Transmission function. Tutorial: Transient and impulse characteristics of electrical circuits Sampling of the input signal and impulse response

Academy of Russia

Department of Physics

Lecture

Transient and impulse characteristics electrical circuits

Eagle 2009

Educational and educational goals:

Explain to students the essence of the transient and impulse characteristics of electrical circuits, show the connection between the characteristics, pay attention to the use of the characteristics under consideration for the analysis and synthesis of electrical circuits, and aim at high-quality preparation for practical training.

Distribution of lecture time

Introductory part……………………………………………………5 min.

Study questions:

1. Transient characteristics of electrical circuits………………15 min.

2. Duhamel integrals……………………………………………………………...25 min.

3. Pulse characteristics of electrical circuits. Relationship between characteristics………………………………………….………...25 min.

4. Convolution integrals………………………………………….15 min.

Conclusion……………………………………………………5 min.

1. Transient characteristics of electrical circuits

The transient response of a circuit (like the impulse response) refers to the temporary characteristics of the circuit, i.e., it expresses a certain transient process under predetermined influences and initial conditions.

To compare electrical circuits by their response to these influences, it is necessary to place the circuits in the same conditions. The simplest and most convenient are zero initial conditions.

Transient response of the circuit is the ratio of the reaction of a chain to a stepwise impact to the magnitude of this impact at zero initial conditions.

A-priory ,

where is the chain response to stepwise action;

– the magnitude of the step effect [B] or [A].

Since and is divided by the magnitude of the impact (this is a real number), it is actually the reaction of the circuit to a single step impact.

If the transient response of the circuit is known (or can be calculated), then from the formula you can find the reaction of this circuit to a stepwise effect at zero NL

.

Let us establish a connection between the operator transfer function of a circuit, which is often known (or can be found), and the transient response of this circuit. To do this, we use the introduced concept of operator transfer function:

.

The ratio of the Laplace-transformed reaction of the chain to the magnitude of the impact is the operator transient characteristic of the chain:

Hence .

From here the operator transition characteristic of the circuit is found using the operator transfer function.

To determine the transient response of the circuit, it is necessary to apply the inverse Laplace transform:

using the correspondence table or (preliminarily) the decomposition theorem.

Example: determine the transient response for the voltage response on a capacitor in a series circuit (Fig. 1):

Here is the reaction to a stepwise effect of magnitude:

,

where does the transition characteristic come from:

.

The transient characteristics of the most frequently encountered circuits are found and given in reference literature.

2. Duhamel integrals

The transient response is often used to find the response of a circuit to a complex stimulus. Let us establish these relations.

Let us agree that the influence is a continuous function and is applied to the circuit at time , and the initial conditions are zero.

A given impact can be represented as the sum of a stepwise impact applied to the circuit at an instant and indefinitely large number infinitely small step effects, continuously following each other. One of these elementary impacts corresponding to the moment of application is shown in Figure 2.

Let's find the value of the chain reaction at some point in time.

A stepwise effect with a difference at the moment of time causes a reaction equal to the product of the difference by the value of the transient characteristic of the circuit at , i.e. equal to:

An infinitesimal stepwise effect with a difference causes an infinitesimal reaction , where is the time elapsed from the moment of application of the influence to the moment of observation. Since by condition the function is continuous, then:

In accordance with the principle of superposition, the reaction will be equal to the sum of reactions caused by the totality of influences preceding the moment of observation, i.e.

.

Usually in the last formula they simply replace it with , since the found formula is correct for any time values:

.

Or, after some simple transformations:

.

Any of these relationships solves the problem of calculating the response of a linear electrical circuit to a given continuous action using a known transient response of the circuit. These relations are called Duhamel integrals.

3. Pulse characteristics of electrical circuits

Impulse response of the circuit is the ratio of the reaction of a circuit to a pulsed action to the area of ​​this action under zero initial conditions.

A-priory ,

where is the circuit’s response to impulse action;

– impact pulse area.

Using the known impulse response of the circuit, one can find the response of the circuit to a given influence: .

A single impulse effect, also called the delta function or Dirac function, is often used as an impact function.

A delta function is a function equal to zero everywhere except , and its area is equal to unity ():

.

The concept of delta function can be arrived at by considering the limit of a rectangular pulse with height and duration when (Fig. 3):

Let us establish a connection between the transfer function of a circuit and its impulse response, for which we use the operator method.

A-priory:

.

If the influence (original) is considered for the most general case in the form of the product of the impulse area and the delta function, i.e. in the form, then the image of this influence according to the correspondence table has the form:

.

Then, on the other hand, the ratio of the Laplace-transformed reaction of the circuit to the area of ​​the impact impulse is the operator impulse response of the circuit:

.

Hence, .

To find impulse response circuit, it is necessary to apply the inverse Laplace transform:

That is, actually.

Generalizing the formulas, we obtain a connection between the operator transfer function of the circuit and the operator transient and impulse characteristics of the circuit:

Thus, knowing one of the characteristics of the circuit, you can determine any others.

Let us carry out the identical transformation of equality by adding to the middle part.

Then we will have .

Since it is an image of the derivative of the transition characteristic, the original equality can be rewritten as:

Moving to the area of ​​originals, we obtain a formula that allows us to determine the impulse response of a circuit from its known transient response:

If, then.

The inverse relationship between these characteristics has the form:

.

Using the transfer function, it is easy to determine the presence of a term in the function.

If the powers of the numerator and denominator are the same, then the term in question will be present. If the function is a proper fraction, then this term will not exist.

Example: determine the impulse characteristics for voltages and in the series circuit shown in Figure 4.

Let's define:

Using the correspondence table, let's move on to the original:

.

The graph of this function is shown in Figure 5.

Rice. 5

Transmission function :

According to the correspondence table we have:

.

The graph of the resulting function is shown in Figure 6.

Let us point out that the same expressions could be obtained using relations establishing a connection between and .

The impulse response in its physical meaning reflects the process of free oscillations and for this reason it can be argued that in real circuits the following condition must always be satisfied:

4. Convolution (overlay) integrals

Let us consider the procedure for determining the response of a linear electrical circuit to a complex influence if the impulse response of this circuit is known. We will assume that the impact is a piecewise continuous function shown in Figure 7.

Let it be required to find the value of the reaction at some point in time. Solving this problem, let us imagine the impact as a sum of rectangular pulses of infinitesimal duration, one of which, corresponding to the moment in time, is shown in Figure 7. This pulse is characterized by duration and height.

From the previously discussed material it is known that the reaction of a circuit to a short pulse can be considered equal to the product of the impulse response of the circuit and the area of ​​the impulse action. Consequently, the infinitesimal component of the reaction due to this impulse action at the moment of time will be equal to:

since the area of ​​the pulse is equal to , and time passes from the moment of its application to the moment of observation.

Using the principle of superposition, the total reaction of a circuit can be defined as the sum of an infinitely large number of infinitesimal components caused by a sequence of infinitesimals in area impulse influences, preceding the moment in time.

Thus:

.

This formula is true for any values, so usually the variable is simply denoted. Then:

.

The resulting relation is called the convolution integral or the superposition integral. The function that is found as a result of calculating the convolution integral is called the convolution and .

You can find another form of the convolution integral if you change variables in the resulting expression for:

.

Example: find the voltage across the capacitance of a serial circuit (Fig. 8), if an exponential pulse of the form acts at the input:

Let's use the convolution integral:

.

Expression for was received previously.

Hence, , And .

The same result can be obtained by applying the Duhamel integral.

Literature:

Beletsky A.F. Theory of linear electrical circuits. – M.: Radio and Communications, 1986. (Textbook)

Bakalov V.P. et al. Theory of electrical circuits. – M.: Radio and Communications, 1998. (Textbook);

Kachanov N. S. et al. Linear radio engineering devices. M.: Military. published, 1974. (Textbook);

Popov V.P. Fundamentals of circuit theory - M.: Higher School, 2000. (Textbook)

2.3 General properties of the transfer function.

The stability criterion for a discrete circuit coincides with the stability criterion for an analog circuit: the poles of the transfer function must be located in the left half-plane of the complex variable, which corresponds to the position of the poles within the unit circle of the plane

Circuit transfer function general view is written, according to (2.3), as follows:

where the signs of the terms are taken into account in the coefficients a i, b j, with b 0 =1.

It is convenient to formulate the properties of the transfer function of a general circuit in the form of requirements for the physical realizability of a rational function of Z: any rational function of Z can be implemented in the form of a transfer function of a stable discrete chain accurate to the factor H 0 × H Q if this function satisfies the requirements:

1. coefficients a i, b j are real numbers,

2. roots of the equation V(Z)=0, i.e. the poles of H(Z) are located within the unit circle of the Z plane.

The H 0 × Z Q multiplier takes into account the constant amplification of the H 0 signal and the constant shift of the signal along the time axis by the value QT.

2.4 Frequency characteristics.

Complex transfer function of a discrete circuit

determines the frequency characteristics of the circuit

Frequency response, - Phase response.

Based on (2.6), the transfer function complex of general form can be written as follows:

Hence the formulas for frequency response and phase response

The frequency characteristics of a discrete circuit are periodic functions. The repetition period is equal to the sampling frequency w d.

Frequency characteristics are usually normalized along the frequency axis to the sampling frequency

where W is the normalized frequency.

In calculations using a computer, frequency normalization becomes a necessity.

Example. Determine the frequency characteristics of the circuit whose transfer function

H(Z) = a 0 + a 1 ХZ -1 .

Transfer function complex: H(jw) = a 0 + a 1 e -j w T .

taking into account normalization by frequency: wT = 2p Х W.

H(jw) = a 0 + a 1 e -j2 p W = a 0 + a 1 cos 2pW - ja 1 sin 2pW .

Frequency response and phase response formulas

H(W) =, j(W) = - arctan .

graphs of the frequency response and phase response for positive values ​​of a 0 and a 1 under the condition a 0 > a 1 are shown in Fig. (2.5, a, b.)

Logarithmic frequency response scale - attenuation A:

; . (2.10)

The zeros of the transfer function can be located at any point in the Z plane. If the zeros are located within the unit circle, then the characteristics of the frequency response and phase response of such a circuit are related by the Hilbert transform and can be uniquely determined from one another. Such a circuit is called a minimum-phase type circuit. If at least one zero appears outside the unit circle, then the circuit belongs to a nonlinear-phase type circuit, for which the Hilbert transform is not applicable.

2.5 Impulse response. Convolution.

The transfer function characterizes a circuit in the frequency domain. In the time domain, the circuit is characterized by an impulse response h(nT). The impulse response of a discrete circuit is the response of the circuit to a discrete d - function. The impulse characteristic and the transfer function are system characteristics and are interconnected by the Z - transformation formulas. That's why impulse response can be considered as a certain signal, and the transfer function H(Z) - Z is an image of this signal.

The transfer function is the main characteristic in design if the standards are set relative to the frequency characteristics of the system. Accordingly, the main characteristic is the impulse response if the norms are specified in the time domain.

The impulse response can be determined directly from the circuit as the response of the circuit to the d - function, or by solving the difference equation of the circuit, assuming x(nT) = d (t).

Example. Determine the impulse response of the circuit, the diagram of which is shown in Fig. 2.6, b.

The difference circuit equation is y(nT)=0.4 x(nT-T) - 0.08 y(nT-T).

Solving the difference equation in numerical form under the condition that x(nT)=d(t)

n=0; y(0T) = 0.4 x(-T) - 0.08 y(-T) = 0;

n=1; y(1T) = 0.4 x(0T) - 0.08 y(0T) = 0.4;

n=2; y(2T) = 0.4 x(1T) - 0.08 y(1T) = -0.032;

n=3; y(3T) = 0.4 x(2T) - 0.08 y(2T) = 0.00256; etc. ...

Hence h(nT) = (0; 0.4; -0.032; 0.00256; ...)

For a stable circuit, the impulse response counts tend to zero over time.

The impulse response can be determined from a known transfer function using

A. inverse Z-transform,

b. decomposition theorem,

V. delay theorem to the results of dividing the numerator polynomial by the denominator polynomial.

The last of the listed methods refers to numerical methods for solving the problem.

Example. Determine the impulse response of the circuit in Fig. (2.6,b) using the transfer function.

Here H(Z) = .

Divide the numerator by the denominator

Applying the delay theorem to the result of division, we obtain

h(nT) = (0; 0.4; -0.032; 0.00256; ...)

By comparing the result with the calculations using the difference equation in the previous example, you can verify the reliability of the calculation procedures.

It is proposed to independently determine the impulse response of the circuit in Fig. (2.6,a), using sequentially both methods considered.

In accordance with the definition of the transfer function, Z - the image of the signal at the output of the circuit can be defined as the product of Z - the image of the signal at the input of the circuit and the transfer function of the circuit:

Y(Z) = X(Z)ХH(Z). (2.11)

Hence, according to the convolution theorem, convolution of the input signal with the impulse response gives a signal at the output of the circuit

y(nT) =x(kT)Хh(nT - kT) =h(kT)Хx(nT - kT). (2.12)

Determining the output signal using the convolution formula is used not only in calculation procedures, but also as an algorithm for the functioning of technical systems.

Determine the signal at the output of the circuit, the diagram of which is shown in Fig. (2.6,b), if x(nT) = (1.0; 0.5).

Here h(nT) = (0; 0.4; -0.032; 0.00256; ...)

Calculation according to (2.12)

n=0: y(0T) = h(0T)x(0T) = 0;

n=1: y(1T) = h(0T)x(1T) + h(1T) x(0T) = 0.4;

n=2: y(2T)= h(0T)x(2T) + h(1T) x(1T) + h(2T) x(0T) = 0.168;

Thus y(nT) = ( 0; 0.4; 0.168; ... ).

IN technical systems Instead of linear convolution (2.12), circular or cyclic convolution is more often used.

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Carrier and amplitude-phase modulation with single sideband (AFM-SBP). 3. Selection of the duration and number of elementary signals used to generate the output signal In real communication channels, a signal of the form is used to transmit signals over a frequency-limited channel, but it is infinite in time, so it is smoothed according to the cosine law. , Where - ...

The time and frequency characteristics of the circuit are related to each other by Fourier transform formulas. Using the transient response found in paragraph 2.1, the impulse response of the circuit is calculated (Figure 1)

The result of the calculations coincides with the formula H(jш) obtained in section 2.2

Input signal and impulse response sampling

Let it be taken as the upper limit of the spectrum of the input signal. Then, according to Kotelnikov’s theorem, the sampling frequency is kHz. Where does the sampling period T=0.2ms come from?

Using the graph shown in Fig. 2, we determine the values ​​of discrete samples of the input signal U 1 (n) for t sampling moments.

The discrete values ​​of the impulse response are calculated using the formula

where T=0.0002 s; n=0, 1, 2,…., 20.

Table 3. Discrete values ​​of the input signal function and impulse response

The discrete signal values ​​at the output of the circuit are calculated for the first 8 samples using the discrete convolution formula.

Table 4. Discrete signal at the output of the circuit.

A comparison of the calculation results with the data in Table 1 shows that the difference in the values ​​of U 2 (t) calculated using the Duhamel integral and by sampling the signal and impulse response differ by several tenths, which is an acceptable deviation for these initial parameters.

Figure 9. Value discrete signal at the input of the circuit.

Figure 10. The value of the discrete signal at the output of the circuit.

Figure 11. The value of discrete samples of the impulse response of the circuit H(n).

This dynamic characteristic is used to describe single-channel systems

with zero initial conditions

Step response h(t) is the response of the system to a single input step action at zero initial conditions.

The moment of occurrence of the input influence

Fig.2.4. Transient response of the system

Example 2.4:

Transient characteristics for various values ​​of active resistance in an electrical circuit:

To determine the transient response analytically, the differential equation must be solved under zero initial conditions and u(t)=1(t).

For a real system, the transient response can be obtained experimentally; in this case, a stepwise effect should be applied to the input of the system and the reaction at the output should be recorded. If the step effect is different from unity, then the output characteristic should be divided by the value of the input effect.

Knowing the transient response, you can determine the system's response to an arbitrary input action using the convolution integral

Using the delta function, a real input effect such as an impact is modeled.

Fig.2.5. Impulse response of the system

Example 2.5:

Pulse characteristics for various values ​​of active resistance in an electrical circuit:

The transition function and the impulse function are uniquely related to each other by the relations

Transition matrix is the solution to the matrix differential equation

Knowing the transition matrix, you can determine the system response

to an arbitrary input influence under any initial conditions x(0) by expression

If the system has zero initial conditions x(0)=0, That

,

(2.17)

For linear systems with constant parameters transition matrix Ф(t) represents the matrix exponent

For small sizes or simple matrix structure A expression (2.20) can be used to accurately represent the transition matrix using elementary functions. In the case of a large matrix dimension A should be used existing programs to calculate the matrix exponential.

Transmission function

Along with ordinary differential equations in theory automatic control various transformations are used. For linear systems, it is more convenient to write these equations in symbolic form using the so-called differentiation operator

which makes it possible to transform differential equations as algebraic and introduce a new dynamic characteristic - the transfer function.

Consider this transition for multi-channel systems type (2.6)

Let us write the equation of state in symbolic form:

px = Ax + Bu,

which allows us to determine the state vector

It is a matrix with the following components:

(2.27)

Where - scalar transfer functions , which represent the ratio of the output quantity to the input quantity in symbolic form under zero initial conditions

Own transfer functions i th channel are the components of the transfer matrix , which are on the main diagonal. Components located above or below the main diagonal are called cross-link transfer functions between channels.

The inverse matrix is ​​found by the expression

Example 2.6.

Determine the transfer matrix for the object

Let us use the expression for the transfer matrix (2.27) and first find inverse matrix(2.29). Here

The transposed matrix has the form

a det(pI-A) = p -2p+1, .

where is the transposed matrix. As a result, we obtain the following inverse matrix:

and the transfer matrix of the object

Most often, transfer functions are used to describe single-channel systems of the form

where is the characteristic polynomial.

Transfer functions are usually written in standard form:

,

(2.32)

where is the transmission coefficient;

The transfer matrix (transfer function) can also be determined using Laplace or Carson-Heaviside images. If we subject both sides of the differential equation to one of these transformations and find the relationships between the input and output quantities under zero initial conditions, we will obtain the same transfer matrix (2.26) or function (2.31).

In order to further distinguish between transformations of differential equations, we will use the following notation:

Differentiation operator;

Laplace transform operator.

Having received one of the dynamic characteristics of an object, you can determine all the others. The transition from differential equations to transfer functions and back is carried out using the differentiation operator p.

Let's consider the relationship between transition characteristics and transfer function. The output variable is found through the impulse function in accordance with expression (2.10),

Let's expose him Laplace transform,

,

and we get y(s) = g(s)u(s). From here we define the impulse function:

(2.33)

Thus, the transfer function is the Laplace transform of the impulse function.

Example 2.7.

Determine the transfer function of an object whose differential equation has the form

Using the differentiation operator d/dt = p, we write the equation of the object in symbolic form

on the basis of which we determine the desired transfer function of the object

Modal characteristics

Modal characteristics correspond to the free component of the motion of the system (2.6) or, in other words, reflect the properties of an autonomous system of type (2.12)

The system of equations (2.36) will have a non-zero solution with respect to if

.

(2.37)

Equation (2.37) is called characteristic and has n-roots, which are called eigenvalues matrices A. Substituting the eigenvalues ​​into (2.37) we obtain

.

where are the eigenvectors,

The set of eigenvalues ​​and eigenvectors is modal characteristics of the system .

For (2.34), only the following exponential solutions can exist

To obtain the characteristic equation of the system, it is sufficient to equate the common denominator of the transfer matrix (transfer function) to zero (2.29).

Frequency characteristics

If a periodic signal of a given amplitude and frequency is applied to the input of an object, then the output will also have a periodic signal of the same frequency, but in the general case of a different amplitude with a phase shift. The relationship between the parameters of periodic signals at the input and output of the object is determined frequency characteristics . Most often they are used to describe single-channel systems:

and is presented in the form

.

(2.42)

The components of the generalized frequency response have their own meaning and the following names:

The frequency response according to expression (2.42) can be constructed on the complex plane. In this case, the end of the vector corresponding to the complex number, when changing from 0 to, draws a curve on the complex plane, which is called amplitude-phase characteristic (AFH).

Fig.2.6. An example of the amplitude-phase characteristic of a system

Phase-frequency response (PFC) - graphic display dependence of the phase shift between the input and output signals depending on frequency,

To determine the numerator and denominator W(j) can be factorized no higher than second order

,

Then , where the "+" sign refers to i=1,2,...,l(numerator of the transfer function), sign "-" -k i=l+1,...,L(denominator of the transfer function).

Each of the terms is defined by the expression

Along with the AFC, all other frequency characteristics are also constructed separately. So the frequency response shows how a signal of different frequencies passes through a link; wherein the transmission estimate is the ratio of the amplitudes of the output and input signals. The phase response shows the phase shifts introduced by the system at various frequencies.

In addition to the considered frequency characteristics, the theory of automatic control uses logarithmic frequency response . The convenience of working with them is explained by the fact that the operations of multiplication and division are replaced by addition and subtraction operations. The frequency response plotted on a logarithmic scale is called logarithmic amplitude frequency response (LACHH)

This value is expressed in decibels (db). When depicting the LFC, it is more convenient to plot the frequency on the abscissa axis on a logarithmic scale, that is, expressed in decades (dec).

Fig.2.7. Example of logarithmic amplitude frequency response

The phase response can also be depicted on a logarithmic scale:

Fig.2.8. Example of logarithmic phase frequency response

Example 2.8.

LFC, real and asymptotic LFC of the system, the transfer function of which has the form:

.

(2.44)

.

Rice. 2.9. Real and asymptotic LFC of the system

.

Rice. 2.10. LFH systems

STRUCTURAL METHOD

3.1. Introduction

3.2. Proportional link (reinforcing, inertia-free)

3.3. Differentiating link

3.4. Integrating link

3.5. Aperiodic link

3.6. Forcing link (proportional - differentiating)

3.7. 2nd order link

3.8. Structural transformations

3.8.1. Serial connection links

3.8.2. Parallel connection of links

3.8.3. Feedback

3.8.4. Transfer rule

3.9. Transition from transfer functions to equations of state using block diagrams

3.10. Scope of applicability of the structural method

Introduction

For calculation various systems automatic control they are usually divided into individual elements, the dynamic characteristics of which are differential equations of no higher than second order. Moreover, elements different in their physical nature can be described by the same differential equations, therefore they are classified into certain classes called standard links .

The image of a system in the form of a set of typical links indicating the connections between them is called a block diagram. It can be obtained both on the basis of differential equations (Section 2) and transfer functions. This method and is the essence of the structural method.

First, let's take a closer look at the typical links that make up automatic control systems.

Proportional link

(amplifying, inertia-free)

Proportional called a link that is described by the equation

and the corresponding one structural scheme shown in Fig. 3.1.

Pulse function has the form:

g(t) = k .

There are no modal characteristics (eigenvalues ​​and eigenvectors) for the proportional link.

Replacing in the transfer function p on j we obtain the following frequency characteristics:

The amplitude frequency response (AFC) is determined by the relation:

This means that the amplitude of the periodic input signal is amplified by k- times, but there is no phase shift.

Differentiating link

Differentiating is called a link that is described by the differential equation:

Its transfer function has the form:

Let us now obtain the frequency characteristics of the link.

AFH : W(j) = jk, coincides with the positive imaginary semi-axis on the complex plane;

VChH: R() = 0,

MCHH: I() = k,

frequency response: ,

FCHH: , that is, for all frequencies the link introduces a constant phase shift;

Integrating link

This is a link whose equation is:

and then to its transfer function

Let us determine the frequency characteristics of the integrating link.

AFH: ; ; VChH: ;

MCH:

it looks like a straight line on a plane (Fig. 3.9).

Characteristic equation

A(p) = p = 0

has a single root, , which represents the modal characteristic of the integrating link. Aperiodic link

Aperiodic

is called a link whose differential equation has the form d/dt on p, let's move on to the symbolic notation of the differential equation,

and determine the transfer function of the aperiodic link:)=20lg(k).