One of the ways to solve differential equations (systems of equations) with constant coefficients is the method of integral transformations, which allows the function of a real variable (original function) to be replaced by a function of a complex variable (image of a function). As a result, the operations of differentiation and integration in the space of original functions are transformed into algebraic multiplication and division in the space of image functions. One of the representatives of the method of integral transformations is the Laplace transform.

Continuous Laplace transform- an integral transformation linking a function of a complex variable (image of a function) with a function of a real variable (original of a function). In this case, the function of a real variable must satisfy the following conditions:

The function is defined and differentiable on the entire positive semiaxis of a real variable (the function satisfies the Dirichlet conditions);

The value of the function until the initial moment is equated to zero ;

The growth of the function is limited by the exponential function, i.e. for a function of a real variable, there exist such positive numbers M and with , what at, where c - abscissa of absolute convergence (some positive number).

Laplace transform (direct integral transform) function of a real variable is called a function of the following form (function of a complex variable):

The function is called the original of the function, and the function is called its image. Complex variable is called the Laplace operator, where is the angular frequency, is some positive constant number.

As a first example, we define an image for a constant function

As a second example, we define an image for the cosine function ... Taking into account the Euler formula, the cosine function can be represented as the sum of two exponentials .

In practice, to perform the direct Laplace transform, transformation tables are used, in which originals and images of typical functions are presented. Some of these functions are presented below.

Original and Image for Exponential Function

Original and image for cosine function

Original and image for sine function

Original and image for exponentially decaying cosine

Original and image for exponentially decaying sine

It should be noted that the function is a Heaviside function that takes a value of zero for negative values ​​of the argument and takes a value equal to one for positive values ​​of the argument.

Laplace transform properties

Linearity theorem

The Laplace transform is linear, i.e. any linear relationship between originals of a function is valid for the images of these functions.

The linearity property makes it easier to find the originals of complex images, since it allows the image of a function to be represented as a sum of simple terms, and then to find the originals of each represented term.

Differentiation theorem of the original functions

The differentiation of the original function matches multiplication

For nonzero initial conditions:

With zero initial conditions (special case):

Thus, the operation of differentiating the function is replaced by an arithmetic operation in the image space of the function.

Integration theorem of the original functions

The integration of the original function matches division function images onto the Laplace operator.

Thus, the operation of integrating the function is replaced by an arithmetic operation in the image space of the function.

Similarity theorem

Changing the argument of the function (compression or expansion of the signal) in the time domain leads to the opposite change in the argument and the ordinate of the function image.

An increase in the pulse duration causes a compression of its spectral function and a decrease in the amplitudes of the harmonic components of the spectrum.

Delay theorem

The delay (shift, shift) of the signal by the argument of the original function by the interval leads to a change in the phase-frequency function of the spectrum (phase angle of all harmonics) by a given amount without changing the modulus (amplitude function) of the spectrum.

The resulting expression is valid for any

Displacement theorem

The delay (shift, shift) of the signal by the argument of the function image leads to the multiplication of the original function by an exponential factor

From a practical point of view, the displacement theorem is used to determine the images of exponential functions.

Convolution theorem

Convolution is a mathematical operation applied to two functions and, resulting in a third function. In other words, having a response of a certain linear system to an impulse, you can use convolution to calculate the response of the system to the entire signal.

Thus, the convolution of the originals of two functions can be represented as a product of images of these functions. The reconciliation theorem is used when considering transfer functions, when the system response (output signal from a four-port network) is determined when a signal is applied to the input of a four-port network with an impulse transient response.

Linear quadrupole

Inverse Laplace transform

The Laplace transform is reversible, i.e. the function of a real variable is uniquely determined from the function of a complex variable . For this, the inverse Laplace transform formula is used(Mellin's formula, Bromwich integral), which has the following form:

In this formula, the limits of integration mean that the integration goes along an infinite straight line that is parallel to the imaginary axis and intersects the real axis at a point. Considering that the latter expression can be rewritten as follows:

In practice, to perform the inverse Laplace transform, the image of the function is decomposed into the sum of the simplest fractions by the method of undefined coefficients, and for each fraction (in accordance with the linearity property) the original of the function is determined, including taking into account the table of typical functions. This method is valid for displaying a function that is a correct rational fraction. It should be noted that the simplest fraction can be represented as a product of linear and quadratic factors with real coefficients, depending on the type of roots of the denominator:

If there is a zero root in the denominator, the function is decomposed into a fraction like:

If there is a zero n -fold root in the denominator, the function is decomposed into a fraction of the type:

If there is a real root in the denominator, the function is decomposed into a fraction like:

If there is a real n-multiple root in the denominator, the function is decomposed into a fraction like:

If there is an imaginary root in the denominator, the function is decomposed into a fraction like:

In the case of complex conjugate roots in the denominator, the function is decomposed into a fraction like:

∙ In general if the image of the function is a regular rational fraction (the degree of the numerator is less than the degree of the denominator of the rational fraction), then it can be expanded into the sum of the simplest fractions.

∙ In a particular case if the denominator of the function image is decomposed only into simple roots of the equation, then the function image can be decomposed into the sum of the simplest fractions as follows:

Unknown coefficients can be determined using the undefined coefficient method or in a simplified way using the following formula:

The value of the function at the point;

The value of the derivative of the function at a point.

Transcript

1 Laplace transform Brief information The Laplace transform, which is widely used in circuit theory, is an integral transform applied to time functions 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 thus defined can be analytically continued to the entire plane of the complex variable = +, with the exception of individual singular points. Most often this continuation is carried out by extending the formula obtained by calculating the integral to the entire plane of the complex variable. The function F, which is analytically continued to the entire complex plane, is called the Laplace image of the time function f or simply the image. The function f in relation 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> The integral on the right side is a contour integral along a straight line parallel to the ordinate axis The value is chosen so that there are no singular points of the function F in the half-plane R>. are the inverse Laplace transform and are denoted by the symbol f L (F) L 7

2 Consider some properties of the Laplace transform Linearity This property can be written as the equality L (ff) L (f) L (f) Laplace transform of the derivative of a function df L () d df d F fdf 3 Laplace transform of the integral: L (fd) df 8 fdd F df: dffdd Consider the simplest application of the Laplace transform in circuit theory Figure shows the simplest elements of circuits: resistance, inductance and capacitance The instantaneous voltage drop across the resistance is 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 diu L, d i.e. there is no direct proportionality, Ohm's law does not hold here After the Laplace transform, we obtain U = LI LI +

3 If, as is often the case, I + =, then the relation takes the form U = LI Thus, for the images of voltage and current, Ohm's law is again valid. The role of resistance is played by the quantity L, which is called the inductance resistance For capacitance, we have the relation between the instantaneous values ​​of voltage and inductance uid C After the Laplace transform, this ratio takes the form UI, C te has the form of Ohm's law, and the capacitive resistance is equal to C Let's compose a table of direct and inverse Laplace transforms of elementary functions found in circuit theory a unit step is determined by the equalities: at; 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 (LL) (L 8) cos (L 9) (F dff L! Ndnnnn L! Nnn L Now consider the inverse transformation of the rational fraction, namely, the transformation of the image bbbb BF nnnnmmmm Let 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 way Let's decompose the image into simple fractions and multiply by: nn KKKKB Let us now strive Then only K remains on the right side: lim BK On the right we have an uncertainty of the form, which is expanded according to L'Hôpital's rule: "BK Substituting, we get" n BB Inverse transformation of a simple fraction known: L Therefore, "n BBL Interest is a special case when one of the roots of the denominator is equal to zero: BF In this case, the decomposition of F into simple fractions will have the form, as follows from the previous," n BBB and B has no roots at zero

6 3 Hence, the inverse Laplace transform of the function F will have the form: n B B B "L Consider another 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, one can compose P + Q equations for P currents in the branches and Q nodal potentials One of the Q nodal potentials is taken to be zero But the number of equations can be reduced on Q, if we use loop currents as alternating currents In this case, the first Kirchhoff's law is automatically fulfilled, since each current enters and leaves the node, that is, it gives a total current equal to zero, and, in addition, Q of the node potentials are expressed through the loop currents The total number of equations, and, therefore, independent loops becomes equal to P + QQ = PQ + Independent equations can be written directly if the loop currents are taken as unknowns. one of the other contours Fig For each of the contours, equations are drawn up according to the second Kirchhoff law a Generally, the branch resistance is equal to i R i C i L where i, =, n, n is the number of independent circuits The equations of loop currents are as follows: 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 EMF included in the i-th circuit. i-th contours Resistances ii represent the sum of the resistances included in the i-th contour Resistance i is part of the resistance of the i-th 33 Fig. Example of independent contours

8 The equation for the m-th circuit will have the form: a circuit that is also included in the th circuit It is obvious that for a passive circuit the equality i = i is true Consider how the equations of circuit currents for active circuits containing transistors are modified, fig mi mi mn I n Em I i Transferring the second term from the right-hand side to the left-hand side, we transform this equation as follows: mi mi I i mn I n Em unknowns, nodal potentials are also used, counted from the potential of one of the nodes, taken as zero. 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, YUY U Y nu n I, Y Y Y Y n

9 The system of equations for the 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 which contains dependent current sources Let us now consider the solutions of the circuit equations The solution of the system of equations of loop currents has the form for the th current: I, where the main determinant of the system is the same determinant in which the ith column is replaced by electromotive forces from the right sides E, E, E n Suppose that there is only one EMF E in the circuit, included in the input circuit, to which the first number is assigned.The equations should be composed so that only one circuit current passes through the branch of interest to us, Fig. 4 Then the input current is equal to IE, where the corresponding algebraic complement determinant i Fig 4 Circuit with EMF in the input circuit 35

10 The ratio EI is called the input resistance.In contrast, this resistance takes into account the influence of all the circuits For the second output circuit, we will have I 36 E, where the corresponding algebraic addition The TIE relationship is called the transmission resistance from the first circuit to the second. 5 Fig 5 A circuit with a current source at the input "UI" I, Y "Y" and the transmission conductance from the first node to the second: U "I" IYT, YT "" where I is the current supplied to the first node, U and U voltage, obtained at the first and second nodes, "is 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 chain, we had = Therefore, the main determinant of the system is symmetric It follows that the algebraic complements are equal: = Therefore, are equal and the transmission resistance T = T This property is called the property of reciprocity. This, as we can see, is the symmetry of the resistance matrix.The property of reciprocity is formulated as follows in Fig. 6: if the EMF located 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 re 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 Fig. 6 Behavior of a circuit with the property of reciprocity 7 UE Fig 7 Voltage transfer coefficient then As follows from the diagram in Fig 7: UUI n; ; K n E T E; I T U n Similarly, the current transfer coefficient can be determined I K I Fig. 8: I Hence I U Yn 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 conductivity resistance, transmission conductivity resistance, etc. For circuits with lumped parameters, any circuit function is rational with respect to the variable and is a fraction m Ф B bnmnbmmnn 38 bb and the coefficients are real Otherwise, it can be represented in the form Ф bmnm, "" "where, m,", "," n roots of the equations mbnmnbmnm, nbb The values ​​=, m are called the zeros of the function Ф, and the values ​​= ",", "n are called poles Φ Obviously, two rational functions, whose zeros and poles coincide, can differ only by constant factors.In other words, the nature of the dependence of the parameters of the chain on frequency is completely determined by the zeros and poles of the chain function. the polynomial acquires the conjugate value * = * and B * = B * It follows that if the polynomial it If there is a complex root, then it will also be a root Thus, the zeros and poles of the chain function can be either real or form complex conjugate pairs Let Ф is the function of the chain Consider its values ​​at =: Ф Ф Ф Since the coefficients in the numerator and denominator Ф are real, then Ф Ф n,

13 No Ф Ф Ф, Ф Ф Ф Comparing these equalities taking into account the equality given above, we obtain that Ф Ф, Ф Ф, that is, the real part of the circuit function is an even function of frequency, and the imaginary odd function of frequency 3 Stability and physical feasibility Consider the equality that determines the current in the input resistance caused by the voltage U: UIB Let U be a unit step, and Then I, B where and B are polynomials from Using the expansion formula, you can get i BB "where zeros of the polynomial B and, therefore, zeros of the resistance function and zeros of the main determinant: = If at least one zero has a positive real part, then i will increase indefinitely. Thus, the resistance, at least one zero of which is in the right half-plane, corresponds to an unstable system, 39

14 me The same conclusion can be made regarding the transmission resistance T, the input conductivity Y, the transmission conductivity YT Definition A circuit function is called physically feasible if it corresponds to a circuit consisting of real elements, and none of the natural vibrations of which has an amplitude that increases indefinitely with The chain specified in the definition is called stable The zeros of the main determinant of the physically realizable stable function of the chain and, therefore, the zeros of the resistance and conductivity functions should be located only in the left half-plane of the variable or on the axis of real frequencies. 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 If, at the same time, =, and m>, then the corresponding solution increases indefinitely o coefficient e transmission, then everything said above refers not to zeros, but to the poles of the function of the transmission coefficient circuit. In fact: n K Zeros of T are the poles of the function K, and the load resistance is passive; its zeros certainly lie in the right plane. From the above, it follows that the physically realizable functions of the chain have the following properties: while the zeros and poles of the chain function are either real or form complex conjugate pairs; b the real and imaginary parts of the chain function are, at real frequencies, an even and odd frequency function, respectively; in the zeros of the main determinant, and, consequently, the conductivity resistance and the transmission conductivity resistance cannot lie in the right half-plane, and multiple zeros neither in the right half-plane nor on the axis of real frequencies T 4

15 3 Transient processes 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 (KE) Of greatest interest is the transient process with an input signal in the form of a step Reaction The system's response to a single step is called the transition function. Knowing the transition function, one can find the response of the system to an input signal of arbitrary shape.The image of a single step has the form, therefore, the response of the system to a single step is: K h L The inverse Laplace transform can be written as: h LKK 4 d At the same time>, since the path of integration should lie to the right of the pole = Of great interest is the definition Fig 3 The contour of the transient function of the amplifier by the type of its integration with the frequency response For this, the path of calculating the transient integration should be combined with the axis of the function of real frequencies = Pole in t point = in this case, you should go around a circle of small radius r Fig. 3: h r K d K r r K r d d r r

16 4 Let's go to the limit r Then we have d KVKK d KV h Here, the expression V with the integral means the main value of this integral The resulting formula allows you to find the transition function through the frequency response of the gain On the basis of this formula, some general conclusions can be drawn. Replace the variable in h with: d KVK h But h, as follows from the principle of causality, since the signal appears at> The gain function K is complex and can be represented as the sum of the real and imaginary parts: K = K + K r Substituting into the expression for h, we obtain d KKVK r Differentiating with respect to, we obtain d KK r or cos sin sin cos d KKKK rr

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 realizable transfer coefficient must satisfy has the form: K cos K sin dr at This condition, as we have seen, follows 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 investigate the integral K hd for< и >Let us introduce two closed contours and B, shown in Fig. 3 Fig. 3 Integration contours: at< ; B при > 43

18 44 Consider a function where the integral is taken over a closed contour Due to the Cauchy integral theorem, the integral is equal to zero, since in the right half-plane the integrand is analytic by the condition.The integral can be written as a sum of integrals over individual sections of the integration contour: sin cos R r R rr RR d RRK rdrr K d K d K h Since cos> at /< < /, 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 RB "which we already had earlier K lim, 45 lim B Example Consider the scheme of the integrating chain shown in Fig. 33 For this chain, the transfer coefficient and its imaginary and real parts have the form: K; K; K r, where RC Let us prove that according to the condition of causality given above, the equality must be satisfied. Equality is known cos sin d cos d Differentiate the right and left sides by: sin d Multiplying the left and right sides of this equality by, we get: sin d, Fig. 33 Scheme of the integrating circuit from which follows the equality that needs to be proved Having the system's transient function, one can find its response to any input signal For this, we approximately represent the input signal as a sum of unit steps Fig. 34

20 Fig. 34 Representation of the input signal This representation can be written as: uuu Next uu "The response to a unit step will be equal to h Therefore, the output signal can be approximately represented as: uuhu" h Passing to the limit at, instead of the sum, we obtain the integral uuhu "hd This one of the forms of the Duhamel integral By integrating by parts, we can obtain another form of the Duhamel integral: uuhuh "d And, finally, by changing the variable =", we can obtain two more forms of the Duhamel integral: uuhu "hd; 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-terminal networks 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 axis of real frequencies Since Y, then the zeros of Y correspond to the poles and vice versa. the function of the input conduction resistance cannot have poles in the right half-plane and multiple poles on the axis of real frequencies. the following asymptotic equality holds: bm mn Since there should not be multiple zeros and poles on the axis of real frequencies, it follows that mn te the powers of the polynomials of the numerator and denominator cannot differ by more than one. Considering the behavior of wb lisi = 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-pole device 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 a complex circuit containing active resistances, capacitances and inductive

If a sinusoidal voltage is applied to the terminals of a two-terminal, then some power is dissipated in the two-terminal, the average value of which P characterizes the dissipation of energy Electric and magnetic energy is stored in capacitors and inductors, the average values ​​of which will be denoted by WE and WH We calculate these values ​​using the equations of loop currents We write directly the expressions for the above quantities by analogy with the simplest cases So, for the resistance R, the average dissipated power is equal to PRII Similarly, for a circuit containing several branches, the average power can be expressed through the loop currents: P i R i I i I Average energy stored in inductance, equal to WHLII For a complex circuit, we express this value through the loop currents: WH 4 i L i I The average energy stored in the capacitor is But Therefore, WEWE i ICUUIUCIIC 4 IIC 48

23 Based on this ratio, you can write an expression for the total average electrical energy: W E 4 Ii I i Ci Let us find out how these quantities are related to the input voltages and currents To do this, write down the equations of loop currents I R I L I E; C I 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 II i E i Ci H 4 W Substituting into the above equality, we obtain E * IP 4 WH 4 WE P 4 WH WE functions

24 Telledzhen's theorem allows you to find expressions for the resistance and conductivity of Y in terms of energy functions: EIEIIIIIEYEEE 5 P WH WIIP WH WEE Some conclusions can be drawn from the expressions obtained for and Y in terms of energy functions.The input resistance and conductance of a passive circuit have a non-negative real part on the axis of real frequencies It is identical is zero only if there are no energy losses in the circuit The stability conditions require that Y also have no zeros and poles in the right half-plane. The absence of poles means that Y are analytic functions in the right half-plane. that if a function is analytic in some region, then its real and imaginary parts reach their smallest and largest values ​​on 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 on the boundary of this region on the axis of real frequencies reaches the smallest value But on the axis of real frequencies the real part is nonnegative, therefore, it is positive in the entire right half-plane. In addition, the functions and Y take real values ​​for real values, since they are the quotient of the division of polynomials with real coefficients A function that takes real values ​​for real 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 device does not contain reactive elements or the average reserves of magnetic and EE;

25 electrical energies in a two-terminal network are the same This is the case with resonance; the frequency at which this takes place is called the resonant frequency.It should be noted that when deriving the energy ratios for and Y, the reciprocity property of the absence of dependent sources was essentially used.For circuits that do not satisfy the principle of reciprocity and contain dependent sources, this formula may turn out to be incorrect. Figure 4 shows a diagram of a series resonant circuit Let's see what the energy formula gives in this simplest case. Power dissipated in resistance R when current I flows is equal to PIR Average reserves of electrical and magnetic energies are equal: WHLICU; W E The voltage U across the capacitor when current I is flowing is From here W E I U C I C Substituting in the energy formula for, we get L I I R I

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 parentheses can be neglected S slope of the lamp Then the input impedance will then be S I E RC E RC I S S RC where Req; Leq SS Fig. 4 Electronic resistance RC SR eq L eq, It is obvious that the calculation of the input resistance using energy functions in this case will give an incorrect result Indeed, there is no magnetic energy reserve in this circuit, which determines the inductance. The reason for the unsuitability of the energy formula for this circuit is the presence in the circuit of a dependent source By selecting the required phase shift in the circuit of the control grid of the lamp, 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. resistance or conductivity of a passive circuit is non-negative on the axis of real 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 in the presence of 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 constant value can be subtracted from the function of resistance or conductivity without violating the conditions of physical feasibility so that the real part, remaining non-negative, turns to zero at some frequency. of poles in the right half-plane of the variable, that is, it is analytic in this region, then its real part has a minimum value at its boundary, that is, on the imaginary axis Therefore, subtracting this minimum value leaves the real part positive in the right half-plane. The function of the input conduction resistance is called a function of the type minimum - the active resistance of conduction, if its real part vanishes on the axis of real frequencies, so that a decrease in this component is impossible without violating the conditions of passivity. then the zero of the real part on the axis of real frequencies has a multiplicity of at least , c and non-minimally active type d In Fig. 43, and the circuit has an input resistance of the 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 minimum active conductivity In Fig. 43, b, the circuit is a circuit of minimum active resistance, since the real part of the resistance vanishes at an infinite frequency 53

28 In Fig. 43, the circuit is a circuit of minimum active resistance R = at the resonance frequency of the series circuit. the circuit in the 3rd circuit has a finite resistance at the resonance frequency 44 Input conductivity resistances of active two-terminal networks Fig. 44 Two-terminal devices: a with an EMF source, b with the addition of resistance R Input conductivity resistances of active, unlike passive two-terminal devices, are not positive functions, and therefore 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 does not have poles there. Consider the circuit shown in Fig. 44, and place exponentially increasing solutions, i.e. two-pole nick is unstable when powered from an EMF source, or, otherwise, when its terminals are short-circuited.On the other hand, since it has no poles in the right half-plane, it is an analytical function in this half-plane It follows that the real part reaches a minimum at the boundary of the right half-plane , i.e., the axes of real frequencies This minimum is negative, since in the opposite case 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 during a short circuit Fig. 44, b.

29 Conductivity Y has zeros in the right half-plane, but has no poles there.This is the case opposite to the previous one, since it means that = / Y has poles in the right half-plane, but does not have zeros there.In this case, stability is investigated in a circuit with a current source Fig. 45, a If Y has zeros in the right half-plane, then the two-terminal network is unstable during no-load operation. Further, we can apply the arguments presented above. Since Y has no poles in the right half-plane, the function Y can be made a real positive function by adding a positive real conductivity G Gmin Thus way of a two-terminal device, in which the conductivity Y has zeros in the right half-plane, but does not have poles there, can be made stable by adding a sufficiently large real conductivity. from voltage source 3 The function has zeros and poles in the right half-plane.In this case, for solving 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 has no 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 device 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 the parallel connection of a negative resistance R with a capacitance C Fig. 46 RCR Here R RC CI 55 Y b G Fig. 45 Two-pole networks: a with a current source; b with the addition of conductivity Y Y Fig. 46 Two-pole with negative resistance I

30 As you can see, it does not have zeros in the right half-plane, therefore such a circuit is stable when powered from a voltage source But it is unstable at no-load Let's add inductance L in series Then Fig. 47 Equivalent circuit of a tunnel diode RRL 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 in series R Fig. 47 Then R LCR RRC LRRLR RC RC The stability condition consists in the absence of numerator zeros in the right half-plane For this, all coefficients of the trinomial in the numerator must be positive: RR CL; RR These two inequalities can be written as: L CR RR Obviously, such inequalities are possible if LLR or R RC C R under the condition R The circuit in Fig. 47 is equivalent to the C circuit of the tunnel diode.

31 possibilities of stabilizing the operating mode of a tunnel diode using an external resistance Example Consider an LC circuit with a parallel connected negative resistance Fig. 48 Find the conditions for the stability of the circuit at no load To do this, calculate the conductivity: 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 45 certain limits without violating the conditions of passivity Physically, this change in the real component by a constant value means the addition or exclusion of a 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 realizability oddness of the imaginary component of the circuit function Physically, this is explained by the fact that there are no elements with a purely reactive frequency-independent conductivity resistance.However, a change in the reactive component without a change in the active component possible in the case when the conductivity resistance has poles on the axis of real frequencies. Due to the conditions of physical feasibility, such poles should be simple and complex conjugate

32 Let the resistance have poles at frequencies Then we can distinguish the simple fractions MNBB It is easy to see that NNMMN r MB r 58 B * M, MM Consider the behavior of one of the fractions, for example, M / near = Then MMM r M r M Near the frequency, the real component changes 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 put = +, and> Then the fraction takes on the value M /, which must be greater than zero, since the fraction must be in the right half-plane a real positive function So, M = N> Thus, if it has complex-conjugate poles on the axis of real frequencies, then it can be represented in the form: MM, B and satisfies the conditions of physical feasibility if they are satisfied Really , has no poles in the right half-plane, since it does not have poles there.Therefore, it is an analytic function in the right half-plane.On the other hand, the first term takes on The axes of real frequencies are purely imaginary values ​​Therefore, they have the same real parts on the axes of real frequencies Separation of the first term does not affect the real part on the axes of real frequencies It follows that in the right half-plane is also a positive function r

33 In addition, it takes real real values ​​in the right half-plane for real values ​​Consequently, it is a real positive function M Resistance is possessed by a parallel resonant circuit without losses: LCCC, LC LC and LC and MC : M "Y, YM" where the expression represents the conductivity of the series resonant circuit: YCLLCL In addition to the poles at the points ±, that is, at finite frequencies, poles are possible at zero and infinite frequencies. These poles correspond to the terms :, L, Y, YC, CL t does not correspond to capacitance or inductance The following statement is true Input impedance the conductance of the passive circuit continues to satisfy the conditions of physical feasibility if 59

34 subtract from it the conductivity reactance corresponding to the poles located on the axis of real frequencies. poles of resistance and conductivity at no 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 46 Properties of circuits composed of purely reactive elements It often happens that a circuit is composed from elements with small losses In this case, the influence of losses can sometimes be neglected It is of interest to find out the properties of circuits without losses, as well as to find out under what conditions losses can be neglected 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 of losses is equal to 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, 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 axis of real frequencies. Physically, this is understandable, since in a circuit without losses, free oscillations do not damp.It follows that using the method of identifying the poles lying on the axis of real frequencies, it is possible to reduce the functions and Y to the following form: bnbnb Y In other words, a two-pole device with resistance can be represented as the following diagram in Fig. 49 of Foster's form:; Fig. 49 The first Foster form Accordingly, Y can be represented in the form of the -th Foster form Fig. 4 Fig. 4 The second Foster form It can be shown that zeros and poles on the axis of real frequencies should alternate be only simple, then near zero the function can be represented in the form M o, where o is a quantity of higher order of smallness compared to Near in the right half-plane, the real quantity must be positive, and this is possible only if M is real 6

36 is a magnitude, and M> Therefore, near zero = the imaginary component can change only with a positive derivative, changing the sign from to "+" there must be a discontinuity, which for circuits with lumped elements can only be a pole All that has been said also applies to the conductivity Y Zeros are called points of resonances, the poles are points of antiresonances 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 the points of resonances and at points of antiresonances, the average reserves of electric and magnetic energies are equal to each other Indeed, at the points of resonances =, i.e. WHWE = At ​​points of antiresonances Y =, therefore, WEWH = Let us now show that in the case of circuits without losses the following formulas take place, I give dependence of resistance and conductivity on frequency Let's represent resistance and conductivity in the form: X, Y B Then: dx WH W d I db WH WE d E For proof, consider the definition of resistance E I 6 E; Let E = cons Let us differentiate by frequency: d E di d I d Suppose that E is a real value Then for a circuit without losses 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 obtained also in p 4 for lossless circuits: i, L i I Ii ii, IIC ii E Differentiating by frequency at E = cons, we get: III id Li I Ii Li IdIi i, i, Ci i, I di di IL di IE di CC iiiii, ii, i, i di I di IL di IL di I niiiiiii, i, Ci i, i, Ci E di E di, since E is a real value by assumption It also follows from the above that: i, LI i di ii, IdI C ii E di di i 63

38 Substituting into the total, we get: di, L i I Ii i, IIC ii 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 was found in section n 4, equals i, L i I Ii i, Ii IC i 4 WHWE di Substituting in the expression for the derivative of the resistance function, we get: d E di WH W d I d I Similarly, you can prove the second equality dy W d 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.Depending on the presence of zeros and poles at zero and infinite frequencies, the graph of the dependence of X and B can have one of the following types, shown in Fig. 4 Finally, we will try to find out how the presence of small losses affects the resistance of a circuit composed of reactive elements.<<, <, 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.

40 Since we are interested in the values ​​on the axis of real frequencies, it should be replaced by In the numerator, we can discard, small in comparison with by the condition: This expression can be transformed as follows :, Qx "where; Q; x; The quantity Q >> is called the quality factor, the quantity x is called the relative detuning Near resonance In addition, we have: The value C x QQ;; QQCC is called the characteristic impedance of the resonant circuit. Consider how the real and imaginary parts of the resistance near resonance depend on the frequency: QQ x R; Im Q x Q x 66

41 Near resonance Im increases, but at resonance it passes through zero with a negative derivative The real part of R at resonance has a maximum. The graphs Im and R depending on the frequency are shown in Fig. 4. speaking, the area under the resonance curve R does not depend on the Q factor. With increasing Q factor, 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, that is, it changes in the same way as in the case of a lossless contour

42 So, the dependence on frequency with the introduction of small losses changes little at frequencies spaced from the resonance frequency by the amount >>. Near the frequency, the course changes significantly. The conduction pole Y, ie the conductivity of the series resonant circuit corresponds to a relation similar to the pole: where Q; 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 axis of real frequencies as follows: Qx x, Y gq Q where = / g changes near zero in the same way as before 68

43 5 Quadrupoles 5 Basic equations of a quadripole A quadrupole is a circuit that has two pairs of terminals: the input to which the signal source is connected and the 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, and T changes It is desirable to have equations and parameters that characterize the four-port network itself. The coefficient is the reciprocal of the transmission conductivity at idle on the output pair of terminals: 69 II; Fig. 5 Turning on the four-port network I Here U and U are the voltages at the input and output terminals, I and I are currents flowing through the input and output terminals towards the four-port network, see Fig. 5 Coefficients of the system of equations connecting voltages and currents have a simple meaning.The value is the proportionality coefficient between I and U at a current at the output terminals I =, that is, at no load at the output terminals; in other words, this is the input resistance at no-load at the output = x Similarly, this is the input resistance from the side of the output terminals at no-load at the first pair of terminals = x The coefficient has the meaning of the value opposite to the transmission conductance at idle at the first pair of terminals, i.e. at zero current input terminals U and IYT x YT x

44 I U; Y T x Y T x Note that for a passive four-port network, both transmission conductivities are equal to each other due to the principle of reciprocity. Therefore, = = / Y Tx The system of equations given above can be written as: I U x I; YT x IU x I YT x I, since the current in this case is directed from a four-port network, that is, in the opposite direction compared to the one adopted above Substituting U into the second equation, we get whence I, I n I x I YTx IY x Tx Substituting I into the first equation, we get UI x Y Tx n From here we find the input impedance in n x U x IY By analogy, you can also write an expression for the output resistance, swapping the indices and: T x n x 7

45 out x YT x n x 5 Characteristic parameters of a four-pole device Of 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 in the expressions for in and out , we get the equations that allow us to find c and c: cc x x YT x YT x 7 cc This system is solved as follows From the first equation we find: whence cc x x; x, Y Tx c x x YT x x YTx x c x kz c x kz x

46 Note that short-circuit and short-circuit 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.Load equal to the characteristic impedance c is called matched. With any number of four-port networks switched on in this way, the matching is preserved in any cross-section. UI c I c ln I c U cg ln U The real part of the characteristic transmission coefficient for real frequencies is called the characteristic attenuation, and the imaginary part is called the characteristic phase constant get also the ratio: I g I; U c g U U U I I

47 The characteristic transmission coefficient is convenient in that with a matched cascade connection of two-port networks, the resulting transmission coefficient is equal to the sum of the transmission coefficients of individual four-port networks.The characteristic transmission coefficient can be found from the relations: gc kz c kz xx c xx cc kz c kz xx c xx c The characteristic impedances c and c, generally speaking, depend on frequency Therefore, the use of characteristic parameters is not always convenient for representing the transmission resistance T. a four-terminal network 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 UI ln, UI where U "and I" are and the current that the generator is capable of developing at a resistance equal to the internal resistance of the generator, i.e.: EU, IE, R 73 EUI, 4R U and I voltage and load current In this case, U = IR Substituting, we obtain for the operating transmission coefficient ln From here we get 4R ERI ln ERRTIRR

48 The value is a function of the complex variable For real frequencies =: = + B, where the operating attenuation, B is the phase constant The operating attenuation is equal to ln TRR 74 ln PP mx, since P mx is the maximum power that the generator can give to the input of the four-port network, and P is the power, allocated on the load RP mx EPIR 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 in the right half-plane. analyticity, in this case on the axis of real frequencies The inverse value reaches the smallest value on this axis For a passive four-port on the axis of real frequencies, therefore R> in the entire right half-plane Further T ln 4R R Function T is the quotient of dividing two polynomials with real coefficients, and T takes real positive e values ​​for real Therefore, it is also real for real values ​​Thus, we can conclude that a real positive function The problem of synthesis of a four-port network with a given operating transmission coefficient in the general case is best solved with the help of the so-called crossed four-port network, which under certain conditions has T

4.11. Laplace transform properties. 1) One-to-one correspondence: s (S И (2) Linearity of the Laplace transform: s И () И 1 (s2 (S1 S2 (and also 3) Analyticity S И (): 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 drawing up equations of state 4 Conclusions Equations of state of electrical

4 .. Properties of the Laplace transform.) One-to-one correspondence: S И () 2) Linearity of the Laplace transform: s (s () И () И 2 S S2 (), and also 3) Analyticity S И (): if satisfies condition

64 Lecture 6 OPERATIONAL METHOD OF ANALYSIS OF ELECTRIC CIRCUITS Plan Laplace transform Properties of Laplace transform 3 Operator method of analyzing electrical circuits 4 Determination of the original by the known

2.2. Operator method for calculating transients. Theoretical information. Calculation of transient processes in complex circuits by the classical method is very often difficult to find the integration constants.

70 Lecture 7 OPERATOR FUNCTIONS OF CIRCUITS Plan Operator input and transfer functions Poles and zeros of circuit functions 3 Conclusions Operator input and transfer functions An 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 changes over time according to the law of a harmonic (sinusoidal) function. The sources of harmonic EMF 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

Transient 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 when the currents, like the voltage drops on individual elements, are constant over 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 vibrations. 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 changes over time according to the law of a harmonic (sinusoidal) function.

Lecture 3. Deductions. The main theorem on residues The residue of a 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 Oscillatory circuit a circuit consisting of inductance coils connected in series, a capacitor of capacitance C and a resistor

1 5 Electrical oscillations 51 Oscillatory circuit Oscillations in physics are called not only periodic movements of bodies but also any periodic or almost periodic process in which the values ​​of one or

Passive circuits Introduction The problems consider the calculation of amplitude-frequency, phase-frequency and transient characteristics in passive - circuits. To calculate the named characteristics, you need to know

STUDY OF FREE AND FORCED VIBRATIONS IN AN OSCILLATORY CIRCUIT Free electrical vibrations in an oscillatory circuit Consider an oscillatory circuit consisting of series-connected capacitors

Lecture 3 Topic Oscillatory systems Sequential oscillatory circuit. Resonance of voltages A series oscillating circuit is a circuit in which a coil and a capacitor are connected in series

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Materials for self-study in the discipline "Theory of electrical circuits" for students of specialties: -6 4 s "Industrial electronics" (part), -9 s "Modeling and computer design

Complex amplitude method Harmonic voltage fluctuations at the terminals of the elements R or cause the flow of 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 preparation for laboratory work 6, 7, 8 in the laboratory "Electricity and Magnetism"

54 Lecture 5 Fourier transform and the spectral method for the analysis of electrical circuits Plan Spectra of aperiodic functions and the Fourier transform Some properties of the Fourier transform 3 Spectral method

Exam Voltage resonance (continued) i iω K = K = ω = = ω => r + iω + r + i ω iω r + ω K = ω r + ω The denominator is minimal at the frequency ω 0, such that ω0 = 0 => ω0 ω 0 = this frequency is called resonant

Chapter 2. Methods for calculating transient processes. 2.1. The classical method of calculation. Theoretical information. In the first chapter, methods for calculating a circuit in a steady state were considered, that is

Yastrebov NI KPI RTF cafe TOP wwwystrevkievu Schematic functions The apparatus of circuit functions is applicable both for the analysis of circuits on direct and harmonic currents and for an arbitrary type of influence In a steady state

4.9. 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 Suppose that K jω possesses the Fourier transform h K j If there exists IH 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 inhomogeneous equations Definition 9 Linear

Methodical development Problem solving by 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 FOR CALCULATION OF TRANSIENTS Section CALCULATION OF TRANSIENTS WITH RANDOM INPUTS USING OVERLAY INTEGRALS 9 CONTROL ISSUES7

4 ELECTROMAGNETIC VIBRATIONS AND WAVES An oscillatory circuit is an electrical circuit composed of capacitors and coils in which an oscillatory process of capacitor recharge is possible.

3.5. Complex parallel oscillatory circuit I A circuit in which at least one parallel branch contains reactivities of both signs. I С С I I There is no magnetic connection between and. Resonance condition

LECTURE N38. The behavior of an analytic function at infinity. Special points. Residues of a function ... a neighborhood of an infinitely distant point ... a Laurent expansion in a neighborhood of an infinitely distant point .... 3. Behavior

4 Lecture 3 FREQUENCY CHARACTERISTICS OF ELECTRIC CIRCUITS Complex transfer functions Logarithmic frequency characteristics 3 Conclusion Complex transfer functions (complex frequency characteristics)

Fluctuations. Lecture 3 Alternator To explain the principle of operation of an alternator, let us first consider what happens when a flat turn of a wire rotates in a uniform magnetic

DIFFERENTIAL EQUATIONS General concepts

Calculation of the source of harmonic oscillations (GCI) Provide the initial circuit of the GCI relative to the primary winding of the transformer with an equivalent voltage source. Determine its parameters (EMF and internal

Work 11 STUDY OF FORCED VIBRATIONS AND THE PHENOMENA OF RESONANCE IN AN OSCILLATING CIRCUIT In a circuit containing an inductor and a capacitor, electrical oscillations can occur. The work is studying

Topic 4 .. AC circuits Topic questions .. AC circuit with inductance .. AC circuit with inductance and active resistance. 3. AC circuit with capacity. 4. Chain variable

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 circuit section 4 Conclusions The task of analyzing electrical

Variant 708 A source of sinusoidal EMF e (ωt) sin (ωt ψ) operates in the electrical circuit. The circuit diagram shown in Fig .. The effective value of the EMF E 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 Kirgoff's laws (constant voltage) 1. Looking for nodes Node point , in which three (or more) conductors are connected

LECTURE Oscillation. Forced oscillations Fig. The oscillation source M athcale feeds a series oscillatory circuit consisting of a resistance R, an inductor L and a capacitor with a capacitance

Exam Resonance of voltages (continued) We will assume that the voltage across one circuit is the voltage across the entire oscillatory circuit, and the voltage at the output of the circuit is the voltage across 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 the spectral method for the analysis of electrical circuits Plan Spectra of aperiodic functions and the Fourier transform 2 Some properties of the Fourier transform 3 Spectral method

Topic: The laws of alternating current Electric current is the ordered movement of charged particles or macroscopic bodies.A variable is a current that changes its value over time

Exam Impedance Impedance Impedance or complex impedance is by definition equal to the ratio of the complex voltage to the complex current: Z ɶ Note that the impedance is also equal to the ratio

Table of contents Introduction. Basic concepts .... 4 1. Integral equations of Volterra ... 5 Variants of homework .... 8 2. Resolvent of the integral equation of Volterra. 10 Homework options ... 11

Chapter II Integrals Antiderivative function and its properties The function F () is called the antiderivative of a continuous function f () on the interval a b, if F () f (), a; b (;) For example, for the function f () antiderivatives

The classic method. Fig. 1- the initial diagram of the electrical circuit Circuit parameters: E = 129 (V) w = 10000 (rad / s) R1 = 73 (Ohm) R2 = 29 (Ohm) R3 = 27 (Ohm) L = 21 (mgn) C = 0.97 (μF) Inductance reactance:

Methods for calculating complex linear electrical circuits Basis: the ability to compose and solve systems of linear algebraic equations - compiled either for a direct current circuit, or after symbolization

A SPECIFIC INTEGRAL. Integral Sums and a Defined Integral Let there be given a function y = f () defined on the interval [, 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 circuit functions 3 Conclusions Operator input and transfer functions An operator function of a chain is a relation

68 Lecture 7 TRANSITION 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. SINUSOIDAL CURRENT GENERATION PRINCIPLE 4.1.012. Sinusoidal current is called instantaneous

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~ ~ FKP Derivative of the function of a complex variable FKP of the Cauchy - Riemann condition the concept of regularity of the FKP Image and form of a complex number Form of the FKP: where the real function of two variables is real

This is the name of another type of integral transforms, 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.

Complex frequency concept.

Spectral methods, as is already known, are based on the fact that the signal under investigation is represented as a sum of an infinitely large number of elementary terms, each of which periodically changes in time according to the law.

The natural generalization of this principle lies in the fact that instead of complex exponential signals with purely imaginary indicators, 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, in this case

Depending on the choice of the real and imaginary parts of the complex frequency, various real signals can be obtained. So, if, but you get the usual harmonic oscillations of the form If, then, depending on the sign, you get either increasing or decreasing exponential oscillations in time. Such signals acquire a more complex form when. Here, the multiplier describes an envelope that changes exponentially over time. Some typical signals are shown in fig. 2.10.

The concept of a complex frequency turns out to be very useful, first of all, 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 Ch. eight.

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 of the complex frequency.

Basic relationships.

Let be some signal, real or complex, defined at t> 0 and equal to zero at negative time values. The Laplace transform of this signal is a function of a complex variable given by an integral:

The signal is called the original, and the function is called its Laplace image (for short, just the image).

The condition that ensures the existence of the integral (2.54) is as follows: the signal must have no more than an exponential growth rate, i.e., 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, at 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 Fourier transform, it is possible, knowing the image, to restore the original. For this, in the inverse Fourier transform formula

an analytical continuation should be performed, passing from the imaginary variable to the complex argument a. On the plane of the complex frequency, the integration is carried out along an infinitely long vertical axis located to the right of the abscissa of absolute convergence. Since at is the 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 from the point of view 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, as a rule, are poles, single or multiple. Therefore, to calculate integrals of the form (2.55), one can use flexible methods of the theory of residues.

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 Appendices to there is such a table that allows you to solve a fairly wide range of problems.

Examples of calculating Laplace transforms.

Image computation methods have much in common with what has already been studied in relation to the Fourier transform. Let's consider the most typical cases.

Example 2.4, Image of the 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 will vanish when the upper limit is substituted. As a result, we get the correspondence

As a special case of formula (2.56), you can find the image of a real exponential video pulse:

and a complex exponential signal:

Finally, putting in (2.57), we find the image of the Heaviside function:

Example 2.5. Delta function image.

Previously, we considered the integral Fourier transform with kernel K (t, О = е 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 get rid of this constraint. Definition 1. Function an original will mean any complex-valued function f (t) of a real argument t, satisfying the following conditions: a finite interval of axes * of such points can be only a finite number; 2.function f (t) is equal to zero for negative values ​​of t, f (t) = 0 for 3. as t increases, the modulus f (t) increases no faster than an exponential function, i.e., there exist numbers M> 0 and s such that for all t It is clear that if inequality (1) holds for some s = aj, then it will also hold for ANY 82> 8]. = infs for which inequality (1) , is called the growth rate of the function f (t). Comment. In the general case, the inequality does not hold, but the estimate is valid where e> 0 is any. So, the function has an exponent of growth в0 = 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 an original function: it has an infinite order of growth, “o = + oo. The simplest original function is the so-called unit function. If some 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), having agreed that all functions that we will consider are equal to zero for negative t, so if we are talking about some function f (t), for example, o 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) be the original function. The image of the function f (t ) by Laplace 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 for operational calculus Duhamel's formula Integration of systems of linear differential equations with constant coefficients Solution of integral equations where the integral is taken over the positive semiaxis t. The function F (p) is also called the Laplace transform of the function / (/); the kernel of the transformation K (t) p) = e ~ pt. The fact that the function has its image F (p), we will write Example 2. Find the image of the unit function r) (t). The function is an original function with a growth rate of 0 - 0. By virtue of formula (2), the image of the function rj (t) will be the function If then for, the integral on the right-hand side of the last equality will converge, and we will get 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 exponent z0, the image F (p) is defined in the half-plane R ep = s > s0 and is an analytic function in this half-plane (Fig. 3). Let To prove the existence of the image F (p) in the indicated half-plane, it is sufficient to establish that the improper integral (2) converges absolutely for a> Using (3), we obtain which proves the absolute convergence of the 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 that 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 which implies the absolute convergence of the integral (5). (The non-integral term, 0., - has a zero limit for t + oo). 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 valid. Since the derivative F "(p) exists, the Laplace transform F (p) everywhere in the half-plane Rep = 5> 5о is an analytical function. Inequality (4) implies Corollary. If thin 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 f (() is equal to a. > a, but also at all points p, except for the point p = a, where this image has a simple pole. In the future, we will more than once encounter a similar situation when the image F (p) is an analytic function in the entire plane of the complex variable p, for excluding isolated singular points. There is no contradiction with Theorem 1. The latter only asserts that in the half-plane Rep> «o the function F (p) has no singular points: they all turn out to lie either to the left of the line Rep = so, or on this line itself. Notice not. In operational calculus, the Heaviside image of the function f (f) is sometimes used, which is defined by equality and differs from the Laplace image by the factor p. §2. Properties of the Laplace transform In what follows we will denote the original functions, and through - their images according to Laplace. £ biw dee are continuous functions) have the same image, then they are identically equal. Teopewa 3 (n "yeyiost * transforming Laplace). If the functions are original, then for any complex constants of the air The validity of the statement follows from the linearity property of the integral that determines the image:, are the growth rates of 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> 0 Putting at = m, we have Using this theorem, from formulas (5) and (6) we obtain Theorem 5 ( on the differentiation of the original). Let be the original function with the image F (p) and let - also be the original functions, and where is the growth rate of the function Then and in general Here, we mean the right limiting value Let. Let us find the image We have Integrating by parts, we obtain The non-integral term on the right-hand side of (10) vanishes at k. For Rc p = s> h, we have the substitution t = Odet - / (0). The second term on the right in (10) is equal to pF (p). Thus, relation (10) takes the form and formula (8) is proved. In particular, if To find the image f (n \ t) we write whence, integrating n times by parts, we get Example 4. Using the theorem on the differentiation of the original, find the image of the function f (t) = sin2 t. Let Therefore, Theorem 5 establishes a remarkable property of the Laplace integral transform: it (like the Fourier transform) transforms the operation of differentiation into an algebraic operation of multiplication by p. Inclusion formula. If they are original functions, then Indeed, By virtue of the corollary to Theorem 1, every image tends to zero as. Hence, whence the inclusion formula follows (Theorem 6 (on the differentiation of the image). Differentiation of the image is reduced to multiplication by the original, Since the function F (p) in the half-plane so is analytical, it can be differentiated with respect to p. We have the latter just means that Example 5. Using Theorem 6, find the image of function 4 As you know, Hence (Again applying Theorem 6, we find, in general, Theorem 7 (integration of the original). Integration of the original is reduced to dividing the image by that if there is an original function, then it will be an original function, moreover. Let. By virtue 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 In this case, so that Therefore, Theorem 8 (image integration) .If the integral also converges, it serves as an image of the function ^: LAPLACE TRANSFORM Basic definitions Properties Convolution of functions Multiplication theorem Finding the original by the image Using the inverse theorem of operational calculus Duhamel's formula Integration of systems of linear differential equations with constant coefficients Solution integral equations Indeed, assuming that the path of the integro lie on the half-plane so, we can change the order of integration. The last equality means that it is an image of a function Example 7. Find an image of a function M As is known,. Therefore, since we put, we obtain £ = 0, for. Therefore, relation (16) takes the form Example. Find the image of the function f (t), given graphically (Fig. 5). Let's write the expression for the function f (t) as follows: This expression can be obtained as follows. Consider the function and subtract the function from it. The difference will be equal to one for. We add the function to the resulting difference. As a result, we obtain the function f (t) (Fig. 6c), so that From here, using the delay theorem, we find Theorem 10 (displacement). then for any complex number p0 Indeed, the theorem allows, from known images of functions, to find images of the same functions multiplied by an exponential function, for example, 2.1. Convolution of functions. Multiplication theorem Let the functions f (t) u be defined and continuous for all t. The convolution of these functions is a new function of t defined by equality (if this integral exists). For original functions, the operation is always collapsible, and (17) 4 Indeed, 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 interval. For finite continuous functions, the convolution operation is satisfiable, and we obtain the formula It is easy to verify that the convolution operation is commutative, Theorem 11 (multiplication). If, then the convolution t) has an image It is easy to check that the convolution (of the original functions is the original function with the growth index "where, are the growth indexes of the functions, respectively. such an operation is legal) and applying the lagging theorem, we obtain Thus, from (18) and (19) we find that the multiplication of images corresponds to the folding of 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 f (t) be a periodic function with period T. Show that its Laplace image F (p) is given by formula 3. Finding the original from the image The problem is posed 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) 1) analytic in the half-plane so tends to zero for in any half-plane R s0 uniformly with respect to arg p; 2) the integral converges absolutely, then F (p) is an image of some original function Problem. Can the function F (p) = serve as an image of some original function? Here are some ways to find the original from the 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) is a fractional rational function of argument p, it is decomposed into elementary fractions and the appropriate properties of the Laplace transform are used. Example 1. Find the original for Let us 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 Let us write F (p) as Hence 3.2. Use of the inversion theorem and its consequences. Theorem 13 (inversion). If the function fit) is an 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 holds 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, suppose, for example, f (t) is piecewise smooth on every finite segment (\ displaystyle F (s) = \ varphi), so φ (z 1, z 2,…, z n) (\ displaystyle \ varphi (z_ (1), \; z_ (2), \; \ ldots, \; z_ (n))) analytic about each z k (\ displaystyle z_ (k)) and is equal to zero for z 1 = z 2 =… = z n = 0 (\ displaystyle z_ (1) = z_ (2) = \ ldots = z_ (n) = 0), and F k (s) = L (fk (x)) (σ> σ ak: 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 forward transformation has the absolute convergence abscissa.

Note: these are sufficient conditions for existence.

• Convolution theorem

Main article: Convolution theorem

• Differentiating and integrating the original

The Laplace image of the first derivative of the original with respect to the argument is the product of the image by 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 using a simple relation. This is, for example, used to analyze the stability of the trajectory of a dynamical 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).)

Multiplication by a 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 a table of Laplace transform 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 region for causal systems
1	perfect lag	δ (t - τ) (\ displaystyle \ delta (t- \ tau) \)	e - τ s (\ displaystyle e ^ (- \ tau s) \)
1a	single impulse	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	sedate 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	sedate 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	unit function	H (t) (\ displaystyle H (t) \)	1 s (\ displaystyle (\ frac (1) (s)))	s> 0 (\ displaystyle s> 0)
2b	lag unit function	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) - ns 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) - ns 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 arsh (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)
Notes to the table: H (t) (\ displaystyle H (t) \); α (\ displaystyle \ alpha \), β (\ displaystyle \ beta \), τ (\ displaystyle \ tau \) and ω (\ displaystyle \ omega \) - Relationship with other transformations Fundamental connections Mellin transform The Mellin transform and the inverse Mellin transform are related to the two-sided Laplace transform by a simple change of variables. If in the Mellin transform G (s) = M (g (θ)) = ∫ 0 ∞ θ sg (θ) θ d θ (\ displaystyle G (s) = (\ mathcal (M)) \ left \ (g (\ theta) \ right \) = \ int \ limits _ (0) ^ (\ infty) \ theta ^ (s) (\ frac (g (\ theta)) (\ theta)) \, d \ theta) put θ = e - x (\ displaystyle \ theta = e ^ (- x)), then we get a two-sided Laplace transform. Z-transform Z (\ displaystyle Z)-transform is the Laplace transform of a lattice function, produced by changing variables: z ≡ e s T, (\ displaystyle z \ equiv e ^ (sT),) Borel transform The integral form of the Borel transform is identical to the Laplace transform, there is also a generalized Borel transform, with the help of which the use of the Laplace transform is extended to a wider class of functions. Bibliography Van der Pol B., Bremer H. Operational calculus based on the two-sided Laplace transform. - M.: Publishing house of foreign literature, 1952. - 507 p. Ditkin V.A., Prudnikov A.P. Integral transformations and operational calculus. - M.: Main edition of physical and mathematical literature of the publishing house "Nauka", 1974. - 544 p. Ditkin V.A., Kuznetsov P.I. Operational Calculus Handbook: Fundamentals of Theory and Formula Tables. - M.: State publishing house of technical and theoretical literature, 1951. - 256 p. Carslow H., Jaeger D. Operational Methods in Applied Mathematics. - M.: Publishing house of foreign literature, 1948. - 294 p. Kozhevnikov N.I., Krasnoshchekova T.I., Shishkin N.E. Fourier series and integrals. Field theory. Analytical and special functions. Laplace transforms. - M.: Nauka, 1964 .-- 184 p. M. L. Krasnov, G. I. Makarenko Operational calculus. Stability of movement. - M.: Nauka, 1964 .-- 103 p. Mikusinsky Y. Operator calculus. - M.: Publishing house of foreign literature, 1956. - 367 p. Romanovsky P.I. Fourier series. Field theory. Analytical and special functions. Laplace transforms. - M.: Nauka, 1980 .-- 336 p.