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Textbook: Transient and impulse characteristics of electrical circuits. Calculation of the transient and impulse response of the circuit Impulse and transient response of rc circuits

The calculation of the circuit response in many cases can be simplified if the input signal is represented by the sum of elementary actions in the form of rectangular pulses of short duration. To do this, first consider the relationship between the functions and shown in Fig.5.8a,6, which can be written as:

The second function is a single impulse, which we considered in Section 2.4. As you can see, the function is a derivative of the function, i.e. . Let us carry out the passage to the limit in these functions as . In this case, the function will go into an identity function, and the function into a function. Then, by virtue of equality, it follows that the unit impulse, or - the function is the derivative of the unit function.

For linear circuit hence we conclude that its response to a single impulse, called impulse response oh circuit, is the derivative of the transient response of the circuit, i.e. or

The dimension of the impulse response is equal to the dimension of the transient response divided by the time.

Finding the impulse response is in most cases easier than finding the transient response. Indeed, as shown in Section 2.4, the spectral function of a unit impulse, and therefore for the impulse response using the Fourier integral, we obtain the expression

It follows from this expression that the spectral function of the characteristic is equal to the complex gain of the circuit, i.e. or, using the direct Fourier transform, we write:

That is, the impulse response of the circuit, as well as the transient response, is determined through the transfer coefficient, but for the impulse response, in most cases, the integrand in the Fourier integral turns out to be simpler.

As an example, we apply relation (5.14) to determine the spectrum of the impulse response of an integrating circuit, the transient response of which is For the impulse response, we get

Using expression (5.14) here, it is necessary to take into account that the transient response at is identically equal to zero, and therefore the lower limit in the integral of expression (5.14) will be zero. Then the spectral function of the impulse response is

those. obtained the transfer coefficient of the integrating circuit, corresponding to the previously obtained expression (3.16).

Knowing the impulse response, you can find the response of the circuit to the impact of a signal of any shape, either by first finding the transient response from relation (5.12), and then using one of the expressions for the Duhamel integral, or directly through the function. In the latter case, the input function, i.e. the acting signal must be represented as a sum of impulses, as shown in Fig. 5.9.

Such a representation of the function will be more accurate if, i.e. if it is represented by the sum of an infinitely large number of infinitely small in duration impulses, which are here elementary influences. If the elementary action were a single impulse, the area of which is equal to unity, then the response of the circuit to such an impulse, appearing at a moment in time, would be an impulse response. In the case under consideration, the elementary impulse has a value equal to the instantaneous value of the function at the moment and a duration equal to, i.e. its area is equal. Then the response to the elementary impact will be a value. The response of the circuit to the action specified by the function will be the sum of the responses to all elementary actions, the time position of which corresponds to the interval from 0 to, i.e.

This expression, which is another form of writing the Duhamel integral, is also called the convolution of functions. It coincides in appearance with the original convolution of images of two functions in formula (4.21).

The impulse response of a circuit can be obtained experimentally by observing the circuit's response (output voltage) on an electronic oscilloscope. It is necessary to apply a pulse of a very short duration to the input of the circuit. For example, consider the impulse response of a series oscillatory circuit, assuming that the output voltage is removed from the capacitance C. Above in paragraph 1.6, we considered the transient process when constant voltage to such a circuit. If the value of the applied voltage is equal to one, then the voltage on the capacitance, which is the transient response of the circuit, is, according to (1.33),

This transient response is shown in Figure 5.10a. Then the impulse response of the circuit

Considering the quality factor of the circuit to be large, we assume that the first term can be neglected:

This characteristic is shown in Figure 5.10b. It corresponds to the oscillogram of free oscillations in the circuit, which we considered in Section 1.5.

Thus, in order to experimentally observe the impulse response of the circuit, it is necessary to apply a pulse of short duration to the input of the circuit, i.e. (as explained in paragraph 2.4) so that its duration satisfies the condition.

Consider a linear electrical circuit that does not contain independent sources of current and voltage. Let the external action on the chain be

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 impact 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 response of the circuit to the effect of a single current or voltage jump . The dimension of the transient response is equal to the ratio of the dimension of the response to the dimension external influence, so the transient response can have the dimensions of resistance, conductivity, or be a dimensionless quantity.

Let the external action on the circuit have the form of an infinitely short pulse of infinitely high height and finite area А И :

and .

We denote the chain response to this action under zero initial conditions

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 high height and finite area to the area of this pulse under zero initial conditions:

⁄ and .

As follows from expression (6.109), the impulse response of the circuit is numerically equal to the response of the circuit to the action of a single impulse(AI = 1). 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 action and time.

Like the complex frequency and operator responses of a circuit, the transient and impulse responses establish a relationship between the external action on the circuit and its response; however, in contrast to the complex frequency and operator responses, the argument of the transient and impulse responses is the time t rather than the angular ω or complex p frequency. Since the characteristics of the circuit, whose argument is time, are called temporal, and whose argument is frequency (including complex) - frequency characteristics

sticks (see module 1.5), the transient and impulse responses are related to the timing response of the circuit.

Each pair of "external influence on the circuit - reaction of the circuit" can be associated with a certain complex frequency

To establish a connection between these characteristics, we find the operator images of the transient and impulse responses. Using Expressions

(6.108), (6.109), we write



		Operator images of the circuit reaction to external
impact. expressing			through operator images of external
impacts				ai	; we get

	0 operator images of transient and impulsive character
stick have a particularly simple form:
So the impulse response of the circuit is						This is a function,

which, according to Laplace, is the operator characteristic of the value

between the frequency and time characteristics of the circuit. Knowing, for example, the impulse response, one can use direct conversion Laplace to find the corresponding operator characteristic of the chain

Using expressions (6.110) and the differentiation theorem (6.51), it is easy to establish a connection between the transient and impulse responses:

Therefore, the impulse response of the circuit is equal to the first derivative of the transient response with respect to time. Due to the fact that the transient response of the circuit g (t-t 0 ) is numerically equal to the response of the circuit to the effect of a single voltage or current jump applied to the circuit with zero initial conditions, the values of the function g (t-t 0 ) at t< t 0 равны нулю. Поэтому, строго говоря, переход ную характеристику цепи следует записывать как g (t-t 0 ) ∙ 1(t-t 0 ), а не g (t-t 0 ). За меняя в выражении (6.112) g (t-t 0 ) на g (t-t 0 ) ∙ 1(t-t 0 ) и используя соотношение (6.104), получаем

Expression (6.113) is known as generalized derivative formulas. 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 δ function and the value of the transient response at the point t = t 0 . If at t \u003d t 0 the function g (t-t 0) changes abruptly, then the impulse response of the circuit contains a δ function multiplied by the jump height of the transient characteristic at the point t \u003d t 0. If the function g (t-t 0) does not undergo a break at t \u003d t 0, i.e., the value of the transition characteristic at the point t \u003d t 0 is zero, then the expression for the generalized derivative coincides with the expression for the ordinary derivative.

Methods for determining time characteristics

To determine the temporal characteristics of a linear circuit, in the general case, it is necessary to consider transient processes that take place in a given circuit when it is exposed to a single jump (single pulse) of current or voltage. This can be done using the classical or operator method of transient analysis. In practice, to find the temporal characteristics of linear circuits, it is convenient to use another way based on the use of relations that establish a relationship between the frequency and temporal characteristics. The definition of time characteristics in this case begins with the composition

the operator characteristic of the chain and applying relations (6.110) or (6.111), determine the required time characteristics.

giving the circuit a certain energy. In this case, the inductance currents and capacitance voltages change abruptly by a value corresponding to the energy supplied to the circuit. At the second stage (at), the action of the external action applied to the circuit ended (in this case, the corresponding energy sources are switched off, i.e., internal resistances), and free processes occur in the circuit, proceeding due to the energy stored in the reactive elements at the first stage of the transient process. Thus, the impulse response of the circuit, numerically equal to the response to the action of a single current or voltage pulse, characterizes free processes in the circuit under consideration.

Example 6.7. For a circuit whose diagram is shown in fig. 3.12, a, we find the transient and impulse responses in the idle mode on the clamps 2–2".

voltage on the circuit ― voltage on the clamps 1―1"		Circuit reaction - clamp voltage

The operator characteristic of this circuit, corresponding to the given pair “external action on the circuit - reaction of the circuit”, was obtained in Example 6.5:

x ⁄ .

Consequently, the operator images of the transient and impulse characteristics of the circuit have the form

⁄ ;

1 ⁄ 1 ⁄ .

Using the tables of the inverse Laplace transform, see Appendix 1, we pass from the images of the desired time characteristics to the originals of Fig. 6.20, a, b:

Note that the expression for the impulse response of the circuit can also be obtained using formula 6.113 applied to the expression for the transient response of the circuit g t .

For a qualitative explanation of the type of transient and impulse characteristics of the circuit in this inclusion, Fig. 6.20, a, b connect an independent voltage source to the 1-1 "clamps Fig. 6.20, c. The transient response of this circuit is numerically equal to the voltage at the 2-2" clamps when exposed to a single voltage surge

1 In and zero initial conditions. At the initial moment of time after commutation

tion, the resistance of the inductance is infinitely large, therefore, at t

at the output of the circuit is equal to the voltage at the terminals 1-1 ": u 2 | t 0

u 1| t0

1 V. Over time

as the voltage across the inductor decreases, tending to zero at t

∞ . According to

Depending on this, the transient response starts from the value g 0

1 and tends to zero

The impulse response of the circuit is numerically equal to the voltage at the terminals 2 - 2 "

when a single voltage pulse e t is applied to the circuit input

Academy of Russia

Department of Physics

Lecture

Transient and impulse characteristics of electrical circuits

Eagle 2009

Educational and educational goals:

Explain to the audience the essence of the transient and impulse characteristics of electrical circuits, show the relationship between the characteristics, pay attention to the use of the considered characteristics for the analysis and synthesis of EC, aim at high-quality preparation for a practical lesson.

Lecture Time Allocation

Introductory part……………………………………………………5 min.

Study questions:

1. Transient characteristics of electrical circuits………………15 min.

2. Duhamel integrals………………………………………………...25 min.

3. Impulse characteristics of electrical circuits. Relationship between characteristics………………………………………….………...25 min.

4. Convolution integrals…………………………………………………….15 min.

Conclusion……………………………………………………………5 min.

1. Transient characteristics of electrical circuits

The transient response of the circuit (as well as the impulse response) refers to the temporal characteristics of the circuit, i.e., it expresses a certain transient process under predetermined influences and initial conditions.

To compare electrical circuits in terms of their response to these influences, it is necessary to put the circuits in the same conditions. The simplest and most convenient are zero initial conditions.

Transient response of the circuit is the ratio of the chain response to a step action to the value of this action at zero initial conditions.

By definition ,

where is the response of the circuit to a step action;

- the magnitude of the step action [B] or [A].

Since and is divided by the magnitude of the impact (this is a real number), then in fact - the reaction of the chain to a single step impact.

If the transient response of the circuit is known (or can be calculated), then from the formula one can find the response of this circuit to a step action at zero NL

Let us establish a connection between the operator transfer function of a circuit, which is often known (or can be found), and the transient response of this circuit. To do this, we use the introduced concept of an operator transfer function:

The ratio of the Laplace-transformed chain reaction to the magnitude of the action is the operator transient response of the chain:

Consequently .

From here, the operator transient response of the circuit is found from the operator transfer function.

To determine the transient response of the circuit, it is necessary to apply the inverse Laplace transform:

using the correspondence table or (preliminarily) the decomposition theorem.

Example: determine the transient response for the response of voltage to capacitances in a series circuit (Fig. 1):

Here is the response to the step action with the value :

whence the transient response:

The transient characteristics of the most common circuits are found and given in the reference literature.

2. Duhamel integrals

The transient response is often used to find the response of a circuit to a complex action. Let's establish these ratios.

Let us agree that the action is a continuous function and is applied to the circuit at time , and the initial conditions are zero.

The given action can be represented as the sum of the step action applied to the circuit at the moment and an infinite number of infinitely small step actions that continuously follow each other. One of these elementary actions corresponding to the moment of application is shown in Figure 2.

Let's find the value of the chain reaction at some point in time .

A step action with a drop by the time causes a reaction equal to the product of the drop and the value of the transient response of the circuit at , i.e. equal to:

An infinitely small step action with a difference causes an infinitely small reaction , where is the time elapsed from the moment the impact was applied to the moment of observation. Since the function is continuous, then:

In accordance with the principle of superposition, the reaction will be equal to the sum of the reactions due to the totality of influences preceding the moment of observation, i.e.

Usually, in the last formula, they simply replace it with, since the found formula is correct for any time values:

Or, after simple transformations:

Any of these ratios solves the problem of calculating the response of a linear electrical circuit to a given continuous action according to the known transient response of the circuit. These relations are called Duhamel integrals.

3. Impulse characteristics of electrical circuits

Impulse response circuit is the ratio of the response of the circuit to an impulsive action to the area of this action at zero initial conditions.

By definition ,

where is the response of the circuit to an impulse action;

is the area of the impact impulse.

According to the known impulse response of the circuit, one can find the reaction of the circuit to a given action: .

As the action function, a single impulse action is often used, also called the delta function or the Dirac function.

The delta function is a function equal to zero everywhere, except, and its area is equal to one ():

The concept of a delta function can be arrived at by considering the limit of a rectangular pulse with height and duration when (Fig. 3):

Let's establish a connection between the transfer function of the circuit and its impulse response, for which we use the operator method.

By definition:

If the impact (original) is considered for the most general case in the form of the product of the pulse area and the delta function, i.e. in the form , then the image of this effect 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 action impulse is the operator impulse response of the circuit:

Consequently, .

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

That is, in fact.

Generalizing the formulas, we obtain a relationship between the operator transfer function of the circuit and the operator transient and impulse responses of the circuit:

Thus, knowing one of the characteristics of the circuit, you can determine any others.

Let's make an identical transformation of equality by adding to the middle part .

Then we will have .

Since it is an image of the derivative of the transient response, the original equality can be rewritten as:

Moving to the realm of the originals, we obtain a formula that allows us to determine the impulse response of the circuit from its known transient response:

If , then .

The inverse relationship between the indicated characteristics has the form:

According to the transfer function, it is easy to establish the presence of a term in the composition of the function.

If the degrees 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 responses for voltages and in the series circuit shown in Figure 4.

Let's define:

According to the table of correspondences, let's move on to the original:

The graph of this function is shown in Figure 5.

Rice. 5

Transmission function :

According to the table of correspondences, 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 .

Impulse response according to physical meaning reflects the process of free oscillations and for this reason it can be argued that in real circuits the following condition must always be satisfied:

4. Convolution integrals (overlays)

Consider the procedure for determining the response of a linear electrical circuit to a complex effect, 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, we represent the impact as a sum of rectangular pulses of infinitesimal duration, one of which, corresponding to the time moment , is shown in Figure 7. This pulse is characterized by duration and height .

From the previously considered material, it is known that the response of a circuit to a short impulse can be considered equal to the product of the impulse response of the circuit and the area of the impulse action. Consequently, the infinitely small component of the reaction, due to this impulsive action, at the moment of time will be equal to:

since the area of the pulse is , and the time elapses from the moment of its application to the moment of observation.

Using the principle of superposition, the total response of the circuit can be defined as the sum of an infinitely large number of infinitely small components , caused by a sequence of infinitely small in area impulse actions preceding the moment of time .

In this way:

This formula is true for any value, so the variable is usually denoted simply. Then:

The resulting relation is called the convolution integral or the overlay integral. The function , which is found as a result of calculating the convolution integral, is called the convolution and .

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

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

Let's use the convolution integral:

Expression for was received earlier.

Consequently, , and .

The same result can be obtained by applying the Duhamel integral.

Literature:

Beletsky A.F. Theory of linear electrical circuits. - M .: Radio and communication, 1986. (Textbook)

Bakalov V. P. et al. Theory of electrical circuits. - M .: Radio and communication, 1998. (Textbook);

Kachanov N. S. et al. Linear radio engineering devices. M.: Voen. ed., 1974. (Textbook);

Popov V.P. Fundamentals of the theory of circuits - M .: Higher school, 2000. (Textbook)

Academy of Russia

Department of Physics

Lecture

Transient and impulse characteristics of electrical circuits

Eagle 2009

Educational and educational goals:

Lecture Time Allocation

Introductory part……………………………………………………5 min.

Study questions:

1. Transient characteristics of electrical circuits………………15 min.

2. Duhamel integrals………………………………………………...25 min.

3. Impulse characteristics of electrical circuits. Relationship between characteristics………………………………………….………...25 min.

4. Convolution integrals…………………………………………………….15 min.

Conclusion……………………………………………………………5 min.

1. Transient characteristics of electrical circuits

Transient response of the circuit is the ratio of the chain response to a step action to the value of this action at zero initial conditions.

By definition ,

– chain reaction to step action; - the magnitude of the step action [B] or [A]. and is divided by the magnitude of the impact (this is a real number), then in fact - the reaction of the chain to a single step impact.

If the transient response of the circuit is known (or can be calculated), then from the formula one can find the response of this circuit to a step action at zero NL

The ratio of the Laplace-transformed chain reaction to the magnitude of the impact

represents the operator transient response of the circuit:

Consequently .

From here, the operator transient response of the circuit is found from the operator transfer function.

To determine the transient response of the circuit, it is necessary to apply the inverse Laplace transform:

using the correspondence table or (preliminarily) the decomposition theorem.

Example: Determine the step response for the voltage response across capacitances in a series

-chains (Fig. 1):

Here, the response to the step action is

whence the transient response:

The transient characteristics of the most common circuits are found and given in the reference literature.

2. Duhamel integrals

The transient response is often used to find the response of a circuit to a complex action. Let's establish these ratios.

We agree that the impact

is a continuous function and is applied to the circuit at time , and the initial conditions are zero.

Target exposure

can be represented as the sum of the step action applied to the circuit at the moment and an infinitely large number of infinitely small step actions that continuously follow each other. One of these elementary actions corresponding to the moment of application is shown in Figure 2.

Find the value of the chain reaction at some point in time

Step action with a difference

by the time it causes a reaction equal to the product of the drop and the value of the transient response of the circuit at , i.e. equal to:

An infinitely small step action with a difference

, causes an infinitesimal reaction , where is the time elapsed from the moment of applying the impact to the moment of observation. Since the function is continuous, then:

In accordance with the principle of reaction superposition

will be equal to the sum of the reactions due to the totality of influences preceding the moment of observation, i.e.

Usually in the last formula

simply replace with , since the formula found is true for any time values:

5. Secondary (characteristic) parameters of quadripoles coordinated mode of a quadripole.

6. Non-sinusoidal currents. Fourier series expansion. The frequency spectrum of a non-sinusoidal function of voltage or current.

7. Maximum, average and effective values of non-sinusoidal current.

8. Resonance in a non-sinusoidal current circuit.

9. The power of the non-sinusoidal current circuit.

10. Higher harmonics in three-phase circuits. The simplest frequency tripler.

11. Occurrence of transient processes in linear circuits. Switching laws.

12. Classical method for calculating transients. Formation of the calculation equation, the degree of the calculation equation. Border conditions.

Classical method for calculating transients

13. Free and forced modes. The time constant of the circuit, the definition of the duration of the transient.

14. Periodic charge of a capacitor. Natural frequency of oscillations of the contour. critical resistance.

15. "Incorrect" initial conditions. Features of the calculation. Do such conditions exist in real circuits?

16. 0Determination of the roots of the characteristic equation. Justify.

17. Turning on a passive two-terminal network under the action of a piecewise continuous voltage. Duhamel formula.

The sequence of calculation using the Duhamel integral

Transient and impulse response

19. Application of Laplace transformations to the calculation of transient processes. Basic properties of Laplace functions.

20. Operator equivalent circuits. Justify.

21. Calculation of transient processes by the method of state variables. Formation of calculation equations. Computer calculation.

22. Fourier transform and its main properties. Frequency spectra of impulse signals, differences from frequency spectra of periodic non-sinusoidal signals.

23. Calculation of the frequency characteristics of the circuit. Determination of the transient response by real frequency.

24. Features of the application of the frequency method of calculation in the study of the passage of a signal through a quadripole.

25. Equations of a long line in partial derivatives. Primary parameters of a long line.

26. Solution of the equations of a long line with a sinusoidal voltage. Secondary parameters of the long line.

27. Wave processes in a long line. Incident and reflected waves. Reflection coefficient. input impedance.

Long Line Differential Equations

Running parameters

Traveling and standing wave coefficients

28. Lossless line. standing waves.

29. Input impedance line without loss. Simulation of inductances and capacitances.

31. Wave processes in a lossless line loaded with active resistance. Standing and traveling wave coefficients.

32. Features of the current-voltage characteristics of nonlinear elements. Linear equivalent circuits for static and differential parameters.

33. Calculation of voltage and current stabilization schemes, determination of the stabilization coefficient for a linear equivalent circuit.

34. Approximation of nonlinear characteristics. Analytical calculation method.

35. Features of periodic processes in electrical circuits with inertial elements.

36. Spectral composition of the current in a circuit with a non-linear resistor when exposed to a sinusoidal voltage. Combination vibrations.

37. Method of equivalent sinusoids. Methods for calculating non-linear circuits by effective values. Method of equivalent sinusoid.

Method for calculating non-linear AC circuits by equivalent effective values

38. The shape of the curves of current, magnetic flux and voltage in a nonlinear ideal coil. Equivalent circuit, vector diagram.

Calculation of the current of a coil with steel, taking into account losses in the core

40. Stress ferroresonance. trigger effect.

42. Fundamentals of the harmonic balance method. Give an example.

43. The method of piecewise linear approximation of the characteristics of nonlinear elements. Calculation of circuits with valves. Scheme of a half-wave and full-wave rectifier.

Circuits with valve resistors

44. Calculation of the circuit of a half-wave rectifier with a capacitance.

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

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

Function Graph 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 single 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).

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

This is a special function or mathematical abstraction, but it 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.

The action in the form of a unit impulse function can be considered as a shock action (a sufficiently large amplitude and an infinitely short time of action). 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 bounded by it, we get the following result:

Rice. 2.4.

Obviously, the integration interval can be anything, as long as the point t= 0. The integral of the shifted unit impulse function d ( t-t) is also equal to 1 (if the point t= t). If we take the integral of the unit impulse function multiplied by some coefficient BUT 0 , then obviously the result of integration will be equal to this coefficient. Therefore, the coefficient BUT 0 before d ( t) determines the area bounded by the function BUT 0 d( t).

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

Transient and impulse response

transient response h(t) is called the reaction of the chain to the action in the form of a unit step function 1 (t). impulse response g(t) is called the reaction of the circuit to the action in the form of a unit impulse function d ( t). Both characteristics are determined under zero initial conditions.

The transient and impulse functions characterize the circuit in transient mode, since they are reactions to jumps, i.e. quite heavy for any impact system. In addition, as will be shown below, the response of the circuit to an arbitrary action can be determined using the transient and impulse responses. The transient and impulse responses are related to each other in the same way as the corresponding actions are related to each other. The unit impulse function is the derivative of the unit step function (see (2.2)), so the impulse response is the derivative of the transient response, and at h(0) = 0 . (2.3)

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

Of the two characteristics under consideration, the transient one is most simply determined, since it can be calculated from the response 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 the impulse) response can have the dimension of resistance, conductivity, or be a dimensionless quantity, depending on the dimension of the action and response.

Example . Define transitional h(t) and impulse g(t) characteristics of a series RC circuit.

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

This problem was solved in Section 1.6, where it was obtained u 2 (t) = u C (t) = In this way, h(t) = u 2 (t) / U 0 = The impulse response is determined by (2.3) .