Relationship between impulse and transfer function. Transfer function and impulse response of the circuit

Impulse (weight) response or impulse function chains - this is its generalized characteristic, which is a time function, numerically equal to the response of the circuit to a single pulse action at its input under zero initial conditions (Fig. 13.14); in other words, it is the response of a circuit free of initial energy reserve to the Diran delta function
at its entrance.

Function
can be determined by calculating the transition
or gear
circuit function.

Function calculation
using the circuit's transient function. Let at the input influence
the reaction of a linear electric circuit is
. Then, due to the linearity of the circuit with an input action equal to the derivative
, the reaction of the chain will be equal to the derivative
.

As noted, when
, chain reaction
, and if
, then the chain reaction will be
, i.e. impulse function

According to the sampling property
work
. Thus, the impulse function of the circuit

. (13.8)

If
, then the impulse function has the form

. (13.9)

Therefore, the dimension of the impulse response is equal to the dimension of the transient response divided by time.

Function calculation
using the circuit transfer function. According to expression (13.6), when acting on the function input
, the response of the function will be the transition function
type:

.

On the other hand, it is known that the image of the derivative of a function with respect to time
, at
, is equal to the product
.

Where
,

or
, (13.10)

those. impulse response
chain is equal to the inverse Laplace transform of its transfer
functions.

Example. Let us find the pulse function of the circuit whose equivalent circuits are shown in Fig. 13.12, A; 13.13.

Solution

The transition and transfer functions of this circuit were obtained earlier:

Then, according to expression (13.8)

Where
.

Impulse response plot
the circuit is shown in Fig. 13.15.

conclusions

Impulse response
introduced for the same two reasons as the step response
.

1. Single impulse impact
– an abrupt and therefore quite heavy external influence for any system or circuit. Therefore, it is important to know the reaction of the system or circuit under such an influence, i.e. impulse response
.

2. Using some modification of the Duhamel integral, we can, knowing
calculate the response of a system or circuit to any external disturbance (see further paragraphs 13.4, 13.5).

4. Imposition integral (duhamel).

Let an arbitrary passive two-terminal network (Fig. 13.16, A) is connected to a source that continuously changes from the moment
voltage (Fig. 13.16, b).

Need to find the current (or voltage) in any branch of the two-terminal network after the switch is closed.

We will solve the problem in two stages. First, we find the required value when turning on a two-terminal network for a single voltage jump, which is specified by a single step function
.

It is known that the reaction of a circuit to a single jump is step response (function)
.

For example, for
– circuit current transient function
(see clause 2.1), for
– circuit voltage transient function
.

In the second stage, continuously changing voltage
replace with a step function with elementary rectangular jumps
(see Fig. 13.16 b). Then the process of voltage change can be represented as switching on at
DC voltage
, and then as the inclusion of elementary constant voltages
, shifted relative to each other by time intervals
and having a plus sign for the increasing and minus sign for the decreasing branch of the given voltage curve.

Component of the desired current at the moment from constant voltage
is equal to:

.

Component of the desired current from an elementary voltage surge
, switched on at the moment of time is equal to:

.

Here the argument of the transition function is time
, since an elementary voltage surge
takes effect temporarily later than the closing of the key or, in other words, since the time interval between the moment the beginning of the action of this jump and the moment of time equals
.

Elementary power surge

,

Where
– scale factor.

Therefore, the required current component

Elementary voltage surges are included in the time interval from
until the moment , for which the required current is determined. Therefore, summing up the current components from all jumps, moving to the limit at
, and taking into account the current component from the initial voltage surge
, we get:

The last formula for determining the current with a continuous change in the applied voltage

(13.11)

called superposition integral or Duhamel integral (the first form of writing this integral).

The problem of connecting a circuit and a current source is solved in a similar way. According to this integral, the reaction of the chain, in general,
at some point after the start of exposure
determined by the entire part of the impact that took place before the point in time .

By replacing variables and integrating by parts, we can obtain other forms of writing the Duhamel integral, equivalent to expression (13.11):

The choice of the form of writing the Duhamel integral is determined by the convenience of calculation. For example, in case
is expressed by an exponential function, formula (13.13) or (13.14) turns out to be convenient, which is due to the ease of differentiation of the exponential function.

At
or
It is convenient to use a form of notation in which the term before the integral vanishes.

Voluntary influence
can also be presented as a sum of sequentially connected pulses, as shown in Fig. 13.17.

For infinitesimal pulse durations
we obtain formulas for the Duhamel integral similar to (13.13) and (13.14).

The same formulas can be obtained from relations (13.13) and (13.14), replacing them with the derivative function
impulse function
.

Conclusion.

Thus, based on the formulas of the Duhamel integral (13.11) – (13.16) and the time characteristics of the circuit
And
time functions of circuit responses can be determined
to voluntary influences
.

Pulse transient function (weight function, impulse response) - the output signal of a dynamic system as a reaction to the input signal in the form of the Dirac delta function. IN digital systems the input signal is a simple pulse of minimum width (equal to the sampling period for discrete systems) and maximum amplitude. When applied to signal filtering, it is also called filter core. It has wide application in control theory, signal and image processing, communication theory and other fields of engineering.

Definition [ | ]

Impulse response of a system is called its response to a single impulse under zero initial conditions.

Properties [ | ]

Application [ | ]

Systems Analysis [ | ]

Restoring frequency response[ | ]

An important property of the impulse response is the fact that on its basis a complex frequency response can be obtained, defined as the ratio of the complex spectrum of the signal at the system output to the complex spectrum of the input signal.

Complex frequency response (CFC) is an analytical expression of a complex function. The CFC is built on the complex plane and represents the trajectory curve of the end of the vector in the operating frequency range, called hodograph KCHH. To construct a frequency response, 5-8 points are usually required in the operating frequency range: from the minimum realizable frequency to the cutoff frequency (the end frequency of the experiment). The CFC, as well as the time characteristic, will provide complete information about the properties of linear dynamic systems.

The frequency response of the filter is defined as the Fourier transform (discrete Fourier transform in the case digital signal) from the impulse response.

H (j ω) = ∫ − ∞ + ∞ h (τ) e − j ω τ d τ (\displaystyle H(j\omega)=\int \limits _(-\infty )^(+\infty )h( \tau)e^(-j\omega \tau )\,d\tau )

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. Define frequency characteristics circuits 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 response and transfer function are system characteristics and are interconnected by Z - transformation formulas. Therefore, the 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.

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.

Student group 220352 Chernyshev D. A. Certificate - report on patent and scientific and technical research Graduation topic qualifying work: television receiver with digital signal processing. Start of search 2.02.99. End of search 25.03.99 Subject of search Country, Index (MKI, NKI) No. ...

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 - ...

Duhamel integral.

Knowing the response of the circuit to a single disturbing influence, i.e. transient conductivity function and/or transient voltage function, you can find the response of the circuit to an influence of an arbitrary shape. The method, the calculation method using the Duhamel integral, is based on the principle of superposition.

When using the Duhamel integral to separate the variable over which the integration is performed and the variable that determines the moment of time at which the current in the circuit is determined, the first is usually denoted as , and the second as t.

Let at the moment of time to the circuit with zero initial conditions (passive two-terminal network PD in Fig. 1) a source with a voltage of arbitrary shape is connected. To find the current in the circuit, we replace the original curve with a step one (see Fig. 2), after which, taking into account that the circuit is linear, we sum up the currents from the initial voltage jump and all voltage steps up to moment t, which come into effect with a time delay.

At time t, the component of the total current determined by the initial voltage surge is equal to .

At the moment of time there is a voltage surge , which, taking into account the time interval from the beginning of the jump to the time point of interest t, will determine the current component.

The total current at time t is obviously equal to the sum of all current components from individual voltage surges, taking into account , i.e.

Replacing the finite time increment interval with an infinitesimal one, i.e. passing from the sum to the integral, we write

.

(1)

Relationship (1) is called Duhamel integral.

It should be noted that voltage can also be determined using the Duhamel integral. In this case, instead of the transition conductivity, (1) will include the transition voltage function.

Calculation sequence using
Duhamel integral

As an example of using the Duhamel integral, we determine the current in the circuit in Fig. 3, calculated in the previous lecture using the inclusion formula.

Initial data for calculation: , , .

1. Transient conductivity

.

18. Transfer function.

The relation of the influence operator to its own operator is called the transfer function or transfer function in operator form.

A link described by an equation or equations in a symbolic or operator form can be characterized by two transfer functions: a transfer function for the input value u; and the transfer function for the input quantity f.

And

Using transfer functions, the equation is written as . This equation is a conditional, more compact form of writing the original equation.

Along with the transfer function in operator form, the transfer function in the form of Laplace images is widely used.

Transfer functions in the form of Laplace images and operator form coincide up to notation. The transfer function in the form, Laplace images can be obtained from the transfer function in operator form, if the substitution p=s is made in the latter. IN general case this follows from the fact that the differentiation of the original - the symbolic multiplication of the original by p - under zero initial conditions corresponds to the multiplication of the image by a complex number s.

The similarity between transfer functions in the form of the Laplace image and in the operator form is purely external, and it occurs only in the case of stationary links (systems), i.e. only under zero initial conditions.

Let's consider a simple RLC (series) circuit, its transfer function W(p)=U OUT /U IN

Fourier integral.

Function f(x), defined on the entire number line is called periodic, if there is a number such that for any value X equality holds . Number T called period of the function.

Let us note some properties of this function:

1) Sum, difference, product and quotient of periodic functions of period T is a periodic function of period T.

2) If the function f(x) period T, then the function f(ax)has a period.

3) If f(x) - periodic function of period T, then any two integrals of this function, taken over intervals of length T(in this case the integral exists), i.e. for any a And b equality is true .

Trigonometric series. Fourier series

If f(x) is expanded on a segment into a uniformly convergent trigonometric series: (1)

Then this expansion is unique and the coefficients are determined by the formulas:

Where n=1,2, . . .

Trigonometric series (1) of the type considered with coefficients is called trigonometric Fourier series.

Complex form of the Fourier series

The expression is called the complex form of the Fourier series of the function f(x), if defined by equality

, Where

The transition from the Fourier series in complex form to the series in real form and back is carried out using the formulas:

(n=1,2, . . .)

The Fourier integral of a function f(x) is an integral of the form:

, Where .

Frequency functions.

If you apply to the input of a system with a transfer function W(p) harmonic signal

then after the transition process is completed, harmonic oscillations will be established at the output

with the same frequency, but different amplitude and phase, depending on the frequency of the disturbing influence. From them one can judge the dynamic properties of the system. Dependencies connecting the amplitude and phase of the output signal with the frequency of the input signal are called frequency characteristics(CH). Analysis of the frequency response of a system in order to study its dynamic properties is called frequency analysis.

Let's substitute expressions for u(t) And y(t) into the dynamics equation

(aоp n + a 1 pn - 1 + a 2 p n - 2 + ... + a n)y = (bоp m + b 1 p m-1 + ... + b m)u.

Let's take into account that

pnu = pnU m ejwt = U m (jw)nejwt = (jw)nu.

Similar relationships can be written for the left side of the equation. We get:

By analogy with the transfer function, we can write:

W(j), equal to the ratio of the output signal to the input signal when the input signal changes according to the harmonic law, is called frequency transfer function. It is easy to see that it can be obtained by simply replacing p by j in the expression W(p).

W(j) is a complex function, therefore:

where P() - real frequency response (RFC); Q() - imaginary frequency response (ICH); A() - amplitude frequency response (AFC): () - phase frequency response (PFC). The frequency response gives the ratio of the amplitudes of the output and input signals, the phase response gives the phase shift of the output quantity relative to the input:

;

If W(j) is represented as a vector on the complex plane, then when changing from 0 to + its end will draw a curve called vector hodograph W(j), or amplitude-phase frequency response (APFC)(Fig. 48).

The AFC branch when changing from - to 0 can be obtained by mirroring this curve relative to the real axis.

TAU is widely used logarithmic frequency characteristics (LFC)(Fig.49): logarithmic amplitude frequency response (LAFC) L() and logarithmic phase frequency response (LPFC) ().

They are obtained by taking the logarithm of the transfer function:

LAC is obtained from the first term, which is multiplied by 20 for scaling reasons, and not the natural logarithm is used, but the decimal one, that is, L() = 20lgA(). The value of L() is plotted along the ordinate axis in decibels.

A change in signal level by 10 dB corresponds to a change in its power by a factor of 10. Since the power of the harmonic signal P is proportional to the square of its amplitude A, a change in the signal by 10 times corresponds to a change in its level by 20 dB, since

log(P 2 /P 1) = log(A 2 2 /A 1 2) = 20log(A 2 /A 1).

The abscissa axis shows the frequency w on a logarithmic scale. That is, unit intervals along the abscissa axis correspond to a change in w by a factor of 10. This interval is called decade. Since log(0) = -, the ordinate axis is drawn arbitrarily.

The LPFC obtained from the second term differs from the phase response only in the scale along the axis. The value () is plotted along the ordinate axis in degrees or radians. For elementary links it does not go beyond: - +.

Frequency characteristics are comprehensive characteristics of the system. Knowing the frequency response of the system, you can restore its transfer function and determine its parameters.

Feedback.

It is generally accepted that a link is covered by feedback if its output signal is fed to the input through some other link. Moreover, if the feedback signal is subtracted from the input action (), then the feedback is called negative. If the feedback signal is added to the input action (), then the feedback is called positive.

Transmission function closed circuit with negative feedback - a link covered by negative feedback - is equal to the forward circuit transfer function divided by one plus the open circuit transfer function

The closed-loop transfer function with positive feedback is equal to the forward-loop transfer function divided by one minus the open-loop transfer function

22. 23. Quadrupoles.

When analyzing electrical circuits in problems of studying the relationship between variables (currents, voltages, powers, etc.) of two branches of a circuit, the theory of four-terminal networks is widely used.

Quadrupole- This is a part of a circuit of any configuration that has two pairs of terminals (hence its name), usually called input and output.

Examples of a four-terminal network are a transformer, amplifier, potentiometer, power line and other electrical devices in which two pairs of poles can be distinguished.

In general, quadripoles can be divided into active, whose structure includes energy sources, and passive, branches of which do not contain energy sources.

To write the equations of a four-terminal network, we select in an arbitrary circuit a branch with a single energy source and any other branch with some resistance (see Fig. 1, a).

In accordance with the principle of compensation, we replace the original resistance with a source with voltage (see Fig. 1, b). Then, based on the superposition method for the circuit in Fig. 1b can be written

Equations (3) and (4) are the basic equations of the quadripole; they are also called quadripole equations in A-form (see Table 1). Generally speaking, there are six forms of writing the equations of a passive quadripole. Indeed, a four-terminal network is characterized by two voltages and and two currents and. Any two quantities can be expressed in terms of the others. Since the number of combinations of four by two is six, then six forms of writing the equations of a passive quadripole are possible, which are given in Table. 1. Positive directions of currents for various forms of writing equations are shown in Fig. 2. Note that the choice of one or another form of equations is determined by the area and type of problem being solved.

Table 1. Forms of writing the equations of a passive quadripole

Form	Equations	Connection with the coefficients of the basic equations
A-shape	; ;
Y-shape	; ;	; ; ; ;
Z-shape	; ;	; ; ; ;
H-shape	; ;	; ; ; ;
G-shape	; ;	; ; ; ;
B-shape	; .	; ; ; .

Characteristic impedance and coefficient
propagation of a symmetrical quadripole

In telecommunications, the operating mode of a symmetrical four-terminal network is widely used, in which its input resistance is equal to the load resistance, i.e.

.

This resistance is designated as and called characteristic resistance symmetrical four-port network, and the operating mode of the four-port network, for which it is true

,

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 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:

Impulse characteristics for different 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, various transformations of them are used in the theory of automatic control. 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.

Let us consider this transition for multichannel systems of the form (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's use the expression for the transfer matrix (2.27) and first find the 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 transient 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 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)- graphical display of the phase shift between the input and output signals depending on the 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)

,

(2.43)

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. Series connection of 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

To calculate various automatic control systems, they are usually divided into separate 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 block diagram is 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:

y = k.

(3.6)

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

where , is the link transmission coefficient.

Replacing in (3.18) d/dt on p, let's move on to the symbolic notation of the differential equation,

(Tp+1)y = ku,

(3.19)

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