Almost any periodic function can be decomposed into simple harmonics using a trigonometric series (Fourier series):
f(x) = + (a n cos nx + b n sin nx), (*)
Let's write down this row as a sum of simple harmonics, assuming the coefficients are equal a n= A n sin j n, b n= A n cos j n. We get: a n cos j n + b n sin j n = A n sin( nx+ j n), where
A n= , tg j n = . (**)
Then the series (*) in the form of simple harmonics takes the form f(x) = 
.
The Fourier series represents a periodic function as the sum of an infinite number of sinusoids, but with frequencies that have a certain discrete value.
Sometimes n th harmonic is written as a n cos nx + b n sin nx = A n cos( nx – j n) , where a n= A n cos j n , b n= A n sin j n .
Wherein A n and j n are determined by the formulas (**). Then the series (*) will take the form
f(x) = 
.
Definition 9. Periodic Function Representation Operation f(x) next to Fourier is called harmonic analysis.
The expression (*) is also found in another, more common form:
Odds a n, b n are determined by the formulas:
magnitude C 0 expresses the average value of the function over the period and is called the constant component, which is calculated by the formula:
In the theory of oscillations and spectral analysis, the representation of the function f(t) in a Fourier series is written as:
(***)
those. a periodic function is represented by the sum of terms, each of which is sinusoidal oscillation with amplitude C n and initial phase j n, that is, the Fourier series of a periodic function consists of individual harmonics with frequencies that differ from each other by a constant number. Moreover, each harmonic has a certain amplitude. Values C n and j n must be properly chosen in order for the equality (***) to hold, that is, they are determined by the formulas (**) [ C n = A n].
Let us rewrite the Fourier series (***) as 
where w 1 is the main frequency. From this we can conclude: a complex periodic function f(t) is determined by the set of quantities C n and j n .
Definition 10. Set of quantities C n, that is, the dependence of the amplitude on frequency, is called amplitude spectrum of the function or amplitude spectrum.
Definition 11. Set of quantities j n is called phase spectrum.
When they say simply “spectrum”, they mean exactly the amplitude spectrum, in other cases they make the appropriate reservations. The periodic function has discrete spectrum(that is, it can be represented as individual harmonics).
The spectrum of a periodic function can be represented graphically. For this we choose the coordinates C n and w = nw one . The spectrum will be depicted in this coordinate system by a set of discrete points, since each value nw 1 corresponds to one specific value With n. A graph consisting of individual points is inconvenient. Therefore, it is customary to depict the amplitudes of individual harmonics as vertical segments of the appropriate length (Fig. 2).
Rice. 2.
This discrete spectrum is often referred to as a line spectrum. He is a harmonic spectrum, i.e. consists of equally spaced spectral lines; harmonic frequencies are in simple multiple ratios. Separate harmonics, including the first one, may be absent, i.e. their amplitudes may be equal to zero, but this does not violate the harmonicity of the spectrum.
Discrete or line spectra can belong to both periodic and non-periodic functions. In the first case, the spectrum is necessarily harmonic.
The Fourier series expansion can be generalized to the case of a non-periodic function. To do this, we must apply the passage to the limit as T®∞, considering a non-periodic function as the limiting case of a periodic function with an indefinitely increasing period. Instead of 1/ T introduce the circular fundamental frequency w 1 = 2p/ T. This value is the frequency interval between adjacent harmonics, the frequencies of which are equal to 2p n/T. If a T® ∞, then w 1® dw and 2p n/T® w, where w is the current frequency, which changes continuously, dw- its increment. In this case, the Fourier series will turn into the Fourier integral, which is the expansion of a non-periodic function in an infinite interval (–∞;∞) into harmonic oscillations, the frequencies of which w change continuously from 0 to ∞:
A non-periodic function has a continuous or continuous spectrum, i.e. instead of individual points, the spectrum is depicted as a continuous curve. This is obtained as a result of passing to the limit from the series to the Fourier integral: the intervals between the individual spectral lines decrease indefinitely, the lines merge, and instead of discrete points, the spectrum is represented by a continuous sequence of points, i.e. continuous curve. Functions a(w) and b(w) give the law of distribution of amplitudes and initial phases depending on the frequency w.
The Fourier transform is the most widely used means of converting an arbitrary function of time into a set of its frequency components in the complex number plane. This transformation can be applied to aperiodic functions to determine their spectra, in which case the complex operator s can be replaced by /co:
In order to determine the most interesting frequencies, numerical integration on the complex plane can be used.
To get acquainted with the basics of the behavior of these integrals, we consider several examples. On Fig. 14.6 (left) shows the unit area pulse in the time domain and its spectral composition; in the center - a pulse of the same area, but of greater amplitude, and on the right - the amplitude of the pulse is infinite, but its area is still equal to unity. The right picture is especially interesting because the zero-width pulse spectrum contains all frequencies with equal amplitudes.
Rice. 14.6. Spectra of pulses of the same width, along the same piaosrdi
In 1822 the French mathematician J. B. J. Fourier (J. B. J. Fourier) showed in his work on thermal conductivity that any periodic function can be decomposed into initial components, including the repetition frequency and a set of harmonics of this frequency, and each of the harmonics has its own amplitude and phase with respect to repetition rate. The basic formulas used in the Fourier transform are:
where A() represents the component direct current, and A p and B p - harmonics of the fundamental frequency of the order and, respectively, in phase and antiphase with it. The function f(*) is thus the sum of these harmonics and Lo-
In cases where f(x) is symmetric with respect to mc/2, i.e. e. f(x) on the region from n to 2n = -f(x) on the region from 0 to n, and there is no DC component, the Fourier transform formulas are simplified to:
where n = 1, 3.5, 7…
All harmonics are sinusoids, only some of them are in phase, and some are out of phase with the fundamental frequency. Most of the waveforms found in power electronics, can be decomposed into harmonics in this manner.
If the Fourier transform is applied to rectangular pulses with a duration of 120°, then the harmonics will be a set of order k = bi ± 1, where n is one of the integers. The amplitude of each harmonic h with respect to the first one is related to its number by the relation h = l//e. In this case, the first harmonic will have an amplitude 1.1 times greater than the amplitude of a rectangular signal.
The Fourier transform gives the amplitude value for each harmonic, but since they are all sinusoidal, the rms value is obtained simply by dividing the corresponding amplitude by the root of 2. The rms value of a complex signal is the square root of the sum of the squares of the rms values of each harmonic, including the first.
When dealing with repetitive impulse functions, it is useful to consider the duty cycle. If the repeated pulses in Fig. 14.7 have an rms value X at time A, then the rms value at time B will be X(A/B) 1 2 . Thus, the RMS value of the repetitive pulses is proportional to the square root of the duty cycle value. Applying this principle to a 120° (duty cycle 2/3) unit amplitude rectangular pulse gives the RMS value (2/3) 1/2 = 0.8165.
Rice. 14.7. Determining the Root Mean Square (RMS) for Repeated
impulses
It is interesting to check this result by summing the harmonics corresponding to the mentioned square wave train. In Table. 14.2 shows the results of this summation. As you can see, everything matches.
Table 14.2. The results of the summation of harmonics corresponding to
periodic signal with duty cycle 2/3 and unit amplitude
|
Harmonic number |
Harmonic amplitude |
Total RMS |
For comparison purposes, any set of harmonics can be grouped together and the corresponding overall level of harmonic distortion determined. In this case, the mean square value of the signal is determined by the formula
where h\ is the amplitude of the first (fundamental) harmonic, and h„ is the amplitude of harmonics of order n > 1.
The components responsible for distortion can be written separately as
where n > 1. Then
where Fund is the first harmonic and the total harmonic distortion (THD) is equal to D/Fund.
Although square wave analysis is interesting, it is rarely used in the real world. Switching effects and other processes make rectangular pulses more like trapezoidal, or, in the case of transducers, with a rising edge described by the expression 1 cos(0) and a falling edge described by the dependence cos(0), where 0< 0 on a logarithmic scale, the slope of the corresponding sections of this graph is -2 and -1. For systems with typical reactance values, the slope change occurs approximately at frequencies from the 11th to the 35th harmonic of the mains frequency, and with an increase in reactance or current in the system, the frequency of slope change decreases . The practical result of all this is that higher harmonics are less important than one might think. Although increasing the reactance helps to reduce the higher order harmonics, this is usually not feasible. It is more preferable to reduce the harmonic components in the consumed current by increasing the number of pulses during rectification or voltage conversion, achieved by phase shift. With regard to transformers, this topic was touched upon in Chap. 7. If the thyristor converter or rectifier is fed from the transformer windings connected by a star and a delta, and the outputs of the converter or rectifier are connected in series or in parallel, then a 12-pulse rectification is obtained. The harmonic numbers in the set are now k = \2n ± 1 instead of k = 6u + 1, where n is one of the integers. Instead of harmonics of the 5th and 7th order, harmonics of the 11th and 13th orders now appear, the amplitude of which is much less. It is quite possible to use even more ripples, and, for example, 48-pulse systems are used in large power supplies for electrochemical installations. Since large rectifiers and converters use sets of diodes or thyristors connected in parallel, the additional cost of phase-shifting windings in a transformer mainly determines its price. On Fig. 14.8 shows the advantages of a 12-pulse circuit over a 6-pulse one. The 11th and 13th order harmonics in a 12-pulse circuit have a typical amplitude value of about 10% of the first harmonic. In circuits with a large number of ripples, the harmonics are of the order k = pn + 1, where p is the number of ripples. For interest, we note that pairs of harmonic sets that are simply shifted relative to each other by 30° do not cancel each other out in a 6-pulse scheme. These harmonic currents flow back through the transformer; thus, an additional phase shift is required to obtain the possibility of their mutual annihilation. Not all harmonics are in phase with the first. For example, in a three-phase harmonic set corresponding to a 120° square wave train, the phases of the harmonics change according to the sequence -5th, +7th, -11th, +13th, and so on. When unbalanced in a three-phase circuit, single-phase components can occur, which entails a tripling of harmonics with zero phase shift. Rice. 14.8. Spectra of 6 and 12 pulsation transducers Isolation transformers are often seen as a panacea for harmonic problems. These transformers add some reactance to the system and thereby help to reduce higher harmonics, however, apart from suppression of zero-sequence currents and electrostatic isolation, they are of little use. Decomposition of periodic non-sinusoidal functions General definitions Part 1. Theory of linear circuits (continued) ELECTRICAL ENGINEERING THEORETICAL BASIS Textbook for students of electric power specialties T. Electrical circuits of periodic non-sinusoidal current As you know, in the electric power industry, a sinusoidal form is adopted as a standard form for currents and voltages. However, in real conditions, the shapes of the curves of currents and voltages may differ to some extent from sinusoidal ones. Distortions in the shapes of the curves of these functions in receivers lead to additional energy losses and a decrease in their efficiency. The sinusoidal shape of the generator voltage curve is one of the indicators of the quality of electrical energy as a commodity. The following reasons for the distortion of the shape of the curves of currents and voltages in a complex circuit are possible: 1) the presence in the electrical circuit of non-linear elements, the parameters of which depend on the instantaneous values of current and voltage [ R, L, C=f(u, i)], (for example, rectifiers, electric welding units, etc.); 2) the presence in the electrical circuit of parametric elements, the parameters of which change over time [ R, L, C=f(t)]; 3) the source of electrical energy (three-phase generator), due to design features, cannot provide an ideal sinusoidal shape of the output voltage; 4) influence in the complex of the factors listed above. Nonlinear and parametric circuits are discussed in separate chapters of the TOE course. This chapter investigates the behavior of linear electrical circuits when exposed to energy sources with a non-sinusoidal waveform. It is known from the course of mathematics that any periodic function of time f(t) that satisfies the Dirichlet conditions can be represented by the harmonic Fourier series: Here BUT 0 - constant component, - k-th harmonic component or abbreviated k I am a harmonica. The 1st harmonic is called the fundamental, and all subsequent harmonics are called the highest. Amplitudes of individual harmonics A to do not depend on the way the function is expanded f(t) into a Fourier series, while the initial phases of individual harmonics depend on the choice of the origin of time (the origin of coordinates). The individual harmonics of the Fourier series can be represented as the sum of the sine and cosine components: Then the whole Fourier series will take the form: The ratios between the coefficients of the two forms of the Fourier series are: If a k th harmonic and its sine and cosine components are replaced by complex numbers, then the ratio between the coefficients of the Fourier series can be represented in complex form: If a periodic non-sinusoidal function of time is given (or can be expressed) analytically in the form of a mathematical equation, then the coefficients of the Fourier series are determined by the formulas known from the mathematics course: In practice, the investigated non-sinusoidal function f(t) is usually set in the form of a graphic diagram (graphically) (Fig. 118) or in the form of a table of coordinates of points (tabular) in the interval of one period (Table 1). To perform a harmonic analysis of such a function according to the above equations, it must first be replaced by a mathematical expression. Replacing a function given graphically or tabularly by a mathematical equation is called function approximation. To check if the program is working correctly, we will form an array of readings as the sum of two sinusoids sin(10*2*pi*x)+0.5*sin(5*2*pi*x) and slip it into the program. The program drew the following: fig.1 Graph of the time function of the signal fig.2 Graph of signal spectrum On the spectrum graph there are two sticks (harmonics) 5 Hz with an amplitude of 0.5 V and 10 Hz - with an amplitude of 1 V, all as in the formula of the original signal. Everything is fine, well done programmer! The program is working correctly. This means that if we apply a real signal from a mixture of two sinusoids to the ADC input, we will get a similar spectrum consisting of two harmonics. Total, our real measured signal, duration 5 sec, digitized by the ADC, i.e. represented discrete counts, has discrete non-periodic spectrum. Something is not right! The 10 Hz harmonic is drawn normally, but instead of a 5 Hz stick, several incomprehensible harmonics appeared. We look on the Internet, what and how ... In, they say that zeros must be added to the end of the sample and the spectrum will be drawn normal. fig.5 Finished zeros up to 5 seconds fig.6 We got the spectrum Still not what it was at 5 seconds. You have to deal with the theory. Let's go to Wikipedia- source of knowledge. K - number of trigonometric function (number of harmonic component, harmonic number) What does it mean to "represent a function as a sum of a series"? This means that by adding the values of the harmonic components of the Fourier series at each point, we will get the value of our function at this point. (More strictly, the standard deviation of the series from the function f(x) will tend to zero, but despite the root-mean-square convergence, the Fourier series of the function, generally speaking, is not required to converge pointwise to it. See https://ru.wikipedia.org/ wiki/Fourier_Series .) This series can also be written as: (2), The relationship between the coefficients (1) and (3) is expressed by the following formulas: Note that all these three representations of the Fourier series are completely equivalent. Sometimes, when working with Fourier series, it is more convenient to use the exponents of the imaginary argument instead of sines and cosines, that is, to use the Fourier transform in complex form. But it is convenient for us to use formula (1), where the Fourier series is represented as a sum of cosine waves with the corresponding amplitudes and phases. In any case, it is wrong to say that the result of the Fourier transform of the real signal will be the complex amplitudes of the harmonics. As the wiki correctly states, "The Fourier transform (?) is an operation that maps one function of a real variable to another function, also of a real variable." Total: The Fourier transform allows us to represent a continuous function f(x) (signal) defined on the segment (0, T) as the sum of an infinite number (infinite series) of trigonometric functions (sine and/or cosine) with certain amplitudes and phases, also considered on the segment (0, T). Such a series is called a Fourier series. We note some more points, the understanding of which is required for the correct application of the Fourier transform to signal analysis. If we consider the Fourier series (the sum of sinusoids) on the entire X-axis, then we can see that outside the segment (0, T) the function represented by the Fourier series will periodically repeat our function. For example, in the graph in Fig. 7, the original function is defined on the segment (-T \ 2, + T \ 2), and the Fourier series represents a periodic function defined on the entire x-axis. This is because the sinusoids themselves are periodic functions, respectively, and their sum will be a periodic function. fig.7 Representation of a non-periodic original function by a Fourier series In this way: Our original function is continuous, non-periodic, defined on some interval of length T. The periods of the harmonic components are multiples of the segment (0, T) on which the original function f(x) is defined. In other words, the harmonic periods are multiples of the duration of the signal measurement. For example, the period of the first harmonic of the Fourier series is equal to the interval T on which the function f(x) is defined. The period of the second harmonic of the Fourier series is equal to the interval T/2. And so on (see Fig. 8). fig.8 Periods (frequencies) of the harmonic components of the Fourier series (here T = 2?) Accordingly, the frequencies of the harmonic components are multiples of 1/T. That is, the frequencies of the harmonic components Fk are equal to Fk= k\T, where k ranges from 0 to?, for example, k=0 F0=0; k=1 F1=1\T; k=2 F2=2\T; k=3 F3=3\T;… Fk= k\T (at zero frequency - constant component). Let our original function be a signal recorded for T=1 sec. Then the period of the first harmonic will be equal to the duration of our signal T1=T=1 sec and the frequency of the harmonic is 1 Hz. The period of the second harmonic will be equal to the duration of the signal divided by 2 (T2=T/2=0.5 sec) and the frequency is 2 Hz. For the third harmonic T3=T/3 sec and the frequency is 3 Hz. And so on. The step between harmonics in this case is 1 Hz. Thus, a signal with a duration of 1 sec can be decomposed into harmonic components (to obtain a spectrum) with a frequency resolution of 1 Hz. There is a way to artificially increase the duration of the signal by adding zeros to the array of samples. But it does not increase the real frequency resolution. The usual scheme for measuring and digitizing a signal is as follows. fig.9 Scheme of the measuring channel The signal from the measuring transducer arrives at the ADC during a period of time T. The signal samples (sample) obtained during the time T are transferred to the computer and stored in memory. fig.10 Digitized signal - N readings received in time T What are the requirements for signal digitization parameters? A device that converts an input analog signal into a discrete code (digital signal) is called an analog-to-digital converter (ADC, English Analog-to-digital converter, ADC) (Wiki). One of the main parameters of the ADC is the maximum sampling rate (or sampling rate, English sample rate) - the frequency of taking samples of a signal continuous in time during its sampling. Measured in hertz. ((Wiki)) According to the Kotelnikov theorem, if a continuous signal has a spectrum limited by the frequency Fmax, then it can be completely and uniquely restored from its discrete samples taken at time intervals , i.e. with frequency Fd ? 2*Fmax, where Fd - sampling frequency; Fmax - maximum frequency of the signal spectrum. In other words, the signal sampling rate (ADC sampling rate) must be at least 2 times the maximum frequency of the signal that we want to measure. And what will happen if we take readings with a lower frequency than required by the Kotelnikov theorem? In this case, the effect of "aliasing" (aka stroboscopic effect, moire effect) occurs, in which a high-frequency signal after digitization turns into a low-frequency signal that does not actually exist. On fig. 5 high frequency red sine wave is the real signal. The lower frequency blue sine wave is a dummy signal resulting from the fact that during the sampling time more than half a period of the high-frequency signal has time to pass. Rice. 11. The appearance of a false low-frequency signal when the sampling rate is not high enough To avoid the effect of aliasing, a special anti-aliasing filter is placed in front of the ADC - LPF (low-pass filter), which passes frequencies below half the ADC sampling frequency, and kills higher frequencies. In order to calculate the spectrum of a signal from its discrete samples, the discrete Fourier transform (DFT) is used. We note once again that the spectrum of a discrete signal is "by definition" limited by the frequency Fmax, which is less than half the sampling frequency Fd. Therefore, the spectrum of a discrete signal can be represented by the sum of a finite number of harmonics, in contrast to the infinite sum for the Fourier series of a continuous signal, the spectrum of which can be unlimited. According to the Kotelnikov theorem, the maximum harmonic frequency must be such that it has at least two samples, so the number of harmonics is equal to half the number of samples of the discrete signal. That is, if there are N samples in the sample, then the number of harmonics in the spectrum will be equal to N/2. Consider now the discrete Fourier transform (DFT). Comparing with the Fourier series We see that they coincide, except that the time in the DFT is discrete and the number of harmonics is limited to N/2 - half the number of samples. The DFT formulas are written in dimensionless integer variables k, s, where k are the numbers of signal samples, s are the numbers of spectral components. Returning to the results obtained at the beginning. As mentioned above, when expanding a non-periodic function (our signal) into a Fourier series, the resulting Fourier series actually corresponds to a periodic function with a period T. (Fig. 12). fig.12 Periodic function f(x) with period Т0, with measurement period Т>T0 As can be seen in Fig. 12, the function f(x) is periodic with period Т0. However, due to the fact that the duration of the measurement sample T does not coincide with the period of the function T0, the function obtained as a Fourier series has a discontinuity at the point T. As a result, the spectrum of this function will contain a large number of high-frequency harmonics. If the duration of the measurement sample T coincided with the period of the function T0, then only the first harmonic (a sinusoid with a period equal to the sample duration) would be present in the spectrum obtained after the Fourier transform, since the function f(x) is a sinusoid. In other words, the DFT program "does not know" that our signal is a "piece of a sine wave", but is trying to represent a periodic function as a series, which has a gap due to the inconsistency of the individual pieces of the sine wave. As a result, harmonics appear in the spectrum, which in total should represent the form of the function, including this discontinuity. Thus, in order to obtain the "correct" spectrum of the signal, which is the sum of several sinusoids with different periods, it is necessary that an integer number of periods of each sinusoid fit on the signal measurement period. In practice, this condition can be met for a sufficiently long duration of the signal measurement. Fig.13 An example of the function and spectrum of the signal of the kinematic error of the gearbox With a shorter duration, the picture will look "worse": Fig.14 An example of the function and spectrum of the rotor vibration signal In practice, it can be difficult to understand where are the “real components” and where are the “artifacts” caused by the non-multiplicity of the periods of the components and the duration of the signal sample or the “jumps and breaks” of the waveform. Of course, the words "real components" and "artifacts" are not in vain quoted. The presence of many harmonics on the spectrum graph does not mean that our signal actually “consists” of them. It's like thinking that the number 7 "consists" of the numbers 3 and 4. The number 7 can be represented as the sum of the numbers 3 and 4 - this is correct. So is our signal ... or rather, not even “our signal”, but a periodic function compiled by repeating our signal (sampling) can be represented as a sum of harmonics (sinusoids) with certain amplitudes and phases. But in many cases important for practice (see the figures above), it is indeed possible to relate the harmonics obtained in the spectrum to real processes that are cyclic in nature and make a significant contribution to the signal shape. 2. The signal is represented by a set of real values and its spectrum is represented by a set of real values. The harmonic frequencies are positive. The fact that it is more convenient for mathematicians to represent the spectrum in a complex form using negative frequencies does not mean that “it is right” and “it should always be done this way”. 3. The signal measured on the time interval T is determined only on the time interval T. What happened before we began to measure the signal, and what will happen after that, is unknown to science. And in our case - it is not interesting. The DFT of a time-limited signal gives its "real" spectrum, in the sense that, under certain conditions, it allows you to calculate the amplitude and frequency of its components. Used materials and other useful materials. 2.1. Spectra of Periodic Signals
A periodic signal (current or voltage) is called such a type of influence when the waveform repeats after a certain time interval T which is called the period. The simplest form of a periodic signal is a harmonic signal or a sine wave, which is characterized by amplitude, period, and initial phase. All other signals will inharmonious or non-sinusoidal. It can be shown, and practice proves it, that if the input signal of the power supply is periodic, then all other currents and voltages in each branch (output signals) will also be periodic. In this case, the waveforms in different branches will differ from each other. There is a general technique for studying periodic non-harmonic signals (input actions and their reactions) in an electrical circuit, which is based on the decomposition of signals into a Fourier series. This technique consists in the fact that it is always possible to select a number of harmonic (i.e. sinusoidal) signals with such amplitudes, frequencies and initial phases, the algebraic sum of the ordinates of which at any time is equal to the ordinate of the studied non-sinusoidal signal. So, for example, the voltage u in fig. 2.1. can be replaced by the sum of stresses and , since at any time the identical equality takes place: For the example under consideration, we have the period of the first harmonic coinciding with the period of the non-harmonic signalT 1
=
T, and the period of the second harmonic is two times smallerT 2
=
T/2, i.e. instantaneous values of harmonics should be written as: Here, the amplitudes of harmonic oscillations are equal to each other ( Rice. 2.1. Example of addition of the first and second harmonics non-harmonic signal In electrical engineering, a harmonic component whose period is equal to the period of a non-harmonic signal is called first or basic signal harmonics. All other components are called higher harmonic components. A harmonic whose frequency is k times greater than the first harmonic (and the period, respectively, k times less) is called k - th harmonic. Allocate also the average value of the function for the period, which is called null harmonica. In the general case, the Fourier series is written as the sum of an infinite number of harmonic components of different frequencies: (2.1)
where k is the harmonic number; - angular frequency of the k - th harmonic; ω 1 \u003d ω \u003d 2 π / T- angular frequency of the first harmonic; - zero harmonic. For commonly occurring waveforms, a Fourier series expansion can be found in the specialized literature. Table 2 shows the expansions for eight waveforms. It should be noted that the expansions given in Table 2 will take place if the origin of the coordinate system is chosen as indicated in the figures on the left; when changing the origin of time t the initial phases of the harmonics will change, while the amplitudes of the harmonics will remain the same. Depending on the type of signal under study, V should be understood as either a value measured in volts if it is a voltage signal, or a value measured in amperes if it is a current signal. Fourier series expansion of periodic functions table 2 Schedule f(t)
Fourier series of functionsf(t)
Note k=1,3,5,... k=1,3,5,... k=1,3,5,... k=1,2,3,4,5 k=1,3,5,... k=1,2,3,4,5 S=1,2,3,4,.. k=1,2,4,6,.. Signals 7 and 8 are generated from a sinusoid by gate circuits. The set of harmonic components that form a non-sinusoidal signal is called the spectrum of this non-harmonic signal. From this set of harmonics, they distinguish and distinguish amplitude and phase spectrum. The amplitude spectrum is a set of amplitudes of all harmonics, which is usually represented by a diagram in the form of a set of vertical lines, the lengths of which are proportional (in the chosen scale) to the amplitude values of the harmonic components, and the place on the horizontal axis is determined by the frequency (harmonic number) of this component. Similarly, phase spectra are considered as a set of initial phases of all harmonics; they are also shown to scale as a set of vertical lines. It should be noted that it is customary to measure the initial phases in electrical engineering in the range from -180 0 to +180 0. Spectra consisting of individual lines are called lined or discrete. Spectral lines are at a distance f apart, where f- frequency interval equal to the frequency of the first harmonic f.Thus, the discrete spectra of periodic signals have spectral components with multiple frequencies - f,
2f, 3f, 4f, 5f etc. Example 2.1. Find the amplitude and phase spectrum for a rectangular signal, when the durations of the positive and negative signals are equal, and the average value of the function over the period is zero u(t) = Vat 0<t<T/2
u(t) = -Vat T/2<t<T
For signals of simple, frequently used forms, it is advisable to find a solution using tables. Rice. 2.2. Linear amplitude spectrum of a rectangular signal From the Fourier expansion of a rectangular signal (see Tables 2 - 1) it follows that the harmonic series contains only odd harmonics, while the amplitudes of the harmonics decrease in proportion to the harmonic number. The amplitude line spectrum of harmonics is shown in fig. 2.2. When constructing, it is assumed that the amplitude of the first harmonic (here voltage) is equal to one volt: B; then the amplitude of the third harmonic will be equal to B, the fifth - B, etc. The initial phases of all harmonics of the signal are equal to zero, therefore, the phase spectrum has only zero values of the ordinates. Problem solved. Example 2.2.Find the amplitude and phase spectrum for a voltage that varies according to the law: at - T/4<t<T/4; u(t) = 0 for T/4<t<3/4T. Such a signal is formed from a sinusoid by eliminating (by circuitry using valve elements) the negative part of the harmonic signal. Rice. 2.3. The line spectrum of a half-wave rectification signal: a) amplitude; b) phase For a half-wave rectification signal of a sinusoidal voltage (see Tables 2 - 8), the Fourier series contains a constant component (zero harmonic), the first harmonic, and then a set of only even harmonics, the amplitudes of which rapidly decrease with increasing harmonic number. If, for example, we put the value V = 100 B, then, multiplying each term by the common factor 2V/π , we find(2.2)
The amplitude and phase spectra of this signal are shown in Fig. 2.3a,b. Problem solved. In accordance with the theory of Fourier series, the exact equality of a non-harmonic signal to the sum of harmonics takes place only for an infinitely large number of harmonics. The calculation of harmonic components on a computer allows you to analyze any number of harmonics, which is determined by the purpose of the calculation, the accuracy and form of non-harmonic effects. If the duration of the signalt
regardless of its shape, much less period T, then the amplitudes of the harmonics will decrease slowly, and for a more complete description of the signal, it is necessary to take into account a large number of terms in the series. This feature can be traced for the signals presented in Tables 2 - 5 and 6, provided that the condition τ
<<T. If the non-harmonic signal is close to a sinusoid in shape (for example, signals 2 and 3 in Table 2), then the harmonics decrease rapidly, and for an accurate description of the signal, it is enough to limit ourselves to three to five harmonics of the series.
From a mathematical point of view, how many mistakes are there in this phrase?
Now the authorities have decided we decided that 5 seconds is too long, let's measure the signal in 0.5 seconds. 
fig.3 Graph of the function sin(10*2*pi*x)+0.5*sin(5*2*pi*x) for a measurement period of 0.5 sec
fig.4 Function spectrum2. A continuous function and its representation by a Fourier series
Mathematically, our signal with a duration of T seconds is a certain function f(x) given on the segment (0, T) (X in this case is time). Such a function can always be represented as a sum of harmonic functions (sine or cosine) of the form:
T - segment where the function is defined (signal duration)
Ak - amplitude of k-th harmonic component,
?k - initial phase of k-th harmonic component
where , k-th complex amplitude.
The mathematical basis of the spectral analysis of signals is the Fourier transform.
The spectrum of this function is discrete, that is, it is presented as an infinite series of harmonic components - the Fourier series.
In fact, a certain periodic function is defined by the Fourier series, which coincides with ours on the segment (0, T), but this periodicity is not essential for us.
To increase the resolution by 2 times to 0.5 Hz, it is necessary to increase the measurement duration by 2 times - up to 2 seconds. A signal with a duration of 10 seconds can be decomposed into harmonic components (to obtain a spectrum) with a frequency resolution of 0.1 Hz. There are no other ways to increase the frequency resolution.3. Discrete signals and discrete Fourier transform
With the development of digital technology, the ways of storing measurement data (signals) have also changed. If earlier the signal could be recorded on a tape recorder and stored on tape in analog form, now the signals are digitized and stored in files in the computer's memory as a set of numbers (counts).
The value of s shows the number of full oscillations of the harmonic in the period T (the duration of the signal measurement). The discrete Fourier transform is used to find the amplitudes and phases of harmonics numerically, i.e. "on the computer"Some results
1. The real measured signal, duration T sec, digitized by the ADC, that is, represented by a set of discrete samples (N pieces), has a discrete non-periodic spectrum, represented by a set of harmonics (N/2 pieces). 
. Each of the terms is a sinusoid, the oscillation frequency of which is related to the period T integer ratios.
), and the initial phases are equal to zero.
a) b)
