The following presentation is a part of the level 4 module -- Electrical and Electronic Principles. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme. The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.

1. Introduction to Fourier Analysis Electrical and Electronic Principles © University of Wales Newport 2009 This work is licensed under a Creative Commons Attribution 2.0 License .

3. It was discovered by Joseph Fourier, that any periodic wave (any wave that repeats itself after a given time, called the period) can be generated by an infinite sum of sine wave which are integer multiples of the fundamental (the frequency which corresponds to the period of the repetition. Lets look at a periodic wave. Here is an example plot of a signal that repeats every second.

4. This wave is not sinusoidal and therefore it is difficult to imagine that we can add sine waves together to produce this waveform. Fourier did show that such a waveform can be generated using sine and cosine wave of different sizes or sine waves with different phase angles. Fourier showed how it was possible to determine the sizes of the sine and cosine waves using integration techniques. Introduction to Fourier Analysis

5. It has been shown that any complex periodic waveform can be written as a infinite sum of sinusoidal waves of different amplitude and phase shift which are all integer frequency multiples of the fundamental waveform which is the frequency of the original complex periodic waveform. V = V 1 Sin (  t   1 ) + V 2 Sin (2  t   2 ) + V 3 Sin (3  t   3 ) + V 4 Sin (4  t   4 ) + … To remove the requirement for the angle  we can say that each component has a sin part and a cos part:  1 V 1 A 1 B 1

6. e.g. V 1 Sin (  t   1 ) = A 1 Cos (  t) + B 1 Sin (  t) The values of A and B can be positive, negative or zero. In addition we must allow for a D.C. component to the wave and this is quantified using A 0 . We can therefore write the complex periodic waveform as: Introduction to Fourier Analysis

7. The constants can be determined using the following integrals: The values of the constants are called the Fourier Coefficients and the whole is called the Fourier Series . Introduction to Fourier Analysis

8. By observation we can make certain decisions as to the “make up” of the Fourier Series. See below. these will have only A constants (all B = 0) the series will have only Cos functions. There is no A 0 component – balanced about 0. Even Functions Odd Functions these will have only B constants (all A = 0) the series will have only Sin functions. There is no A 0 component – balanced about 0. Introduction to Fourier Analysis

9. Half Wave Symmetrical these will have no even harmonics. There is no A 0 component – balanced about 0. Example Introduction to Fourier Analysis 5v -1v T T/2

10. Value of A 0 We have two parts Introduction to Fourier Analysis

11. Value of A n Again we have two parts We can replace  by using: Introduction to Fourier Analysis

12. Now we put the values of n in: n=1 n=2 All A values are zero! Introduction to Fourier Analysis

13. Value of B n Once again we have two parts We can again replace  by using: Introduction to Fourier Analysis

14. Now we put the values of n in: n=1 n=2 Introduction to Fourier Analysis

15. n=3 n=4 n=5 From the results so far we can see that all even harmonics are 0 and that we have a common value of 12/  with this being divided by the harmonic, i.e. 1, 3, 5, 7, etc. Introduction to Fourier Analysis

16. Fundamental only Fundamental + 3 rd Harmonic

17. This includes the 7 th , 9 th , and 11 th harmonics

18. By observation we should have been able to deduce that the waveform was ODD and only calculated the B (Sin only) values and also that it was half wave symmetrical which meant that we would have known that B 2 , B 4 , B 6 etc would be 0. Example Is the wave EVEN, ODD or Half Wave Symmetrical? NO Introduction to Fourier Analysis 5v T 3T/4

19. Can we modify it to make it fit into one of the categories? EVEN with two areas to integrate EVEN with one area to integrate 5v T 3T/8 5T/8 5v T T/8 7T/8

20. Value of A 0 Introduction to Fourier Analysis

21. Value of A n

22. Now we put the values of n in: n=1 n=2 n=3 n=4 n=5 Introduction to Fourier Analysis

23. Introduction to Fourier Analysis

25. Example By observation the function is EVEN and Half Wave Symmetrical. It is also obvious that the average value is 5 A 0 = 5v Introduction to Fourier Analysis 10v T/2 T

26. Numerical Integration can be used if the waveform is complicated and the results need not be too accurate. Angle (Rad) Function FnxCos(angle) FnxCos(3xangle) Fnxcos(5xangle) 0 0 0 0 0 0.31415927 1 0.951056516 0.587785252 6.12574E-17 0.62831853 2 1.618033989 -0.618033989 -2 0.9424778 3 1.763355757 -2.853169549 -5.51317E-16 1.25663706 4 1.236067977 -3.236067977 4 1.57079633 5 3.06287E-16 -9.18861E-16 1.53144E-15 1.88495559 6 -1.854101966 4.854101966 -6 2.19911486 7 -4.114496766 6.657395614 -3.00161E-15 2.51327412 8 -6.472135955 2.472135955 8 2.82743339 9 -8.559508647 -5.290067271 4.96185E-15 3.14159265 10 -10 -10 -10 3.45575192 9 -8.559508647 -5.290067271 9.92273E-15 3.76991118 8 -6.472135955 2.472135955 8 4.08407045 7 -4.114496766 6.657395614 -6.86007E-15 4.39822972 6 -1.854101966 4.854101966 -6 4.71238898 5 -9.18861E-16 2.75658E-15 -1.34761E-14 5.02654825 4 1.236067977 -3.236067977 4 5.34070751 3 1.763355757 -2.853169549 -2.20494E-15 5.65486678 2 1.618033989 -0.618033989 -2 5.96902604 1 0.951056516 0.587785252 -2.94025E-15 6.28318531 0 Sum -40.86345819 -4.851839996 -2 Average -2.043172909 -0.242592 -0.1 Value -4.086345819 -0.485184 -0.2

27. A 0 = 5 A 1 = -4.086 A 2 = -0.485 A 3 = -0.2 This produces

28. Introduction to Fourier Analysis This resource was created by the University of Wales Newport and released as an open educational resource through the Open Engineering Resources project of the HE Academy Engineering Subject Centre. The Open Engineering Resources project was funded by HEFCE and part of the JISC/HE Academy UKOER programme. © 2009 University of Wales Newport This work is licensed under a Creative Commons Attribution 2.0 License . The JISC logo is licensed under the terms of the Creative Commons Attribution-Non-Commercial-No Derivative Works 2.0 UK: England & Wales Licence. All reproductions must comply with the terms of that licence. The HEA logo is owned by the Higher Education Academy Limited may be freely distributed and copied for educational purposes only, provided that appropriate acknowledgement is given to the Higher Education Academy as the copyright holder and original publisher. The name and logo of University of Wales Newport is a trade mark and all rights in it are reserved. The name and logo should not be reproduced without the express authorisation of the University.