View of AN EXCELLENT TECHNIQUE TO MEASURE PHASE DIFFERENCE OF SINUSOIDAL SIGNALS

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VOLUME: 08, Issue 05, Paper id-IJIERM-VIII-V, October 2021

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AN EXCELLENT TECHNIQUE TO MEASURE PHASE DIFFERENCE OF SINUSOIDAL SIGNALS

Surendra Kumar Rai M. Tech Scholar, JEC Jabalpur

Prashant Kumar Jain Prof. JEC Jabalpur

Abstract - The work reported in this paper addresses the issue of measurement of phase difference between two sinusoidal signals. In case, the signals having very small phase difference, accurate measurement of such phase difference is a big challenge. Conventional methods have been found unsuitable for such applications. Measurement of phase difference between primary and secondary voltages in a potential transformer and measurement of phase difference between primary and secondary currents for a current transformer are some applications of proposed algorithm. The work reported in this paper proposes the development of MATLAB program based digital measurement of phase difference with high resolution. The two signals have been taken into account and samples are formed at predefined sampling frequency. On the collected samples DFS algorithm is applied. This leads to recovery of amplitudes and phase difference of the signals. The proposed work thoroughly analyses the effects of sampling frequency and harmonic contents in the signals. MATLAB simulations have been carried out to evaluate above aspects. The results obtained have been thoroughly discussed. The interpretations of results have been used for deciding optimum sampling frequencies.

Keywords: DFS, Phase difference, sampling frequency, Harmonics 1 INTRODUCTION

Accurate, reliable and economic solution for the measurement of phase difference between two sinusoids, with high resolution has been a challenge for the engineers. Measurement of ratio error and phase error in potential transformers and current transformers in electric power systems is one of such important applications. Use of calibrated AC potentiometers has been a standard method for such applications.

Fabrication and calibration of these AC potentiometers seems to be very crucial issue. Besides this due to drift in the reactive components, frequent calibration is necessary.

Enormous progress in the field of programmable devices, the development of user friendly integrated development environment for these devices has made it possible to apply these microcontrollers for the implementation of complex algorithms and fabrication of hardware to solve such complex problems. The modern development tools have reduced the time and efforts needed to develop such products. Mathematical developments in the field of digital signal processing and their implementation on real time processors have attracted the researchers and industries. Many such

types of equipment are being developed and manufactured these days to meet the challenging requirements of industries and users.

2 DISCRETE FOURIER SERIES

The current and voltage signals are periodic analog signals. These signals are first discritized. This means that the original continuous signal is discritized in to samples.

Discrete Fourier series is applicable only in discrete signals. A discrete time signal can be expanded using DFS to recover the phase angle and amplitude of the signal. Let x(n) is a discrete time signal then Fourier coefficients for this signal are expressed as:

3 WORK STUDY

Traditionally Cathode Ray Oscilloscopes have been used for the measurement of phase difference. The zero crossovers of waveforms and their time shift are measured manually to estimate the phase difference. Besides a manual method, it suffers from following disadvantages:

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VOLUME: 08, Issue 05, Paper id-IJIERM-VIII-V, October 2021

2 (1) It is difficult to locate the zero

crossover of a waveform on CRO screen.

(2) In the presence of harmonics locating zero crossovers becomes even more complex and prone to errors.

(3) Reading the time shifts on CRO screen is quite difficult and prone to reading errors.

(4) Being a manual method, this method is not suitable for automation.

Measurement of amplitudes and phase differences has been a crucial issue for electrical power supply industry and protection of electrical equipment. The history of such measurements dates back to the era of onset of power industry.

Various researchers have attempted to measure these parameters meeting the accuracy and resolution needed.

4 PROPOSED METHODOLOGY

In this research work phase difference of two sinusoidal signals is being measured at very high resolution. Very small phase difference measurement is a big challenge to the engineers. In this thesis work three simulations has been done in MATLAB environment. In first simulation basic algorithm is used in which two sinusoidal signals have taken. In second simulation effect of harmonics is studied and in third one resolution of ADC is added. All three simulations have carried out in MATLAB and results obtained in digital form.

Algorithm 1

Step1: Define the sinusoidal signals a1 and a2. Here it is assumed that a2 has a relative phase difference of Φ radians with respect to a1.

Step 2: Discretize the signals a1 and a2 for given number of sample per cycle.

Step 3: Recover signals by applying DFS on discrete signals a1 and a2.

Step 4: Recover amplitudes and phases.

Step 5: Determine the phase difference.

Step 6: Repeat step 1 to 5 for different number of sample per cycle.

Algorithm 2

Step1: Define the sinusoidal signals a1 and a2;

Step 2: Add harmonics to signals a1 and a2.

Step 4: Recover signals for fundamentals, by applying DFS on discrete signals a1 and a2.

Step 7: Repeat step 1 to 6 for different number of samples per cycle.

Algorithm 3

Step1: Define the sinusoidal signals a1 and a2;

Step 2: Add harmonics to signals a1 and a2.

Step 4: Apply resolution of ADC and produce a digitized signal with limited resolution.

Step 5: Recover signals by applying DFS on discrete signals a1 and a2.

Step 8: Repeat step 1 to 7 for different number of sample per cycle.

Step 9: Repeat step 1 to 8 for different value of resolution.

5 RESULTS

Fig.1: Phase difference v/s no. of sample per cycle (without harmonics)

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360 1440 2520 3600 4680 5760 6840 7920 9000

Phase difference

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VOLUME: 08, Issue 05, Paper id-IJIERM-VIII-V, October 2021

3 Fig.2: Phase difference v/s no. of

sample per cycle (with harmonics).

Fig 3: Comparison between results of various ADCs

6 CONCLUSION

An effective method of measuring the phase difference between two signals with very small phase difference has been presented. Simulation results show that the algorithms have very high accuracy and are easy to implement and they are therefore useful tools for measuring precise phase difference. Effects of sampling rate, harmonics and resolution of ADC used have been evaluated. Effect of sampling rate has been seen in different aspects. It has been found that, on increasing the number of samples per cycle more accurate phase difference can be measured. Effect of harmonics is also studied and simulated. It is well known that practically that all physical do possess some harmonics. That is why it is necessary to add harmonics to get practically true results. It has been seen that without adding the harmonics phase difference is more precise. Difference between results before and after adding harmonics is very quite small. Although it can be said that addition of harmonics

to the continuous signal affects the phase angle measured by DFS algorithm. Most important parameter which has been noted from the simulation result is resolution of ADC. As resolution of ADC increases, it yields more accurate phase difference (see fig 2.6). Results of 10-bit ADC are more accurate than results of 8- bit ADC but not better than results of 12- bit ADC. Similarly results for 8-bit, 10-bit, 12-bit, 14-bit and 16-bit ADCs are compared and it is found that 16-bit ADC is producing best results. The results produced by 12 bit ADC are quite acceptable.

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4 implementation of a new phase

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