How to experimentally measure the time characteristics of linear circuits. Calculation of timing characteristics of linear electrical circuits

MINISTRY OF EDUCATION OF UKRAINE

Kharkov State Technical University of Radio Electronics

Settlement and explanatory note

for course work

in the course “Fundamentals of Radio Electronics”

Topic: Calculation of frequency and time characteristics of linear circuits

Option No. 34

INTRODUCTION

Knowledge of fundamental basic disciplines in the preparation and formation of a future design engineer is very great.

The discipline “Fundamentals of Radio Electronics” (FRE) is one of the basic disciplines. When studying this course theoretical knowledge and practical skills are acquired in using this knowledge to calculate specific electrical circuits.

The main goal of the course work is to consolidate and deepen knowledge in the following sections of the electronics training course:

calculation of linear electrical circuits under harmonic influence using the complex amplitude method;

frequency characteristics of linear electrical circuits;

timing characteristics of circuits;

methods for analyzing transient processes in linear circuits (classical, superposition integrals).

Coursework consolidates knowledge in the relevant field, and those who do not have any knowledge are encouraged to obtain it by a practical method - solving assigned problems.

Option No. 34

1. Determine the complex input resistance of the circuit.

2. Find the module, argument, active and reactive components of the complex resistance of the circuit.

3. Calculation and construction of frequency dependences of the module, argument, active and reactive components of the complex input resistance.

4. Determine the complex transmission coefficient of the circuit, plot graphs of amplitude-frequency (AFC) and phase-frequency (PFC) characteristics.

5. Determine the transient response of the circuit using the classical method and construct its graph.

6. Find the impulse response of the circuit and plot it.

1 CALCULATION OF THE COMPLEX INPUT RESISTANCE OF THE CIRCUIT

1.1 Determination of the complex input impedance of a circuit

(1)

After substitution numerical values we get:

(2)

Specialists who design electronic equipment. Coursework in this discipline is one of the stages independent work, which allows you to determine and study the frequency and time characteristics of election circuits, establish a connection between the limiting values ​​of these characteristics, and also consolidate knowledge on spectral and time methods for calculating the response of the circuit. 1. Calculation...

T, μs m=100 1.982*10-4 19.82 m=100000 1.98*10-4 19.82 The timing characteristics of the circuit under study are shown in Fig. 6, Fig. 7. Frequency characteristics are shown in Fig. 4, fig. 5. TIME METHOD OF ANALYSIS 7. DETERMINING THE RESPONSE OF A CIRCUIT TO AN IMPULSE Using the Duhamel integral, you can determine the response of a circuit to a given impact even in the case when an external impact on...

Previously, we considered frequency characteristics, and time characteristics describe the behavior of a circuit over time for a given input action. There are only two such characteristics: transient and impulse.

Step response

The transient response - h(t) - is the ratio of the circuit's response to an input step action to the magnitude of this action, provided that before it there were no currents or voltages in the circuit.

The graph has a stepwise effect:

1(t) - single step effect.

Sometimes a step function is used that does not start at moment “0”:

To calculate the transient response, a constant EMF (if the input action is voltage) or a constant current source (if the input action is current) is connected to a given circuit and the transient current or voltage specified as a reaction is calculated. After this, divide the result by the source value.

Example: find h(t) for u c with input action in the form of voltage.

Example: solve the same problem with input action in the form of current

Impulse response

The impulse response - g(t) - is the ratio of the circuit's response to an input influence in the form of a delta function to the area of ​​this influence, provided that before connecting the influence there were no currents or voltages in the circuit.

d(t) - delta function, delta impulse, unit impulse, Dirac impulse, Dirac function. This is the function:

It is extremely inconvenient to calculate g(t) using the classical method, but since d(t) is formally a derivative, it can be found from the relation g(t) = h(0) d(t) + dh(t)/dt.

To experimentally determine these characteristics, one has to act approximately, that is, it is impossible to create the exact required effect.

A sequence of pulses similar to rectangular ones fall at the input:

t f - duration of the leading edge (rise time of the input signal);

t and - pulse duration;

These impulses have certain requirements:

a) for the transient response:

T pause should be so large that by the time the next pulse arrives, the transition process from the end of the previous pulse is practically over;

T should be so large that the transient process caused by the occurrence of a pulse also practically has time to end;

T f should be as small as possible (so that during t cf the state of the circuit practically does not change);

X m should, on the one hand, be so large that using the existing equipment it would be possible to register the reaction of the chain, and on the other hand, it should be so small that the chain under study retains its properties. If all this is true, record the circuit reaction graph and change the scale along the ordinate axis by X m times (X m = 5V, divide the ordinate by 5).

b) for impulse response:

t pause - the requirements are the same for X m - the same, there are no requirements for t f (because even the pulse duration t f itself must be so short that the state of the circuit practically does not change. If all this is so, record the reaction and change the scale along the ordinate axis by the area of ​​the input pulse.

Results using the classical method

The main advantage is the physical clarity of all quantities used, which allows you to check the progress of the solution from the point of view of physical meaning. In simple circuits it is possible to get the answer very easily.

Disadvantages: as the complexity of the problem increases, the complexity of the solution quickly increases, especially at the stage of calculating the initial conditions. Not all problems are convenient to solve using the classical method (almost no one looks for g(t), and everyone has problems when calculating problems with special contours and special sections).

Before switching, .

Consequently, according to the commutation laws, u c1 (0) = 0 and u c2 (0) = 0, but from the diagram it is clear that immediately after closing the key: E= u c1 (0)+u c2 (0).

In such problems it is necessary to use a special procedure for searching for initial conditions.

These shortcomings can be overcome in the operator method.

Linear circuits

Test No. 3

Self-test questions

1. List the main properties of the probability density of a random variable.

2. How are the probability density and the characteristic function of a random variable related to each other?

3. List the basic laws of distribution of a random variable.

4. What is the physical meaning of the dispersion of an ergodic random process?

5. Give several examples of linear and nonlinear, stationary and nonstationary systems.

1. A random process is called:

a. Any random change in some physical quantity over time;

b. A set of time functions that obey some statistical pattern common to them;

c. A set of random numbers that obey some statistical pattern common to them;

d. A set of random functions of time.

2. Stationarity of a random process means that over the entire period of time:

a. The mathematical expectation and variance are unchanged, and the autocorrelation function depends only on the difference in time values t 1 and t 2 ;

b. The mathematical expectation and dispersion are unchanged, and the autocorrelation function depends only on the start and end times of the process;

c. The mathematical expectation is unchanged, and the variance depends only on the difference in time values t 1 and t 2 ;

d. The variance is unchanged, and the mathematical expectation depends only on the start and end times of the process.

3. An ergodic process means that the parameters of a random process can be determined by:

a. Multiple final implementations;

b. One final implementation;

c One endless realization;

d. Several infinite implementations.

4. The power spectral density of the ergodic process is:

a. Limit on the spectral density of a truncated implementation divided by time T;

b. Spectral density of the final realization with duration T, divided by time T;

c. Limit of spectral density of truncated implementation;

d. Spectral density of the final realization with duration T.

5. The Wiener–Khinchin theorem is the relationship between:

a. Energy spectrum and mathematical expectation of a random process;

b. Energy spectrum and dispersion of a random process;

c. Correlation function and dispersion of a random process;

d. Energy spectrum and correlation function of a random process.

The electrical circuit converts the signals arriving at its input. Therefore, in the very general case mathematical model circuits can be specified in the form of a relationship between the input influence S in (t) and output reaction S out (t) :

S out (t)=TS in (t),

Where T– chain operator.

Based on the fundamental properties of the operator, we can draw a conclusion about the most essential properties of the circuits.

1. If the chain operator T does not depend on the amplitude of the influence, then the circuit is called linear. For such a circuit, the principle of superposition is valid, reflecting the independence of the action of several input influences:

T=TS in1 (t)+TS in2 (t)+…+TS inn (t).

It is obvious that when linear transformation signals in the response spectrum do not oscillate with frequencies different from the frequencies of the impact spectrum.

The class of linear circuits is formed by both passive circuits, consisting of resistors, capacitors, inductances, and active circuits, which also include transistors, lamps, etc. But in any combination of these elements, their parameters should not depend on the amplitude of the influence.

2. If a time shift in the input signal leads to the same shift in the output signal, i.e.

S out (t t 0)=TS in (t t 0),

then the circuit is called stationary. The property of stationarity does not apply to circuits containing elements with time-varying parameters (inductors, capacitors, etc.).

Unit functions and their properties An important place in the theory of linear circuits is occupied by the study of the reaction of these circuits to idealized external influences, described by the so-called unit functions. A unit step function (Heaviside function) is the function: The graph of the function 1(t-t 0) has the form of a step or jump, the height of which is 1. A jump of this type will be called unit.

Unit functions and their properties Due to the fact that the product of any bounded time function f(t) by 1(t-t 0) is equal to zero at t

Unit functions and their properties If at t=t 0 a source of harmonic current or voltage is included in the circuit, then the external influence on the circuit can be represented as: If the external influence on the circuit at time t=t 0 changes abruptly from one fixed value X 1 to another X 2, then

Unit functions and their properties External influence on the circuit, which has the form of a rectangular pulse of height X and duration t and (Fig.), can be represented as the difference between two identical jumps shifted in time by t

Unit functions and their properties Consider a rectangular pulse of duration and height 1/ t (Fig.). Obviously, the area of ​​this pulse is equal to 1 and does not depend on t. As the duration of the pulse decreases, its height increases, and as t→ 0 it tends to infinity, but the area remains equal to 1. A pulse of infinitely short duration, infinitely large height, the area of ​​which is 1, will be called a unit pulse. The function defining the unit impulse is denoted (t-t 0) and is called the δ-function or Dirac function.

Unit functions and their properties Using the δ-function, you can select the values ​​of the function f(t) at arbitrary times t 0. This feature of the δ-function is usually called the filtering property. At t 0 =0, operator images of unit functions have a particularly simple form:

Transient and impulse characteristics of linear circuits The transient response g(t-t 0) of a linear circuit that does not contain independent energy sources is the ratio of the reaction of this circuit to the influence of a non-unit current or voltage jump to the height of this jump under zero initial conditions: The transient response of the circuit is numerically equal to the reaction of the circuit on the impact of a single current or voltage surge. The dimension of the transient characteristic is equal to the ratio of the response dimension to the dimension of the external influence, therefore the transient characteristic can have the dimension of resistance, conductivity, or be a dimensionless quantity.

Transient and impulse characteristics of linear circuits The impulse response h(t-t 0) of a linear circuit that does not contain independent energy sources is the ratio of the reaction of this circuit to the action of an infinitely short pulse of infinitely large height and finite area to the area of ​​this impulse under zero initial conditions: Impulse response of the circuit is numerically equal to the reaction of the circuit to the action of a single impulse. The dimension of the impulse response is equal to the ratio of the dimension of the circuit response to the product of the dimension of the external influence and time.

Transient and impulse characteristics of linear circuits Like the complex frequency and operator characteristics of a circuit, the transient and impulse characteristics establish a connection between the external influence on the circuit and its reaction, however, unlike the complex frequency and operator characteristics, the argument of the transient and impulse characteristics is time t, and not angular ω or complex p frequency. Since the characteristics of a circuit whose argument is time are called time characteristics, and whose argument is frequency (including complex) are called frequency characteristics, then transient and impulse characteristics refer to the time characteristics of the circuit.

Transient and impulse characteristics of linear circuits Thus, impulse response chain hkv(t) is a function whose image, according to Laplace, is the operator characteristic of the chain Hkv(p), and the transition characteristic of the chain gkv(t) is a function whose operator image is equal to Hkv(p)/p.

Determination of the reaction of a chain to an arbitrary external influence External influence on the circuit is presented in the form of a linear combination of the same type of elementary components: and the reaction of the chain to such an influence is found in the form of a linear combination of partial reactions to the influence of each of the elementary components of the external influence separately: You can choose as elementary components external influences, the most widespread are elementary (test) influences in the form of a harmonic function of time, a single jump and a single impulse.

Determining the response of a circuit to an arbitrary external influence by its transient response Let us consider an arbitrary linear electrical circuit that does not contain independent energy sources, the transient response g(t) of which is known. Let the external influence on the circuit be given in the form of an arbitrary function x=x(t), equal to zero at t

Determining the response of a circuit to an arbitrary external influence by its transient characteristic The function x(t) can be approximately represented as a sum of non-unit jumps or, what is the same, as a linear combination of single jumps, shifted relative to one another by: In accordance with the definition of the transient characteristic the response of the circuit to the influence of a non-unit jump applied at time t= k is equal to the product of the jump height and the transient response of the circuit g(t- k). Consequently, the response of the circuit to the impact represented by the sum of non-unit jumps (6.114) is equal to the sum of the products of the jump heights and the corresponding transient characteristics:

Determining the response of a circuit to an arbitrary external influence by its transient response Obviously, the accuracy of representing the input action in the form of a sum of non-unit jumps, as well as the accuracy of representing the response of the circuit, increases with decreasing time step. When → 0, summation is replaced by integration: The expression is known as the Duhamel integral (superposition integral). Using this expression, you can find the exact value of the circuit's response to a given impact x=x(t) at any time t after switching. Integration in is carried out over the interval t 0

Determining the reaction of a chain to an arbitrary external influence by its transient characteristic Using the Duhamel integral, you can determine the reaction of a chain to a given influence even in the case when the external influence on the chain is described by a piecewise continuous function, i.e. a function that has a finite number of finite breaks . In this case, the integration interval must be divided into several intervals in accordance with the intervals of continuity of the function x=x(t) and take into account the reaction of the circuit to finite jumps of the function x=x(t) at the break points.