Passage of random processes through linear and nonlinear chains. Analysis of the passage of random signals through linear circuits The passage of signals through linear circuits

Goal of the work:

    study of the processes of passage of harmonic signals and rectangular signals through linear circuits, such as differentiating and integrating circuits, series and parallel oscillatory circuits, transformer;

    study of transient processes in linear circuits;

    gaining skills in working with measuring instruments;

    learn to perform calculations of RCL circuits using the symbolic method;

    processing and analysis of the obtained experimental data.

Tasks:

    measure the amplitude-frequency characteristics of seven linear circuits;

    measure the phase-frequency characteristics of the above listed linear circuits;

    obtain and study the transient characteristics of seven linear circuits;

1 Linear circuits

In radio electronics, electrical circuits are a collection of connected circuit elements such as resistors, capacitors, inductors, diodes, transistors, operational amplifiers, current sources, voltage sources and others.

Circuit elements are connected using wires or printed busbars. Electrical circuits composed of idealized elements are classified according to a number of criteria:

By energy characteristics:

    active (containing power supplies);

    passive circuits (do not contain sources of current and (or) voltage);

According to topological features:

    planar (flat);

    non-planar;

    branched;

    unbranched;

    simple (single-, double-circuit);

    complex (multi-circuit, multi-node);

By the number of external pins:

    bipolar;

    quadripoles;

    multi-port networks;

From the frequency of the measuring field:

    circuits with lumped parameters (in circuits with lumped parameters, only the resistor has resistance, only the capacitor has capacitance, and only the inductor has inductance);

    circuits with distributed parameters (in circuits with distributed parameters, even connecting wires have capacitance, conductivity and inductance, which are distributed along their length; this approach is most typical for circuits in the microwave region);

From element type:

    linear circuits, if they consist of linear idealized elements;

    nonlinear circuits, if the circuit includes at least one nonlinear element;

This paper examines passive circuits consisting of three circuit elements. Elements
– are called idealized circuit elements. The current flowing through such elements is a linear function of the applied voltage:

for resistor
:
;

for capacitor :
;

for inductor :

Therefore, chains consisting of
elements are called linear.

Strictly speaking, in practice not all
the elements are linear, but in many cases the deviation from linearity is small and the actual element can be taken as an idealized linear one. Active resistance can be considered as a linear element only if the current flowing through it is so small that the heat generated does not lead to a noticeable change in the value of its resistance. Similar considerations can be made for the inductor and capacitor. If the parameters
circuits remain unchanged during the time when the electrical process being studied occurs, then we speak of a circuit with constant parameters.

Since processes in linear circuits are described by linear equations, the principle of superposition is applicable to them. This means that the result of an action in a linear circuit of a signal of a complex shape can be found as the sum of the results of actions of simpler signals into which the original, complex signal is decomposed.

Two methods are used to analyze linear circuits: the frequency response method and the transient response method.

MINISTRY OF EDUCATION AND SCIENCE OF RUSSIA

FEDERAL STATE BUDGET EDUCATIONAL INSTITUTION OF HIGHER PROFESSIONAL EDUCATION

"MORDOVIAN STATE UNIVERSITY named after. N. P. OGAREVA"

FACULTY OF ELECTRONIC ENGINEERING

Department of AUTOMATION

M. V. ILYIN

With. With. Kapitonov

Compiled by: Head of the Department of Automation, Ph.D. tech. Sciences, Associate Professor Department of Automation, Ph.D. tech. sciences, teacher Department of Automation , Associate Professor of the Department of Automation.

Passage of signals various shapes through linear R.C.-chains: laboratory workshop / N. N. Bespalov, M. V. Ilyin, . - Saransk: Kovylk. typ., 2012. - 24 p.

ISBN___________

Contained theoretical information and methodological instructions for performing laboratory work “Passage of signals of various shapes through linear R.C.-circuits” in the course “Electronic circuits and microcircuitry”. Designed for students in the fields of study “Electronics and nanoelectronics”, “Infocommunication technologies and communication systems”, “Power engineering and electrical engineering” and “Instrument making”. However, this manual will be able to be used by students of other specialties related to electrical engineering, electronics and radio engineering.

Published by decision of the scientific and methodological council of Mordovian State University. Eve.

UDC 621.391.3.011.71(076)

BBK B534

PREFACE

This laboratory workshop contains a description of the first laboratory work, which is carried out when students study full-time and part-time forms of study of pulse circuits as part of the course “Electronic circuits and microcircuitry”.

The main goal of this work is to study the processes of transmission of pulses of various shapes through R.C.-chains.

Since the implementation of laboratory work in the course being studied often precedes the lecture presentation of the relevant sections, the description of the work includes theoretical applications that can serve as teaching aids for the relevant sections of the course, as well as manuals on course design and standard calculations.

However, using only one theoretical application to prepare for laboratory work is insufficient. It is necessary to study the relevant sections in the literature given at the end of the collection.

When preparing for the next job, the student is required to familiarize himself with the job description, theoretical manual, specified literature, and also complete a preliminary calculation task.

The report on the work must contain the schemes studied, the preliminary calculation task performed and the results obtained. The report must be neatly prepared on standard A4 size sheets, and also submitted in electronic form.

The procedure for completing this laboratory work is as follows.

1. A group of students starting to perform laboratory work must be instructed on the general rules of behavior in this laboratory and on safety rules, which is recorded in the appropriate journal signed by each student.

2. Before the next lesson, each student takes a colloquium on current work. If a student is not ready to work or has not completed the preliminary calculation task, then he is not allowed to work.

3. At the next lesson after completing the work, the student must present a completed report on the work completed and defend the work.

Students who have not defended two papers by the time the next paper is completed are not allowed to attend classes. Each student prepares a work report.

All laboratory work for the course being studied is designed for a four-hour classroom lesson and four-hour home preparation.

1 BRIEF THEORETICAL INFORMATION

Linear circuits are circuits consisting of a set of linear elements, i.e. elements whose nominal values ​​do not depend on the flowing current or applied voltage. The principle of superposition applies to all linear circuits. For example, to describe processes in linear circuits, you can use methods based on the application of the Duhamel integral, or methods of harmonic analysis. Considered R.C.-circuits are used in many practical circuits as functional converters. Depending on the structure and ratio of the parameters of the elements R.C.-networks can be used for differentiation (high-pass filter) or integration (filter low frequencies) input signals.


To analyze transient processes in pulsed circuits, classical, operator, frequency methods, as well as the Duhamel integral method (superposition method) are used.

Classic method. When calculating transient processes using this method, the input signal is represented as a function Uinput(t), and the RC circuit under study is described by a differential equation (DE), which establishes the relationship between the output and input voltages, the parameters of the circuit elements and the external influence. When compiling a control system, a number of laws and theorems are used that determine the relationship between voltages and currents. The main ones are Ohm's law, commutation, Kirchhoff and the equivalent generator theorem.

In many cases, when analyzing transient processes, the equivalent circuit of the circuit under study is described by a first-order DE with a constant right-hand side:

Where τ - time constant characterizing the inertia of the circuit; x(t)-required quantity (current, voltage); Z 0 - external disturbing influence.

The general solution to equation (1) has the form:

https://pandia.ru/text/78/069/images/image003_175.gif" width="93" height="29 src=">,

Where A- constant of integration (found from the initial conditions); R- root of the characteristic equation https://pandia.ru/text/78/069/images/image005_134.gif" width="63" height="48 src=">.

Thus, the general solution of DE (1) will be written as:

https://pandia.ru/text/78/069/images/image007_114.gif" width="40" height="20"> and, let's find:

https://pandia.ru/text/78/069/images/image011_83.gif" width="123" height="24 src=">.

Therefore, the solution to DE (1) can be written in the form

https://pandia.ru/text/78/069/images/image013_87.gif" width="181" height="60 src=">. (3)

For a specific RC circuit, the operator transmission coefficient is determined K(r), then find the image of the output voltage and by function Uout(R) determine the original Uout(t) , using the inverse Laplace transform:

https://pandia.ru/text/78/069/images/image008_110.gif" alt="*" width="12" height="23 src=">is determined by the formula:

https://pandia.ru/text/78/069/images/image017_73.gif" width="248" height="56 src=">.

If the denominator of the image Uout(R) has, along with simple roots R 1, R 2 …, R n root R n+1 multiplicity a, i.e. image Uout(R) is written as a fraction:

https://pandia.ru/text/78/069/images/image008_110.gif" alt="*" width="12" height="23">there will be a function:

https://pandia.ru/text/78/069/images/image008_110.gif" alt="*" width="12 height=23" height="23"> Frequency method. When using this method, the input signal Uinput(t)based direct conversion Fourier is represented as a frequency spectrum Uinput(jw). Then the complex transfer coefficient is found TO(jw)https://pandia.ru/text/78/069/images/image020_61.gif" width="244" height="60 src=">.

https://pandia.ru/text/78/069/images/image008_110.gif" alt="*" width="12" height="23 src=">complex shape. The output voltage is found from the expression:

https://pandia.ru/text/78/069/images/image022_36.jpg" width="507" height="353 src=">

Figure 1 - Passage of a voltage step through R.C.-chain.

The input signal can be written as

0 at t < 0

Uinput(t)= Um at t > 0.

When using the classical method, it is necessary to draw up a control system R.C.-chains. According to Kirchhoff's second law, we can write:

Uout(t) = U c( t) + Uinput(t). (4)

When applying an input signal through a capacitance WITH current flows i(t) and voltage on the capacitance https://pandia.ru/text/78/069/images/image025_52.gif" width="237" height="60 src=">.

Considering that Ri(t) = Uout(t), and differentiating the right and left sides of this equation, we get:

https://pandia.ru/text/78/069/images/image027_46.gif" width="212" height="43 src=">.

Substituting the value into the resulting equation Uinput(t), for the output voltage we get:

https://pandia.ru/text/78/069/images/image029_48.gif" width="289 height=49" height="49">.

To find the expression Uout(t) in this case, you can use equation (3), which will be written in the form:

https://pandia.ru/text/78/069/images/image031_40.gif" width="67" height="25 src="> - output voltage at t= ∞ (after the end of the transition process, i.e. at = 0); Uout(0) - output voltage at t= 0, (at the moment of switching, when U out(0) = Um).

Therefore, the output voltage will be determined as:

https://pandia.ru/text/78/069/images/image033_39.gif" width="104" height="52">. Operator transmission coefficient TO(R) for a given RC circuit is determined as follows:

https://pandia.ru/text/78/069/images/image028_48.gif" width="129" height="47">.

Passing throughR.C. - rectangular pulse circuit. Figure 2a shows R.C.- a circuit to the input of which a rectangular pulse with an amplitude is supplied Um and duration. The input signal can be represented in the form of two opposite-polar voltage drops of magnitude Um, shifted relative to each other for a time tAnd(Figure 2b).

At 0< t < tAnd

Uinput(p)= https://pandia.ru/text/78/069/images/image039_37.gif" width="18" height="151 src=">.gif" width="151" height="72 src="> when tAnd > 0,

and then, using the inverse Laplace transform, we find the time function Uout(t):

At 0< t < tAnd

Uout(t)= at tAnd > 0.

The shape of the output pulse depends on the ratio tAnd And τ . Figure 3a shows the shape of the output signal when τ << tAnd, and Figure 3b shows the output signal at τ >> tAnd. From the figure it is clear that if R.C.- the circuit must transmit a rectangular pulse without distortion, then you need to choose the ratio τ >> tAnd. To estimate the distortion of the pulse apex, the relative decay of the pulse apex Δ is used:

https://pandia.ru/text/78/069/images/image046_20.jpg" width="597" height="285 src=">

Figure 3 - Output signal shape for various t.

Similarly, you can determine the shape of the output signal for R.C.-circuit shown in Figure 4a (integrating R.C.-chain). From Figure 4b it is clear that to transmit a pulse with minimal front distortion it is necessary to choose τ << tAnd.

https://pandia.ru/text/78/069/images/image048_18.jpg" width="376" height="261">

Figure 5 - To determine the duration of the pulse front.

Passing throughR.C. - linearly increasing voltage circuit. Figure 6 shows R.C.- a circuit whose input receives a linearly increasing voltage Uinput(t) =kt, Where k= tgα- proportionality coefficient.

https://pandia.ru/text/78/069/images/image050_24.gif" width="221" height="25 src=">.gif" width="31 height=43" height="43"> can be represented as a series:

https://pandia.ru/text/78/069/images/image054_22.gif" width="323" height="55 src=">.

From this it is clear that for small values t (t<<τ ) the output voltage practically coincides with the input, i.e. . Uout(t) ≈ kt.

Output waveform distortion:

https://pandia.ru/text/78/069/images/image056_21.gif" width="141" height="48 src="> - lower cut-off frequency, determined when the frequency response rolls off equal to 3 dB. For example, to transmit a sweep voltage with a duration of 2 ms and a deviation from linearity of no more than 0.1%, from the last equation we find that it is necessary to have fn < 0,16 Гц или R.C. = τ > 1s.

At t >> τ output voltage tends to a constant value kτ. Capacitance voltage WITH can be found like this:

https://pandia.ru/text/78/069/images/image058_11.jpg" width="369" height="314">

Figure 7 - Representation of trapezoidal voltage in the form of four linearly increasing signals.

Resistor dividers with multiple inputs. An example of a multi-input divider circuit is shown in Figure 8.

https://pandia.ru/text/78/069/images/image060_22.gif" width="269" height="64 src=">,

In the particular case when https://pandia.ru/text/78/069/images/image064_18.gif" width="253" height="60 src=">,

https://pandia.ru/text/78/069/images/image067_19.gif" width="21" height="25 src=">, but also on the number of voltage components, the ratio of the values ​​of communication resistance and load resistance.

Figure 9 - Resistor divider loaded with capacitance C.

When a pulse is transmitted through such a divider, its fronts are stretched due to the processes of charging and discharging the capacitor WITH, and a decrease in its amplitude due to the presence of a divider (https://pandia.ru/text/78/069/images/image072_18.gif" width="165" height="29 src=">

and amplitude:

DIV_ADBLOCK157">


https://pandia.ru/text/78/069/images/image075_17.gif" width="128" height="49 src=">.

Resistor-capacitance dividers. In some cases, to transmit differences in input voltage, the resistor output https://pandia.ru/text/78/069/images/image077_4.jpg" width="511" height="377 src=">

Figure 9 - Passage of a rectangular pulse through a resistor-capacitance divider.

Let a rectangular voltage pulse with amplitude be applied to the input of such a divider E, and we will assume that the source of input pulses is ideal, devoid of internal resistance, and, therefore, capable of developing infinitely large power.

At the moment of switching ( t= 0) there is an infinitely large current jump through the capacitors https://pandia.ru/text/78/069/images/image079_17.gif" width="24" height="23">, and as a result instantaneous final power surges and https://pandia.ru/text/78/069/images/image082_18.gif" width="273" height="55 src=">,

where and are the charges on the capacitors and at the moment t. At t= 0 = , since when t= 0 current passes only through capacities https://pandia.ru/text/78/069/images/image079_17.gif" width="24" height="23 src="> then:

https://pandia.ru/text/78/069/images/image088_12.gif" width="336" height="60 src=">,

https://pandia.ru/text/78/069/images/image091_11.gif" width="205" height="55 src=">. and up to the initial ones (if t> 0) voltage levels.

In some devices (for example, in multivibrators) in the resistor-capacitance divider there is a resistor https://pandia.ru/text/78/069/images/image111_9.gif" width="64" height="23 src=">.

In practice, resistor-capacitance dividers with multiple inputs are also used.

2 WORK TASK

Goal of the work: Study of the influence of parameters R.C.- circuits to distort the shape of transmitted pulses.

1. According to the instructions of the teacher, for one of the circuits shown in Figure 10 and the selected values ​​of the element parameters, calculate the relative decline of the peak and the duration of the front of the output signal when a unipolar rectangular pulse is applied to the input.

2. For the selected R.C.- circuit and parameters of its elements, calculate the distortion of the output signal shape when a linearly increasing voltage (sawtooth pulse) is applied to the input.

3. Create a model in Multisim for the selected circuit. Experimentally, using a virtual oscilloscope, determine the values ​​of the parameters of the output pulses given in paragraphs 1 and 2, and compare them with the calculated values. Save as graphic files oscillograms of input and output pulses for subsequent report generation.

4. In the model created in step 3, replace the input signal source with a signal source of a complex shape. Variants of complex signals are shown in Figure 11. The signal shape is set by the teacher. The simulation results should be presented in the report in the form of oscillograms of the input and output signals.

https://pandia.ru/text/78/069/images/image113_3.jpg" width="604" height="527 src=">

Figure 11 - Input signals of various shapes.

3 TEST QUESTIONS

1. Formulate the basic principles of the classical method of analyzing transient processes in pulsed circuits.

2. Formulate the basic principles of the operator method for analyzing transient processes in pulsed circuits.

3. State the basic principles frequency method analysis of transient processes in pulse circuits.

4. What circuits are called linear?

5. What is the principle of superposition when analyzing signals of complex shapes?

BIBLIOGRAPHICAL LIST

1. Ulakhovich theory of linear electrical circuits /. - St. Petersburg. : BHV-Petersburg, 2009. - 816 p.

2. Beletsky linear electrical circuits. Edition 2 / - M.: Lan, 2011. - 544 p.

3. Kolontaevsky: Textbook. manual for vocational technical schools / . - M.: Higher. school, 1988. -304 p.

4 Radio engineering: Tutorial for students of physics and mathematics. fak. ped. in-tov / , . - M.: Education, 1986. -319 p.

5. Goldenberg devices / . M.: Radio and communication, 1981. - 221 p.

6. Honorovsky circuits and signals: Textbook for universities. 4th ed., translated. and additional / . M.: Radio and communication, 1988. -512 p.

preface……………………………………………………….

Brief theoretical information …………………………………

Work assignment………………………………………………………

Control questions ……………………………………………

Bibliography………………………………………

PASSAGE OF SIGNALS

DIFFERENT SHAPE

VIA LINEARR.C. -CHAINS

Laboratory workshop

in the course “Electronic circuits and microcircuitry”

Educational edition

B. I. PETROV
Contributing authors: N. N. BESPALOV, M. V. ILYIN,

S. S. KAPITONOV, .

Printed as provided

original layout

Delivered for recruitment on __.11.2012. Signed for publication on __.12.2012.

Times typeface. Offset printing. Format 60x84 1/16.

Academic ed. l. 0.00 Cond. oven l. ___. Circulation 100 copies.

Mordovian State University named after. Eve

Printed at the Kovylkino Printing House of the Ministry of Press and Information of the Republic of Mordovia

Electrical circuits are an integral part electronic elements automation that performs a large number of various specific functions. The main difference between electrical circuits and electronic circuits is that they are a set of passive linear elements, i.e. those whose current-voltage characteristics obey Ohm’s law, and they do not amplify input signals. Because of this, electrical circuits electronic devices more often called linear devices for converting and generating electrical signals.

Functionally, linear devices for generating and converting electrical signals can be divided into the following main groups:

Integrating circuits used to integrate signals, and sometimes to expand (increase the duration) of pulses;

Differentiating (shortening) circuits used to differentiate signals, as well as to shorten pulses (receiving pulses of a given duration);

Resistor and resistor-capacitance dividers used to change the amplitude of electrical signals;

Pulse transformers used to change the polarity and amplitude of pulses, for galvanic isolation of pulse circuits, to form a positive feedback in generators and pulse shapers, for matching circuits according to the load, for receiving pulses from several output windings;

Electrical filters designed to isolate frequency components located in a given region from a complex electrical signal and to suppress frequency components located in all other frequency regions.

Depending on the elements on which linear devices are implemented, they can be divided into RC, RL and RLC circuits. In this case, linear devices may include a linear resistor R, a linear capacitor C, a linear inductor L, pulse transformer without core saturation. The word “linear” emphasizes that we mean only those types of elements that have volt-ampere characteristics of a linear type, or, in other words, the nominal value of a parameter (resistance, capacitance, etc.) for which is constant and does not depend on the current current or applied voltage. For example, a conventional capacitor with mica dielectric spacers is considered linear over a wide voltage range, but the value of the pn junction capacitance depends on the applied voltage and cannot be attributed to linear elements. In addition, there are always limitations on the amplitude or power of the signal under which the element retains its linear properties. For example, the permissible voltage on a capacitor should not exceed the breakdown value. Other elements have similar restrictions, and they must be taken into account when assigning an element to a particular class.

The most important property of linear devices is their ability to accumulate and release energy in capacitive and inductive elements and thereby convert input signals into time-varying output intervals. This property underlies the operation of generators, impulse noise suppression devices and “competitions” in digital circuits, arising during the passage of an electrical signal through circuits with different time delays.

It should be noted that there are certain difficulties in the use of linear electrical circuits in integrated technology. This is due to the presence of a number of technological difficulties in manufacturing resistors and capacitors, not to mention inductors, in an integral design.

A frequency independent voltage divider is designed to reduce the signal source voltage to the required value. DN is used to match the input stage with the signal source in terms of voltage, to set the operating point of the transistor in the amplifier, to form a reference (more often called “reference”) voltage. The circuit of the simplest voltage divider is shown in the figure just above

When analyzing real electronic circuits To avoid gross errors, it is always necessary to take into account the electrical characteristics of the signal source and load. The most important of them are:

The magnitude and polarity of the EMF of the signal source;

Internal resistance of the signal source (Rg);

Frequency response and phase response of the signal source;

Load resistance (Rн);

The following figure shows the types of voltage dividers.


Figure (a) shows a voltage divider using a variable resistor. Used to adjust the sensitivity of the EI. In the same place, figure b depicts a divider with several output voltages. Such a pattern is used, for example, in a cascode amplifier. In some cases, when the resistance Rн is low, it is used as the lower arm of the divider. For example, when building an amplifier with an OE, the position of the operating point is set by the divider formed by Rb and the resistance of the base junction of the transistor rbe.

An important place in electronics is occupied by voltage dividers, in which the upper or lower arm is formed by variable resistance. If the divider is powered with a constant stable voltage, and, say, a resistance is placed in the lower arm, the value of which depends on temperature, pressure, humidity and other physical parameters, then a voltage proportional to temperature, pressure, humidity, etc. can be removed from the output of the voltage divider . A special place is occupied by dividers, in which one of the resistances depends on the frequency of the supply voltage. They form large group various filters for electrical signals.

Further improvement of the voltage divider led to the appearance of a measuring bridge, which consists of two dividers. In such a circuit, you can pick up a signal both between the midpoint and the common wire, and between two midpoints. In the second case, the range of the output signal doubles with the same change in variable resistances. Electrical signal amplifiers are also a voltage divider, in which the role of a variable resistance is played by a transistor controlled by the input voltage

The simplest integrating chain is a voltage divider in which the role of the lower arm of the divider is performed by capacitor C

Differentiating linear circuits


The simplest differentiating chain is a voltage divider in which the role of the upper arm of the divider is performed by capacitor C

When exposed to continuous random signals, the integrating and differentiating links behave as, respectively, low and high pass filters, elements R1 and C2 form a filter low frequencies, and C1 and R2 are high pass filter

Consider a linear inertial system with a known transfer function or impulse response. Let the input of such a system be a stationary random process with given characteristics: probability density, correlation function or energy spectrum. Let us determine the characteristics of the process at the output of the system: and

The easiest way to find the energy spectrum of the process is at the output of the system. Indeed, individual implementations of the input process are deterministic functions, and the Fourier apparatus is applicable to them. Let

truncated implementation of the duration T of the random process at the input, and

Its spectral density. Output spectral density linear system will be equal

The energy spectrum of the process at the output according to (1.3) will be determined by the expression

those. will be equal to the energy spectrum of the process at the input, multiplied by the square of the amplitude-frequency characteristic of the system, and will not depend on the phase-frequency characteristic.

The correlation function of the process at the output of the linear system can be defined as the Fourier transform of the energy spectrum:

Consequently, when a random stationary process acts on a Linear system, the output also produces a stationary random process with an energy spectrum and correlation function defined by expressions (2.3) and (2.4). The process power at the system output will be equal to

As a first example, consider passing white noise with spectral density through an ideal low-pass filter for which

According to (2.3), the energy spectrum of the process at the output will have a uniform spectral density in the frequency band, and the correlation function will be determined by the expression

The power of the random process at the output of an ideal low-pass filter will be equal to

As a second example, consider the passage of white noise through an ideal bandpass filter, the amplitude-frequency response of which for positive frequencies (Fig. 1.6) is determined by the expression:

We define the correlation function using the Fourier cosine transform:

The correlation function graph is shown in Fig. 1.7

The considered examples are indicative from the point of view that they confirm the connection established in § 3.3 between the correlation functions of low-frequency and narrow-band high-frequency processes with the same shape of the energy spectrum. The process power at the output of an ideal bandpass filter will be equal to



The law of probability distribution of a random process at the output of a linear inertial system differs from the law of distribution at the input, and determining it is a very difficult task, with the exception of two special cases, which we will focus on here.

If a random process acts on a narrow-band linear system, the bandwidth of which is much less than its spectral width, then the phenomenon occurs at the output of the system normalization distribution law. This phenomenon lies in the fact that the distribution law at the output of a narrowband system tends to normal, regardless of what distribution the broadband random process at the input has. Physically this can be explained as follows.

The process at the output of an inertial system at some point in time is a superposition of individual responses of the system to the chaotic influences of the input process at different points in time. The narrower the system bandwidth and the wider the spectrum of the input process, the greater the number of elementary responses that form the output process. According to the central limit theorem of probability theory, the distribution law of a process, which is the sum of a large number of elementary responses, will tend to normal.

From the above reasoning follows the second particular, but very important case. If the process at the input of a linear system has a normal (Gaussian) distribution, then it remains normal at the output of the system. In this case, only the correlation function and the energy spectrum of the process change.

In radio electronics one has to deal with various signals and different circuits; when signals pass through such circuits, transient processes occur, as a result of which the shape transmitted signal can change. Most devices contain a combination of linear and nonlinear elements, which complicates the rigorous analysis of signal flow. However, there is a fairly wide range of problems that can be successfully solved by linear methods, even if there is a nonlinear element in the circuit. This applies to devices in which the signals are so small in amplitude that the nonlinearity of the characteristics nonlinear element can be neglected, so it can also be considered linear.

Most methods for analyzing the passage of signals through a linear circuit are based on a fundamental principle - the principle of superposition, in which the response of a circuit to a complex influence can be defined as the sum of reactions to simpler signals into which the complex influence can be decomposed. The response of a linear circuit to a known simple (test) influence is called systemic (i.e., dependent only on the circuit) transmission characteristics of the circuit. The transfer characteristic itself can be determined:

A) classic a method in which the circuit is described by a system of linear differential equations, on the right side of which the test effect is written; this method most often determines reactions to a unit step function or delta function, the so-called transient and impulse characteristics of the circuit, which are the transfer characteristics of the circuit for the superposition method (or the Duhamel integral method); Using the classical method, with fairly simple circuits and influences, the analysis problem can be immediately solved, i.e. finding the circuit's response to the input signal;

b) comprehensive method, if a harmonic oscillation is used as a test signal; in this case, the transfer characteristic of the circuit is determined as frequency characteristic that is the basis of the frequency analysis method;

V) operator a method in which the Laplace transform apparatus is used, as a result of which it is determined control room transfer characteristic of the circuit, since the operator method uses a signal of the form e pt, Where p=s + jw, then when replacing in the operator transfer characteristic p on jw the frequency transfer characteristic is obtained; in addition, as will be shown below, the original from the operator transfer characteristic is the impulse response of the circuit.

Therefore, it is possible to classify methods for analyzing the passage of complex signals into

A) frequency, used mainly for the analysis of steady-state processes;

b) temporary, using the transient or impulse response of the circuit, used in cases of rapidly changing (pulse) signals, when transient processes in the circuit are important.

When analyzing the passage of signals through narrow-band selective circuits, the same methods can be used not for instantaneous signal values, but for a slowly changing envelope.