Transmission function. Impulse response and transfer function Relationship between impulse and transfer function

The timing characteristics of a circuit are called responses to typical components of the original signal.

The transient response of a circuit is the response of a circuit with zero initial conditions to the influence of a unit function (Heaviside function). The transition response is determined from the operator transfer function by dividing it by the operator and finding the original from the resulting image using the inverse Laplace transform through residues.

The impulse response of a circuit is the response of the circuit to the influence of the delta function. - an infinitely short duration and infinitely large amplitude pulse of a unit area. The impulse response is determined by finding the residues from the transfer function of the circuit.

We will also look for the time characteristics of the circuit using the operator method. To do this, we need to find the operator image of the input signal, multiply it by the transmission coefficient in operator form and find the original from the resulting expression, i.e., knowing the transmission coefficient of the circuit, we can find a response to any influence.

Finding the impulse response comes down to finding the circuit's response to the delta function. It is known that for the delta function the image is 1. Using the inverse Laplace transform, we find the impulse response.

.

Let us select the integer part for the transfer function of the circuit, since the degrees of the leading coefficients in the numerator and denominator are equal:

Let us find the singular points of the transfer function by equating the denominator to zero.

We have only one special point, now we take a residue at this special point.

The expression for the impulse response will be written as follows:

Let us similarly find the transient response of the circuit, knowing that for the Heaviside function the image is the function .

; , ;

Transient and impulse characteristics are interconnected, as well as input influences:

Let's check the fulfillment of the limiting relationships between the frequency and time characteristics of the circuit, i.e. fulfillment of the following conditions:

We substitute specific expressions for the characteristics of circuits into the system.

.

As you can see, the conditions are met, which indicates the correctness of the formulas found.

Let us write down the final formulas for the time characteristics, taking into account the normalization

Using the above formulas, we will construct graphs of these functions.

Fourier signal analog linear

Figure 2.5 - Impulse response of a prototype analog filter

Figure 2.6 - Transition response of the prototype analog filter

Temporal characteristics exist only for , since responses cannot precede impacts.

Our circuit is differentiating, so the transient response behaves like this. The differentiating circuit sharpens the transient process and passes the leading edge. The high frequencies that have passed through are responsible for the “throw”, and the low frequencies that have not passed through are responsible for the blockage.

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Temporary and frequency characteristics the chains are interconnected 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. The value of a 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).

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.

Consider a simple RLC (series) circuit, its Transmission 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 the 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

,

A multipath communication channel, like any linear system, is uniquely determined by its IR in the time domain and/or transfer function in the frequency domain. The IR channel and its transfer function make it possible to determine the relationship between the output and input signals and their spectra, respectively. The multipath channel is shown in Fig. 2.4.

Rice. 2.4. Multipath channel

IN multipath channel the signal travels along many paths, and n The th path (beam) is characterized by a signal delay t n(t) and complex transfer coefficient a n(t). If a signal is transmitted s(t), then a signal is observed at the receiver input x(t), which is the sum of signals propagating in different ways. This signal can be written as follows:

, (2.3.1)

The vast majority of communication systems use narrowband signals, which can be represented in the form (1.1.2). Substituting (1.1.2) into (2.3.1), we obtain that

It follows that the complex amplitude of the received low-frequency signal is equal to

Further we will assume that during the passage of the delay signal t n(t) and complex transfer coefficients a n(t) for all rays remain unchanged and equal to t n and a n.

By definition, the IR of a linear system with fixed parameters is the response of the system to the input d-pulse. Therefore, we will obtain the IR channel if we apply signal (1.1.2) with a complex amplitude equal to . As a result, we will have that

To obtain the channel transfer function, it is necessary to take a harmonic signal of unit frequency amplitude f, i.e. substitute the signal into (2.3.1). Then we get that

. (2.3.5)

As an example, consider the properties of a two-beam channel. Let us assume that there is a direct signal and a signal reflected by a local object. The direct signal arrives without distortion and has a delay during propagation from the transmitter to the receiver. In addition, its amplitude decreases and depends on the distance between the transmitter and receiver. These changes in signal parameters are not of fundamental importance for our consideration. Therefore, the beginning of the time count is compatible with the moment of arrival of the direct signal at the receiving antenna, and the amplitude of the direct signal is normalized so that it is equal to unity. Let us take the phase of the direct signal equal to zero. In this case, from (2.3.4) we obtain that the channel can be characterized by IM

Where – complex coefficient of signal reflection from a local object, – phase difference between the first and second signals due to the delay t 2 second signals relative to the first, A 2 – complex amplitude of the second signal relative to the first.

The IR of a two-beam channel is shown in Fig. 2.5.

Rice. 2.5. Dual-beam channel: a) direct signals come to the receiver input s 1 and reflected s 2
signals; b) THEIR double-beam channel

Note that the IR channel (2.3.6) does not provide information about the direction of arrival of the second signal. It is usually assumed that the second signal has a smaller amplitude, i.e. .

We find the channel transfer function from (2.3.5). We get that

The power transmission coefficient of a channel is defined as the square of the modulus of the transfer function, i.e.

An example of this function is shown in Fig. 2.6 for | a 2 |=0.8, t 2 =1, arga 2 =p/6. It can be seen that the power transmission coefficient of the channel has maxima and minima, that is, harmonic signals with some frequencies are weakened, while with other frequencies they are amplified. Minima are observed for frequencies where n=0, ±1.¼. The distance between the minima on the frequency axis does not depend on the phase of the reflection coefficient a 2 and is equal to . The average power transfer coefficient is 1+| a 2 | 2 and shown in Fig. 2.6 with a dashed line, the minimum is (1-| a 2 |) 2 , and the maximum is (1+| a 2 |) 2 . If the amplitude of the direct signal is equal to the amplitude of the delayed signal, then a complete loss of signal at the receiver input may be observed.

Rice. 2.6. Power transmission coefficient of a two-beam channel

A change in the level of the received signal caused by the interference of signals passing through different paths in the channel is usually called fading of the received signal or fading. If the receiver's bandwidth is , then all spectral components of the signal within the receiver's frequency band will experience consistent fading. In this case, it is customary to say that the channel is flat(flat channel). If another condition is met, then different spectral components of the signal experience different fading. In this case the channel is said to be frequency selective(frequency selective channel).

The phase of the reflected signal in (2.3.7) can change significantly even with very small changes in the delay t 2 of this signal. In fact, a phase change of 2p radians occurs when the delay t 2 changes by 1/ f. For example, if the carrier frequency f c=900 MHz, then the value is 1/ f is only 1.1 nanoseconds, which corresponds to a change in the signal propagation path by 33 cm, that is, by wavelength. Thus, if the path difference between the direct and reflected signals changes by only 16.5 cm, the phase difference between them will change by 180 degrees. This example shows that the signal can experience deep and rapid fading even when the subscriber is moving at walking speed.

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:

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, 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 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 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).