Transient and impulse characteristics of the rl circuit. Transient response. Impulse response. Impulse characteristics of electrical circuits. Ministry of Education and Science of Ukraine

Impulse (weight) characteristic or impulse function chains - this is its generalized characteristic, which is a time function, numerically equal to the reaction of the circuit to a single impulse action at its input at zero initial conditions (Fig. 13.14); in other words, this is the response of the circuit, free of the initial energy supply, to the Diran delta function
at its entrance.

Function
can be determined by calculating the transition
or gear
chain function.

Function calculation
using the transient function of the circuit. Let at the input action
the reaction of a linear electric circuit is
... Then, due to the linearity of the circuit at the input action equal to the derivative
, the chain reaction will be equal to the derivative
.

As noted, at
, chain reaction
, what if
, then the chain reaction will be
, i.e. impulse function

According to the sampling property
work
... Thus, the impulse function of the circuit

. (13.8)

If
, then the impulse function has the form

. (13.9)

Therefore, the dimension impulse response and is equal to the dimension of the transient response divided by time.

Function calculation
using transfer function chains. According to expression (13.6), when acting on the input of the function
, the response of the function will be the transient function
kind:

On the other hand, it is known that the image of the time derivative of a function
, at
, is equal to the product
.

Where
,

or
, (13.10)

those. impulse response
circuit is equal to the inverse Laplace transform of its transmission
functions.

Example. Find impulse function circuit, the equivalent circuits of which are shown in Fig. 13.12, a; 13.13.

Solution

The transition and transfer functions of this circuit were obtained earlier:

Then, according to expression (13.8)

where
.

Impulse response graph
circuit is shown in Fig. 13.15.

conclusions

Impulse response
introduced for the same two reasons as the transient response
.

1. Single impulse action
- an abrupt and therefore quite heavy external influence for any system or circuit. Therefore, it is important to know the reaction of a system or a chain precisely under such an action, i.e. impulse response
.

2. With the help of some modification of the Duhamel integral, one can, knowing
calculate the response of the system or circuit to any external disturbance (see further Sections 13.4, 13.5).

4. Integral overlay (Duhamel).

Let an arbitrary passive two-terminal network (Fig.13.16, a) connects to a source that is continuously changing from the moment
stresses (fig.13.16, b).

It is required to find the current (or voltage) in any branch of the two-pole after the key is closed.

We will solve the problem in two stages. First, we find the desired value when the two-terminal network is turned on for a single voltage jump, which is set by a single step function
.

It is known that the reaction of a chain to a unit jump is transient response (function)
.

For example, for
- circuit current transient function
(see clause 2.1), for
- circuit voltage transient function
.

At the second stage, the continuously changing voltage
replace with a step function with elementary rectangular jumps
(see fig.13.16 b). Then the process of voltage change can be represented as switching on at
constant voltage
, and then as the inclusion of elementary constant voltages
offset relative to each other by time intervals
and having a plus sign for the rising and minus for the falling branch of the given voltage curve.

The component of the required current at the moment from constant voltage
is equal to:

The component of the required current from an elementary voltage jump
included at the moment of time is equal to:

Here, the argument of the transition function is time
, since the elementary voltage jump
begins to act for a while later than the closure of the key, or, in other words, since the time interval between the moment the beginning of the action of this jump and the moment of time is equal to
.

Elementary voltage surge

where
- scale factor.

Therefore, the sought-for component of the current

Elementary voltage surges are switched on in the time interval from
until the moment , for which the sought current is determined. Therefore, summing up the components of the current from all jumps, passing to the limit at
, and taking into account the current component from the initial voltage jump
, we get:

The last formula for determining the current with a continuous change in the applied voltage

(13.11)

called integral of superposition (superposition) or the Duhamel integral (the first form of writing this integral).

The problem is solved in a similar way when the circuit is connected to the current source. According to this integral, the reaction of the chain, in general form,
at some point after the start of exposure
is determined by all that part of the impact that took place up to the point in time .

By substituting variables and integrating by parts, we can obtain other forms of writing the Duhamel integral, equivalent to expression (13.11):

The choice of the notation form for the Duhamel integral is determined by the convenience of the calculation. For example, if
is expressed by an exponential function, formula (13.13) or (13.14) turns out to be convenient, which is due to the simplicity of differentiating the exponential function.

At
or
it is convenient to use the notation in which the term before the integral vanishes.

Arbitrary impact
can also be represented as a sum of pulses connected in series, as shown in Fig. 13.17.

With an infinitely short pulse duration
we obtain the Duhamel integral formulas similar to (13.13) and (13.14).

The same formulas can be obtained from relations (13.13) and (13.14), replacing the derivative of the function
impulse function
.

Output.

Thus, based on the Duhamel integral formulas (13.11) - (13.16) and the time characteristics of the chain
and
the timing functions of the circuit responses can be defined
on arbitrary influences
.

3. Impulse characteristics of electrical circuits

Impulse response of the circuit is called the ratio of the reaction of the chain to an impulse action to the area of this action at zero initial conditions.

A-priory ,

where is the reaction of the circuit to the impulse action;

- the area of the impulse of the impact.

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

A single impulse action, also called the delta function or the Dirac function, is often used as an action function.

A delta function is a function equal to zero everywhere, except for, 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 us establish a connection between the transfer function of the circuit and its impulse response, for which we use the operator method.

A-priory:

If the impact (original) is considered for the most general case in the form of the product of the pulse area by the delta function, that is, in the form, then the image of this impact according to the correspondence table has the form:

Then, on the other hand, the ratio of the Laplace-transformed chain reaction to the magnitude of the impact area of the impulse is the operator impulse response of the circuit:

Hence, .

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

, i.e., actually .

Generalizing the formulas, we obtain the relationship between the operator transfer function of the chain and the operator transient and impulse characteristics of the chain:

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

Let's make the identity transformation of equality, adding to the middle part.

Then we will have.

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

Passing to the area of originals, we obtain a formula that allows us to determine the impulse response of the circuit according to its known transient response:

If, then.

The inverse relationship between these characteristics is as follows:

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

If the degrees of the numerator and denominator are the same, then the term under consideration will be present. If the function is a regular fraction, then this term will not exist.

Example: Determine the impulse response for voltages and in a series -circuit shown in Figure 4.

Let's define:

Let's go to the original according to the table of correspondences:

The graph of this function is shown in Figure 5.

Rice. 5

Transmission function :

According to the correspondence table, we have:

The graph of the resulting function is shown in Figure 6.

We point out that the same expressions could be obtained using the relations establishing the connection between and.

The impulse response in its physical meaning reflects the process of free oscillations and for this reason it can be argued that in real circuits the condition must always be met:

4. Integrals of convolution (overlays)

Consider the procedure for determining the reaction of a linear electric 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 a certain moment of time. Solving this problem, we represent the impact as a sum of rectangular impulses of infinitely short duration, one of which, corresponding to a moment in time, is shown in Figure 7. This impulse is characterized by its duration and height.

It is known from the previously considered material 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 caused by this impulse effect at the moment of time will be equal to:

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

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

Thus:

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

The resulting relationship is called the convolution integral or the superposition integral. The function that is found as a result of calculating the convolution integral is called convolution and.

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

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

circuit is associated with: a change in the energy state ... (+0) ,. Uc (-0) = Uc (+0). 3. Transitional characteristic electric chains this: Response to a single step ...

Study chains second order. Search for input and output specifications

Coursework >> Communication and Communication

3. Transitional and impulse specifications chains Laplace image transitional specifications has the form. To receive transitional specifications in ... A., Zolotnitsky V.M., Chernyshev E.P. Fundamentals of the theory electrical chains.-SPb.: Lan, 2004. 2. Dyakonov V.P. MATLAB ...

The main provisions of the theory transitional processes

Abstract >> Physics

Laplace; - temporary, using transitional and impulse specifications; - frequency based on ... the classical method of analysis transitional fluctuations in electrical chains Transitional processes in electrical chains are described by equations, ...

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 application of the characteristics under consideration for the analysis and synthesis of EC, aim at high-quality preparation for a practical lesson.

Allocation of lecture time

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. Integrals of convolution ……………………………………………… .15 min.

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

1. Transient characteristics of electrical circuits

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

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

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

A-priory ,

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

If the transient characteristic of the circuit is known (or can be calculated), then from the formula it is possible to find the reaction of this circuit to the step action at zero NL

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

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

is the operator transient response of the circuit:

Hence .

From here, the operator transient response of the circuit is found in terms of 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 the (preliminary) decomposition theorem.

Example: Determine the transient response for the response voltage across a capacitor in a series

-chains (fig. 1):

Here the reaction to a stepwise action of the magnitude

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 chain to a complex stimulus. Let us establish these relations.

Let us agree that the impact

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

Specified impact

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, continuously following each other. One of such elementary actions corresponding to the moment of application is shown in Figure 2.

Find the value of the chain reaction at a certain moment in time

Step action with differential

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

An infinitesimal step effect with a drop

, causes an infinitely small reaction, where there is time elapsed from the moment of application of the influence to the moment of observation. Since by condition the function is continuous, then:

According to the superimposed reaction principle

will be equal to the sum of the reactions caused by the set of influences preceding the moment of observation, i.e.

Usually in the last formula

are simply replaced by, since the found formula is correct for any time value:

Duhamel integral.

Knowing the reaction of the chain to a single disturbing effect, i.e. transient conductance function or / and voltage transient function, you can find the circuit's response to an arbitrary shape. The method - the method of calculation using the Duhamel integral - is based on the superposition principle.

When using the Duhamel integral to separate the variable over which the integration is performed and the variable that determines the moment in time at which the current in the circuit is determined, the first is usually denoted as, and the second as t.

Let at the moment of time to the circuit with zero initial conditions (passive two-terminal PD in fig. 1) a source with an arbitrary voltage is connected. To find the current in the circuit, we replace the original curve with a step one (see Fig. 2), after which, taking into account that the circuit is linear, we sum up the currents from the initial voltage jump and all voltage steps to the moment t, which come into effect with a time lag.

At time t, the component of the total current, determined by the initial voltage jump, is equal to.

At the moment of time, there is a voltage jump , which, taking into account the time interval from the start of the jump to the moment of interest t, will determine the current component.

The total current at time t is obviously equal to the sum of all current components from individual voltage surges, taking into account, i.e.

Replacing the finite interval of the time increment with an infinitesimal one, i.e. passing from the sum to the integral, we write

(1)

Relation (1) is called the Duhamel integral.

It should be noted that stress can also be determined using the Duhamel integral. In this case, in (1), instead of the transient conductance, there will be a voltage transient function.

Sequence of calculation using
the Duhamel integral

As an example of using the Duhamel integral, we define the current in the circuit in Fig. 3 calculated in the previous lecture using the inclusion formula.

Initial data for the calculation: , , .

Transient conductance

18. Transfer function.

The ratio of the action operator to its own operator is called the transfer function or the transfer function in operator form.

A link described by an equation or equations in symbolic or operator form can be characterized by two transfer functions: a transfer function for the input value u; and the transfer function for the input value f.

and

Using the transfer functions, the equation is written in the form ... This equation is a conditional more compact notation form of the original equation.

Along with the transfer function in the operator form, the transfer function in the form of Laplace images is widely used.

Transfer functions in the form of Laplace images and in operator form coincide up to notation. The transfer function in the form, the Laplace image can be obtained from the transfer function in the operator form, if the substitution p = s is made in the latter. In the general case, this follows from the fact that the differentiation of the original - the symbolic multiplication of the original by p - with zero initial conditions corresponds to the multiplication of the image by a complex number s.

The similarity between transfer functions in the form of the Laplace image and in the operator form is purely external, and it takes place only in the case of stationary links (systems), i.e. only with zero initial conditions.

Consider a simple RLC (in series) circuit, its transfer function W (p) = U OUT / U IN

Fourier integral.

Function f(x), defined on the whole number axis is called periodic if there is a number such that for any value NS equality holds ... Number T called period of the function.

Let us note some of the features of this function:

1) Sum, difference, product and quotient of periodic functions of the period T there is a periodic function of the period T.

2) If the function f(x) period T, then the function f(ax) has a period.

3) If f(x) - periodic function of the period T, then any two integrals of this function taken over intervals of length are equal T(in this case, the integral exists), i.e., for any a and b fair equality .

Trigonometric series. Fourier series

If f(x) decomposes on a segment into a uniformly converging trigonometric series: (1)

Then this expansion is unique and the coefficients are determined by the formulas:

where n=1,2, . . .

Trigonometric series (1) of the considered form with coefficients is called trigonometric Fourier series.

Complex form of the Fourier series

The expression is called the complex form of the Fourier series of the function f(x) if defined by the equality

, where

The transition from the Fourier series in complex form to the series in real form and vice versa is carried out using the formulas:

(n=1,2, . . .)

The Fourier integral of a function f (x) is an integral of the form:

, where .

Frequency functions.

If you apply to the input of a system with a transfer function W (p) harmonic signal

then after the completion of the transient process, harmonic oscillations will be established at the output

with the same frequency, but different amplitude and phase, depending on the frequency of the disturbing effect. They can be used to judge the dynamic properties of the system. The dependences linking the amplitude and phase of the output signal with the frequency of the input signal are called frequency characteristics(CH). Analysis of the frequency response of a system in order to study its dynamic properties is called frequency analysis.

Substitute expressions for u (t) and y (t) into the equation of dynamics

(aоp n + a 1 pn - 1 + a 2 p n - 2 + ... + a n) y = (bоp m + b 1 p m-1 + ... + b m) u.

Let's take into account that

pnu = pnU m ejwt = U m (jw) nejwt = (jw) nu.

Similar relationships can be written for the left side of the equation. We get:

By analogy with the transfer function, you can write:

W (j), equal to the ratio of the output signal to the input signal when the input signal changes according to the harmonic law, is called frequency transfer function... It is easy to see that it can be obtained by simply replacing p with j in the expression W (p).

W (j) is a complex function, therefore:

where P () - real frequency response (high frequency response); Q () - imaginary frequency response (MChH); A() - amplitude frequency response (frequency response): () - phase frequency response (phase frequency response)... The frequency response gives the ratio of the amplitudes of the output and input signals, the phase response is the phase shift of the output value relative to the input:

;

If W (j) is depicted as a vector on the complex plane, then when changing from 0 to +, its end will draw a curve called vector hodograph W (j), or amplitude - phase frequency response (AFC)(fig. 48).

The AFFC branch when changing from - to 0 can be obtained by mirroring this curve about the real axis.

In TAU are widely used logarithmic frequency characteristics (LFC)(fig. 49): logarithmic amplitude frequency response (LFC) L () and logarithmic phase frequency response (LPFC) ().

They are obtained by taking the logarithm of the transfer function:

The LFC is obtained from the first term, which is multiplied by 20 for scaling reasons, and not the natural logarithm, but the decimal one, that is, L () = 20lgA (). The value L () is plotted along the ordinate in decibels.

A change in the signal level by 10 dB corresponds to a change in its power by a factor of 10. Since the power of a harmonic signal P is proportional to the square of its amplitude A, a 10-fold change in the signal corresponds to a change in its level by 20 dB, since

log (P 2 / P 1) = log (A 2 2 / A 1 2) = 20 log (A 2 / A 1).

The abscissa shows the frequency w on a logarithmic scale. That is, unit intervals along the abscissa axis correspond to a 10-fold change in w. This interval is called decade... Since lg (0) = -, the ordinate axis is drawn arbitrarily.

The LPFC obtained from the second term differs from the phase response only in the scale along the axis. The value () is plotted along the ordinate in degrees or radians. For elementary links, it does not go beyond: - +.

Frequency Responses are comprehensive characteristics of the system. Knowing the frequency response of the system, you can restore its transfer function and determine the parameters.

Feedbacks.

It is generally accepted that the link is covered by feedback if its output signal is fed to the input through some other link. Moreover, if the feedback signal is subtracted from the input action (), then the feedback is called negative. If the feedback signal is added to the input action (), then the feedback is called positive.

The transfer function of a closed loop with negative feedback - the link covered by negative feedback - is equal to the transfer function of the forward chain divided by one plus the transfer function of the open circuit

The closed loop transfer function with positive feedback is equal to the forward loop transfer function divided by one minus the open loop transfer function

22.23. Quadripoles.

In the analysis of electrical circuits in the tasks of studying the relationship between the alternating (currents, voltages, powers, etc.) of some branches of the circuit, the theory of four-poles is widely used.

Quadrupole- This is a part of a circuit of an arbitrary configuration, which has two pairs of terminals (hence its name), usually called input and output.

Examples of a four-port network are a transformer, amplifier, potentiometer, power line, and other electrical devices, in which two pairs of poles can be distinguished.

In the general case, four-pole networks can be divided into active, the structure of which includes energy sources, and passive, whose branches do not contain energy sources.

To write the equations of a two-port network, we select in an arbitrary circuit a branch with a single energy source and any other branch with some resistance (see Fig. 1, a).

In accordance with the principle of compensation, we replace the initial resistance with a voltage source (see Fig. 1, b). Then, based on the superposition method for the circuit in Fig. 1, b can be written

Equations (3) and (4) represent the basic equations of a four-port network; they are also called A-form equations of a two-port network (see Table 1). Generally speaking, there are six forms of writing the equations of a passive two-port network. Indeed, a four-pole network is characterized by two voltages and and two currents and. Any two quantities can be expressed in terms of the rest. Since the number of combinations from four to two is equal to six, then six forms of writing the equations of a passive four-port network are possible, which are given in table. 1. Positive directions of currents for various forms of writing the equations are shown in Fig. 2. Note that the choice of one form or another of the equations is determined by the area and type of the problem being solved.

Table 1. Forms of writing the equations of a passive two-port network

The form	Equations	Relationship with the coefficients of the basic equations
A-form	; ;
Y-shape	; ;	; ; ; ;
Z-shape	; ;	; ; ; ;
H-form	; ;	; ; ; ;
G-shape	; ;	; ; ; ;
B-shape	; .	; ; ; .

Characteristic resistance and coefficient
propagation of a symmetrical two-port network

In telecommunications, the operation mode of a symmetrical two-port network is widely used, in which its input impedance is equal to the load impedance, i.e.

This resistance is denoted as it is called characteristic resistance symmetrical four-terminal network, and the operating mode of the four-terminal network, for which it is true

Academy of Russia

Department of Physics

Lecture

Transient and impulse characteristics of electrical circuits

Eagle 2009

Educational and educational goals:

Allocation of lecture time

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. Integrals of convolution ……………………………………………… .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, that is, it expresses a certain transient process under predetermined influences and initial conditions.

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

A-priory ,

where is the reaction of the chain to the step effect;

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

Since it 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 action.

If the transient characteristic of the circuit is known (or can be calculated), then from the formula it is possible to find the reaction of this circuit to the step action at zero NL

The ratio of the Laplace-transformed chain reaction to the magnitude of the impact is the operator transient characteristic of the chain:

Hence .

From here, the operator transient response of the circuit is found in terms of 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 the (preliminary) decomposition theorem.

Example: Determine the transient response for the response voltage across capacitors in a series -circuit (Fig. 1):

Here is the reaction to a stepwise action by the magnitude:

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 chain to a complex stimulus. Let us establish these relations.

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

A given action can be represented as the sum of the stepwise action applied to the circuit at the moment and an infinitely large number of infinitely small step actions, continuously following each other. One of such elementary actions corresponding to the moment of application is shown in Figure 2.

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

A stepwise action with a drop by the time instant causes a reaction equal to the product of the drop by the value of the transient characteristic of the circuit at, that is, equal to:

An infinitely small stepwise effect with a drop causes an infinitely small reaction , where is the time elapsed from the moment of application of the influence to the moment of observation. Since by condition the function is continuous, then:

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

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

Or, after some simple transformations:

Any of these ratios solves the problem of calculating the reaction of a linear electric circuit to a given continuous action using the known transient characteristic of the circuit. These relations are called Duhamel integrals.

3. Impulse characteristics of electrical circuits

Impulse response of the circuit is called the ratio of the reaction of the chain to an impulse action to the area of this action at zero initial conditions.

A-priory ,

where is the reaction of the circuit to the impulse action;

- the area of the impulse of the impact.

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

A single impulse action, also called the delta function or the Dirac function, is often used as an action function.

A delta function is a function equal to zero everywhere, except for, 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 us establish a connection between the transfer function of the circuit and its impulse response, for which we use the operator method.

A-priory:

Then, on the other hand, the ratio of the Laplace-transformed chain reaction to the magnitude of the impact area of the impulse is the operator impulse response of the circuit:

Hence, .

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

That is, in fact.

Generalizing the formulas, we obtain the relationship between the operator transfer function of the chain and the operator transient and impulse characteristics of the chain:

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

Let's make the identity transformation of equality, 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:

Passing to the area of originals, we obtain a formula that allows us to determine the impulse response of the circuit according to its known transient response:

If, then.

The inverse relationship between these characteristics is as follows:

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

If the degrees of the numerator and denominator are the same, then the term under consideration will be present. If the function is a regular fraction, then this term will not exist.

Example: Determine the impulse response for voltages and in a series -circuit shown in Figure 4.

Let's define:

Let's go to the original according to the table of correspondences:

The graph of this function is shown in Figure 5.

Rice. 5

Transmission function :

According to the correspondence table, we have:

The graph of the resulting function is shown in Figure 6.

We point out that the same expressions could be obtained using the relations establishing the connection between and.

The impulse response in its physical meaning reflects the process of free oscillations and for this reason it can be argued that in real circuits the condition must always be met:

4. Integrals of convolution (overlays)

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

Thus:

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

The resulting relationship is called the convolution integral or the superposition integral. The function that is found as a result of calculating the convolution integral is called convolution and.

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

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

Let's use the convolution integral:

Expression for was received earlier.

Hence, , and .

The same result can be obtained using the Duhamel integral.

Literature:

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

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

Kachanov NS and other Linear radio engineering devices. M .: Military. publ., 1974. (Textbook);

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