Impulse response of the circuit. Transition characteristic. Impulse response. Impulse characteristics of electrical circuits Reaction of the circuit to the delta function

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 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 of the Laplace image 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 mirror image of a given curve relative to the real axis.

Widely used in TAU 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.

The transfer function of a closed circuit with negative feedback - the 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 the 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	Relationship 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 characteristic resistance symmetrical four-port network, and the operating mode of the four-port network, for which it is true

,

3. Pulse characteristics of electrical circuits

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

By definition,

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 impact: .

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 of 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.

By definition:

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 the impulse response of a circuit, it is necessary to apply the inverse Laplace transform:

, i.e. 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 is an image of the derivative of the transition characteristic, then 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

Transfer 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 free vibrations 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 infinitely small area pulsed actions 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 convolution and .

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

.

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

chain is connected: with a change in energy state... (+0),. Uc(-0) = Uc(+0). 3. Transitional characteristic electrical chains is: Response to a single step...

A remarkable feature of linear systems - the validity of the superposition principle - opens a direct path to the systematic solution of problems about the passage of various signals through such systems. The dynamic representation method (see Chapter 1) allows you to represent signals in the form of sums of elementary pulses. If it is possible in one way or another to find the reaction at the output that arises under the influence of an elementary impulse at the input, then the final stage of solving the problem will be the summation of such reactions.

The intended path of analysis is based on the temporal representation of the properties of signals and systems. Equally applicable, and sometimes much more convenient, is analysis in the frequency domain, when signals are specified by Fourier series or integrals. The properties of systems are described by their frequency characteristics, which indicate the law of transformation of elementary harmonic signals.

Impulse response.

Let some linear stationary system be described by the operator T. For simplicity, we will assume that the input and output signals are one-dimensional. By definition, the impulse response of a system is a function that is the system’s response to an input signal. This means that the function h(t) satisfies the equation

Since the system is stationary, a similar equation will exist if the input action is shifted in time by the derivative value:

It should be clearly understood that the impulse response, as well as the delta function that generates it, is the result of a reasonable idealization. From a physical point of view, the impulse response approximates the response of a system to an input pulse signal of an arbitrary shape with a unit area, provided that the duration of this signal is negligible compared to the characteristic time scale of the system, for example, the period of its own oscillations.

Duhamel integral.

Knowing the impulse response of a linear stationary system, one can formally solve any problem about the passage of a deterministic signal through such a system. Indeed, in ch. 1 it was shown that the input signal always admits a representation of the form

The output reaction corresponding to it

Now let us take into account that the integral is the limiting value of the sum, therefore the linear operator T, based on the principle of superposition, can be included under the sign of the integral. Further, the operator T “acts” only on quantities that depend on the current time t, but not on the integration variable x. Therefore, from expression (8.7) it follows that

or finally

This formula, which is of fundamental importance in the theory of linear systems, is called the Duhamel integral. Relationship (8.8) indicates that the output signal of a linear stationary system is a convolution of two functions - the input signal and the impulse response of the system. Obviously, formula (8.8) can also be written in the form

So, if the impulse response h(t) is known, then further stages of the solution are reduced to completely formalized operations.

Example 8.4. Some linear stationary system, internal structure which is not significant, has an impulse response, which is a rectangular video pulse with a duration T. The pulse occurs at t = 0 and has an amplitude

Determine the output response of this system when a step signal is applied to the input

When applying the Duhamel integral formula (8.8), you should pay attention to the fact that the output signal will look different depending on whether or not the current value exceeds the duration of the impulse response. When we have

If then at the function vanishes, therefore

The found output reaction is displayed in a piecewise linear graph.

Generalization to the multidimensional case.

Until now, it has been assumed that both the input and output signals are one-dimensional. In the more general case of a system with inputs and outputs, partial impulse responses should be introduced, each of which represents the signal at the output when a delta function is applied to the input.

The set of functions forms a matrix of impulse responses

The Duhamel integral formula in the multidimensional case takes the form

where is -dimensional vector; - -dimensional vector.

Condition of physical realizability.

Whatever the specific type of impulse response of a physically feasible system, the most important principle must always be satisfied: the output signal corresponding to the impulse input action cannot arise until the moment the impulse appears at the input.

This leads to a very simple restriction on the type of permissible impulse characteristics:

This condition is satisfied, for example, by the impulse characteristic of the system considered in Example 8.4.

It is easy to see that for a physically realizable system, the upper limit in the Duhamel integral formula can be replaced by the current value of time:

Formula (8.13) has a clear physical meaning: a linear stationary system, processing the signal arriving at the input, carries out a weighted summation of all its instantaneous values ​​that existed “in the past” at - The role of the weighting function is played by the impulse response of the system. It is fundamentally important that a physically implemented system is under no circumstances capable of operating with “future” values ​​of the input signal.

A physically realizable system must, in addition, be stable. This means that its impulse response must satisfy the condition of absolute integrability

Transition characteristic.

Let a signal represented by the Heaviside function act at the input of a linear stationary system.

Output reaction

is usually called the transient characteristic of the system. Since the system is stationary, the transient response is invariant with respect to the time shift:

The previously stated considerations about the physical realizability of the system are completely transferred to the case when the system is excited not by a delta function, but by a single jump. Therefore, the transient response of a physically realizable system is different from zero only at while at t There is a close connection between the impulse and transient characteristics. Indeed, since then based on (8.5)

The differentiation operator and the linear stationary operator T can change places, so

Using the dynamic representation formula (1.4) and proceeding in the same way as when deriving relation (8.8), we obtain another form of the Duhamel integral:

Frequency transmission coefficient.

In the mathematical study of systems, of particular interest are those input signals that, being transformed by the system, remain unchanged in form. If there is equality

then is an eigenfunction of the system operator T, and the number X, in the general case complex, is its eigenvalue.

Let us show that a complex signal at any frequency value is an eigenfunction of a linear stationary operator. To do this, we use the Duhamel integral of the form (8.9) and calculate

This shows that the eigenvalue of the system operator is a complex number

(8.21)

called the frequency gain of the system.

Formula (8.21) establishes a fundamentally important fact - the frequency transmission coefficient and the impulse response of a linear stationary system are related to each other by the Fourier transform. Therefore, always, knowing the function, you can determine the impulse response

We have come to the most important point of the theory of linear stationary systems - any such system can be considered either in the time domain using its impulse or transient characteristics, or in the frequency domain, setting the frequency transmission coefficient. Both approaches are equivalent and the choice of one of them is dictated by the convenience of obtaining initial data about the system and the ease of calculations.

In conclusion, we note that the frequency properties of a linear system having inputs and outputs can be described by a matrix of frequency transfer coefficients

There is a connection law between matrices, similar to that, which is given by formulas (8.21), (8.22).

Amplitude-frequency and phase-frequency characteristics.

The function has a simple interpretation: if a harmonic signal with a known frequency and complex amplitude is received at the input of the system, then the complex amplitude of the output signal

In accordance with formula (8.26), the modulus of the frequency transfer coefficient (AFC) is an even, and the phase angle (PFC) is an odd function of frequency.

It is much more difficult to answer the question of what the frequency transmission coefficient should be in order for the conditions of physical realizability (8.12) and (8.14) to be satisfied. Let us present without proof the final result, known as the Paley-Wiener criterion: the frequency transfer coefficient of a physically realizable system must be such that the integral exists

Let's consider concrete example, illustrating the properties of the frequency gain of a linear system.

Example 8.5. Some linear stationary system has the properties of an ideal low-pass filter, i.e. its frequency transmission coefficient is given by the system of equalities:

Based on expression (8.20), the impulse response of such a filter

The symmetry of the graph of this function relative to the point t = 0 indicates the impracticability of an ideal low-pass filter. However, this conclusion directly follows from the Paley-Wiener criterion. Indeed, integral (8.27) diverges for any frequency response that vanishes at some finite segment of the frequency axis.

Despite the impracticability of an ideal low-pass filter, this model is successfully used for an approximate description of the properties frequency filters, assuming that the function contains a phase factor linearly dependent on frequency:

As is easy to check, here is the impulse response

The parameter, equal in magnitude to the slope coefficient of the phase response, determines the time delay of the maximum of the function h(t). It's clear that this model the more accurately it reflects the properties of the implemented system, the larger the value

Ministry of Education and Science of Ukraine

Donetsk National University

Report

on the topic: Radio engineering circuits and signals

3rd year full-time student of NF-3

Developed by a student:

Alexandrovich S. V.

Checked by the teacher:

Dolbeshchenkov V.V.

INTRODUCTION

"Radio Engineering Circuits and Signals" (RTC and S)– a course that is a continuation of the course “Fundamentals of Circuit Theory”. Its goal is to study the fundamental principles associated with the reception of signals, their transmission through communication channels, processing and conversion in radio circuits. The methods of analysis of signals and radio engineering circuits presented in the course "RTC and C" use mathematical and physical information, mainly known to students from previous disciplines. An important goal of the course "RTC and S" is to teach students to choose a mathematical apparatus that is adequate to the problem encountered, and to show how this apparatus works when solving specific problems in the field of radio engineering. It is equally important to teach students to see the close connection between the mathematical description and the physical side of the phenomenon under consideration, and to be able to draw up mathematical models of the processes being studied.

Main sections studied in the course "Radio Engineering Circuits and Signals":

1. Timing analysis of circuits based on convolution;

2. Spectral analysis of signals;

3. Radio signals with amplitude and angle modulation;

4. Correlation analysis of signals;

5. Active linear circuits;

6. Analysis of the passage of signals through narrow-band circuits;

7. Negative feedback in linear circuits;

8. Filter synthesis;

9. Nonlinear circuits and methods of their analysis;

10. Circuits with variable parameters;

11. Principles of generating harmonic oscillations;

12. Principles of processing discrete time signals;

13. Random signals;

14. Analysis of the passage of random signals through linear circuits;

15. Analysis of the passage of random signals through nonlinear circuits;

16. Optimal filtration deterministic signals in noise;

17. Optimal filtering of random signals;

18. Numerical methods calculation of linear circuits.

TIMING CIRCUIT ANALYSIS BASED ON CONVOLUTION

Step and impulse response

The time method is based on the concept of transient and impulse characteristics of a circuit. Step response chains are the response of a chain to an influence in the form of a unit function. Indicates the transient response of a circuit g(t).Impulse response circuits are called the response of a circuit to a single impulse function (d-function). Denotes impulse response h(t). Moreover, g(t) And h(t) are determined at zero initial conditions in the circuit. Depending on the type of reaction and type of impact (current or voltage), transient and impulse characteristics can be dimensionless quantities, or have dimensions A/B or V/A.

Using the concepts of transient and impulse characteristics of a circuit allows us to reduce the calculation of the circuit response from the action of a non-periodic signal of arbitrary shape to the determination of the circuit response to the simplest impact such as a single 1( t) or impulse function d( t), with the help of which the original signal is approximated. In this case, the resulting reaction of a linear chain is found (using the superposition principle) as the sum of the reactions of the chain to elementary influences 1( t) or d( t).

Between the transitional g(t) and pulse h(t) there is a certain connection between the characteristics of a linear passive circuit. It can be established if we represent a unit impulse function through the passage to the limit of the difference of two unit functions of magnitude 1/t, shifted relative to each other by time t:

i.e., the unit impulse function is equal to the derivative of the unit function. Since the circuit under consideration is assumed to be linear, the relationship remains the same for impulse and transient reactions of the circuit

i.e., the impulse response is a derivative of the step response of the circuit.

The equation is valid for the case when g(0) = 0 (zero initial conditions for the circuit). If g(0) ¹ 0, then presenting g(t) in the form g(t) = , where = 0, we obtain the coupling equation for this case:

To find the transient and impulse characteristics of a circuit, you can use both classical and operator methods. The essence of the classical method is to determine the time response of the circuit (in the form of voltage or current in individual branches of the circuit) to the influence of a single 1( t) or impulse d( t) functions. It is usually convenient to determine the transient response using the classical method g(t), and the impulse response h(t) find using coupling equations or the operator method.

It should be noted that the value I(r)V equation is numerically equal to the image of transient conductivity. A similar image of the impulse response is numerically equal to the operator conductivity of the circuit

For example, for RС-chains we have:

Applying to Y(p) expansion theorem, we obtain:

In table 1.1 summarizes the values ​​of transient and impulse characteristics for current and voltage for some first- and second-order circuits.

18. Reaction of linear circuits to unit functions. Transient and impulse characteristics of the circuit, their connection.

Unit step function (on function) 1 (t) is defined as follows:

Graph of a function 1 (t) is shown in Fig. 2.1.

Function 1 (t) is equal to zero for all negative values ​​of the argument and one for t³ 0 . Let us also introduce into consideration the shifted unit step function

This effect is activated at the moment of time t= t..

The voltage in the form of a unit step function at the input of the circuit will be when a constant voltage source is connected U 0 =1 V at t= 0 using an ideal key (Fig. 2.3).

Unit impulse function (d - function, Dirac function) is defined as the derivative of the unit step function. Because at the moment t= 0 function 1 (t) undergoes a discontinuity, then its derivative does not exist (turns to infinity). Thus, the unit impulse function

It is a special function or mathematical abstraction, but is widely used in the analysis of electrical and other physical objects. Functions of this kind are considered in the mathematical theory of generalized functions.

An impact in the form of a single impulse function can be considered as an impact impact (a fairly large amplitude and an infinitesimal impact time). A unit impulse function is also introduced, shifted by time t= t

The unit impulse function is usually depicted graphically as a vertical arrow at t= 0, and shifted at - t= t (Fig. 2.4).

If we take the integral of the unit impulse function, i.e. determine the area limited by it, we get the following result:

Rice. 2.4.

Obviously, the integration interval can be any, as long as the point falls there t= 0. Integral of the shifted unit impulse function d ( t-t) is also equal to 1 (if the point falls within the limits of integration t= t). If we take the integral of the unit impulse function multiplied by a certain coefficient A 0 , then obviously the result of integration will be equal to this coefficient. Therefore, the coefficient A 0 before d( t) determines the area bounded by the function A 0 d ( t).

For the physical interpretation of the d-function, it is advisable to consider it as a limit to which a certain sequence of ordinary functions tends, for example

Transient and impulse characteristics

Step response h(t) is called the response of a circuit to an impact in the form of a unit step function 1 (t). Impulse response g(t) is called the response of the circuit to an impact in the form of a unit impulse function d ( t). Both characteristics are determined under zero initial conditions.

The transition and impulse functions characterize the circuit in the transition mode, since they are reactions to stepwise, i.e. quite heavy for any impact system. In addition, as will be shown below, using the transient and impulse characteristics, the response of the circuit to an arbitrary influence can be determined. The transient and impulse characteristics are interconnected in the same way as the corresponding influences are interconnected. The unit impulse function is a derivative of the unit step function (see (2.2)), therefore the impulse response is a derivative of the step response and at h(0) = 0 . (2.3)

This statement follows from general properties linear systems that are described by linear differential equations, in particular, if its derivative is applied to a linear chain with zero initial conditions instead of an effect, then the reaction will be equal to the derivative of the initial reaction.

Of the two characteristics under consideration, the transient one is most simply determined, since it can be calculated from the reaction of the circuit to the inclusion of a constant voltage or current source at the input. If such a reaction is known, then to obtain h(t) it is enough to divide it by the amplitude of the input constant action. It follows that the transient (as well as impulse) response can have the dimensions of resistance, conductivity, or be a dimensionless quantity, depending on the dimension of the impact and reaction.

Example . Define transition h(t) and pulse g(t) characteristics of a serial RC circuit.

The influence is the input voltage u 1 (t), and the reaction is the voltage across the capacitance u 2 (t). According to the definition of transient response, it should be defined as the output voltage when a constant voltage source is connected to the input of the circuit U 0

This problem was solved in section 1.6, where we obtained u 2 (t) = u C (t) = Thus, h(t) = u 2 (t) / U 0 = The impulse response is determined by (2.3) .