Determination of the impulse response. Transient and impulse characteristics of linear circuits. Timing methods
18. Reaction linear circuits on unit functions... Transient and impulse characteristics of the circuit, their connection.
Single step function (enable function) 1 (t) is defined as follows:
Function graph 1 (t) is shown in Fig. 2.1.
Function 1 (t) is zero for all negative values of the argument and one for t ³ 0. We also introduce into consideration the shifted unit step function
Such an impact is turned on 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 a perfect key (fig. 2.3).
Single impulse function (d - function, Dirac function) is defined as the derivative of a unit step function. Since at the moment of time t= 0 function 1 (t) undergoes a discontinuity, then its derivative does not exist (turns to infinity). Thus, the unit impulse function
It 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.
An impact in the form of a single impulse function can be considered as a shock impact (a sufficiently large amplitude and an infinitely short exposure time). A unit impulse function is also introduced, shifted by time t= t
It is customary to depict a single impulse function in the form of 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 any, as long as the point gets there t= 0. The integral of the displaced 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 A 0 , then obviously the result of integration will be equal to this coefficient. Therefore, the coefficient A 0 before d ( t) defines the area bounded by the function A 0 d ( t).
For the physical interpretation of the d - function, it is advisable to consider it as the limit to which a certain sequence of ordinary functions should strive, for example
Transient and impulse response
Transient response h (t) is called the reaction of the chain to the impact in the form of a single step function 1 (t). Impulse response g (t) is called the reaction of the chain to the action in the form of a unit impulse function d ( t). Both characteristics are determined with zero initial conditions.
The transient and impulse functions characterize the circuit in the transient mode, since they are responses to jump-like, i.e. quite heavy for any impact system. In addition, as will be shown below, using the transient and impulse characteristics, the response of the circuit to an arbitrary action can be determined. The transient and impulse characteristics are related to each other as well as the corresponding influences are related to each other. The unit impulse function is the derivative of the unit step function (see (2.2)), therefore 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 to a linear chain with zero initial conditions instead of an action, then the reaction will be equal to the derivative of the initial reaction.
Of the two considered characteristics, the transient one is most simply determined, since it can be calculated from the circuit's reaction to the switching on 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. Hence it follows that the transient (as well as the impulse) characteristic can have the dimension of resistance, conductivity, or be a dimensionless quantity, depending on the dimension of the action and reaction.
Example ... Define transitional h (t) and impulse g(t) characteristics of the serial RC-circuit.
Impact is input voltage u 1 (t), and the reaction is the voltage across 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)
=
Thus, h (t)
= u 2
(t)
/ U 0
= The impulse response is determined by (2.3) 
.
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 impulse area by the delta function, i.e. 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 pulse area is the operator's impulse response of the chain:
.
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 circuit, 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:
.
By 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.
Impulse response in physical sense, it reflects the process of free vibrations and for this reason it can be argued that in real circuits the condition must always be satisfied:
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 pulses of infinitely short duration, one of which, corresponding to the moment in time, is shown in Figure 7. This pulse is characterized by 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 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:
chain 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 Communication3. 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 >> PhysicsLaplace; - 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, ...
A remarkable feature of linear systems - the validity of the principle of superposition - opens a direct path to the systematic solution of problems of the passage of various signals through such systems. The dynamic representation method (see Ch. 1) allows the signals to be represented as sums of elementary impulses. If it is possible in one way or another to find the response at the output, which arises under the influence of an elementary impulse at the input, then the final stage in solving the problem will be the summation of such reactions.
The intended path of analysis is based on a temporal representation of the properties of signals and systems. The analysis in the frequency domain is equally applicable, and sometimes much more convenient, when the signals are given by Fourier series or integrals. In this case, the properties of systems are described by their frequency characteristics, which indicate the law of transformation of elementary harmonic signals.
Impulse response.
Let some linear stationary system be described by the operator T. For simplicity, we will assume that the input and output signals are one-dimensional. By definition, the impulse response of a system is a function that is a response of the system to an input signal.This means that the function h (t) satisfies the equation
Since the system is stationary, a similar equation will also be in the case when the input action is shifted in time by a derivative value:
It should be clearly understood that the impulse response, as well as the delta function that generates it, is the result of a reasonable idealization. From a physical point of view, the impulse response approximately reflects the response of the system to an input pulse signal of arbitrary shape with a unit area, provided that the duration of this signal is negligible compared to the characteristic time scale of the system, for example, the period of its natural oscillations.
Duhamel integral.
Knowing the impulse response of a linear stationary system, one can formally solve any problem of the passage of a deterministic signal through such a system. Indeed, in Ch. 1 it was shown that the input signal always admits a representation of the form
The corresponding output reaction
Now we will take into account that the integral is the limiting value of the sum, therefore the linear operator T, based on the principle of superposition, can be introduced under the integral sign. Further, the operator T "acts" only on quantities that depend on the current time t, but not on the variable of integration x. Therefore, from expression (8.7) it follows that
or finally
This formula, which is of fundamental importance in the theory of linear systems, is called the Duhamel integral. Relation (8.8) indicates that the output signal of a linear stationary system is a convolution of two functions - the input signal and the impulse response of the system. Obviously, formula (8.8) can also be written in the form
So, if the impulse response h (t) is known, then the further stages of the solution are reduced to fully formalized operations.
Example 8.4. Some linear stationary system, the internal structure of which is insignificant, has an impulse response, which is a rectangular video pulse of duration T. The pulse arises at t = 0 and has an amplitude
Determine the output response of this system when a step signal is applied to the input
Applying the Duhamel integral formula (8.8), you should pay attention to the fact that the output signal will look different depending on whether the duration of the impulse response exceeds the current value or not. For we have
If then for the function vanishes, therefore
The found output response is displayed in a piecewise-linear graph.
Generalization to the multidimensional case.
Until now, it has been assumed that both the input and output signals are one-dimensional. In a more general case of a system with inputs and outputs, partial impulse responses should be introduced, each of which represents the signal at the output when a delta function is applied to the input.
The set of functions forms the impulse response matrix
In the multidimensional case, the Duhamel integral formula takes the form
where is -dimensional vector; - -dimensional vector.
Condition of physical realizability.
Whatever the specific type of impulse response of a physically feasible system, the most important principle must always be fulfilled: the output signal corresponding to the impulse input action cannot occur until the moment the impulse appears at the input.
This implies a very simple restriction on the type of permissible impulse responses:
This condition is satisfied, for example, by the impulse characteristic of the system considered in Example 8.4.
It is easy to see that for a physically realizable system, the upper limit in the Duhamel integral formula can be replaced by the current time value:
Formula (8.13) has a clear physical meaning: a linear stationary system, processing the input signal, carries out the operation of weighted summation of all its instantaneous values that existed "in the past" at - The role of the weighting function is performed by the impulse response of the system. It is fundamentally important that a physically realizable system is under no circumstances capable of operating with “future” values of the input signal.
A physically realizable system must, moreover, be stable. This means that its impulse response must satisfy the condition of absolute integrability
Transient response.
Let the signal represented by the Heaviside function act at the input of a linear stationary system.
Output reaction
it is customary to call the transient response of the system. Since the system is stationary, the transient response is invariant with respect to the time shift:
The considerations expressed earlier about the physical realizability of the system are completely transferred to the case when the system is excited not by a delta function, but by a single jump. Therefore, the transient response of a physically realizable system differs from zero only at, while at t There is a close relationship between the impulse and transient characteristics. Indeed, since on the basis of (8.5)
The differentiation operator and the linear stationary operator T can change places, therefore
Using the dynamic representation formula (1.4) and proceeding in the same way as in deriving relation (8.8), we obtain another form of the Duhamel integral:
Frequency transmission coefficient.
In the mathematical study of systems, such input signals are of particular interest, which, being transformed by the system, remain unchanged in shape. If there is equality
then is an eigenfunction of the system operator T, and the number X, which is generally complex, is its eigenvalue.
Let us show that the complex signal for any value of the frequency is an eigenfunction of the linear stationary operator. To do this, we use the Duhamel integral of the form (8.9) and calculate
Hence it is seen that the eigenvalue of the system operator is the complex number
(8.21)
called the frequency gain of the system.
Formula (8.21) establishes a fundamentally important fact - the frequency transmission coefficient and the impulse response of a linear stationary system are interconnected by the Fourier transform. Therefore, always, knowing the function, you can determine the impulse response
We have come to the most important position of the theory of linear stationary systems - any such system can be considered either in the time domain using its impulse or transient characteristics, or in the frequency domain, setting the frequency transmission coefficient. Both approaches are equivalent and the choice of one of them is dictated by the convenience of obtaining initial data about the system and the simplicity of calculations.
In conclusion, we note that the frequency properties of a linear system with inputs and outputs can be described by the matrix of frequency transmission coefficients
There is a connection law between matrices, similar to that given by formulas (8.21), (8.22).
Amplitude-frequency and phase-frequency characteristics.
The function has a simple interpretation: if a harmonic signal with a known frequency and complex amplitude arrives at the input of the system, then the complex amplitude of the output signal
In accordance with formula (8.26), the modulus of the frequency transfer coefficient (AFC) is even, and the phase angle (FFC) is an odd function of frequency.
It is much more difficult to answer the question of what the frequency transmission coefficient should be in order to satisfy the conditions of physical realizability (8.12) and (8.14). We present without proof the final result known as the Paley - Wiener criterion: the frequency transfer coefficient of a physically realizable system must be such that the integral
Consider a specific example that illustrates the properties of the frequency gain of a linear system.
Example 8.5. Some linear stationary system has the properties of an ideal low-pass filter, i.e., its frequency transmission coefficient is set by the system of equalities:
Based on expression (8.20), the impulse response of such a filter
The symmetry of the graph of this function relative to the point t = 0 indicates that an ideal low-pass filter is unrealizable. However, this conclusion directly follows from the Paley - Wiener criterion. Indeed, the integral (8.27) diverges for any frequency response, which vanishes on some finite segment of the frequency axis.
Despite the unrealizability of an ideal low-pass filter, this model is successfully used for an approximate description of the properties frequency filters, assuming that the function contains a phase factor linearly dependent on frequency:
As it is easy to check, here the impulse response
The parameter equal in magnitude to the phase-frequency characteristic slope coefficient determines the time delay of the maximum of the function h (t). It is clear that the higher the value
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
The transient response of the circuit (like 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.
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 ,
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
.
Let us establish a connection 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 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 impact 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 effects, 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:
.
If the impact (original) is considered for the most general case in the form of the product of the impulse area by the delta function, i.e. 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 pulse area is the operator's impulse response of the chain:
.
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 circuit, 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)
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 pulses of infinitely short duration, one of which, corresponding to the moment in time, is shown in Figure 7. This pulse is characterized by 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 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)
The transient response is used to calculate the response of a linear electrical circuit when a pulse is applied to its input. 
free form. In this case, the input pulse 
approximated by a set of steps and determine the reaction of the chain to each step, and then find the integral circuit 
, as the sum of responses to each component of the input pulse 
.
Transient response or transient function 
chains -
this is its generalized characteristic, which is a time function that is numerically equal to the circuit's response to a single voltage or current jump at its input, with zero initial conditions (Fig. 13.11);
|
|
in other words, this is the response of a circuit free of the initial energy supply to the function 
at the entrance.
Transient response expression
depends only on the internal structure and the values of the parameters of the circuit elements.
From the definition of the transient characteristic of the circuit, it follows that with the input action 
chain reaction 
(fig.13.11).
Example. Let the circuit connect to a constant voltage source 
... Then the input action will have the form, the reaction of the circuit -, and the transient voltage characteristic of the circuit - 
... At 
.
Chain reaction multiplication 
per function 
or 
means that the transition function
at 
and 
at 
which reflects principle of causality
in linear electrical circuits, i.e. the response (at the output of the circuit) cannot appear before the moment the signal is applied to the input of the circuit.
Types of transient characteristics.
There are the following types of transient response:
|
|
- voltage transient response of the circuit; |
|
|
- the transient characteristic of the circuit in terms of current; |
|
|
- transient resistance of the circuit, Ohm; |
|
|
- transient conductivity of the circuit, Cm, |
(13.5)
where 
- the levels of the input step signal.
Transient function 
for any passive two-terminal network can be found by the classical or operator method.
Calculation of the transient response by the classical method. Example.
Example. We calculate the voltage transient response for the circuit (Fig.13.12, a) with parameters.
|
|
Solution
We will use the result obtained in Section 11.4. According to the expression (11.20), the voltage across the inductance
where 
.
We carry out the scaling according to expression (13.5) and the construction of the function 
(fig.13.12, b):
.
Calculation of the transient response by the operator method
The complex equivalent circuit of the original circuit will take the form in Fig. 13.13.
|
|
The voltage transfer function of this circuit is:
where 
.
At 
, i.e. at 
, image 
, and the image of the voltage on the coil 
.
In this case, the original 
Images 
is the voltage transient function of the circuit, i.e.
or in general view:
,
(13.6)
those. transient function
circuit is equal to the inverse Laplace transform of its transfer function
multiplied by the unit jump image
.
In the considered example (see Fig.13.12) the voltage transfer function:
where 
and the function 
has the form.
Note
.
If voltage is applied to the circuit input 
, then in the formula of the transition function 
time 
must be replaced with the expression 
... In the considered example, the lagging voltage transfer function has the form:
conclusions
The transient response was introduced mainly for two reasons.
1. Single step 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. transient response 
.
2. With a known transient response 
using the Duhamel integral (see subsections 13.4, 13.5 below), it is possible to determine the response of a system or chain to any form of external influences.
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