APPLICATION OF FOURIER SERIES IN THE ANALYSIS OF NON-SINUSOIDAL ALTERNATING VALUES

Subject review Analyses of alternating electrical circuits in electrical engineering are usually based on assumption that currents and voltages are sinusoidal values. Such assumption allows analysis of electrical networks by symbolic mathematical calculation. It also allows the application of vector diagrams in representing relationship of the analyzed electrical values. In practice, periodic non sinusoidal values are sometimes found. For solving such values symbolic mathematical calculation and vector diagrams cannot be used. In this paper is shown how such non sinusoidal periodic values can be separated, by applying Fourier series, into infinite series which contains a constant term and infinitely many harmonic components. That way allows all the acquired knowledge and methods for solving sinusoidal periodic values to still be used.


Introduction
In various mathematical applications composite functions are approximated by simpler functions that are more suitable for further calculation [1,2].In that way, to solve numerous technical and physical problems, periodic functions need to be converted in series according to trigonometric functions, i.e. to display them in the form of a sum of sine and cosine functions of different amplitudes and frequencies [3].
For solving and the analysis of alternating electrotechnical networks it is suitable to use complex calculation, wherein the voltage and the current are shown by phasors.Phasors are composite numbers that are used to display sinusoidal values in a complex plane.The length of those vectors that rotate counter-clockwise with angle velocity represents the maximal value of voltage or current, while the projection of that vector (phasor) on the vertical axis gives the current value of voltage or current.The possibility of application of phasors in the analysis of electrical values is directly dependent on their sinusoidal nature.In case of presence of periodic signals that do not have sinusoidal shape, the conversion of all nonsinusoidal values into sinusoidal is the necessary precondition for using complex calculation in the analysis of alternating circuits [4].
The inner product of functions and  on the space  2 [−, ] is defined by formula Functions and  are orthogonal if their inner product is zero, i.e. if Trigonometrics series of the form are used for the approximation of the periodic functions.
From space  2 [−, ] the sequence of functions ( 5) is orthogonal over an interval [−, ] which means that the integral of the product of any two different functions in (5) over the interval [−, ] is zero, while the integral of the square of each function in (5) is different from zero.Coefficients  0 ,   and   ,  ≥ 1 of trigonometrics series (4) are determined by using the orthogonality of functions (5) and are calculated by formulas Numbers  0 ,   and   , ≥ 1 are called Fourier coefficient of , while trigonometrics series (4) with those coefficients is called Fourier series of .
In a case of an even or odd function defined on simetric interval [−, ] above formulas are simplifying.Namely, if  is an even function, that is, (−) = (), for all , then   = 0, for all .Fourier series of function in that case contains only cosine functions as follows where and If  is an odd function, that is, (−) = −(), for all , then   = 0, for all .Fourier series of an odd function  on interval [−, ] contains only sine functions as follows where All of the above formulas are derived for the case of the periodic function observed on simetric interval [−, ].For the general case, when the function is observed on interval [, ], Fourier series of periodic function with period  =  −  is while the Fourier coefficients are calculated according to formulas More detailed information about Fourier series the reader may find in [5,6,7].

Transformation of non-sinusoidal values into infinite series
As mentioned in the introduction, alternating networks can be solved by using complex calculation, wherein the voltage and the current are shown by phasors, which cannot be directly used if value does not have sinusoidal shape.Therefore, all non sinusoidal periodic values have to be expanded into infinite series of form which contains a constant term and infinitely many harmonic components of different amplitudes  max ,  ≥ 1, frequencies  and phases   ,  ≥ 1.With increasing frequency, amplitudes of the sinusoidal terms are smaller, so the nonsinusoidal value can be written with a first few terms of the infinite series with sufficient accuracy.According to addition theorem Eq. ( 18) becomes where Constants in equations ( 20) and (21) are, by formulas (15), ( 16) and (17), calculated according to while and

Analysis of square wave signal
Distinctive non sinusoidal patterns of voltages are square and triangular.Those voltages are composed of a larger number of sinusoidal voltages with different frequencies and amplitudes (primary wave and higher harmonics).Primary wave has the same frequency as analysed non sinusoidal signal, while the other sinusoidal terms have significantly greater frequencies [8,9].
In this paper will be shown how, by applying Fourier's series, square wave voltage signal can be analysed, what is shown in Fig. 1.According to formula ( 22), what can be seen from Fig. 1, this voltage does not contain DC component.Due to that, constant in formula (20), is zero.

Calculation of constant terms
Since  1max = 0, it is necessary to calculate only Since all the amplitudes of the sinusoidal terms with increase in frequency are declining, it is easy to mathematically prove the fact that amplitude of the third harmonic term is three times smaller than amplitude of the primary term, and amplitude of the fifth harmonic term is five times smaller than the primary term and so on.
It is evident that amplitudes of the higher harmonics are declining rapidly, therefore, non sinusoidal dimensions can be recorded, with decent accuracy, with only first few terms of the infinite series.Thus, Fourier's series for voltage, according to Fig. 1, is  =

Transformation of unipolar signal
For the practice, unipolar square wave voltage signal is also interesting, which is shown in Fig. 4. • cos(3) + ⋯�. ( DC terms is represented as  0 .From the formula (30) it can be seen that such function contains constant term, so as cosine harmonic terms with both positive and negative signs.
A short example will be used as a display of the calculations of the DC term, as well as a few higher harmonics of such Fourier's series.
Let  = 10 V. DC term is 5 V.The amplitude of primary harmonic term is  1max =

Conclusion
In electrical circuits we often find non sinusoidal alternating values that are extremely important in transmission and analysis of different signals.Unlike sinusoidal alternating values, they cannot be displayed by using vectors and complex numbers, but only as continuous time (or discrete) functions.
In this paper it is shown that such functions can be, using Fourier transformation, distinguished to DC component and to a series of alternating components (harmonics).Since nowadays the importance of application of Fourier transformation in the analysis of electrical networks is emphasized [11], the introductory chapters of this paper mathematically explain Fourier series and the calculation of its coefficients.
Unlike discrete Fourier transformations [12,13,14], in this paper is shown, based on the example of continuous time non sinusoidal functions (square wave signals), the usage of continuous time Fourier transformation.Such transformed signal is possible to analyze by using conventional complex/vector methods and available software for the analysis of electrical networks [15].

Therefore 2 �
(−) = −() is non sinusoidal dimension, it does not contain cosine terms.Since,  � +  = −(), is non sinusoidal dimension, it only contains odd terms.Therefore, the square wave voltage in Fig. 1 has only sinusoidal odd terms.

2 2 <
Transformation of bipolar signal Let  =  1 •√2•π 4 be on interval 0 <  <  < , where  1 stands for effective value of the primary harmonic term.If we consider only few first terms (until ninth harmonic; Fig. 2) graphical representation in Fig. 3 looks like square wave voltage.(Simulation has been made by Software: Electronics Workbench EWB 4.1).

Figure 4
Figure 4 Representation of unipolar voltage signal Relation (−) = () shows that the function does not contain sinusoidal terms.Development of this specific function into Fourier's series goes in the following way

= 4 , 5
V. Calculated primary terms and first few harmonics are shown in Tab. 1.

Table 1
Calculated primary terms of Fourier's series