First harmonic of the Fourier series. Decomposition of periodic non-sinusoidal curves in trigonometric Fourier series

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NON-SINUSOID CURRENT CIRCUITS

Until now, we have studied sinusoidal current circuits, but the law of current variation in time may differ from sinusoidal. In this case, non-sinusoidal current circuits take place. All non-sinusoidal currents are divided into three groups: periodic, i.e. having a period T(Figure 6.1, a), non-periodic (Figure 6.1, b) and almost periodic, having a periodically changing envelope ( T o) and the pulse repetition period ( T i) (Figure 6.1, c). There are three ways to obtain non-sinusoidal currents: a) a non-sinusoidal EMF acts in the circuit; b) a sinusoidal EMF acts in the circuit, but one or several elements of the circuit are nonlinear; c) a sinusoidal EMF acts in the circuit, but the parameters of one or more circuit elements change periodically in time. In practice, method b) is most often used. The most widespread non-sinusoidal currents are found in devices of radio engineering, automation, telemechanics and computer technology, where impulses of the most varied shape are often found. Non-sinusoidal currents are also encountered in the electric power industry. We will consider only periodic non-sinusoidal voltages and currents that can be decomposed into harmonic components.

Decomposition of periodic non-sinusoidal curves in trigonometric Fourier series

The phenomena that occur in linear circuits at periodic non-sinusoidal voltages and currents are the easiest to calculate and study if the non-sinusoidal curves are decomposed into a trigonometric Fourier series. It is known from mathematics that the periodic function f (ωt) satisfying the Dirichlet conditions, i.e. having on any finite time interval a finite number of discontinuities of only the first kind and a finite number of maxima and minima, can be expanded into a trigonometric Fourier series

f (ωt) = A o +
sinωt +
sin2ωt +
sin3ωt + +
cosωt +
cos2ωt +
cos3ωt + =

A o +
.

Here: A o- constant component or zero harmonic;
- sine component amplitude k th harmonic;
- cosine amplitude k th harmonic. They are determined by the following formulas

Since where, as follows from the vector diagram (Figure 6.2), we obtain

.

The terms included in this expression are called harmonics. Distinguish between even ( k- even) and odd harmonics. The first harmonic is called the fundamental, and the rest are called the highest. The last form of the Fourier series is convenient when you need to know the percentage of each harmonic. The same form of the Fourier series is used when calculating non-sinusoidal current circuits. Although theoretically the Fourier series contains an infinitely large number of terms, it usually converges quickly. and a converging series can express a given function with any degree of accuracy. In practice, it is enough to take a small number of harmonics (3-5) to obtain a calculation accuracy of a few percent.

Features of the Fourier series expansion of curves with symmetry

1. Curves, the average value of which over the period is equal to zero, do not contain a constant component (zero harmonic). 2
f (ωt) = - f (ωt + π), then it is called symmetric about the abscissa axis. This kind of symmetry is easy to determine by the shape of the curve: if you shift it by half a period along the abscissa axis, mirror it and at the same time it will merge with the original curve (Figure 6.3), then there is symmetry. When such a curve is expanded in a Fourier series, the latter lacks a constant component and all even harmonics, since they do not satisfy the condition f (ωt) = - f (ωt + π).

f (ωt) = sin (ωt + ψ 1 ) + sin (3ωt + ψ 3 )+
sin (5ωt + ψ 5 )+···.

3
... If the function satisfies the condition f (ωt) = f (-ωt), then it is called symmetric about the ordinate (even). This type of symmetry is easy to determine by the shape of the curve: if the curve lying to the left of the ordinate axis is mirrored and it merges with the original curve, then there is symmetry (Figure 6.4). When such a curve is expanded into a Fourier series, the latter will lack the sine components of all harmonics ( = f (ωt) = f (-ωt). Therefore, for such curves

f (ωt) = A O +
cosωt +
cos2ωt +
cos3ωt +

4
... If the function satisfies the condition f (ωt) = - f (-ωt), then it is called symmetric about the origin (odd). The presence of this type of symmetry is easy to determine by the shape of the curve: if the curve lying to the left of the ordinate axis is unfolded relative to points origin and it will merge with the original curve, then there is symmetry (Figure 6.5). When such a curve is expanded into a Fourier series, the latter will lack the cosine components of all harmonics (
= 0), since they do not satisfy the condition f (ωt) = - f (-ωt). Therefore, for such curves

f (ωt) =
sinωt +
sin2ωt +
sin3ωt +

If there is any symmetry in the formulas for and you can take the integral for half a period, but double the result, i.e. use expressions

There are several types of symmetry in curves at the same time. To facilitate the question of harmonic components in this case, fill in the table

Symmetry type	Analytic expression
1. Axes of abscissas	f (ωt) = - f (ωt + π)		Odd only
2. Axes of ordinates	f (ωt) = f (-ωt)
3. Origin of coordinates	f (ωt) = - f (-ωt)
4. Axes of abscissas and axes of ordinates	f (ωt) = - f (ωt + π) = f (-ωt)			Odd
5. Axes of abscissas and origin of coordinates	f (ωt) = - f (ωt + π) = - f (-ωt)		Odd

Expanding the curve in a Fourier series, one should first find out if it has any kind of symmetry, the presence of which allows one to predict in advance which harmonics will be in the Fourier series and not perform unnecessary work.

Grapho-analytical Fourier series expansion of curves

When a non-sinusoidal curve is given by a graph or a table and does not have an analytical expression, to determine its harmonics, one resorts to a graph-analytical decomposition. It is based on replacing a definite integral with a sum of a finite number of terms. For this purpose, the period of the function f (ωt) break into n equal parts Δ ωt = 2π / n(Figure 6.6). Then for the zero harmonic

where: R- current index (section number), taking values ​​from 1 to n; f R (ωt) - function value f (ωt) at ωt = pΔ ωt(see figure 6.6) . For the amplitude of the sine component k-Th harmonic

For the amplitude of the cosine component k-Th harmonic

Here sin p kωt and cos p kωt- values sinkωt and coskωt at ωt = p... In practical calculations, it is usually taken n= 18 (Δ ωt = 20˚) or n= 24 (Δ ωt = fifteen). At graphoanalytic decomposition curves in the Fourier series is even more important than in the analytic one to find out whether it has any kind of symmetry, the presence of which significantly reduces the volume computing work... Thus, the formulas for and in the presence of symmetry, take the form

When plotting harmonics on the general graph, it is necessary to take into account that the scale along the abscissa axis for k-Th harmonic in k times more than for the first.

Maximum, average and effective values ​​of non-sinusoidal quantities

Periodic non-sinusoidal values, in addition to their harmonic components, are characterized by maximum, average and effective values. Maximum value BUT m is the largest value of the modulus of the function during the period (Figure 6.7). The modulo average value is determined as follows

.

If the curve is symmetric about the abscissa axis and does not change sign during a half-period, then the value averaged over the absolute value is equal to the average value over a half-period

,

moreover, in this case, the origin of time should be chosen so that f ( 0)= 0. If the function never changes sign for the entire period, then its mean value in absolute value is equal to the constant component. In circuits of non-sinusoidal current, the values ​​of EMF, voltages or currents are understood to mean their effective values, determined by the formula

.

If the curve is expanded in a Fourier series, then its effective value can be determined as follows

Let us explain the receipt of the result. The product of sinusoids of different frequencies ( kω and iω) is a harmonic function, and the integral over the period from any harmonic function is zero. The integral under the sign of the first sum was determined in the circuits of sinusoidal current and its value was shown there. Consequently,

.

It follows from this expression that the effective value of periodic non-sinusoidal quantities depends only on the effective values ​​of its harmonics and does not depend on their initial phases ψ k... Let's give an example. Let be u=120
sin (314 t+ 45˚) -50sin (3 314 t-75˚) B... Its effective value

There are cases when the mean in absolute value and the effective values ​​of non-sinusoidal quantities can be calculated based on the integration of the analytical expression of the function and then there is no need to expand the curve in a Fourier series. In the electric power industry, where curves are predominantly symmetric about the abscissa axis, a number of coefficients are used to characterize their shape. Three of them are most widely used: crest factor k a, aspect ratio k f and distortion factor k and. They are defined like this: k a = A m / A; /A Wed; k and = A 1 /A. For a sinusoid, they have the following meanings: k a =; kφ = π A m / 2A m ≈1.11; 1.D For a rectangular curve (Figure 6.8, a), the coefficients are as follows: k a = 1; k f = 1; k and = 1.26 /. For a curve of a pointed (spike) shape (Figure 6.8, b), the values ​​of the coefficients are as follows: k a> and the higher, the more spiky its shape is; kφ> 1.11 and the higher the sharper the curve; k and<1 и чем более заостренная кривая, тем меньше. Как видим рассмотренные коэффициенты в определенной степени характеризуют форму кривой. Уwe seem to be one of the practical applications of the distortion factor. The voltage curves of industrial networks are usually different from the ideal sinusoid. In the electric power industry, the concept of an almost sinusoidal curve is introduced. According to GOST, the voltage of industrial networks is considered to be almost sinusoidal if the greatest difference between the corresponding ordinates of the true curve and its first harmonic does not exceed 5% of the amplitude of the fundamental harmonic (Figure 6.9). Measurement of non-sinusoidal values ​​with instruments of different systems gives unequal results. Amplitude electronic voltmeters measure maximum values. Magnetoelectric devices react only to the DC component of the measured values. Magnetoelectric devices with a rectifier measure the mean modulo value. Instruments in all other systems measure rms values.

Calculation of non-sinusoidal current circuits

If one or more sources with non-sinusoidal EMF act in the circuit, then its calculation is divided into three stages. 1. Decomposition of EMF sources into harmonic components. How to do this is discussed above. 2. Application of the superposition principle and calculation of currents and voltages in the circuit from the action of each component of the EMF separately. 3. Joint consideration (summation) of the solutions obtained in clause 2. The summation of the components in general form is most often difficult and not always necessary, since on the basis of the harmonic components one can judge both the shape of the curve and the basic quantities that characterize it. O
the main stage is the second. If a non-sinusoidal EMF is represented by a Fourier series, then such a source can be considered as a series connection of a constant EMF source and sources of sinusoidal EMF with different frequencies (Figure 6.10). Applying the superposition principle and considering the action of each EMF separately, it is possible to determine the components of the currents in all branches of the circuit. Let be E o creates I o, e 1 - i 1 , e 2 - i 2, etc. Then the actual current i=I o + i 1 +i 2 +··· . Consequently, the calculation of a non-sinusoidal current circuit is reduced to solving one problem with a constant EMF and a number of problems with sinusoidal EMF. When solving each of these problems, it is necessary to take into account that for different frequencies the inductive and capacitive resistances are not the same. The inductive reactance is directly proportional to the frequency, so it is for k-Th harmonics x Lk = kωL=kx L1, i.e. for k-Th harmonic it is in k times more than for the first. Capacitance is inversely proportional to frequency, so it is for k-Th harmonics xСk = 1 / kωС=x C1 / k, i.e. for k-Th harmonic it is in k times less than for the first. The active resistance, in principle, also depends on the frequency due to the surface effect, however, with small cross-sections of conductors and at low frequencies, the surface effect is practically absent and it is permissible to assume that the active resistance is the same for all harmonics. If a non-sinusoidal voltage is applied directly to the capacitor, then for k-Th harmonic of current

H The higher the harmonic number, the lower the capacitance resistance for it. Therefore, even if the amplitude of the high-order harmonic voltage is a small fraction of the amplitude of the first harmonic, it can still cause a current comparable to or higher than the fundamental current. In this regard, even at a voltage close to sinusoidal, the current in the capacitor may turn out to be sharply nonsinusoidal (Figure 6.11). In this regard, capacitance is said to emphasize high harmonic currents. If a non-sinusoidal voltage is applied directly to the inductance, then for k-Th harmonic of current

.

WITH
an increase in the order of the harmonic increases the inductive reactance. Therefore, in the current through the inductance, higher harmonics are presented to a lesser extent than in the voltage at its terminals. Even with a sharply non-sinusoidal voltage, the current curve in the inductance often approaches a sinusoid (Figure 6.12). Therefore, inductance is said to bring the current curve closer to a sine wave. When calculating each harmonic component of the current, you can use a complex method and build vector diagrams, however, it is unacceptable to perform geometric summation of vectors and the addition of complexes of voltages or currents of different harmonics. Indeed, vectors representing, say, the currents of the first and third harmonics rotate at different speeds (Figure 6.13). Therefore, the geometric sum of these vectors gives the instantaneous value of their sum only for ω t= 0 and does not make sense in the general case.

Non-sinusoidal current power

As in the sinusoidal current circuits, we will talk about the powers consumed by a passive two-pole. Active power is also understood as the average value of the instantaneous power over the period

Let the voltage and current at the input of a two-terminal network be represented by Fourier series

Substitute the values u and i into the formula R

The result was obtained taking into account the fact that the integral over the period from the product of sinusoids of different frequencies is zero, and the integral over the period from the product of sinusoids of the same frequency was determined in the section of sinusoidal current circuits. Thus, the active power of the non-sinusoidal current is equal to the sum of the active powers of all harmonics. It's clear that R k can be determined by any known formula. By analogy with a sinusoidal current, for a non-sinusoidal one, the concept of total power is introduced as the product of the effective values ​​of voltage and current, i.e. S = UI... Attitude R To S is called the power factor and is equal to the cosine of some conditional angle θ , i.e. cos θ =P / S... In practice, very often non-sinusoidal voltages and currents are replaced by equivalent sinusoids. In this case, two conditions must be met: 1) the effective value of the equivalent sinusoid must be equal to the effective value of the replaced quantity; 2) the angle between the equivalent sinusoids of voltage and current θ should be such that UI cos θ would be equal to active power R... Consequently, θ is the angle between the equivalent voltage and current sinusoids. Usually the rms value of the equivalent sinusoids is close to the rms values ​​of the fundamental harmonics. By analogy with sinusoidal current, for non-sinusoidal current, the concept of reactive power is introduced, defined as the sum of reactive powers of all harmonics

For non-sinusoidal current as opposed to sinusoidal S 2 ≠P 2 +Q 2. Therefore, the concept of distortion power is introduced here T, which characterizes the difference between the shapes of the voltage and current curves and is defined as

Higher harmonics in three-phase systems

In three-phase systems, typically the voltage curves in phases B and C accurately reproduce the curve in phase A, offset by one third of the period. So if u A = f (ωt), then u B = f (ωt- 2π/ 3), but u C = f (ωt + 2π/ 3). Let us assume that the phase voltages are non-sinusoidal and are expanded in a Fourier series. Then consider k-Th harmonic in all three phases. Let be u Ak = U km sin ( kωt + ψ k), then we get uВk = U km sin ( kωt + ψ k -k 2π/ 3) and u Ck = U km sin ( kωt + ψ k + k 2π/ 3). Comparing these expressions for different values k, we note that for harmonics that are multiples of three ( k=3n, n- a natural series of numbers, starting from 0) in all phases of the voltage at any moment of time have the same meaning and direction, i.e. form a zero sequence system. At k=3n + 1 harmonics form a voltage system, the sequence of which coincides with the sequence of actual voltages, i.e. they form a direct sequence system. At k=3n- 1 harmonics form a voltage system, the sequence of which is opposite to the sequence of actual voltages, i.e. they form a reverse sequence system. In practice, both the constant component and all even harmonics are most often absent, therefore, in the future, we will restrict ourselves to considering only the odd harmonics. Then the closest harmonic forming the reverse sequence is the fifth. In electric motors, it causes the greatest harm, therefore, it is with it that they are relentlessly fighting. Consider the features of the operation of three-phase systems caused by the presence of harmonics that are multiples of three. one ... When connecting the windings of a generator or transformer in a triangle (Figure 6.14), harmonic currents that are multiples of three flow through the branches of the latter, even in the absence of an external load. Indeed, the algebraic sum of the EMF of harmonics that are multiples of three ( E 3 , E 6, etc.), in the triangle has a triple value, in contrast to the other harmonics, for which this sum is equal to zero. If the phase resistance of the winding for the third harmonic Z 3, then the third harmonic current in the triangle circuit will be I 3 =E 3 /Z 3. Similarly, the sixth harmonic current I 6 =E 6 /Z 6, etc. The effective value of the current flowing through the windings will be
... Since the resistances of the generator windings are small, the current can reach large values. Therefore, if there are harmonics in the phase EMF that are multiples of three, the windings of the generator or transformer are not connected into a triangle. 2 ... If you connect the windings of a generator or transformer in an open triangle (Fig. 6.155, then a voltage will act on its terminals, equal to the sum of the EMF of harmonics, multiples of three, i.e. u BX = 3 E 3m sin (3 ωt + ψ 3)+3E 6m sin (6 ωt + ψ 6)+3E 9m sin (9 ωt + ψ 9)+···. Its effective value

.

An open delta is usually used before connecting the generator windings into a conventional delta to test the possibility of a trouble-free implementation of the latter. 3. Linear voltages, regardless of the connection diagram of the windings of the generator or transformer, do not contain harmonics that are multiples of three. When connected by a triangle, the phase EMFs containing harmonics that are multiples of three are compensated by the voltage drop across the internal resistance of the generator phase. Indeed, according to the second Kirchhoff's law for the third, for example, harmonics for the circuit in Fig. 6.14, we can write U AB3 + I 3 Z 3 =E 3, whence we obtain U AB3 = 0. Similarly for any of the harmonics that are multiples of three. When connected to a star, the line voltages are equal to the difference between the corresponding phase EMF. For harmonics that are multiples of three, when compiling these differences, the phase EMFs are destroyed, since they form a zero-sequence system. Thus, the components of all harmonics and their effective value can be present in the phase voltages. In line voltages, harmonics that are multiples of three are absent, therefore their effective value. In this regard, in the presence of harmonics that are multiples of three, U l / U f<
... 4. In circuits without a neutral wire, harmonic currents that are multiples of three cannot be closed, since they form a zero-sequence system and can close only if the latter is present. In this case, between the zero points of the receiver and the source, even in the case of a symmetrical load, a voltage appears equal to the sum of the EMF of harmonics, multiples of three, which is easy to verify by the equation of Kirchhoff's second law, taking into account that the currents of these harmonics are absent. The instantaneous value of this voltage u 0 1 0 =E 3m sin (3 ωt + ψ 3)+E 6m sin (6 ωt + ψ 6)+E 9m sin (9 ωt + ψ 9)+···. Its effective value
. 5... In a star-star circuit with a neutral wire (Figure 6.16), the latter will close harmonic currents that are multiples of three, even in the case of a symmetrical load, if the phase EMF contains the indicated harmonics. Taking into account that harmonics that are multiples of three form a zero-sequence system, we can write

Decomposition of periodic non-sinusoidal functions

General definitions

Part 1. Theory of linear circuits (continued)

ELECTRICAL ENGINEERING

THEORETICAL BASIS

Study guide for students of electric power specialties

T. Electric 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 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 nonlinear 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 in time [ R, L, C = f(t)];

3) the source of electrical energy (three-phase generator), due to design features, cannot provide an ideal sinusoidal output voltage;

4) the influence of the above factors in the complex.

Nonlinear and parametric circuits are considered in separate chapters of the TOE course. This chapter examines the behavior of linear electrical circuits when exposed to energy sources with a non-sinusoidal curve shape.

It is known from the course of mathematics that any periodic function of time f(t) satisfying the Dirichlet conditions can be represented by a harmonic Fourier series:

Here BUT 0 - constant component, - k-th harmonic component or abbreviated k th harmonic. The 1st harmonic is called the fundamental, and all subsequent ones are called the highest.

Amplitudes of individual harmonics And to do not depend on the way of decomposition of the function f(t) in the Fourier series, while the initial phases of individual harmonics depend on the choice of the time origin (origin).

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 relations between the coefficients of the two forms of the Fourier series are as follows:

If k-th harmonic and its sine and cosine components are replaced by complex numbers, then the relationship 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 course of mathematics:

In practice, the investigated non-sinusoidal function f(t) is usually set in the form of a graphical 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 in a tabular way by a mathematical equation is called function approximation.

The Fourier and Hartley transforms transform functions of time into functions of frequency, containing information about amplitude and phase. Below are the graphs of the continuous function g(t) and discrete g(τ), where t and τ are times.

Both functions start at zero, abruptly reach a positive value, and decay exponentially. By definition, the Fourier transform for a continuous function is an integral over the entire real axis, F(f), and for a discrete function - the sum over a finite set of samples, F(ν):

where f, ν - frequency values, n Is the number of sampled values ​​of the function, and i= √ –1 - imaginary unit. The integral representation is more suitable for theoretical research, and the representation in the form of a finite sum is more suitable for calculations on a computer. Integral and discrete Hartley transforms are defined in a similar way:

Although the only notation difference between Fourier and Hartley's definitions is the presence of a factor in front of the sine, the fact that the Fourier transform has both real and imaginary parts makes the representations of these two transforms completely different. Discrete Fourier and Hartley transforms have essentially the same form as their continuous counterparts.

Although the plots look different, the same amplitude and phase information can be derived from the Fourier and Hartley transforms, as shown below.

The Fourier amplitude is determined by the square root of the sum of the squares of the real and imaginary parts. The Hartley amplitude is determined by the square root of the sum of squares H(–Ν) and H(ν). The Fourier phase is determined by the arctangent of the imaginary part divided by the real part, and the Hartley phase is determined by the sum of 45 ° and the arctangent of H(–Ν) divided by H(ν).

As you know, in the electric power industry, the sinusoidal form is adopted as the standard form for currents and voltages. However, in real conditions, the shapes of the curves of currents and voltages can differ to some extent from sinusoidal ones. Distortions 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 of nonlinear elements in the electrical circuit, the parameters of which depend on the instantaneous values ​​of current and voltage, (for example, rectifiers, electric welding units, etc.);

2) the presence of parametric elements in the electrical circuit, the parameters of which change over time;

3) the source of electrical energy (three-phase generator), due to design features, cannot provide an ideal sinusoidal output voltage;

4) the influence of the above factors in the complex.

Nonlinear and parametric circuits are considered in separate chapters of the TOE course. This chapter examines the behavior of linear electrical circuits when exposed to energy sources with a non-sinusoidal curve shape.

It is known from the course of mathematics that any periodic function of time f (t) satisfying the Dirichlet conditions can be represented by a harmonic Fourier series:

Here A0 is a constant component, Ak * sin (kωt + αk) is the kth harmonic component, or abbreviated as the kth harmonic. The 1st harmonic is called the fundamental, and all subsequent ones are called the highest.

The amplitudes of the individual harmonics Ak do not depend on the method of expanding the function f (t) in the Fourier series, while the initial phases of the individual harmonics αk depend on the choice of the time origin (origin).

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 relations between the coefficients of the two forms of the Fourier series are as follows:

If the k-th harmonic and its sine and cosine components are replaced by complex numbers, then the relationship 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 course of mathematics:

In practice, the investigated non-sinusoidal function f (t) is usually specified in the form of a graphical diagram (graphically) (Fig. 46.1) 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 in a tabular way by a mathematical equation is called function approximation.

Currently, harmonic analysis of non-sinusoidal functions of time f (t) is performed, as a rule, on a computer. In the simplest case, a piecewise linear approximation is used for the mathematical representation of a function. For this, the entire function in the interval of one full period is divided into M = 20-30 sections so that the individual sections are as close as possible to straight lines (Fig. 1). In some sections, the function is approximated by the equation of the straight line fm (t) = am + bm * t, where the approximation coefficients (am, bm) are determined for each section through the coordinates of its end points, for example, for the 1st section we get:

The period of the function T is divided into a large number of integration steps N, the integration step Δt = h = T / N, the current time ti = hi, where i is the ordinal number of the integration step. Certain integrals in the formulas of harmonic analysis are replaced by the corresponding sums, their calculation is performed on a computer using the method of trapeziums or rectangles, for example:

To determine the amplitudes of higher harmonics with sufficient accuracy (δ≤1%), the number of integration steps must be at least 100k, where k is the harmonic number.

In technology, special devices called harmonic analyzers are used to isolate individual harmonics from non-sinusoidal voltages and currents.

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), (*)

We write this series in the form of a sum of simple harmonics, setting the coefficients equal to 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 will take 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 in the form 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 (*) takes the form

f(x) = .

Definition 9... Periodic function representation operation f(x) the Fourier series is called harmonic analysis.

The expression (*) also occurs 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 vibration theory and spectral analysis, the function representation f(t) in the Fourier series is written in the form:

(***)

those. the periodic function is represented by the sum of terms, each of which is a sinusoidal oscillation with an amplitude C n and the 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. The values C n and j n must be properly selected in order for the equality (***) to hold, that is, determined by the formulas (**) [ C n = A n].

We rewrite the Fourier series (***) as where w 1 - fundamental 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 bears the name phase spectrum.

When they simply say “spectrum,” they mean exactly the amplitude spectrum; in other cases, they make 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 displayed graphically. For this we choose the coordinates C n and w = nw one . The spectrum will be displayed in this coordinate system as a set of discrete points, since every 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 linear spectrum. He is a harmonic spectrum, i.e. consists of equally spaced spectral lines; harmonic frequencies are in simple multiples. Individual harmonics, including the first, may be absent, i.e. their amplitudes can be 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. For this, it is necessary to apply the passage to the limit at Т®∞, considering the non-periodic function as the limiting case of the periodic one with an unboundedly increasing period. Instead of 1 / T we 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 T® ∞, then w 1 ® dw and 2p n/T® w, where w- current frequency, changing continuously, dw- its increment. In this case, the Fourier series transforms into a Fourier integral, which is an 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 continuous or continuous spectra, i.e. instead of individual points, the spectrum is shown as a continuous curve. This is obtained as a result of the passage to the limit from the series to the Fourier integral: the intervals between individual spectral lines contract indefinitely, the lines merge, and instead of discrete points, the spectrum is depicted as 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.