State educational institution higher professional education

TOMSK POLYTECHNIC UNIVERSITY

Department 21

Abstract on the topic:

"Laplace Transform"

Completed

student gr.0850

Kiseleva Yu.V.

Checked:

Daneikin Yu.V.

Tomsk, 2008

Introduction

Laplace transform is an integral transform that relates a function

One of the features of the Laplace transform, which predetermined its wide distribution in scientific and engineering calculations, is that many relations and operations on the originals correspond to simpler relations on their images.

1. Direct Laplace transform

Laplace transform of a function of a real variable

Right side This expression is called the Laplace integral.

2. Reverse conversion Laplace

By inverse Laplace transform of a function of a complex variable

3. Two-way Laplace transform

The two-way Laplace transform is a generalization to the case of problems in which for the function

The two-way Laplace transform is defined as follows:

4. Discrete Laplace transform

Used in the field of computer control systems. The discrete Laplace transform can be applied to lattice functions.
Distinguish

lattice function, that is, the values ​​of this function are determined only at discrete times

If we apply the following change of variables:

we get the Z-transform:

5. Properties and theorems

· Absolute convergence

If the Laplace integral converges absolutely at σ = σ 0, that is, there is a limit

then it converges absolutely and uniformly for

· Conditions of existence direct conversion Laplace

Laplace transform

2. Case σ > σ a: the Laplace transform exists if the integral

exists for every finite

3. Case σ > 0 or σ > σ a (whichever boundary is greater): a Laplace transform exists if there is a Laplace transform for the function f"(x) (derivative to f(x)) for σ > σ a.

Conditions for the existence of the inverse Laplace transform

For the existence of the inverse Laplace transform, it is sufficient to satisfy the following conditions:

1. If the image F(s) is an analytic function for

is analytic with respect to each z k and is equal to zero for

then the inverse transformation exists and the corresponding direct transformation has the abscissa of absolute convergence.

Note: this is sufficient conditions existence.

Convolution theorem

The Laplace transform of the convolution of two originals is the product of the images of these originals.

Image multiplication

Lecture No. 12

Topic: Operator method of analysis of transitional

processes.

Study questions

1 Laplace transform and its properties.

2 Ohm's and Kirchhoff's laws in operator form. Operator substitution scheme.

3 Algorithm for analyzing transient processes using the operator method.

4 Identification of the original by its image. Decomposition theorem.

Literature: pp. 331-342.

1 Laplace transform and its properties

Previously reviewed classic method has the following significant disadvantages:

limited application, it is used mainly in cases where the circuit under study has a low order of complexity, and external influence on it after switching is harmonic function time or constantly; if the external influence on the circuit after switching has more complex character, then determining the forced component of the chain reaction becomes significantly more difficult.

bulkiness when analyzing transient processes of circuits of more than second order, since finding the free component and constant integrations requires solving algebraic equations high order.

Free from the listed disadvantages operator method transient analysis based on application Laplace transform.

The operator method does not have physical clarity due to mathematical formalization, but greatly simplifies the calculations. An important feature of the operator method is its applicability for functions that are not absolutely integrable, for example, a single voltage step, a harmonic voltage switched on at some point in time, and other waveforms for which the classical and spectral methods analysis cannot be applied.

The essence of the operator method is that the calculation of the transient process is transferred from the domain of functions of a real variable (time t) to the domain of functions of a complex variable. In this case, the operations of differentiation and integration of functions of time are replaced by the corresponding operations of multiplication and division of functions of a complex variable by the operator p. This significantly simplifies the calculation, since it reduces the system of differential equations to an algebraic system. In the operator method there is no need to determine integration constants. This circumstance explains the widespread use of this method in practice.

The transition from the domain of a real variable to the domain of functions of a complex variable is carried out using direct Laplace transform. After this, algebraic equations are solved for the images of the sought functions. The resulting solution to algebraic equations inverse Laplace transform is transferred to the domain of a real variable.

The mathematical justification of the operator method was first given in 1862. Russian mathematician M.E. Vashchenko-Zakharchenko, who showed the possibility of applying symbolic (operator) calculus to integration differential equations based on the direct Laplace transform.

At the end of the 19th century. English electrical engineers O. Heaviside and D. Carson successfully applied and developed a symbolic method for solving differential equations for calculating transient processes in electrical circuits. However, the operator method received a rigorous justification only in the 20th century. based on general theory functional transformations.

Direct Laplace transform is determined by the equation

where f(t) is a function of a real variable t, defined at
(at t< 0; f(t) = 0) и удовлетворяющая условием граниченного роста:

where the multiplier M and the growth rate C 0 are positive real numbers.

Figure 12.1 shows the domain of definition of the complex variable F(p).

Inverse Laplace transform determined from the solution
(12.1).

The function F(p) defined by equation (12.1) is called Laplace image, and the function f(t) in (12.3) is original.

Consequently, the original and the image are a pair of functions of a real f(t) and a complex variable F(p), related by the Laplace transform and placed in strict correspondence with each other.

To shorten the notation of transformations (12.1) and (12.3), the following symbolism is used:

where L is the Laplace operator.

In what follows, for definiteness, we will use the correspondence sign.

Based on the Laplace transform, one can obtain an image of any functions that satisfy condition (12.2). There are special reference books that contain originals and images of a wide range of functions.

Table 12.1 shows examples of images of simple functions.

Table 12.1 – Images of Laplace functions

Let's look at some properties of the Laplace transform , also called theorems.

Addition theorem or linearity of transformation

Differentiation theorem

.

Integration theorem

.

Delay theorem

The Laplace transform allows you to obtain the relationship between voltage and current in operator form for resistive, inductive and capacitive elements.

Illustration of voltage across a resistive element

U r (t) = r i(t) according to (12.1) will take the form:

The expression U r (p) = r I (p) is called Ohm's law in operator form for a resistive element (Fig. 12.1, a), the operator equivalent circuit of which is presented in Fig. 12.1, b.

Voltage image
on the inductive element (Fig. 12.2, a) according to (12.4) and (12.5) will take the form:

U L (p) = - L i(0) + pLI(p), (12.9)

where i(0) = i(0 -) = i(0 +) is the current in the inductive element at the moment of commutation t = 0, taking into account the initial conditions (according to the first law of commutation).

Expression (12.9) corresponds to the operator equivalent circuit of the inductive element in Fig. 12.2, b.

Voltage on the capacitive element (Fig. 12.3, a), starting from the moment of time t = 0 of the occurrence of the transient process in general case

where U c (0) = U c (0 -) = U c (0 +) is the voltage on the capacitive element corresponding to the initial condition (according to the second commutation law).

Given an image unit function
(Table 12.1) and relations (12.4) and (12.5), we find the image of voltage U c (t):

Expression (12.10) corresponds to the equivalent circuit of the capacitive element in operator form in Fig. 12.3b.

If the initial conditions are zero, i.e. i L (0 -) = 0 and U c (0 -) = 0, then expressions (12.9) and (12.10) will take the form of Ohm’s law in operator form for the inductive element

U L (p) = LpI(p) = Z L (p)I(p), (12.11)

where Z L (p) = Lp – operator resistance of the inductive element, for the capacitive element

Previously, we considered the integral Fourier transform with the kernel K(t, O = e). The Fourier transform is inconvenient in that the condition of absolute integrability of the function f(t) on the entire t axis must be satisfied. The Laplace transform allows us to free ourselves from this limitation. Definition 1. By the function We will call an original any complex-valued function f(t) of a real argument t that satisfies the following conditions: 1. f(t) is continuous on the entire t axis, except for individual points at which f(t) has a discontinuity of the 1st kind, and on each finite interval of the axis * there can be only a finite number of such points; 2. the function f(t) is equal to zero at negative values t, f(t) = 0 for 3. As t increases, the module f(t) increases no faster than the exponential function, i.e. there are numbers M > 0 and s such that for all t It is clear that if inequality (1) is fulfilled for some s = aj, then it will be TRUE for ANY 82 > 8]. The exact infimum s0 of all numbers 3, «o = infs, for which inequality (1) holds, is called the growth index of the function f(t). Comment. In the general case, the inequality does not hold, but the estimate where e > 0 is any is valid. Thus, the function has a growth exponent 0 = For it, the inequality \t\ ^ M V* ^ 0 does not hold, but the inequality |f| ^ Mei. Condition (1) is much less restrictive than condition (*). Example 1. the function does not satisfy condition ("), but condition (1) is satisfied for any s ^ I and A/ ^ I; growth rate 5o = So this is the original function. On the other hand, the function is not the original function: it has an infinite order of growth, “o = +oo. The simplest original function is the so-called unit function. If a certain function satisfies conditions 1 and 3 of Definition 1, but does not satisfy condition 2, then the product is already an original function. For simplicity of notation, we will, as a rule, omit the factor rj(t), stipulating that all functions that we will consider are equal to zero for negative t, so if we're talking about about some function f(t), for example, about sin ty cos t, el, etc., then the following functions are always implied (Fig. 2): n=n(0 Fig. 1 Definition 2. Let f( t) is the original function. The Laplace image of the function f(t) is the function F(p) of a complex variable, defined by the formula LAPLACE TRANSFORM Basic definitions Properties Convolution of functions Multiplication theorem Finding the original from the image Using the inversion theorem of operational calculus Duhamel's formula Integration of linear differential systems equations with constant coefficients Solving integral equations where the integral is taken along the positive semi-axis t. The function F(p) is also called the Laplace transform of the function /(/); transformation kernel K(t) p) = e~pt. The fact that the function has F(p) as its image will be written Example 2. Find the image of the unit function r)(t). The function is the original function with growth exponent 0 - 0. By virtue of formula (2), the image of the function rj(t) will be the function If then when the integral on the right side of the last equality will be convergent, and we will obtain so that the image of the function rj(t) will be function £. As we agreed, we will write that rj(t) = 1, and then the result obtained will be written as follows: Theorem 1. For any original function f(t) with growth index 30, the image F(p) is defined in the half-plane R e = s > s0 and is in this half-plane analytical function(Fig. 3). Let To prove the existence of the image F(p) in the indicated half-plane, it is sufficient to establish that improper integral(2) converges absolutely for a > Using (3), we obtain which proves the absolute convergence of integral (2). At the same time, we obtained an estimate for the Laplace transform F(p) in the half-plane of convergence. Differentiating expression (2) formally under the integral sign with respect to p, we find. The existence of integral (5) is established in the same way as the existence of integral (2) was established. Applying integration by parts for F"(p), we obtain an estimate from which it follows absolute convergence integral (5). (The non-integral term,0.,- at t +oo has a limit, equal to zero). In any half-plane Rep ^ sj > o, integral (5) converges uniformly with respect to p, since it is majorized by a convergent integral independent of p. Consequently, differentiation with respect to p is legal and equality (5) is true. Since the derivative F"(p) exists, the Laplace transform F(p) everywhere in the half-plane Rep = 5 > 5о is an analytic function. Inequality (4) implies the Corollary. If p tends to infinity so that Re p = s increases indefinitely , then Example 3. Let us also find the image of the function, any complex number. The exponent of the function /(() is equal to a. 4 Considering Rep = i > a, we obtain Thus, For a = 0 we again obtain the formula Let us pay attention to the fact that the image of the function eat is an analytic function of the argument p not only in the half-plane Rep > a, but also at all points p, except for the point p = a, where this image has a simple pole. In what follows we will encounter this more than once. similar situation, when the image F(p) will be an analytical function in the entire plane of the complex variable p, with the exception of isolated singular points. There is no contradiction with Theorem 1. The latter only states that in the half-plane Rep > o the function F(p) has no singular points: all of them turn out to lie either to the left of the line Rep = so, or on this line itself. Don't notice. In operational calculus, the Heaviside representation of the function f(f) is sometimes used, which is defined by the equality and differs from the Laplace representation by the factor p. §2. Properties of the Laplace transform In what follows, we will denote the original functions, and their Laplace images. From the definition of an image it follows that if Theorem 2 (unity). £biw dee continuous functions) have the same image, then they are identically equal. Teopewa 3 (p'ieiost* Laplace's transformation). If the functions are originals, then for any complex constants α The validity of the statement follows from the linearity property of the integral that defines the image: , are the growth indicators of the functions, respectively). Based on this property, we obtain Similarly, we find that and, further, Theorem 4 (similarities). If f(t) is the original function and F(p) is its Laplace image, then for any constant a > O. Setting at = m, we have Using this theorem, from formulas (5) and (6) we obtain Theorem 5 ( on differentiation of the original). Let be the original function with the image F(p) and let be also the original functions, and where is the growth index of the function Then and in general Here we mean the right limit value Let. Let's find the image We have Integrating by parts, we obtain The out-of-integral term on the right side of (10) vanishes as k. For Rc р = s > з we have the substitution t = Gives -/(0). The second term on the right in (10) is equal to pF(p). Thus, relation (10) takes the form and formula (8) is proven. In particular, if To find the image f(n\t) we write from where, integrating n times by parts, we obtain Example 4. Using the theorem on differentiation of the original, find the image of the function f(t) = sin2 t. Let Therefore, Theorem 5 establishes wonderful property integral Laplace transform: it (like the Fourier transform) transforms the operation of differentiation into the algebraic operation of multiplication by p. Inclusion formula. If they are original functions, then In fact, By virtue of the corollary to Theorem 1, every image tends to zero at. This means that the inclusion formula follows (Theorem 6 (on differentiation of an image). Differentiation of an image is reduced to multiplication by the original. Since the function F(p) in the half-plane so is analytic, it can be differentiated with respect to p. We have The latter just means, that Example 5. Using Theorem 6, find the image of the function 4 As is known, Hence (Applying Theorem 6 again, we find, in general, Theorem 7 (integration of the original). Integration of the original is reduced to dividing the image by Let It is easy to check that if there is an original function, then it will be the original function, and Let. In view of so that On the other hand, whence F = The latter is equivalent to the proved relation (13) Example 6. Find the image of the function M B. in this case, So. Therefore Theorem 8 (image integration). If the integral also converges, then it serves as an image of the function ^: LAPLACE TRANSFORM Basic definitions Properties Convolution of functions Multiplication theorem Finding the original from an image Using the inversion theorem of operational calculus Duhamel's formula Integrating systems of linear differential equations with constant coefficients Solving integral equations Indeed, Assuming that the path of integration lie on the half-plane so, we can change the order of integration. The last equality means that it is an image of the function Example 7. Find the image of the function M As is known, . Therefore, since we assume that we get £ = 0, when. Therefore, relation (16) takes the form Example. Find the image of the function f(t), specified graphically (Fig. 5). Let us write the expression for the function f(t) in the following form: This expression can be obtained as follows. Consider the function and subtract the function from it. The difference will be equal to one for. To the resulting difference we add the function. As a result, we obtain the function f(t) (Fig. 6 c), so that from here, using the delay theorem, we find Theorem 10 (displacement). then for anyone complex number ro In fact, the Theorem makes it possible to use known images of functions to find images of the same functions multiplied by exponential function, for example, 2.1. Function folding. Multiplication theorem Let the functions f(t) be defined and continuous for all t. The convolution of these functions is called new feature on t, defined by equality (if this integral exists). For original functions, the operation convolve is always feasible, and (17) 4 In fact, the product of original functions as a function of m is a finite function, i.e. vanishes outside some finite interval (in this case, outside the segment. For finite continuous functions, the convolution operation is feasible, and we obtain the formula It is not difficult to verify that the convolution operation is commutative, Theorem 11 (multiplication). If, then the convolution t) has an image It is not difficult to verify that that the convolution (of the original functions is the original function with the growth exponent » where, are the growth exponents of the functions, respectively. Let us find the image of the convolution. Using what we have. Changing the order of integration in the integral on the right (such an operation is legal) and applying the retardation theorem, we obtain Thus Thus, from (18) and (19) we find that the multiplication of images corresponds to the convolution of the originals, Prter 9. Find the image of the function A function V(0) is the convolution of functions. By virtue of the multiplication theorem Problem. Let the function /(ξ) be periodic with period T , is the original function. Show that its Laplace image F(p) is given by formula 3. Finding the original from the image The problem is stated as follows: given the function F(p), we need to find the function /(<)>whose image is F(p). Let us formulate conditions sufficient for the function F(p) of a complex variable p to serve as an image. Theorem 12. If a function F(p) analytic in the half-plane so 1) tends to zero as in any half-plane R s0 uniformly with respect to arg p; 2) the integral converges absolutely, then F(p) is the image of some original function Problem. Can the function F(p) = serve as an image of some original function? We will indicate some ways to find the original from an image. 3.1. Finding the original using image tables First of all, it is worth bringing the function F(p) to a simpler, “tabular” form. For example, in the case when F(p) - fractional rational function argument p, it is decomposed into elementary fractions and the appropriate properties of the Laplace transform are used. Example 1. Find the original for We write the function F(p) in the form Using the displacement theorem and the linearity property of the Laplace transform, we obtain Example 2. Find the original for the function 4 We write F(p) in the form Hence 3.2. Using the inversion theorem and its corollaries Theorem 13 (inversion). If the function fit) is the original function with growth exponent s0 and F(p) is its image, then at any point of continuity of the function f(t) the relation is satisfied where the integral is taken along any straight line and is understood in the sense of the principal value, i.e. as Formula (1) is called the Laplace transform inversion formula, or Mellin's formula. Indeed, let, for example, f(t) be piecewise smooth on each final segment }