How to experimentally record the temporal characteristics of linear circuits. Time characteristics of linear circuits, unit functions and. Self-test questions

Send your good work in the knowledge base is simple. Use the form below

Students, graduate students, young scientists who use the knowledge base in their studies and work will be very grateful to you.

Posted on http://www.allbest.ru/

COURSE WORK

Time and frequency characteristics of linear electrical circuits

Initial data

Circuit diagram of the investigated circuit:

The meaning of the parameters of the elements:

External influence:

u 1 (t) = (1 + e - бt) 1 (t) (B)

As a result of the course work, you need to find:

1. Expression for the primary parameters of a given two-port network as a function of frequency.

2. Find an expression for the complex voltage transfer coefficient K 21 (j w) a four-pole system in idle mode on terminals 2 - 2 ".

3. Amplitude-frequency K 21 (j w) and phase-frequency Ф 21 (j w

4. Operator voltage transmission coefficient K 21 (p) of a four-pole in no-load mode at terminals 2-2 ".

5. Transient response h (t), impulse response g (t).

6. Response u 2 (t) to a given input action in the form u 1 (t) = (1 + e - бt) 1 (t) (B)

1. DefineYparameters for a given two-port network

I1 = Y11 * U1 + Y12 * U2

I2 = Y21 * U1 + Y22 * U2

To find Y22 easier, we find A11 and A12 and express Y22 in terms of them.

Experience 1. XX on clamps 2-2 "

We make the change 1 / jwС = Z1, R = Z2, jwL = Z3, R = Z4

Let's make a circuit equivalent circuit

Z11 = (Z4 * Z2) / (Z2 + Z3 + Z4)

Z33 = (Z2 * Z3) / (Z2 + Z3 + Z4)

U2 = (U1 * Z11) / (Z11 + Z33 + Z1)

Experience 2: short circuit at terminals 2-2 "

By the method of loop currents, we will compose the equations.

a) I1 (Z1 + Z2) - I2 * Z2 = U1

b) I2 (Z2 + Z3) - I1 * Z2 = 0

From equation b) we express I1 and substitute it into equation a).

I1 = I2 (1 + Z3 / Z2) * (Z1 + Z2) - I2 * Z2 = U1

A12 = Z1 + Z3 + (Z1 * Z3) / Z2

Hence we get that

Experience 2: short circuit at terminals 2-2 "

Let's compose an equation using the loop current method:

I1 * (Z1 + Z2) - I2 * Z2 = U1

I2 (Z2 + Z3) - I1 * Z2 = 0

Let us express I2 from the second equation and substitute it into the first:

We express I1 from the second equation and substitute it into the first:

For a mutual four-pole Y12 = Y21

Matrix A of parameters of the considered two-port network

2 . Find the complex voltage transfer coefficientTO 21 (jw ) a four-terminal network in idle mode on terminals 2-2 ".

Complex voltage transfer coefficient K 21 (j w) is determined by the relation:

It can be found from the system of standard basic equations for Y parameters:

I1 = Y11 * U1 + Y12 * U2

I2 = Y21 * U1 + Y22 * U2

So, according to the condition for idling I2 = 0, you can write

We get the expression:

K 21 (j w) = - Y21 / Y22

We make the replacement Z1 = 1 / (j * w * C), Z2 = 1 / R, Z3 = 1 / (j * w * C), Z4 = R, we obtain an expression for the complex voltage transfer coefficient K 21 (j w) in idle mode on clamps 2-2 "

Let us find the complex voltage transfer coefficient K 21 (j w) a four-terminal network in idle mode on clamps 2-2 "in numerical form, substituting the parameter values:

Let us find the amplitude-frequency K 21 (j w) and phase-frequency Ф 21 (j w) characteristics of the voltage transfer coefficient.

Let us write the expression for K 21 (j w) in numerical form:

Let us find the calculation formula for the phase-frequency Ф 21 (j w) characteristics of the voltage transfer coefficient as arctan of the imaginary part to the real one.

As a result, we get:

Let us write the expression for the phase-frequency Ф 21 (j w) characteristics of the voltage transfer coefficient in numerical form:

Resonant frequency w0 = 7 * 10 5 rad / s

Let's build frequency response graphs (Appendix 1) and phase frequency response (Appendix 2)

3. Find the operator voltage transfer coefficientK 21 x (p) a four-pole system in idle mode at terminals 2-2 "

operator voltage pulse circuit

Operator circuit equivalent circuit outward appearance does not differ from a complex equivalent circuit, since the analysis of the electrical circuit is carried out at zero initial conditions. In this case, to obtain the operator voltage transmission coefficient, it is sufficient to replace jw in the expression for the complex transmission coefficient by the operator R:

Let us write the expression for the operator voltage transfer coefficient K21x (p) in numerical form:

Let us find the value of the argument p n for which M (p) = 0, i.e. the poles of the function K21x (p).

Let us find the values of the argument p k for which N (p) = 0, i.e. zeros of the function K21x (p).

Let's compose a pole-zero diagram:

Such a pole-zero diagram testifies to the vibrationally damping nature of transient processes.

This pole-zero diagram contains two poles and one zero.

4. Timing calculation

Let us find the transient g (t) and impulse h (t) characteristics of the circuit.

The operator expression K21 (p) allows you to obtain an image of the transient and impulse characteristics

g (t) h K21 (p) / p h (t) h K21 (p)

We transform the image of the transient and impulse characteristics to the form:

Let us now determine the transient characteristic g (t).

Thus, the image is reduced to the following operator function, the original is in the table:

Thus, we find the transient response:

Let's find the impulse response:

Thus, the image is reduced to the following operator function, the original, which is in the table:

Hence we have

Let's calculate a number of values g (t) and h (t) for t = 0h10 (μs). And we will build graphs of the transient (Appendix 3) and impulse (Appendix 4) characteristics.

For a qualitative explanation of the type of transient and impulse characteristics of the circuit, we connect an independent voltage source e (t) = u1 (t) to the input terminals 1-1. voltage jump e (t) = 1 (t) (V) at zero initial conditions. At the initial moment of time after switching, the voltage across the capacitance is equal to zero, since according to the laws of commutation, at a finite value of the amplitude of the input jump, the voltage across the capacitance cannot change. Therefore, looking at our chain, you can see that u2 (0) = 0 i.e. g (0) = 0. Over time, with t tending to infinity, only direct currents will flow through the circuit, which means that the capacitor can be replaced with a gap, and the coil with a short-circuited section, and looking at our circuit it can be seen that u2 (t) = 0.

The impulse characteristic of the circuit numerically coincides with the output voltage when a single voltage pulse is applied to the input e (t) = 1d (t) V. During the action of a single pulse, the input voltage is applied to the inductance, the current in the inductance increases abruptly from zero to 1 / L, and the voltage across the capacitor does not change and is equal to zero. At t> = 0, the voltage source can be replaced by a short-circuited jumper, and a damped oscillatory process of energy exchange between the inductance and the capacitance arises in the circuit. At the initial stage, the inductance current gradually decreases to zero, charging the capacitance to the maximum voltage value. Subsequently, the capacitance is discharged, and the inductance current gradually increases, but in the opposite direction, reaching the largest negative value at Uc = 0. When t tends to infinity, all currents and voltages in the circuit tend to zero. Thus, the oscillatory character of the voltage across the capacitor, which dies out over time, explains the form of the impulse response, in which case h (?) Is equal to 0

6. Calculation of the response to a given input action

Using the superposition theorem, the action can be represented as partial actions.

U 1 (t) = U 1 1 + U 1 2 = 1 (t) + e - бt 1 (t)

The response U 2 1 (t) coincides with the transient response

The operator response U 2 2 (t) to the second partial action is equal to the product of the operator transmission coefficient of the chain and the Laplace exponential image:

Find the original U22 (p) according to the Laplace transform table:

We define a, w, b, K:

Finally, we get the original response:

Let's calculate a number of values and build a graph (Appendix 5)

Conclusion

In the course of work, the frequency and time characteristics of the circuit were calculated. Expressions are found for the response of the circuit to harmonic action, as well as the basic parameters of the circuit.

The complex-conjugate poles of the operator voltage coefficient indicate the damping nature of the transients in the circuit.

Bibliography

1. Popov V.P. Fundamentals of circuit theory: Textbook for universities - 4th ed., Revised, M. Vyssh. shk., 2003. - 575 p .: ill.

2. Biryukov V.N., Popov V.P., Sementsov V.I. Collection of problems in the theory of circuits / ed. V.P. Popov. M .: Higher. school: 2009, 269 p.

3. Korn G., Korn T., Handbook of mathematics for engineers and university students. Moscow: Nauka, 2003, 831 p.

4. Biryukov VN, Dedyulin KA, Methodological manual №1321. Methodical instruction to the course work for the course Fundamentals of circuit theory, Taganrog, 1993, 40 p.

Posted on Allbest.ru

Similar documents

Determination of the primary parameters of the four-port network, the voltage transmission coefficient in the open mode at the output. Amplitude-frequency and phase-frequency characteristics of the voltage transmission coefficient. Analysis of the response of the circuit to the input stimulus.

term paper, added 07/24/2014

Determination of parameters of a four-port network. Complex voltage transfer ratio. Complex equivalent circuit in case of short circuit at the output of the circuit. Amplitude-frequency and phase-frequency characteristics of the voltage transfer coefficient.

term paper, added 07/11/2012

Analysis of frequency and transient characteristics of electrical circuits. Calculation of the frequency characteristics of an electrical circuit and a linear circuit under impulse action. Complex functions of the frequency of exposure. Formation and generation of electrical impulses.

test, added 01/05/2011

Methods for obtaining the characteristic equation. Transient processes in circuits with one reactive element, with two dissimilar reactive elements. Time characteristics of circuits. Calculation of the reaction of a linear circuit to an input action of an arbitrary type.

test, added 11/28/2010

Calculation of the complex voltage transfer coefficient for a four-port network, Determination of its transient response by the classical and operator method. Calculation of characteristic impedances of a two-port network, as well as its constant transmission.

term paper added 11/26/2014

Construction of passive four-pole circuits, active four-pole circuits, their cascade connection. Finding the voltage transfer coefficient. Calculation of frequency characteristics and transient process in an electrical circuit. Transient circuit analysis.

term paper, added 09/23/2014

Characteristics of methods for analyzing non-stationary operating modes of the circuit. Features of the study of transient processes in linear electrical circuits. Calculation of transient processes, the law of voltage change using the classical and operator method.

test, added 08/07/2013

Determination of the amplitude and phase-frequency characteristics (CH) of the input and transfer functions of the circuit. Calculation of resonant frequencies and resistances. Investigation of a transistor model with generalized and selective load. Automated calculation of the frequency response of the complete model.

term paper, added 12/05/2013

Analysis of the parameters of an active four-pole network, drawing up an equation for the electrical equilibrium of a circuit using the loop currents method. Determination of the voltage transmission coefficient. Transient and impulse characteristics of the circuit. Determination of reversibility conditions.

term paper added 03/21/2014

Calculation of a linear electrical circuit with a periodic non-sinusoidal voltage, active and full power of the network. The procedure for determining the parameters of an asymmetrical three-phase circuit. Calculation of the main transient processes in linear electrical circuits.

The expressions (5.17), (5.18) given in the previous paragraph for the amplification factors can be interpreted as transfer functions of a linear active four-port network. The nature of these functions is determined by the frequency properties of the Y parameters.

Writing in the form of functions, we arrive at the concept of the transfer function of a linear active four-port network. Generally dimensionless, the complex function is an exhaustive characteristic of a two-port network in the frequency domain. It is determined in a stationary mode with harmonic excitation of a four-port network.

The transfer function is often conveniently represented in the form

The module is sometimes called the amplitude-frequency characteristic (AFC) of the four-port network. The argument is called the phase-frequency characteristic (PFC) of the quadrupole.

Another comprehensive characteristic of a four-port network is its impulse response, which is used to describe a circuit in the time domain.

For active linear circuits, as well as for passive ones, the impulse response of the circuit means the response, the response of the circuit to the action, which has the form of a single impulse (delta function). The connection between is easy to establish using the Fourier integral.

If a single EMF pulse (delta function) acts at the input of a four-port network with a spectral density equal to one for all frequencies, then the spectral density of the output voltage is simply. The response to a single impulse, that is, the impulse response of the circuit, is easily determined using the inverse Fourier transform applied to transfer function :

It should be borne in mind that before right side This equality has a factor of 1 with the dimension of the area of the delta function. In the particular case, when the b-pulse of voltage is meant, this dimension will be [volts x second].

Accordingly, the function is the Fourier transform of the impulse response:

In this case, before the integral we mean a factor of one with the dimension [volts x second] ^ - 1.

In what follows, the impulse response will be denoted by a function, by which we can mean not only voltage, but also any other electrical quantity that is a response to the action in the form of a delta function.

As in the representation of signals on the plane of complex frequency (see § 2.14), in the theory of circuits the concept of a transfer function considered as the Laplace transform of the function 8

1. TASK

The circuit of the investigated circuit [Fig. 1] No. 22, in accordance with the option of assignment 22 - 13 - 5 - 4. Parameters of the circuit elements: L = 2 mH, R = 2 kOhm, C = 0.5 nF.

The external influence is given by the function:, where a is calculated by the formula (1) and is equal to.

Figure 1. Wiring diagram of a given circuit

It is necessary to determine:

a) an expression for the primary parameters of a given two-port network as a function of frequency;

b) the complex voltage transmission coefficient of the four-port network in the no-load mode at the terminals;

c) amplitude-frequency and phase-frequency characteristics of the voltage transmission coefficient;

d) the operator's voltage transmission coefficient of the four-port network in the no-load mode at the terminals;

e) the transient response of the circuit;

e) impulse response of the circuit;

g) the response of the circuit to a given input action when the load is disconnected.

2. CALCULATION PART

.1 Determination of the primary parameters of a four-port network

To determine the Z - parameters of the four-terminal network, we will compose the equations of electrical equilibrium of the circuit by the method of loop currents using a complex circuit equivalent circuit [Fig. 2]:

Figure 2. Complex equivalent circuit of a given electrical circuit

Choosing the direction of traversing the contours, as indicated in [Fig. 2], and considering that

we write down the contour equations of the circuit:

Substitute the values and into the resulting equations:

(2)

The resulting equations (2) contain only currents and voltages at the input and output terminals of a four-port network and can be converted to the standard form of writing the basic equations of a four-port network in the form Z:

(3)

Transforming equations (2) to the form (3), we get:

Comparing the resulting equations with equations (3), we obtain:

quadripole voltage idle amplitude

2.2 Determination of the voltage transmission coefficientin idle mode at the output

We find the complex voltage transfer coefficient from terminals to terminals in no-load mode () at the output using the values obtained in paragraph 2.1 expressions for primary parameters:

2.3 Determination of amplitude-frequencyand phase-frequencyvoltage transmission coefficient characteristics

Consider the resulting expression for as the ratio of two complex numbers, find an expression for the frequency response and phase response.

The frequency response will look like:

From formula (4) it follows that the phase-frequency characteristic will have the form:

Where, rad / s is found from the equation

Frequency response and phase response graphs are shown on the next page. [fig.3, fig.4]

Figure 3. Frequency response

Figure 4. Phase response

Limit values and at to control calculations, it is useful to determine without resorting to calculation formulas:

· Given that the resistance of the inductance at constant current is zero, and the resistance of the capacitance is infinitely large, in the circuit [see. fig. 1], you can break the branch containing the capacitance and replace the inductance with a jumper. In the resulting circuit and, because the input voltage is in phase with the voltage at the terminals;

· At an infinitely high frequency, the branch containing the inductance can be broken, because the inductance resistance tends to infinity. Despite the fact that the resistance of the capacitance tends to zero, it cannot be replaced with a jumper, since the voltage across the capacitance is a response. In the resulting circuit [see. Fig. 5], for,, the input current is ahead of the input voltage in phase, and the output voltage coincides in phase with the input voltage, therefore .

Figure 5. Electrical diagram of a given circuit at.

2.4 Determination of the operating voltage transmission ratioquadripole in idle mode on the clamps

The operator circuit of the equivalent circuit in appearance does not differ from the complex equivalent circuit [Fig. 2], since the analysis of the electrical circuit is carried out under zero initial conditions. In this case, to obtain the operator voltage transmission coefficient, it is sufficient to replace the operator in the expression for the complex transmission coefficient:

We transform the last expression so that the coefficients at the highest powers in the numerator and denominator are equal to one:

The function has two complex conjugate poles:; and one real zero: .

Figure 6. Pole-zero function diagram

The pole-zero diagram of the function is shown in Fig. 6. Transient processes in the circuit have an oscillatory damping character.

2.5 Definition of transientand impulsecircuit characteristics

The operator expression allows you to get images of the transient and impulse responses. It is convenient to determine the transient response using the relationship between the Laplace image of the transient response and the operator transmission coefficient:

(5)

The impulse response of the circuit can be obtained from the ratios:

(6)

(7)

Using formulas (5) and (6), we write the expressions for the images of the impulse and transient characteristics:

We transform the images of the transient and impulse responses to a form convenient for determining the originals of the time characteristics using the Laplace transform tables:

(8)

(9)

Thus, all images are reduced to the following operator functions, the originals of which are given in the Laplace transform tables:

(12)

Considering that for this considered case , , , we find the values of the constants for expression (11) and the values of the constants for expression (12).

For expression (11):

And for expression (12):

Substituting the obtained values into expressions (11) and (12), we get:

After transformations, we get the final expressions for the temporal characteristics:

The transient process in this circuit ends after switching for the time , where - is defined as the reciprocal of the absolute minimum value of the real part of the pole. Because , then the decay time is (6 - 10) μs. Accordingly, we choose the interval for calculating the numerical values of the time characteristics ... Transient and impulse response graphs are shown in Figs. 7 and 8.

For a qualitative explanation of the type of transient and impulse characteristics of the circuit to the input terminals, an independent voltage source. The transient response of the circuit numerically coincides with the voltage at the output terminals when a single voltage jump is applied to the circuit at zero initial conditions. At the initial moment of time after switching, the voltage across the capacitor is zero, since, according to the laws of commutation, at a finite value of the jump amplitude, the voltage across the capacitance cannot change abruptly. Hence, that is. When the voltage at the input can be considered constant and equal to 1V, that is. Accordingly, only direct currents can flow in the circuit, therefore the capacitance can be replaced by an open, and the inductance by a jumper, therefore, in the circuit converted in this way, that is. The transition from the initial state to the steady state occurs in an oscillatory mode, which is explained by the process of periodic exchange of energy between inductance and capacitance. Damping of oscillations occurs due to energy losses on resistance R.

Figure 7. Transient response.

Figure 8. Impulse response.

The impulse response of the circuit numerically coincides with the output voltage when a single voltage pulse is applied to the input ... During the action of a single pulse, the capacitance is charged to its maximum value, and the voltage across the capacitance becomes equal to

When the voltage source can be replaced with a short-circuited jumper, and a damped oscillatory process of energy exchange between the inductance and the capacitance occurs in the circuit. At the initial stage, the capacitance is discharged, the capacitance current gradually decreases to 0, and the inductance current increases to its maximum value at. Then the inductance current, gradually decreasing, recharges the capacitor in the opposite direction, etc. When, due to the dissipation of energy in the resistance, all currents and voltages of the circuit tend to zero. Thus, the oscillatory nature of the voltage across the capacitance damping over time explains the form of the impulse response, and and .

The correctness of the impulse response calculation is confirmed qualitatively by the fact that the graph in Fig. 8 passes through 0 at those times when the graph in Fig. 7 has local extremes, and the maxima coincide in time with the inflection points of the graph. And also the correctness of the calculations is confirmed by the fact that the graphs and, in accordance with formula (7), coincide. To check the correctness of finding the transient characteristic of the circuit, we will find this characteristic when a single voltage jump is applied to the circuit using the classical method:

Let us find independent initial conditions ():

Let us find the dependent initial conditions ():

To do this, turn to Fig. 9, which shows a circuit diagram at a time, then we get:

Figure 9. Circuit diagram at time

Let's find the forced component of the response:

To do this, refer to Fig. 10, which shows the circuit diagram after switching. Then we get that

Figure 10. Circuit diagram for.

Let's compose differential equation:

To do this, we first write down the equation of the balance of currents in the node according to the first Kirchhoff's law and write down some equations based on the second Kirchhoff's laws:

Using the component equations, we transform the first equation:

Let us express all unknown voltages in terms of:

Now, differentiating and transforming, we obtain a differential equation of the second order:

Substitute the known constants and get:

5. Let's write down the characteristic equation and find its roots:
to zero. The time constant and the quasi-period of the oscillation of the temporal characteristics coincide with the results obtained from the analysis of the operator gain; The frequency response of the circuit under consideration is close to the frequency response of an ideal low-pass filter with a cutoff frequency .

List of used literature

1. Popov V.P. Fundamentals of circuit theory: Textbook for universities - 4th ed., Rev. - M .: Higher. shk., 2003 .-- 575s.: ill.

Korn, G., Korn, T., A Handbook of Mathematics for Engineers and High School Students. Moscow: Nauka, 1973, 832 p.

MINISTRY OF EDUCATION OF UKRAINE

Kharkiv State Technical University of Radio Electronics

Settlement and explanatory note

to term paper

on the course "Fundamentals of Radio Electronics"

Topic: Calculation of frequency and time characteristics of linear circuits

Option number 34

INTRODUCTION	3
EXERCISE	4
1 CALCULATION OF INTEGRATED INPUT CIRCUIT RESISTANCE	5
1.1 Determination of the complex input impedance of a circuit	5
1.2 Determination of the active component of the complex input resistance of the circuit	6
1.3 Determination of the reactive component of the complex input resistance of the circuit	7
1.4 Determination of the modulus of the complex input impedance of the circuit	9
1.5 Determining the Argument of the Complex Input Impedance of a Circuit	10
2 CALCULATION OF CIRCUIT FREQUENCY CHARACTERISTICS	12
2.1 Determination of the complex transmission coefficient of the circuit	12
2.2 Determination of the frequency response of the circuit	12
2.3 Determination of the phase-frequency characteristic of the circuit	14
3 CALCULATION OF CIRCUIT TIMING	16
3.1 Determination of the transient response of the circuit	16
3.2 Determination of the impulse response of the circuit	19
3.3 Calculation of the response of a circuit to a given action using the Duhamel integral method	22
CONCLUSIONS	27
LIST OF USED SOURCES	28

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" (WEM) is one of the basic disciplines. During the study of this course, theoretical knowledge and practical skills are acquired on 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 WEM course:

calculation of linear electric circuits under harmonic action by the method of complex amplitudes;

frequency characteristics of linear electrical circuits;

timing characteristics of circuits;

methods of analysis of transient processes in linear circuits (classical, superposition integrals).

Course work reinforces knowledge in the relevant field, and those who do not have any knowledge are invited to get it by a practical method - by solving the assigned tasks.

Option number 34

R1, Ohm	4,5	t1, μs	30
R2, Ohm	1590	I1, A	7
R3, Ohm	1100
L, μH	43
C, pF	18,8
Reaction

1. Determine the complex input resistance of the circuit.

2. Find the modulus, 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 impedance.

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

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

6. Find and graph the impulse response of the circuit.

1 CALCULATION OF INTEGRATED INPUT CIRCUIT RESISTANCE

1.1 Determination of the complex input impedance of a circuit

(1)

After substitution of numeric values, we get:

(2)

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

T, μs m = 100 1.982 * 10-4 19.82 m = 100000 1.98 * 10-4 19.82 7. Frequency response is shown in fig. 4, fig. 5. TIME ANALYSIS METHOD 7. DETERMINATION OF THE CIRCUIT RESPONSE TO THE IMPULSE Using the Duhamel integral, it is possible to determine the reaction of the circuit to a given action even in the case when an external action on ...

The time characteristics of circuits include transient and impulse responses.

Consider a linear electrical circuit that does not contain independent sources of current and voltage.

Let the external influence on the circuit be the switch-on function (unit jump) x (t) = 1 (t - t 0).

Transient 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 effect of a single current or voltage jump

The dimension of the transient characteristic is equal to the ratio of the dimension of the response to the dimension of the external influence, therefore the transient characteristic can have the dimension of resistance, conductivity, or be a dimensionless quantity.

Let the external influence on the chain have the form of the -function

x (t) = d (t - t 0).

Impulse response g (t - t 0) a linear chain that does not contain independent energy sources is called the reaction of the chain to an action in the form of an-function with zero initial conditions /

The dimension of the impulse response is equal to the ratio of the dimension of the response of the circuit to the product of the dimension of the external action and time.

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 response, however, unlike the first characteristics, the argument of the latter is time t rather than angular w or complex p frequency. Since the characteristics of the circuit, the argument of which is time, are called temporal, and the characteristics, the argument of which is the frequency (including the complex one), are called frequency, the transient and impulse characteristics refer to the temporal characteristics of the circuit.

Each operator characteristic of the circuit H k n (p) can be associated with the transient and impulse characteristics.

(9.75)

At t 0 = 0 operator images of transient and impulse responses have a simple form

Expressions (9.75), (9.76) establish the relationship between the frequency and time characteristics of the circuit. Knowing, for example, the impulse response, you can use direct conversion Laplace find the corresponding operator characteristic of the chain

and from the known operator characteristic H k n (p) using the inverse Laplace transform, determine the impulse response of the circuit

Using expressions (9.75) and the differentiation theorem (9.36), it is easy to establish a connection between the transient and impulse characteristics

If at t = t 0 the function h (t - t 0) changes abruptly, then the impulse response of the circuit is related to it by the following relation

(9.78)

Expression (9.78) is known as the generalized derivative formula. The first term in this expression is the derivative of the transient response at t> t 0, and the second term contains the product of the d-function and the value of the transient response at the point t = t 0.

If the function h 1 (t - t 0) does not undergo a discontinuity at t = t 0, that is, the value of the transient characteristic at the point t = t 0 is equal to zero, then the expression for the generalized derivative coincides with the expression for the ordinary derivative., The impulse response circuit is equal to the first derivative of the transient response with respect to time

(9.77)

To determine the transient (impulse) characteristics of a linear circuit, two main methods are used.

1) It is necessary to consider the transient processes that take place in a given circuit when exposed to a current or voltage in the form of a switch-on function or a-function. This can be done using classical or operator transient analysis methods.

2) In practice, to find the temporal characteristics of linear circuits, it is convenient to use a path based on the use of relations that establish a relationship between frequency and time characteristics. Determination of temporal characteristics in this case begins with drawing up an operator circuit equivalent circuit for zero initial conditions. Further, using this scheme, the operator characteristic H k n (p) is found corresponding to the given pair: external influence on the chain x n (t) is the reaction of the chain y k (t). Knowing the operator characteristic of the circuit and applying relations (6.109) or (6.110), the sought time characteristics are determined.

It should be noted that when qualitatively considering the reaction of a linear circuit to the effect of a single current or voltage pulse, the transient process in the circuit is divided into two stages. At the first stage (at tÎ] t 0-, t 0+ [) the circuit is under the influence of a single impulse, imparting a certain energy to the circuit. In this case, the currents of inductors and capacitance voltages change abruptly to a value corresponding to the energy supplied to the circuit, while the laws of commutation are violated. At the second stage (at t ³ t 0+) the action of the external influence applied to the circuit has ended (while the corresponding energy sources are turned off, that is, they are represented by internal resistances), and free processes arise in the circuit due to the energy stored in the reactive elements at the first stage of the transient process. Consequently, the impulse response characterizes free processes in the circuit under consideration.