0.40 to 10.10 kg min⁻¹. The performances of the three methods are tested by 10,000 sampling points each time, and the results are shown in Figure 6, Figure 7 and

Table 1. Theoretical versus estimated time delay at different flowrates.

Mass Flowrate /(kg·min−1)	Theoretical Time Delay/μs	Estimated Time Delay/Us
		SGA Method	Relative Error	DTFT Method	Relative Error	Proposed Method	Relative Error
0.40	3.1241	3.1344	0.330%	3.1289	0.154%	3.1286	0.144%
2.28	16.5079	16.5573	0.299%	16.5301	0.134%	16.5299	0.133%
5.88	42.5057	42.7321	0.533%	42.5742	0.161%	42.5733	0.159%
8.50	60.7757	60.9880	0.349%	60.8615	0.141%	60.8603	0.139%
10.10	73.0124	73.2589	0.338%	73.1076	0.130%	73.1054	0.127%

, respectively.

It can be seen from Figure 6 and Figure 7 that the result of SGA method lags behind that of the proposed method, and the accuracy of the proposed method is comparable to that of DTFT, which is consistent with the result shown in Figure 5. It also proves the superiority of the proposed method in practical engineering applications. As shown in Table 1, the time delay of the proposed method is closer to the theoretical value at different flowrates, which also proves that the performance of the proposed method is better than SGA and DTFT. The DTFT method is one of the methods with high measurement accuracy for Coriolis mass flowmeter. The time domain method proposed in this paper can also achieve considerable accuracy. In a word, the proposed method is helpful to improve the precision of phase difference estimation and flow measurement for CMF.

4. Conclusions

In order to improve the accuracy of phase difference estimation, a novel phase difference measurement method based on correlation theory is proposed for CMF signals, which improves the anti-interference ability by using the cross-correlation characteristic. The proposed method can be used to calculate the dynamic phase difference, including only two points, even if the signal length is a non-integer period length. Simulation and experiment results validate that the proposed method has a good performance of phase difference estimation and improves the measurement accuracy of CMF effectively. Notably, if the filtered signal is used for processing, the effect will be better.

This method takes advantage of the fact that the frequency and phase differences of Coriolis Mass flowmeter need to be estimated for flow measurement. However, for other applications where only measuring phase difference is needed, the algorithm is a little complicated. The next step is to focus on a simple algorithm that can directly estimate the phase difference with high accuracy. Notably, in order to further popularize this method, and to prove the validity of this method, the research method will be applied to other energy fields, such as power system analysis, fault diagnosis, and so on.