Section II. Mathematical analysis
E. Yu. Anokhina
HISTORY OF DEVELOPMENT AND FORMATION OF THE THEORY OF THE FUNCTION OF A COMPLEX VARIABLE (TFV) AS A SUBJECT
One of the complex mathematical courses is the TFKT course. The complexity of this course is due, first of all, to the diversity of its interrelationships with other mathematical disciplines, historically expressed in the broad applied orientation of the science of TFKT.
In the scientific literature on the history of mathematics, there is scattered information about the history of the development of the TFCT, they require systematization and generalization.
For this reason, the main goal of this article is short description development of the TFKP and the formation of this theory as an educational subject.
As a result of the study, the following three stages in the development of the TFCT as a science and academic subject were identified:
The stage of emergence and recognition of complex numbers;
Accumulation stage actual material by functions of imaginary quantities;
The stage of formation of the theory of functions of a complex variable.
The first stage in the development of the TFKP (mid-16th century - 18th century) begins with the work of G. Cardano (1545), who published Artis magnae sive de regulis algebraitis (Great Art, or on algebraic rules). The work of G. Cardano had the main task of substantiating the general algebraic methods for solving equations of the third and fourth degrees, recently discovered by Ferro (1465-1526), Tartaglia (1506-1559) and Ferrari (1522-1565). If the cubic equation is reduced to the form
x3 + px + q = 0,
and should be
When (p^Ap V (|- 70) the equation has three real roots, and two of them
are equal to each other. If then the equation has one real and two co-
spun complex roots. Complex numbers appear in the final result, so G. Cardano could do as they did before him: declare the equation to have
one root. When (<7 Г + (р V < (). тогда уравнение имеет три действительных корня. Этот так
The so-called irreducible case is characterized by one feature that was not encountered until the 16th century. The equation x3 - 21x + 20 = 0 has three real roots 1, 4, - 5 which is easy
check with a simple substitution. But ^du + y _ ^20y + ^-21y _ ^ ^ ^; therefore, according to the general formula, x = ^-10 + ^-243 -^-10-4^243 . Complex, i.e. "false", the number is not the result here, but an intermediate term in the calculations that lead to the real roots of the equation in question. G. Cardano encountered a difficulty and realized that in order to preserve the generality of this formula, it is necessary to abandon the complete disregard for complex numbers. J. D'Alembert (1717-1783) believed that it was this circumstance that made G. Cardano and the mathematicians who followed this idea become seriously interested in complex numbers.
At this stage (in the 17th century), two points of view were generally accepted. The first point of view was expressed by Girard, who raised the issue of recognizing the need for unrestricted use of complex numbers. The second - Descartes, who denied the possibility of interpreting complex numbers. Opposite to the opinion of Descartes was the point of view of J. Wallis - about the existence of a real interpretation of complex numbers was ignored by Descartes. Complex numbers began to be “forced” to be used in solving applied problems in situations where the use of real numbers led to a complex result, or the result could not be obtained theoretically, but had a practical implementation.
The intuitive use of complex numbers led to the need to preserve the laws and rules of arithmetic of real numbers on the set of complex numbers, in particular, there were attempts at direct transfer. This sometimes led to erroneous results. In this regard, questions about the justification of complex numbers and the construction of algorithms for their arithmetic have become topical. This was the beginning of a new stage in the development of the TFCT.
The second stage in the development of the TFKP (the beginning of the 18th century - the 19th century). In the XVIII century. L. Euler expressed the idea of the algebraic closure of the field of complex numbers. The algebraic closure of the field of complex numbers C led mathematicians to the following conclusions:
That the study of functions and mathematical analysis in general acquire their proper completeness and completeness only when considering the behavior of functions in the complex domain;
It is necessary to consider complex numbers as variables.
In 1748, L. Euler (1707-1783) in his work "Introduction to the analysis of infinitesimals" introduced a complex variable as the most general concept of a variable, using complex numbers when decomposing functions into linear factors. L. Euler is rightfully considered one of the creators of the TFCT. In the works of L. Euler, elementary functions of a complex variable were studied in detail (1740-1749), conditions for differentiability (1755) and the beginning of the integral calculus of functions of a complex variable (1777) were given. L. Euler practically introduced the conformal mapping (1777). He called these mappings "similar in a small way", and the term "conformal" was first used, apparently, by the St. Petersburg academician F. Schubert (1789). L. Euler also gave numerous applications of functions of a complex variable to various mathematical problems and laid the foundation for their application in hydrodynamics (17551757) and cartography (1777). K. Gauss formulates the definition of an integral in the complex plane, an integral theorem on the expansion of an analytic function into a power series. Laplace uses complex variables to calculate difficult integrals and develops a method for solving linear, difference and differential equations known as the Laplace transform.
Starting from 1799, papers appear in which more or less convenient interpretations of the complex number are given and actions on them are defined. A fairly general theoretical interpretation and geometric interpretation was published by K. Gauss only in 1831.
L. Euler and his contemporaries left a rich heritage to posterity in the form of accumulated, somewhere systematized, somewhere not, but still scattered facts on the TFCT. We can say that the factual material on the functions of imaginary quantities, as it were, required its systematization in the form of a theory. This theory has begun to take shape.
The third stage of the formation of the TFKP (XIX century - XX century). The main achievements here belong to O. Cauchy (1789-1857), B. Riemann (1826-1866), and K. Weierstrass (1815-1897). Each of them represented one of the directions of development of the TFKP.
The representative of the first direction, which in the history of mathematics was called "the theory of monogenic or differentiable functions", was O. Cauchy. He formalized disparate facts on the differential and integral calculus of functions of a complex variable, explained the meaning of the basic concepts and operations with imaginary ones. In the works of O. Cauchy, the theory of limits and the theory of series and elementary functions based on it are stated, a theorem is formulated that completely elucidates the region of convergence of a power series. In 1826, O. Cauchy introduced the term: deduction (literally: remainder). In writings from 1826 to 1829, he created the theory of deductions. O. Cauchy deduced the integral formula; obtained an existence theorem for the expansion of a function of a complex variable into power series (1831). O. Cauchy laid the foundations for the theory of analytic functions of several variables; determined the main branches of multi-valued functions of a complex variable; first used plane cuts (1831-1847). In 1850 he introduces the concept of monodromic functions and singles out the class of monogenic functions.
O. Cauchy's follower was B. Riemann, who also created his own "geometric" (second) direction of development of the TFCT. In his works, he overcame the isolation of ideas about functions of complex variables and formed new departments of this theory, closely related to other disciplines. Riemann made an essentially new step in the history of the theory of analytic functions, he proposed to associate with each function of a complex variable the idea of mapping one region onto another. He distinguished between the functions of a complex and a real variable. B. Riemann laid the foundation for the geometric theory of functions, introduced the Riemann surface, developed the theory of conformal mappings, established the connection between analytic and harmonic functions, introduced the zeta function into consideration.
Further development of TFKP took place in another (third) direction. The basis of which was the possibility of representing functions by power series. This trend has been given the name “analytical” in history. It was formed in the works of K. Weierstrass, in which he brought to the fore the concept of uniform convergence. K. Weierstrass formulated and proved a theorem on the legality of reducing similar terms in a series. K. Weierstrass obtained a fundamental result: the limit of a sequence of analytic functions that converges uniformly inside a certain domain is an analytic function. He was able to generalize Cauchy's theorem on power series expansion of a function of a complex variable and described the process of analytic continuation of power series and its application to the representation of solutions to a system of differential equations. K. Weierstrass established the fact of not only the absolute convergence of the series, but also the uniform convergence. The Weierstrass theorem appears on the expansion of an entire function into a product. He lays the foundations for the theory of analytic functions of many variables, builds the theory of divisibility of power series.
Consider the development of the theory of analytic functions in Russia. Russian mathematicians of the XIX century. for a long time they did not want to devote themselves to a new field of mathematics. Despite this, we can name several names for whom she was not alien, and list some of the works and achievements of these Russian mathematicians.
One of the Russian mathematicians was M.V. Ostrogradsky (1801-1861). About M.V. Little is known about Ostrogradsky in the field of the theory of analytic functions, but O. Cauchy spoke with praise of this young Russian scientist, who applied integrals and gave new proofs of formulas and generalized other formulas. M.V. Ostrogradsky wrote the work "Remarks on Definite Integrals", in which he derived the Cauchy formula for the deduction of a function with respect to the n-th order pole. He outlined the applications of residue theory and Cauchy's formula to the calculation of definite integrals in an extensive public lecture course given in 1858-1859.
A number of works by N.I. Lobachevsky, which are of direct importance for the theory of functions of a complex variable. The theory of elementary functions of a complex variable is contained in his work "Algebra or calculation of finite" (Kazan, 1834). In which cos x and sin x are defined initially for real x as real and
imaginary part of the function ex^. Using the previously established properties of the exponential function and power expansions, all the main properties of trigonometric functions are derived. By-
Apparently, Lobachevsky attached particular importance to such a purely analytical construction of trigonometry, independent of Euclidean geometry.
It can be argued that in the last decades of the XIX century. and the first decade of the 20th century. fundamental research in the theory of functions of a complex variable (F. Klein, A. Poincaré, P. Kebe) consisted in the gradual elucidation of the fact that Lobachevsky's geometry is, at the same time, the geometry of analytic functions of one complex variable.
In 1850, Professor of St. Petersburg University (later Academician) I.I. Somov (1815-1876) published the Foundations of the Theory of Analytic Functions, which were based on Jacobi's New Foundations.
However, the first truly “original” Russian researcher in the field of the theory of analytic functions of a complex variable was Yu.V. Sokhotsky (1842-1929). He defended his master's thesis "Theory of integral residues with some applications" (St. Petersburg, 1868). From the autumn of 1868 Yu.V. Sokhotsky taught courses on the theory of functions of an imaginary variable and on continued fractions with applications to analysis. Master's thesis Yu.V. Sokhotsky is devoted to applications of the theory of residues to the inversion of a power series (Lagrange series) and, in particular, to the expansion of analytic functions into continued fractions, as well as to the Legendre polynomials. In this paper, the famous theorem on the behavior of an analytic function in a neighborhood of an essential singular point is formulated and proved. In Sokhotsky's doctoral dissertation
(1873) for the first time the concept of an integral of Cauchy type is introduced in an expanded form: *r/ ^ & _ where
a and b are two arbitrary complex numbers. The integral is supposed to be taken along some curve (“trajectory”) connecting a and b. In this work, a number of theorems are proved.
A huge role in the history of analytic functions was played by the works of N.E. Zhukovsky and S.A. Chaplygin, who opened up a boundless area of its applications in aero- and hydromechanics.
Speaking about the development of the theory of analytic functions, one cannot fail to mention the studies of S.V. Kovalevskaya, although their main meaning lies outside this theory. The success of her work was due to a completely new formulation of the problem in terms of the theory of analytic functions and the consideration of time t as a complex variable.
At the turn of the XX century. the nature of scientific research in the field of the theory of functions of a complex variable is changing. If earlier most of the research in this area was carried out in terms of the development of one of the three directions (the theory of monogenic or differentiable Cauchy functions, Riemann's geometric and physical ideas, the analytical direction of Weierstrass), now the differences and the controversies associated with them are being overcome, appearing and growing rapidly. the number of works in which a synthesis of ideas and methods is carried out. One of the basic concepts on which the connection and correspondence between geometric representations and the apparatus of power series was clearly revealed was the concept of analytic continuation.
At the end of the XIX century. The theory of functions of a complex variable includes an extensive complex of disciplines: the geometric theory of functions based on the theory of conformal mappings and Riemann surfaces. We received an integral form of the theory of various types of functions: integer and meromorphic, elliptic and modular, automorphic, harmonic, algebraic. In close connection with the last class of functions, the theory of Abelian integrals has been developed. The analytic theory of differential equations and the analytic theory of numbers adjoined this complex. The theory of analytic functions established and strengthened links with other mathematical disciplines.
The wealth of interrelations between the TFCT and algebra, geometry and other sciences, the creation of the systematic foundations of the science of the TFCT itself, its great practical significance contributed to the formation of the TFCT as an academic subject. However, simultaneously with the completion of the formation of the foundations, new ideas were introduced into the theory of analytic functions, significantly changing its composition, nature and goals. Monographs appear containing a systematic exposition of the theory of analytic functions in a style close to axiomatic and also having educational purposes. Apparently, the significance of the results on the TFCT, obtained by scientists of the period under review, prompted them to popularize the TFCT in the form of lecturing and publishing monographic studies in a teaching perspective. It can be concluded that the TFCT appeared as a learning
subject. In 1856, Ch. Briot and T. Bouquet published a small memoir "Investigation of the Functions of an Imaginary Variable", which is essentially the first textbook. General concepts in the theory of the function of a complex variable began to be worked out in lectures. Since 1856, K. Weiersht-rass lectured on the representation of functions by convergent power series, and since 1861 - on the general theory of functions. In 1876, a special work by K. Weierstrass appeared: "On the theory of single-valued analytic functions", and in 1880 "On the doctrine of functions", in which his theory of analytic functions acquired a certain completeness.
Weierstrass's lectures served for many years as a prototype for textbooks on the theory of functions of a complex variable, which began to appear quite often since then. It was in his lectures that the modern standard of rigor in mathematical analysis was basically built and the structure that became traditional was singled out.
REFERENCES
1. Andronov I.K. Mathematics of real and complex numbers. M.: Education, 1975.
2. Klein F. Lectures on the development of mathematics in the XIX century. M.: ONTI, 1937. Part 1.
3. Lavrentiev M.A., Shabat B.V. Methods of the theory of functions of a complex variable. Moscow: Nauka, 1987.
4. Markushevich A.I. Theory of analytic functions. M.: State. publishing house of technical and theoretical literature, 1950.
5. Mathematics of the 19th century. Geometry. Theory of Analytic Functions / ed. A. N. Kolmogorova and A. P. Yushkevich. Moscow: Nauka, 1981.
6. Mathematical Encyclopedia / Chap. ed. I. M. Vinogradov. M.: Soviet encyclopedia, 1977. T. 1.
7. Mathematical Encyclopedia / Chap. ed. I. M. Vinogradov. M.: Soviet Encyclopedia, 1979. Vol. 2.
8. Young V.N. Fundamentals of the doctrine of number in the 18th and early 19th centuries. Moscow: Uchpedgiz, 1963.
9. Rybnikov K.A. History of mathematics. M.: Publishing House of Moscow State University, 1963. Part 2.
NOT. Lyakhova TOUCHING OF PLANE CURVES
The question of the tangency of plane curves, in the case when the abscissas of the common points are found from an equation of the form Рп x = 0, where Р x is some polynomial, is directly related to the question
on the multiplicity of the roots of the polynomial Pn x . In this article, the corresponding statements are formulated for the cases of explicit and implicit assignment of functions whose graphs are curves, and the application of these statements in solving problems is also shown.
If the curves that are graphs of the functions y \u003d f (x) and y \u003d cp x have a common point
M() x0; v0 , i.e. y0 \u003d f x0 \u003d cp x0 and tangents to the indicated curves drawn at the point M () x0; v0 do not coincide, then we say that the curves y = fix) and y - cp x intersect at the point Mo xo;
Figure 1 shows an example of the intersection of function graphs.
transcript
1 Laplace Transform Brief information The Laplace transform, which is widely used in circuit theory, is an integral transformation applied to functions of time f equal to zero at< L { f } f d F, где = + комплексная переменная Величина выбирается так, чтобы интеграл сходился Если функция f возрастает не быстрее, чем экспонента, то интеграл преобразования Лапласа сходится, если >It can be proved that if the Laplace integral converges for some value s, then it defines a function F that is analytic in the entire half-plane > s. The function F defined in this way can be analytically extended to the entire plane of the complex variable = +, except for individual singular points. Most often this continuation is carried out by extending the formula obtained in the calculation of the integral to the entire plane of the complex variable Function F, analytically continued to the whole complex plane, is called the Laplace image of the time function f or simply the image The function f with respect to its image F is called the original If the image F is known, then the original can be found using the inverse Laplace transform f F d for > direct, parallel to the ordinate axis The value is chosen so that in the half-plane R > there are no singular points of the function F Determination of the original from a known image is called the inverse Laplace transform and is denoted by the symbol f L ( F ) L 7
2 Consider some properties of the Laplace transform Linearity This property can be written as an equality L( f f ) L( f ) L( f ) Laplace transform of the derivative of the function df L( ) d df d F f d f 3 Laplace transform of the integral: L( f d) d f 8 f d d F df: d f f d d the equality still has the form of Ohm's law, but already for the images of voltage and current For the instantaneous voltage across the inductance, the relation d i u L takes place, d i e there is no direct proportionality Ohm's law does not hold here After the Laplace transform, we get U = LI LI+
3 If, as is often the case, I + =, then the ratio takes the form U = LI Thus, Ohm's law is again valid for the images of voltage and current. The role of resistance is played by the value L, which is called the resistance of inductance. C After the Laplace transform, this ratio takes the form U I, C t e has the form of Ohm's law, and the capacitance is equal to C Let's make a table of the direct and inverse Laplace transformations of elementary functions encountered in the theory of circuits. at Laplace transform of this function will be L ( ) L ( ) d d 3 L ( ) 4 L ( ) 5 L (sin ) 9
4 3 6 ) (cos L 7 ) ( ) sin ( L L ) ( L 8 ) cos ( L 9 ) ( F d f f L! n d n n n n L! n n n L m< n и знаменатель имеет только простые корни Тогда n n K K K B, где, n корни полинома B, стоящего в знаменателе изображения Коэффициенты K, K, K n могут быть найдены следующим
5 3 Let's decompose the image into simple fractions and multiply by: n n K K K K B Now we tend to Then only K remains on the right side: lim B K is known: L Therefore " n B B L Of interest is the special case when one of the roots of the denominator is equal to zero: B F In this case, the expansion of F into simple fractions will have the form, as follows from the previous one, " n B B B and B has no roots at zero
6 3 From here, the inverse Laplace transform of the function F will have the form: n B B B " L Let's consider one more case when the polynomial in the denominator B has multiple roots. Let m< n и корень кратности l При разложении на простые дроби этому корню соответствует сумма: l l l K K K Обратное преобразование слагаемых этой суммы мы уже имели выше см п:! n n n L Таким образом, обратное преобразование суммы будет иметь вид: M, где M полином от степени l
7 Some general properties of circuits Let a complex circuit contain P branches and Q nodes Then, according to the first and second Kirchhoff laws, P + Q equations can be made for P currents in the branches and Q nodal potentials One of the Q nodal potentials is assumed to be zero But the number of equations can be reduced on Q, if we use the loop currents as alternating currents. In this case, the first Kirchhoff law is automatically satisfied, since each current enters and exits the node, that is, it gives a total current equal to zero, and, in addition, the Q nodal potentials are expressed in terms of the loop currents The total number of equations, and therefore, independent circuits becomes equal to P + Q Q = P Q + Independent equations can be drawn up directly if we take the circuit currents as unknowns. Independent circuits will be such, each of which contains at least one branch that is not included in any one of the other contours Fig. For each of the contours, equations are compiled according to the second Kirchhoff law a In the general case, the resistance of the branch is i R i C i L where i, =, n, n is the number of independent circuits. The equations of the circuit currents are: I I n I n E; I I n I n E; ni n I nn I n En i, Here E i is the sum of all emfs included in i-th circuit Resistances with the same indices ii are called intrinsic resistances of the i-th circuit, and resistances with different indices i are called mutual resistances, or connection resistances of the i-th and -th circuits. Resistances ii are the sum of the resistances included in the i-th circuit. Resistance i is part of the resistance. i-th 33 Fig Example of independent contours
8 The equation for the m-th circuit will look like: a circuit that is also included in the -th circuit Obviously, the equality i = i is valid for a passive circuit Let's consider how the equations of circuit currents change for active circuits containing transistors, Fig I i Transferring the second term from the right side to the left side, we transform this equation as follows: mi mi I i mn I n Em of unknowns, nodal potentials are also used, counted from the potential of one of the nodes, taken as zero Instead of EMF generators, current generators are used. Y which can be rewritten as follows: where Fig Equivalent circuit of a transistor in a complex circuit U YU U YnU U n I, Y U Y U Y nu n I, Y Y Y Y n
9 The system of equations for nodal potentials has the form Y U YU Y nu n I; YU YU Y nu n I; Yn U Yn U YnnU n In where Y i is the conductance of the connection of the i-th and -th nodes: It is obvious that Y i G i L i Yi Y i C This symmetry disappears if the circuit contains transistors, lamps or other active elements, the equivalent circuit which contains dependent current sources Let us now consider the solutions of the circuit equations The solution of the system of equations of the loop currents has the form for the -th current: I, where the main determinant of the system, the same determinant, in which the -th column is replaced by electromotive forces from the right parts E, E, E n Suppose that there is only one EMF E in the circuit, included in the input circuit, which is assigned the first number. The equations must be drawn up in such a way that only one circuit current passes through the branch of interest to us. determinant i Fig 4 Circuit with EMF in the input circuit 35
10 The ratio E I is called the input resistance. In contrast, this resistance takes into account the influence of all circuits. For the second output circuit, we will have I 36 E, where the corresponding algebraic addition. The ratio T I E is called the transfer resistance from the first circuit to the second. Similarly, from the nodal potential equations, you can get the input conductivity fig. 5 Fig 5 Circuit with a current source at the input "U I" I, Y "Y" and the transmission conductance from the first node to the second: U " I " I Y T, Y T " " where I is the current supplied to the first node, U and U are the voltages, obtained at the first and second nodes, " the main determinant of the system of equations of nodal potentials, and " i is the corresponding algebraic complement Between and Y there is a relation Y For a passive circuit, we had = Therefore, the main determinant of the system is symmetrical. It follows that and algebraic additions are equal: = Therefore, the transmission resistances are also equal T = T This property is called the reciprocity property The reciprocity condition, as we see, is the symmetry of the resistance matrix The reciprocity property is formulated as follows Figure 6: if the EMF in the input circuit causes some current in the output circuit , then the same EMF, included in the output circuit, will cause in the input circuit,
11 current of the same value Briefly, this property is sometimes formulated as follows: EMF in the input circuit and the ammeter in the output circuit can be interchanged, while the ammeter reading will not change 7 U E Fig 7 Voltage transfer coefficient then As follows from the diagram in Fig. 7: U U I n; ; K n E T E ; I T U n Similarly, the current transfer coefficient I K I Fig. 8 can be determined: I Hence I U Yн I ; Y ; K n I YT I U Y T I Fig. 8 Current transfer ratio Yn Y T T 37
12 3 More about the general properties of circuit functions Circuit functions are functions of a variable obtained by solving equations, for example, input resistance conduction, resistance conduction transmission, etc. For circuits with lumped parameters, any circuit function is rational with respect to the variable and is a fraction m Ф B b n m n b m m n n 38 b b and the coefficients are real Otherwise, it can be represented as Ф b m n m, " " " where, m, ", ", " n are the roots of the equations m b n m n b m n m, n b b Ф It is obvious that two rational functions whose zeros and poles coincide can differ only by constant factors. In other words, the nature of the dependence of the circuit parameters on the frequency is completely determined by the zeros and poles of the circuit function. Since polynomials have real coefficients, when replaced by a conjugate value * the polynomial acquires the conjugate value * = * and B * = B * It follows that if the polynomial im has a complex root, then it will also be a root Thus, the zeros and poles of the chain function can be either real or make up complex conjugate pairs Let Ф be the chain function Consider its values for = : Ф Ф Ф F F n,
13 But F F F, F F F Comparing these equalities, taking into account the equality given above, we obtain that F F, F F, i.e. the real part of the circuit function is an even function of frequency, and the imaginary odd function of frequency an equality that defines the current in the input resistance caused by the voltage U: U I B Let U be a unit step, and Then I, B where and B are polynomials from Using the expansion formula, you can get i B B" where the zeros of the polynomial B and, therefore, the zeros of the resistance function and the zeros main determinant: = If at least one zero has a positive real part, then i will increase indefinitely
14 me The same conclusion can be drawn regarding the transmission resistance T, the input conductivity Y, the transmission conductivity Y T Definition A circuit function is called physically feasible if it corresponds to a circuit consisting of real elements, and none of the natural oscillations of which has an amplitude that increases indefinitely with time The circuit specified in the definition is called stable. Zeros of the main determinant of a physically feasible stable circuit function and, therefore, zeros of the resistance and conductivity functions, should be located only in the left half-plane of the variable or on the real frequency axis. If two or more zeros coincide, multiple roots, then the corresponding solutions have form: M, where M is a polynomial of degree m, m is the multiplicity of the root o coefficient e transmission, then all of the above applies not to zeros, but to the poles of the function of the transfer coefficient circuit. In fact: n K The zeros of T are the poles of the function K, and the load resistance n is passive; its zeros certainly lie in the right plane From the above it follows that the physically realizable chain functions have the following properties: and the zeros and poles of the chain function are either real or form complex conjugate pairs; b the real and imaginary parts of the circuit function are at real frequencies, respectively, even and odd functions of the frequency; into the zeros of the main determinant, and therefore, the conduction resistance and the transmission conduction resistance cannot lie in the right half-plane, and multiple zeros neither in the right half-plane nor on the real frequency axis T 4
15 3 Transients in amplifiers Solving the system of equations of the circuit gives an image of the output signal for a given input U = KE The function of the circuit in the time domain can be found using the inverse Laplace transform u L ( K E ) Of greatest interest is transition process with an input signal in the form of a step The reaction of the system to a single step is called the transition function Knowing transition function, you can find the response of the system to the input signal of arbitrary shape. lie to the right of the pole = Of great interest is the definition 3: h r K d K r r K r d d r r
16 4 Let us pass to the limit r Then we have d K V K K d K V h frequency response gain From this formula, we can draw some general conclusions Let's replace the variable in h with: d K V K h But h, as follows from the principle of causality, since the signal appears at imaginary part: K = K + K r Substituting into the expression for h, we get d K K V K r Differentiating with respect, we get d K K r or cos sin sin cos d K K K K r r
17 The imaginary part of the integrand is an odd function of frequency, therefore the integral of it is equal to zero Since the real part is an even function of frequency, the condition that the physically realized transfer coefficient must satisfy is: from the causality principle It can be shown that a system whose transmission coefficient can be written as a ratio of polynomials K, B is stable in the sense that all zeros of the polynomial B lie in the left half-plane, satisfies the causality principle. To do this, we study the integral K h d for< и >Let's introduce two closed contours and B, shown in Figure 3 Figure 3 Integration contours: at< ; B при > 43
18 44 Let us consider a function where the integral is taken along a closed contour. Due to Cauchy's integral theorem, the integral is equal to zero, since the integrand in the right half-plane is analytical by condition. The integral can be written as a sum of integrals over individual sections of the integration contour: sin cos R r R r r R R d R R K r d r r K d K d K h< < /, то при < последний интеграл стремится к нулю при R т е h h при R Отсюда следует что h при < Рассмотрим функцию где интеграл берется по контуру B Здесь R вычеты подынтегральной функции относительно полюсов, лежащих в левой полуплоскости Аналогично предыдущему можно показать, что при >holds h B h for R Thus: R h, for >
19 The residue with respect to a simple pole is equal to R B" which we already had earlier K lim, 45 lim B where RC Let's prove that according to the causality condition given above, the equality must be satisfied. The equality cos sin d cos d is known. Differentiate the right and left parts by: sin d Multiplying the left and right parts of this equality by, we get: sin d from which the equality that needs to be proved follows. Having the transition function of the system, you can find its response to any input signal. To do this, we will approximately represent the input signal as a sum of unit steps Fig. 34
20 Fig 34 Presentation input signal This representation can be written as: u u u Next, u u "The response to a unit step will be equal to h Therefore, the output signal can be approximately represented as: u u h u" h Passing to the limit at, instead of the sum, we obtain the integral u u h u" h d by parts, you can get another form of the Duhamel integral: u u h u h" d And finally, using the change of variable = ", you can get two more forms of the Duhamel integral: u u h u" h d ; u u h u h" d 46
21 4 Some properties of two-pole circuits 4 General properties of the input conduction resistance function Two-terminals are completely characterized by the function of the input conduction resistance This function cannot have zeros in the right half-plane, as well as multiple zeros on the real frequency axis Since Y, then the zeros of Y correspond to the poles and vice versa Therefore the function of the input conductance resistance cannot also have poles in the right half-plane and multiple poles on the real frequency axis Passive two-pole networks are always stable, since they do not contain energy sources The expression for the input conductance resistance is: m b n m n b m n 47 m n b b the following asymptotic equality holds: b m mn lizi = similarly, it can be shown that the smallest exponents of the numerator and denominator cannot differ by more than one. The physical meaning of these statements is that at very high and very low frequencies, a passive two-terminal network should behave like a capacitance or inductance or active resistance n, 4 Energy functions of a two-terminal network Suppose that a two-terminal network is some complex circuit containing active resistances, capacitances and inductors.
If a sinusoidal voltage is applied to the terminals of a two-terminal network, then some power is dissipated in the two-terminal network, the average value of which P characterizes the energy dissipation. Electric and magnetic energies are stored in capacitances and inductances, the average values of which are denoted by W E and W H. We calculate these quantities using the equations of loop currents Directly, we write the expressions for the above quantities by analogy with the simplest cases. So, for the resistance R, the average power dissipation is P R I I Similarly, for a circuit containing several branches, average power can be expressed in terms of loop currents: P i R i I i I The average energy stored in the inductance is W H L I I For a complex circuit, this value is expressed in terms of loop currents: W H 4 i L i I 4 I I C 48
23 Based on this relationship, we can write an expression for the total average electrical energy: W E 4 Ii I i Ci Let's find out how these quantities are related to input voltages and currents To do this, we write the equations of the loop currents I R I L I E ; C I R i I Li I ; Ci Multiply each of the equations by the corresponding current 49 Ii and add all I Ii Ri I Ii Li I Ii EI i i i Ci If R i = R i ; L i = L i ; C i = C i, that is, the circuit satisfies the principle of reciprocity, and there are no active elements, then: i i i R I I P ; i i L I I 4W ; i I I i E i Ci H 4 W functions
24 Tellagen's theorem allows you to find expressions for resistance and conductivity Y in terms of energy functions: E I E I I I I I E Y E E E 5 P WH W I I P WH W E E is zero only if there are no energy losses in the circuit. The stability conditions require that both Y have no zeros and no poles in the right half-plane. The absence of poles means that Y are also analytic functions in the right half-plane In the theory of functions of a complex variable, there is a theorem that if a function is analytic in a certain region, then its real and imaginary parts reach their smallest and largest values at the boundary of the region. Since the functions of the input resistance and conductivity are analytic in the right half-plane, then their real part at the boundary this region on the real frequency axis reaches the smallest value But on the real frequency axis the real part is non-negative, therefore, it is positive in the entire right half-plane. In addition, the functions and Y take real values for real values, since they represent the quotient of dividing polynomials with real coefficients A function that takes real values at real values and has a positive real part in the right half-plane is called a positive real function. The input resistance and conductance functions are positive real functions. the function was a positive real function 3 The imaginary part on the real frequency axis is equal to zero if the two-terminal network does not contain reactive elements or the average reserves of magnetic and E E ;
25 electrical energies in a two-terminal network are the same. This takes place at resonance; the frequency at which this occurs is called the resonant frequency. It should be noted that in deriving the energy relations for and Y, the reciprocity property was essentially used. absence of dependent sources. For circuits that do not satisfy the reciprocity principle and contain dependent sources, this formula may turn out to be incorrect. As an example, on Figure 4 shows a diagram of a series resonant circuit. Let's see what the energy formula gives in this simplest case. The power dissipated in the resistance R when current I flows is equal to P I R. The average reserves of electrical and magnetic energies are: W H L I C U ; W E The voltage U across the capacitance when current I is flowing is Hence W E I U C I C Substituting into the energy in the formula for, we get L I I R I Fig 4 Series resonant circuit I C R L C as you would expect for a series circuit
26 Here E E C C S I S E R R RC RC C C Let, S >> C so that the first term in brackets can be neglected S is the slope of the lamp Then the Input resistance will then be S I E RC E RC I S S RC where Req; Leq S S in the dependent source circuit Picking up in the circuit control grid required phase shift, it is possible to obtain an inductive or capacitive phase shift between the voltage and current at the input and, accordingly, the inductive or capacitive nature of the input resistance frequencies It can be equal to zero identically for any frequencies only if all elements of the circuit have no losses, that is, they are purely reactive But even if there are losses, the real part of the resistance or conductivity can vanish at some frequencies 5
27 If it does not vanish anywhere on the imaginary axis, then a certain constant value can be subtracted from the resistance or conductivity function without violating the conditions of physical feasibility so that the real part, remaining non-negative, vanishes at a certain frequency Since the conductivity resistance function does not have poles in the right half-plane of the variable, i.e., is analytical in this region, then its real part has a minimum value at its boundary, i.e., on the imaginary axis. Therefore, subtracting this minimum value leaves the real part positive in the right half-plane. -active resistance of conduction, if its real part vanishes on the real frequency axis, so that a decrease in this component is impossible without violating the passivity conditions. Since the real part of a minimally active circuit vanishes, simultaneously reaching a minimum, then the zero of the real part on the real frequency axis has a multiplicity of at least Example Figure 43 shows the simplest circuits that we analyze for the minimum active conduction resistance R C R C R L R C R C a b c d Fig 43 Circuits: minimum active conductivity a, minimum active resistance b , c and non-minimally active type d In Fig. 43, a, the circuit has an input resistance of a non-minimally active type, since the real part of the resistance does not vanish at any real frequency. At the same time, the real part of the conductivity vanishes at frequency = Therefore, the circuit is a circuit of minimally active conductivity In Fig. 43, b, the circuit is a circuit of minimally active resistance, since the real part of the resistance vanishes at an infinite frequency 53
28 In Fig. 43, c is a circuit of minimum active resistance R = at the resonance frequency of the series circuit. In Fig. 43, d, the circuit is non-minimum active. the circuit in the 3rd circuit has a finite resistance at the resonance frequency such two-terminal networks under certain conditions can be unstable. Consider the possibilities available here. The resistance has zeros in the right half-plane of the variable, but has no poles there. place exponentially growing solutions, i.e. two-pole nick is unstable when powered from an EMF source, or, otherwise, when its terminals short circuit , i.e. the real frequency axis This minimum is negative, since otherwise it would be a positive real function and could not have zeros in the right half-plane The minimum of the real part on the real frequency axis can be increased to zero by adding a positive real resistance In this case, the function + R becomes a positive real function. Therefore, a two-terminal network with the addition of resistance R will be stable in the event of a short circuit.
29 Conductivity Y has zeros in the right half-plane, but has no poles there. This is the opposite case to the previous one, since it means that = /Y has poles in the right half-plane, but has no zeros there. In this case, the stability is investigated in the circuit with a current source Fig. 45, a If Y has zeros in the right half-plane, then the two-terminal network is unstable at idle. Further, the above reasoning can be applied. Since Y has no poles in the right half-plane, then the function Y can be made a real positive function by adding a positive real conductivity G Gmin Thus Thus, a two-terminal network, in which the conductivity Y has zeros in the right half-plane, but has no poles there, can be made stable by adding a sufficiently large real conductivity. from the voltage source 3 The function has zeros and poles in the right half-plane In this case, for resolving the issue of stability requires special consideration. So, we can draw the following conclusions: if an active two-terminal network is stable when powered from a current source, it does not have poles in the right half-plane, then it can be made stable when powered from a voltage source by connecting in series some positive material resistance; if an active two-terminal network is stable when powered from a voltage source Y does not have poles in the right half-plane, then it can be made stable when powered from a current source by connecting a sufficiently large real conductivity in parallel. Example Consider a parallel connection of a negative resistance R with a capacitance C fig 46 R C R C I 55 Y b G Fig 45 Two-terminal networks: a with a current source; b with the addition of conductivity Y Y Fig 46 Two-terminal network with negative resistance I
30 As you can see, it has no zeros in the right half-plane, so such a circuit is stable when powered by a voltage source But it is unstable at idle Let's add the inductance L in series Then Fig 47 Equivalent circuit of the tunnel diode R R L LCR L RC RC This function has zeros in the right half-plane: , RC 4 RC LC Therefore, the circuit is unstable when powered from a voltage source But it also has a pole in the right half-plane Let's try to make it stable by adding some resistance R in series Figure 47 Then R LCR RRC L R R L R RC RC The stability condition is the absence of zeros of the numerator in the right half-plane To do this, all coefficients of the trinomial in the numerator must be positive: RR C L ; R R These two inequalities can be written as: L CR R R Obviously, such inequalities are possible if L L R or R RC C R under the condition R The circuit in Fig. 47 is the equivalent C tunnel diode circuit. Therefore, the found condition is the condition 56
Fig. 48 Let us find the conditions for the stability of the circuit at idle. To do this, calculate the conductivity: Y R R C L 57 LC L R L o o th R or R > R o When the reverse inequality is fulfilled, self-oscillations are excited in the circuit at the frequency of the resonant circuit. some limits without violating the conditions of passivity Physically, this change in the real component by a constant value means the addition or exclusion of real active resistance, ideally independent of frequency Change in the reactive component of the resistance function n conductivity by a constant value is unacceptable, since this violates the conditions of physical feasibility; possible in the case when the conduction resistance has poles on the real frequency axis Due to the conditions of physical feasibility, such poles must be simple and complex conjugate
32 Let the resistance have poles at frequencies Then we can distinguish simple fractions M N B B It is easy to see that N N M M N r M B r 58 B * M, M M sign, which contradicts the conditions of physical realizability Therefore, M r = N r = Then M = N In addition, it can be shown that M = N > Indeed, we set = +, and > Then the fraction takes the value M/, which must be greater than zero, since the fraction must be a real positive function in the right half-plane So, M = N > Thus, if it has complex conjugate poles on the real frequency axis, then it can be represented as: M M, B and satisfies the conditions of physical feasibility, if they are satisfied Real , has no poles in the right half-plane, since it has no poles there. Therefore, it is an analytic function in the right half-plane. On the other hand, the first term takes on axes of real frequencies are purely imaginary values Therefore, and have the same real parts on the axes of real frequencies The selection of the first term does not affect the real part on the axes of real frequencies It follows that is also a positive function of r in the right half-plane
33 In addition, it takes real real values in the right half-plane at real values Therefore, is a real positive function M The resistance has a parallel lossless resonant circuit: L C C C, L C LC with LC and M C Similar reasoning can be carried out for the conductivity function Y, which has poles at points ± : M " Y, Y M " where the expression is the conductivity of the series resonant circuit: Y C L L C L e correspond to capacitance or inductance The following statement is true
34 subtract from it the conductance reactance corresponding to the poles located on the real frequency axis. The conduction resistance, in which all poles are removed in this way, is called the conduction resistance of the minimum reactive type. poles of resistance and conductivity at any real frequencies The presence of such poles would mean the possibility of the existence of free oscillations in them without damping But in many cases, with a good approximation, losses in reactive elements can be neglected. of elements with low losses In this case, the influence of losses can sometimes be neglected It is of interest to find out the properties of lossless circuits, and also to find out under what conditions it is possible to neglect losses Assume that all elements of the circuit are purely reactive It is easy to show that in this case, on the axis of real frequencies, the resistance and conductivity Y take imaginary values. Indeed, in this case, the power loss is zero, therefore: W I 6 H WE W Y E WE ; Since the imaginary part of the resistance or conductivity is an odd function of the circuit, then in this case = Therefore, and in the more general case = The conditions of physical feasibility require that it does not have zeros and poles in the right half-plane But since =, then there should also be no zeros and poles in the left half-plane Therefore H
35 functions and Y can have zeros and poles only on the real frequency axis. Physically, this is understandable, since in a lossless circuit free vibrations do not decay It follows that using the method of selecting poles lying on the axis of real frequencies, it is possible to reduce the functions and Y to the following form: b n b n b Y Figure 49 Foster's first form Accordingly, Y can be represented as the -th Foster form Figure 4 Figure 4 Foster's second form It can be shown that zeros and poles on the real frequency axis must alternate In fact, since zeros and poles on the real frequency axis can be only simple, then near zero the function can be represented in the form M o, where o is a value of higher order of smallness compared to . Near in the right half-plane, the real value must be positive, and this is possible only if M is real
36 value, and M > Therefore, near zero = the imaginary component can only change with a positive derivative, changing the sign from to "+" Further, it will be shown that for a circuit composed of purely reactive elements, the indicated derivative is positive for any frequencies. Therefore, between two adjacent zeros there must be a discontinuity, which for circuits with lumped elements can only be a pole. All of the above also applies to conductivity Y Zeros are called resonance points, poles are antiresonance points Therefore, resonances always alternate with antiresonances For conductivity Y, resonances correspond to poles, and antiresonances to zeros It is easy to see , that both at resonance points and at antiresonance points, the average reserves of electric and magnetic energies are equal to each other Indeed, at resonance points =, t e W H W E = At antiresonance points Y =, therefore, W E W H = losses, the following formulas take place, I give dx WH W d I db WH WE d E Let's consider the definition of resistance E I 6 E ; Let E = cons Differentiate with respect to frequency: d E di d I d Assume that E is a real value Then for a lossless circuit I is a purely imaginary value In this case d E d I di d I I and
37 Let us now turn to the system of equations for loop currents n 4: I Li I Ei, i, n C Assuming that only E, we multiply each of the equations by and add all the equations: i, i I di i Li I di i E di, i, C i, Next, we turn to the relation also obtained in paragraph 4 for lossless circuits: i, L i I Ii i i, I I C i i E i, Ci i, I di di I L di I E di C C i i i i i, i i, i, i di I di I L di I L di I n i i i i i i i, i, Ci i, i, Ci E di E di, since E is by assumption a real value It also follows from the above that: i, L I i di i i, IdI C i i E di di i 63
38 Substituting in the total sum, we get: d i, L i I Ii i, I I C i i E di E Reducing similar terms on the left and right, we find: di I Ii E di d Li I Ii i, i, Ci E The expression in brackets, as was found in Section 4, is equal to i, L i I Ii i, Ii I C i 4 W H W E di E WE From these formulas it follows that with increasing frequency, the reactance and conductivity of a circuit of purely reactive elements can only increase 4 Finally, we will try to find out how the presence of small losses affects the resistance of a circuit composed of reactive elements. When losses are introduced, attenuation appears. We will consider small losses that cause low attenuation that satisfies the condition /<<, <, где = + -й полюс сопротивления Это означает, что полюсы и нули сопротивления смещаются с оси вещественных частот на малую величину затухания H E 64
39 Attenuation can be different for different poles Therefore, it is advisable to consider the behavior of the resistance function near one of the poles. The shift of the pole by an amount to the left can be displayed by replacing the function of the variable with +. Then, near the pole, we will have
40 Since we are interested in the values on the real frequency axis, it should be replaced by In the numerator can be discarded, small compared to according to the condition: This expression can be transformed as follows:, Qx "where; Q; x; the value of x is called the relative detuning Near the resonance In addition, we have: The value C x Q Q ; ; Q Q C C is called the characteristic impedance of the resonant circuit Consider how the real and imaginary parts of the resistance near the resonance depend on the frequency: Q Q x R ; Im Q x Q x 66
41 Near the resonance, Im increases, but at resonance it passes through zero with a negative derivative The real part of R at resonance has a maximum Graphs of Im and R depending on the frequency are shown in Fig. 4 Note that R dx Q Q x dx, i.e. does not depend on the quality factor Otherwise speaking, the area under the resonant curve R does not depend on the quality factor As the quality factor increases, the width of the curve decreases, but the height increases, so that the area remains unchanged Qx >>, the real part decreases rapidly, and the imaginary part is equal to Im x 67, i.e. it changes in the same way as in the case of a lossless circuit
42 So, the dependence on frequency when introducing small losses changes little at frequencies that are far from the resonant frequency by a value \u003e\u003e\u003e\u003e\u003e\u003e\u003e\u003e Near the frequency, the course changes significantly gq Y, Qx g characteristic conductivity; L x Zero corresponds to the conduction pole Y Near zero, therefore, the resistance can be represented on the real frequency axis as follows: Qx x, Y gq Q where = /g Thus, near zero, the introduction of small losses affects the appearance of a small real component in the resistance Imaginary component varies near zero in the same way as before 68
43 5 Quadripoles 5 Basic equations of a quadripole A quadripole is a circuit that has two pairs of terminals: an input to which the signal source is connected, and an output to which the load is connected. transmission resistance Under these conditions, the resistance of the signal source n and the load resistance n are included in T. When they change, T also changes. It is desirable to have equations and parameters characterizing the quadripole itself. The coefficient is the reciprocal of the transfer conductivity at idle on the output pair of clamps: 69 I I ; Fig. 5 Turning on the quadripole I Here, U and U are the voltages at the input and output terminals, I and I are the currents flowing through the input and output terminals towards the quadripole, see Fig. 5 The coefficients of the system of equations relating voltages and currents have a simple meaning. I and U at current at the output terminals I =, i.e. at no load at the output terminals; in other words, this is the input resistance at idle at the output = x Similarly, this is the input resistance from the side of the output terminals at idle at the first pair of terminals = x current input terminals U and I Y T x Y T x
44 I U ; Y Tx Y Tx YT x I U x I YT x I, since the current in this case is directed from the quadripole, that is, in the opposite direction compared to the one adopted above Substituting U into the second equation, we get from where I, I n I x I YTx I Y x Tx Substituting I into the first equation, we get U I x Y Tx n From here we find the input resistance in n x U x I Y By analogy, we can also write the expression for the output resistance by swapping the indices and: T x n x 7
45 out х Y T х н х 5 Characteristic parameters of the two-port network Considerable interest is the case when the generator and the load are simultaneously matched, i.e., when n = c and n = c, the relation in = c and out = c takes place Substituting into the expressions for in and out , we obtain equations that allow us to find c and c: c c x x Y T x Y T x 7 c c This system is solved as follows From the first equation we find: whence c c x x; x, Y Tx c x x Y T x Equating c from the second equation, we have x Y Tx x YTx x c x kz c x kz x
46 Note that kz and kz are input resistances from the side of the first and second pair of terminals, respectively, in case of a short circuit on the other pair of terminals A load equal to the characteristic impedance c is called matched. For any number of quadripoles connected in this way, matching is preserved in any section. As the third characteristic parameter of a quadripole, the characteristic transfer coefficient g ln U U I ln rg I U I 7 U I is often used when the quadripole is connected to a matched load, i.e. to the characteristic impedance In this case, U c I; U I c I c ln I c U c g ln U get also the ratios: I g I ; U c g U U U I I
47 The characteristic transfer coefficient is convenient in that with a coordinated cascade connection of four-terminal networks, the resulting transfer coefficient is equal to the sum of the transmission coefficients of individual four-terminal networks. The characteristic transfer coefficient can be found from the relationships: The characteristic impedances c and c, generally speaking, depend on the frequency. Therefore, the use of characteristic parameters is not always convenient for representing the transmission resistance T. Thus, to study the characteristic coefficient g depending on the frequency, it is necessary to load the quadrilateral on the characteristic impedance, which also depends on the frequency. The most interesting connection is quadripole to a constant real load R with a purely active resistance of the generator R Fig. 53 In this case, the transmission is determined using the operating transmission coefficient U I ln, U I where U "and I" is ie and current that the generator is able to develop on a resistance equal to the internal resistance of the generator, that is: E U, I E, R 73 E U I, 4R U and I load voltage and current In this case, U \u003d I R Substituting, we get for the working transfer coefficient ln From here we get 4R E R I ln E R R T I R R
48 The value of the function of a complex variable For real frequencies = : = + B, where the operating attenuation, B is the phase constant allocated on the load R P mx E P I R 4R Let us show that the real positive function Indeed, since T has no zeros in the right half-plane, the function is analytic in the right half-plane Therefore, the analytic function proportional to it is also analytic in the right half-plane The modulus of the analytic function reaches its maximum value at the boundary of the domain analyticity, in this case on the real frequency axis The reciprocal value reaches the smallest value on this axis For a passive quadripole on the real frequency axis, therefore R > in the entire right half-plane Next T ln 4R R The function T is the quotient of dividing two polynomials with real coefficients, and T takes real positive e values for real Therefore, also real for real values Thus, we can conclude that a real positive function
4.11. Properties of the Laplace transform. 1) One-to-one correspondence: s(S ˆ(2) Linearity of the Laplace transform: s ˆ () ˆ 1(s2(S1 S2(, and also 3) Analyticity of S ˆ() : if s(satisfies
4 Lecture 5 ANALYSIS OF DYNAMIC CIRCUITS Plan Equations of state of electrical circuits Algorithm for the formation of equations of state 3 Examples of compiling equations of state 4 Conclusions Equations of state of electrical
4. Properties of the Laplace transform.) One-to-one correspondence: S ˆ() 2) Linearity of the Laplace transform: s (s () ˆ () ˆ 2 S S2(), and 3) Analyticity of S ˆ() : if satisfies condition
64 Lecture 6 OPERATOR METHOD OF ANALYSIS OF ELECTRIC CIRCUITS Plan Laplace transform Properties of Laplace transform 3 Operator method of analysis of electrical circuits 4 Definition of the original from the known
2.2. Operator method for calculating transient processes. Theoretical information. The calculation of transient processes in complex circuits by the classical method is very often difficult to find the constants of integration.
70 Lecture 7 OPERATOR FUNCTIONS OF CIRCUITS Plan Operator input and transfer functions Poles and zeros of functions of circuits 3 Conclusions Operator input and transfer functions The operator function of a circuit is called
Sinusoidal current "in the palm of your hand" Most of the electrical energy is generated in the form of EMF, which varies in time according to the law of a harmonic (sinusoidal) function. Harmonic EMF sources are
4 Lecture RESONANCE FREQUENCY CHARACTERISTICS OF ELECTRIC CIRCUITS Resonance and its significance in radio electronics Complex transfer functions 3 Logarithmic frequency characteristics 4 Conclusions Resonance and
Transitional processes "in the palm of your hand". You already know the methods for calculating a circuit that is in a steady state, that is, in one where the currents, as well as the voltage drops on individual elements, are unchanged in time.
Resonance in the palm of your hand. Resonance is the mode of a passive two-terminal network containing inductive and capacitive elements, in which its reactance is zero. Resonance Condition
Forced electrical oscillations. Alternating current Consider the electrical oscillations that occur when there is a generator in the circuit, the electromotive force of which changes periodically.
Chapter 3 Alternating current Theoretical information Most of the electrical energy is generated in the form of EMF, which varies over time according to the law of a harmonic (sinusoidal) function
Lecture 3. Deductions. The main residue theorem The residue of the function f () at an isolated singular point a is a complex number equal to the value of the integral f () 2 taken in the positive direction i along the circle
Electromagnetic oscillations Quasi-stationary currents Processes in an oscillatory circuit An oscillating circuit is a circuit consisting of an inductor connected in series, a capacitance capacitor C and a resistor
1 5 Electrical oscillations 51 Oscillating circuit In physics, oscillations are called not only periodic movements of bodies, but also any periodic or almost periodic process in which the values of one or more
Passive circuits Introduction The problems consider the calculation of amplitude-frequency, phase-frequency and transient characteristics in passive circuits. To calculate these characteristics, you need to know
STUDY OF FREE AND FORCED OSCILLATIONS IN AN OSCILLATORY CIRCUIT Free electrical oscillations in an oscillatory circuit
Lecture 3 Topic Oscillatory systems Sequential oscillatory circuit. Voltage resonance A series oscillatory circuit is a circuit in which a coil and a capacitor are connected in series.
Moscow State University M.V. Lomonosov Moscow State University Faculty of Physics Department of General Physics Laboratory practice in general physics (electricity and magnetism) Kozlov
Materials for self-study in the discipline "Theory of electrical circuits" for students of specialties: -6 4 h "Industrial electronics" (part), -9 with "Modeling and computer design
Complex amplitude method Harmonic voltage fluctuations at the terminals of the R or elements cause the flow of a harmonic current of the same frequency. Differentiation integration and addition of functions
Appendix 4 Forced electrical oscillations Alternating current The following theoretical information may be useful in preparing for laboratory work 6, 7, 8 in the laboratory "Electricity and Magnetism"
54 Lecture 5 FOURIER TRANSFORM AND SPECTRAL METHOD OF ANALYSIS OF ELECTRIC CIRCUITS Plan Spectra of aperiodic functions and Fourier transform Some properties of the Fourier transform 3 Spectral method
Stress Resonance exam (continued) i iω K = K = ω = = ω => r+ iω + r+ i ω iω r + ω K = ω r + ω Denominator is minimum at frequency ω 0 such that ω0 = 0 => ω0 ω 0= this frequency is called resonant
Chapter 2. Methods for calculating transient processes. 2.1. Classical calculation method. Theoretical information. In the first chapter, methods for calculating the circuit in steady state were considered, that is,
Yastrebov NI KPI RTF department of TOP wwwystrevkievu Circuit functions
4.9. The transient response of the circuit, its relationship with the impulse response. Consider the function K j K j j > S j j K j S 2 Assume that K jω has a Fourier transform h K j If there exists an IC k K j, then
Lecture 9 Linearization of differential equations Linear differential equations of higher orders Homogeneous equations properties of their solutions Properties of solutions of non-homogeneous equations Definition 9 Linear
Methodical development Solving problems on TFKP Complex numbers Operations on complex numbers Complex plane A complex number can be represented in algebraic and trigonometric exponential
Table of contents INTRODUCTION Section CLASSICAL METHOD OF CALCULATION OF TRANSIENT PROCESSES Section CALCULATION OF TRANSIENT PROCESSES UNDER ARBITRARY INPUT ACTIONS USING SUPERVISION INTEGRALS9 CHECK-OUT QUESTIONS7
4 ELECTROMAGNETIC OSCILLATIONS AND WAVES An oscillatory circuit is an electric circuit composed of capacitors and coils in which an oscillatory process of recharging capacitors is possible. This process
3.5. Complex parallel oscillatory circuit I A circuit in which at least one parallel branch contains reactivity of both signs. I C C I I There is no magnetic connection between and. Resonance condition
LECTURE N38. Behavior of an analytic function at infinity. special points. Function residues..neighborhood of a point at infinity.....Laurent expansion in a neighborhood of a point at infinity.... 3. Behavior
4 Lecture 3 FREQUENCY CHARACTERISTICS OF ELECTRIC CIRCUITS Complex transfer functions Logarithmic frequency responses 3 Conclusion Complex transfer functions (complex frequency responses)
Fluctuations. Lecture 3 Alternator To explain the principle of the alternator, let's first consider what happens when a flat coil of wire rotates in a uniform magnetic
DIFFERENTIAL EQUATIONS General concepts Differential equations have numerous and very diverse applications in mechanics, physics, astronomy, technology, and in other branches of higher mathematics (for example,
Calculation of the source of harmonic oscillations (HHC) Present the original circuit of the HHC with respect to the primary winding of the transformer with an equivalent voltage source Determine its parameters (EMF and internal
Work 11 STUDYING FORCED OSCILLATIONS AND THE PHENOMENA OF RESONANCE IN THE OSCILLATORY CIRCUIT In a circuit containing an inductor and a capacitor, electrical oscillations can occur. The work studies
Topic 4 .. AC circuits Topic questions .. AC circuit with inductance .. AC circuit with inductance and active resistance. 3. AC circuit with capacitance. 4. AC circuit
4 Lecture ANALYSIS OF RESISTIVE CIRCUITS Plan The task of analyzing electrical circuits Kirchhoff's laws Examples of analyzing resistive circuits 3 Equivalent transformations of a section of a circuit 4 Conclusions The task of analyzing electrical
Option 708 A source of sinusoidal EDC e(ωt) sin(ωt ψ) operates in the electrical circuit. The circuit diagram shown in Fig.. The effective value of the EDC E of the source, the initial phase and the value of the circuit parameters
Initial data R1=10 ohm R2=8 ohm R3=15 ohm R4=5 ohm R5=4 ohm R6=2 ohm E1=10 V E2=15 V E3=20 V Kirhoff's laws (DC voltage) 1. Looking for nodes Node point , in which three (or more) conductors are connected
LECTURE OSCILLATION. Forced vibrations
Exam Voltage Resonance (continued) We will assume that the voltage on the ode of the circuit is the voltage on the entire oscillatory circuit, and the voltage at the output of the circuit is the voltage on the capacitor Then Amplitude
Autumn semester of the academic year Topic 3 HARMONIC ANALYSIS OF NON-PERIODIC SIGNALS Direct and inverse Fourier transforms Spectral characteristic of the signal Amplitude-frequency and phase-frequency spectra
Lecture 6. Classification of rest points of a linear system of two equations with constant real coefficients. Consider a system of two linear differential equations with constant real
54 Lecture 5 Fourier transform AND SPECTRAL METHOD FOR ANALYSIS OF ELECTRIC CIRCUITS Plan Spectra of aperiodic functions and Fourier transform 2 Some properties of the Fourier transform 3 Spectral method
Topic: Laws of alternating current Electric current is called the ordered movement of charged particles or macroscopic bodies Variable current is called, which changes its value over time
Exam Complex resistance impedance Impedance or complex resistance is by definition equal to the ratio of complex voltage to complex current: Z ɶ Note that impedance is also equal to the ratio
Title Introduction. Basic concepts.... 4 1. Volterra integral equations... 5 Homework options.... 8 2. Resolvent of the Volterra integral equation. 10 Homework options.... 11
Chapter II Integrals An antiderivative function and its properties The function F() is called an antiderivative continuous function f() on the interval a b if F() f(), a; b (;) For example, for the function f(), the antiderivatives
classical method. Fig. 1 - the initial diagram of the electrical circuit Circuit parameters: E \u003d 129 (V) w \u003d 10000 (rad / s) R1 \u003d 73 (Ohm) R2 \u003d 29 (Ohm) R3 \u003d 27 (Ohm) L = 21 (mH) C = 0.97 (uF) Inductor Reactance:
Methods for calculating complex linear electrical circuits Basis: the ability to compose and solve systems of linear algebraic equations - compiled either for a DC circuit or after symbolization
DEFINITE INTEGRAL. Integral Sums and Definite Integral Let a function y = f () defined on the segment [, b ], where< b. Разобьём отрезок [, b ] с помощью точек деления на n элементарных
8 Lecture 7 OPERATOR FUNCTIONS OF CIRCUITS Operator input and transfer functions Poles and zeros of functions of circuits 3 Conclusions Operator input and transfer functions An operator function of a circuit is a relation
68 Lecture 7 TRANSIENT PROCESSES IN FIRST ORDER CIRCUITS Plan 1 Transient processes in RC circuits of the first order 2 Transient processes in R-circuits of the first order 3 Examples of calculation of transient processes in circuits
4 LINEAR ELECTRIC CIRCUITS OF AC SINUSOIDAL CURRENT AND METHODS OF THEIR CALCULATION 4.1 ELECTRIC MACHINES. PRINCIPLE OF GENERATION OF SINUSOIDAL CURRENT 4.1.012. Sinusoidal current is called instantaneous
Federal Agency for Education State Educational Institution of Higher Professional Education "KUBAN STATE UNIVERSITY" Faculty of Physics and Technology Department of Optoelectronics
~ ~ FCF Derivative of a function of a complex variable FCF of the Cauchy-Riemann condition Concept of regularity of the FCF Depiction and form of a complex number Form of the FCF: where the real function of two variables is real
This is the name of another type of integral transformation, which, along with the Fourier transform, is widely used in radio engineering to solve a wide variety of problems related to the study of signals.
The concept of complex frequency.
Spectral methods, as is already known, are based on the fact that the signal under study is represented as the sum of an unlimited number of elementary terms, each of which periodically changes in time according to the law .
A natural generalization of this principle lies in the fact that instead of complex exponential signals with purely imaginary exponents, exponential signals of the form are introduced into consideration, where is a complex number: called the complex frequency.
Two such complex signals can be used to compose a real signal, for example, according to the following rule:
where is the complex conjugate value.
Indeed, while
Depending on the choice of the real and imaginary parts of the complex frequency, various real signals can be obtained. So, if , but the usual harmonic oscillations of the form If are obtained, then, depending on the sign, either increasing or decreasing exponential oscillations in time are obtained. Such signals acquire a more complex form when . Here, the multiplier describes an envelope that changes exponentially with time. Some typical signals are shown in Fig. 2.10.
The concept of complex frequency turns out to be very useful, primarily because it makes it possible, without resorting to generalized functions, to obtain spectral representations of signals whose mathematical models are not integrable.
Rice. 2.10. Real signals corresponding to different values of the complex frequency
Another consideration is also essential: exponential signals of the form (2.53) serve as a "natural" means of studying oscillations in various linear systems. These questions will be explored in Chap. 8.
It should be noted that the true physical frequency is the imaginary part of the complex frequency. There is no special term for the real part o of the complex frequency.
Basic ratios.
Let - some signal, real or complex, defined for t > 0 and equal to zero for negative values of time. The Laplace transform of this signal is a function of the complex variable given by the integral:
The signal is called the original, and the function is called its Laplace image (for short, just an image).
The condition that ensures the existence of the integral (2.54) is as follows: the signal must have at most an exponential growth rate for ie must satisfy the inequality where are positive numbers.
When this inequality is satisfied, the function exists in the sense that the integral (2.54) converges absolutely for all complex numbers for which the number a is called the abscissa of absolute convergence.
The variable in the main formula (2.54) can be identified with the complex frequency Indeed, for a purely imaginary complex frequency, when formula (2.54) turns into formula (2.16), which determines the Fourier transform of the signal, which is zero at Thus, the Laplace transform can be considered
Just as it is done in the theory of the Fourier transform, it is possible, knowing the image, to restore the original. To do this, in the formula for the inverse Fourier transform
an analytic continuation should be performed by passing from the imaginary variable to the complex argument a On the plane of the complex frequency, integration is carried out along an unlimitedly extended vertical axis located to the right of the abscissa of absolute convergence. Since for differential , the formula for the inverse Laplace transform takes the form
In the theory of functions of a complex variable, it is proved that Laplace images have "good" properties in terms of smoothness: such images at all points of the complex plane, with the exception of a countable set of so-called singular points, are analytic functions. Singular points are usually poles, single or multiple. Therefore, to calculate integrals of the form (2.55), flexible methods of residue theory can be used.
In practice, Laplace transform tables are widely used, which collect information about the correspondence between the originals. and images. The presence of tables made the Laplace transform method popular both in theoretical studies and in engineering calculations of radio engineering devices and systems. In the Annexes to there is such a table, which allows solving a fairly wide range of problems.
Examples of calculating Laplace transforms.
There are many similarities in the methods of computing images with what has already been studied in relation to the Fourier transform. Let's consider the most typical cases.
Example 2.4, Image of a generalized exponential momentum.
Let , where is a fixed complex number. The presence of the -function determines the equality at Using formula (2.54), we have
If then the numerator vanishes when the upper limit is substituted. As a result, we get the correspondence
As a special case of formula (2.56), one can find the image of a real exponential video pulse:
and complex exponential signal:
Finally, putting in (2.57) , we find the image of the Heaviside function:
Example 2.5. An image of a delta function.
Laplace transform- integral transformation relating the function F (s) (\displaystyle \ F(s)) complex variable ( image) with the function f (x) (\displaystyle \f(x)) real variable ( original). It is used to explore the properties dynamic systems and decide differential And integral equations.
One of the features of the Laplace transform, which predetermined its widespread use in scientific and engineering calculations, is that many ratios and operations on originals correspond to simpler ratios on their images. Thus, the convolution of two functions in the space of images is reduced to the operation of multiplication, and linear differential equations become algebraic.
Encyclopedic YouTube
1 / 5
✪ Laplace transform - bezbotvy
✪ Lecture 10: Laplace Transform
✪ Higher mathematics - 4. Laplace transformations. Part 1
✪ Laplace method for DE solution
✪ Lecture 11: Applying the Laplace transform to solving differential equations
Subtitles
Definition
Direct Laplace Transform
lim b → ∞ ∫ 0 b | f(x) | e − σ 0 x d x = ∫ 0 ∞ | f(x) | e − σ 0 x d x , (\displaystyle \lim _(b\to \infty )\int \limits _(0)^(b)|f(x)|e^(-\sigma _(0)x)\ ,dx=\int \limits _(0)^(\infty )|f(x)|e^(-\sigma _(0)x)\,dx,)then it converges absolutely and uniformly for and - analytic function at σ ⩾ σ 0 (\displaystyle \sigma \geqslant \sigma _(0)) (σ = R e s (\displaystyle \sigma =\mathrm (Re) \,s)- real part complex variable s (\displaystyle s)). Exact lower bound σ a (\displaystyle \sigma _(a)) sets of numbers σ (\displaystyle \sigma ), under which this condition is satisfied, is called abscissa absolute convergence Laplace transform for the function .
- Conditions for the existence of the direct Laplace transform
Laplace transform L ( f (x) ) (\displaystyle (\mathcal (L))\(f(x)\)) exists in the sense of absolute convergence in the following cases:
- σ ⩾ 0 (\displaystyle \sigma \geqslant 0): the Laplace transform exists if the integral exists ∫ 0 ∞ | f(x) | d x (\displaystyle \int \limits _(0)^(\infty )|f(x)|\,dx);
- σ > σ a (\displaystyle \sigma >\sigma _(a)): the Laplace transform exists if the integral ∫ 0 x 1 | f(x) | d x (\displaystyle \int \limits _(0)^(x_(1))|f(x)|\,dx) exists for every finite x 1 > 0 (\displaystyle x_(1)>0) And | f(x) | ⩽ K e σ a x (\displaystyle |f(x)|\leqslant Ke^(\sigma _(a)x)) For x > x 2 ≥ 0 (\displaystyle x>x_(2)\geqslant 0);
- σ > 0 (\displaystyle \sigma >0) or σ > σ a (\displaystyle \sigma >\sigma _(a))(which of the bounds is greater): a Laplace transform exists if a Laplace transform exists for the function f ′ (x) (\displaystyle f"(x)) (derivative from f (x) (\displaystyle f(x))) For σ > σ a (\displaystyle \sigma >\sigma _(a)).
Note
- Conditions for the existence of the inverse Laplace transform
For the existence of the inverse Laplace transform, it is sufficient that the following conditions are met:
- If the image F (s) (\displaystyle F(s)) - analytic function For σ ≥ σ a (\displaystyle \sigma \geqslant \sigma _(a)) and has an order less than −1, then the inverse transformation for it exists and is continuous for all values of the argument, and L − 1 ( F (s) ) = 0 (\displaystyle (\mathcal (L))^(-1)\(F(s)\)=0) For t ⩽ 0 (\displaystyle t\leqslant 0).
- Let F (s) = φ [ F 1 (s) , F 2 (s) , … , F n (s) ] (\displaystyle F(s)=\varphi ), So φ (z 1 , z 2 , … , z n) (\displaystyle \varphi (z_(1),\;z_(2),\;\ldots ,\;z_(n))) is analytic with respect to each z k (\displaystyle z_(k)) and equals zero for z 1 = z 2 = … = z n = 0 (\displaystyle z_(1)=z_(2)=\ldots =z_(n)=0), And F k (s) = L ( f k (x) ) (σ > σ a k: k = 1 , 2 , … , n) (\displaystyle F_(k)(s)=(\mathcal (L))\(f_ (k)(x)\)\;\;(\sigma >\sigma _(ak)\colon k=1,\;2,\;\ldots ,\;n)), then the inverse transformation exists and the corresponding direct transformation has an abscissa of absolute convergence.
Note: these are sufficient conditions for existence.
- Convolution theorem
Main article: Convolution theorem
- Differentiation and integration of the original
The image according to Laplace of the first derivative of the original with respect to the argument is the product of the image and the argument of the latter minus the original at zero on the right:
L ( f ′ (x) ) = s ⋅ F (s) − f (0 +) . (\displaystyle (\mathcal (L))\(f"(x)\)=s\cdot F(s)-f(0^(+)).)Initial and final value theorems (limit theorems):
f (∞) = lim s → 0 s F (s) (\displaystyle f(\infty)=\lim _(s\to 0)sF(s)), if all poles of the function s F (s) (\displaystyle sF(s)) are in the left half-plane.The finite value theorem is very useful because it describes the behavior of the original at infinity with a simple relation. This is, for example, used to analyze sustainability trajectories of a dynamic system.
- Other properties
Linearity:
L ( a f (x) + b g (x) ) = a F (s) + b G (s) . (\displaystyle (\mathcal (L))\(af(x)+bg(x)\)=aF(s)+bG(s).)Multiply by number:
L ( f (a x) ) = 1 a F (s a) . (\displaystyle (\mathcal (L))\(f(ax)\)=(\frac (1)(a))F\left((\frac (s)(a))\right).)Direct and inverse Laplace transform of some functions
Below is the Laplace transform table for some functions.
| № | Function | Time domain x (t) = L − 1 ( X (s) ) (\displaystyle x(t)=(\mathcal (L))^(-1)\(X(s)\)) |
frequency domain X (s) = L ( x (t) ) (\displaystyle X(s)=(\mathcal (L))\(x(t)\)) |
Convergence area For causal systems |
|---|---|---|---|---|
| 1 | ideal lag | δ (t − τ) (\displaystyle \delta (t-\tau)\ ) | e − τ s (\displaystyle e^(-\tau s)\ ) | |
| 1a | single pulse | δ (t) (\displaystyle \delta (t)\ ) | 1 (\displaystyle 1\ ) | ∀ s (\displaystyle \forall s\ ) |
| 2 | lag n (\displaystyle n) | (t − τ) n n ! e − α (t − τ) ⋅ H (t − τ) (\displaystyle (\frac ((t-\tau)^(n))(n}e^{-\alpha (t-\tau)}\cdot H(t-\tau)} !} | e − τ s (s + α) n + 1 (\displaystyle (\frac (e^(-\tau s))((s+\alpha)^(n+1)))) | s > 0 (\displaystyle s>0) |
| 2a | power n (\displaystyle n)-th order | t n n ! ⋅ H (t) (\displaystyle (\frac (t^(n))(n}\cdot H(t)} !} | 1 s n + 1 (\displaystyle (\frac (1)(s^(n+1)))) | s > 0 (\displaystyle s>0) |
| 2a.1 | power q (\displaystyle q)-th order | t q Γ (q + 1) ⋅ H (t) (\displaystyle (\frac (t^(q))(\Gamma (q+1)))\cdot H(t)) | 1 s q + 1 (\displaystyle (\frac (1)(s^(q+1)))) | s > 0 (\displaystyle s>0) |
| 2a.2 | single function | H (t) (\displaystyle H(t)\ ) | 1 s (\displaystyle (\frac (1)(s))) | s > 0 (\displaystyle s>0) |
| 2b | single function with delay | H (t − τ) (\displaystyle H(t-\tau)\ ) | e − τ s s (\displaystyle (\frac (e^(-\tau s))(s))) | s > 0 (\displaystyle s>0) |
| 2c | "speed step" | t ⋅ H (t) (\displaystyle t\cdot H(t)\ ) | 1 s 2 (\displaystyle (\frac (1)(s^(2)))) | s > 0 (\displaystyle s>0) |
| 2d | n (\displaystyle n)-th order with frequency shift | t n n ! e − α t ⋅ H (t) (\displaystyle (\frac (t^(n))(n}e^{-\alpha t}\cdot H(t)} !} | 1 (s + α) n + 1 (\displaystyle (\frac (1)((s+\alpha)^(n+1)))) | s > −α (\displaystyle s>-\alpha ) |
| 2d.1 | exponential decay | e − α t ⋅ H (t) (\displaystyle e^(-\alpha t)\cdot H(t)\ ) | 1 s + α (\displaystyle (\frac (1)(s+\alpha ))) | s > − α (\displaystyle s>-\alpha \ ) |
| 3 | exponential approximation | (1 − e − α t) ⋅ H (t) (\displaystyle (1-e^(-\alpha t))\cdot H(t)\ ) | α s (s + α) (\displaystyle (\frac (\alpha )(s(s+\alpha)))) | s > 0 (\displaystyle s>0\ ) |
| 4 | sinus | sin (ω t) ⋅ H (t) (\displaystyle \sin(\omega t)\cdot H(t)\ ) | ω s 2 + ω 2 (\displaystyle (\frac (\omega )(s^(2)+\omega ^(2)))) | s > 0 (\displaystyle s>0\ ) |
| 5 | cosine | cos (ω t) ⋅ H (t) (\displaystyle \cos(\omega t)\cdot H(t)\ ) | s s 2 + ω 2 (\displaystyle (\frac (s)(s^(2)+\omega ^(2)))) | s > 0 (\displaystyle s>0\ ) |
| 6 | hyperbolic sine | s h (α t) ⋅ H (t) (\displaystyle \mathrm (sh) \,(\alpha t)\cdot H(t)\ ) | α s 2 − α 2 (\displaystyle (\frac (\alpha )(s^(2)-\alpha ^(2)))) | s > | α | (\displaystyle s>|\alpha |\ ) |
| 7 | hyperbolic cosine | c h (α t) ⋅ H (t) (\displaystyle \mathrm (ch) \,(\alpha t)\cdot H(t)\ ) | s s 2 − α 2 (\displaystyle (\frac (s)(s^(2)-\alpha ^(2)))) | s > | α | (\displaystyle s>|\alpha |\ ) |
| 8 | exponentially decaying sinus |
e − α t sin (ω t) ⋅ H (t) (\displaystyle e^(-\alpha t)\sin(\omega t)\cdot H(t)\ ) | ω (s + α) 2 + ω 2 (\displaystyle (\frac (\omega )((s+\alpha)^(2)+\omega ^(2)))) | s > − α (\displaystyle s>-\alpha \ ) |
| 9 | exponentially decaying cosine |
e − α t cos (ω t) ⋅ H (t) (\displaystyle e^(-\alpha t)\cos(\omega t)\cdot H(t)\ ) | s + α (s + α) 2 + ω 2 (\displaystyle (\frac (s+\alpha )((s+\alpha)^(2)+\omega ^(2)))) | s > − α (\displaystyle s>-\alpha \ ) |
| 10 | root n (\displaystyle n)-th order | t n ⋅ H (t) (\displaystyle (\sqrt[(n)](t))\cdot H(t)) | s − (n + 1) / n ⋅ Γ (1 + 1 n) (\displaystyle s^(-(n+1)/n)\cdot \Gamma \left(1+(\frac (1)(n) )\right)) | s > 0 (\displaystyle s>0) |
| 11 | natural logarithm | ln (t t 0) ⋅ H (t) (\displaystyle \ln \left((\frac (t)(t_(0)))\right)\cdot H(t)) | − t 0 s [ ln (t 0 s) + γ ] (\displaystyle -(\frac (t_(0))(s))[\ln(t_(0)s)+\gamma ]) | s > 0 (\displaystyle s>0) |
| 12 | Bessel function first kind order n (\displaystyle n) |
J n (ω t) ⋅ H (t) (\displaystyle J_(n)(\omega t)\cdot H(t)) | ω n (s + s 2 + ω 2) − n s 2 + ω 2 (\displaystyle (\frac (\omega ^(n)\left(s+(\sqrt (s^(2)+\omega ^(2) ))\right)^(-n))(\sqrt (s^(2)+\omega ^(2))))) | s > 0 (\displaystyle s>0\ ) (n > − 1) (\displaystyle (n>-1)\ ) |
| 13 | first kind order n (\displaystyle n) |
I n (ω t) ⋅ H (t) (\displaystyle I_(n)(\omega t)\cdot H(t)) | ω n (s + s 2 − ω 2) − n s 2 − ω 2 (\displaystyle (\frac (\omega ^(n)\left(s+(\sqrt (s^(2)-\omega ^(2) ))\right)^(-n))(\sqrt (s^(2)-\omega ^(2))))) | s > | ω | (\displaystyle s>|\omega |\ ) |
| 14 | bessel function second kind zero order |
Y 0 (α t) ⋅ H (t) (\displaystyle Y_(0)(\alpha t)\cdot H(t)\ ) | − 2 a r s h (s / α) π s 2 + α 2 (\displaystyle -(\frac (2\mathrm (arsh) (s/\alpha))(\pi (\sqrt (s^(2)+\alpha ^(2)))))) | s > 0 (\displaystyle s>0\ ) |
| 15 | modified Bessel function second kind, zero order |
K 0 (α t) ⋅ H (t) (\displaystyle K_(0)(\alpha t)\cdot H(t)) | ||
| 16 | error function | e r f (t) ⋅ H (t) (\displaystyle \mathrm (erf) (t)\cdot H(t)) | e s 2 / 4 e r f c (s / 2) s (\displaystyle (\frac (e^(s^(2)/4)\mathrm (erfc) (s/2))(s))) | s > 0 (\displaystyle s>0) |
Table notes:
| ||||
Loading...