Transfer function and impulse response of the circuit. Impulse response: definition and properties Impulse response rc circuit
To judge the capabilities of electrical devices that receive and transmit input actions, they resort to studying their transient and impulse characteristics.
Step response h(t) linear circuit, which does not contain independent sources, is numerically equal to the circuit response to the impact of a single current or voltage jump in the form of a unit step function 1( t) or 1( t – t 0) at zero initial conditions (Fig. 14). The dimension of the transient response is equal to the ratio of the dimension of the reaction to the dimension of the impact. It can be dimensionless, have the dimension of Ohm, Siemens (Cm).
Rice. fourteen
impulse response k(t) of a linear circuit that does not contain independent sources is numerically equal to the circuit response to the action of a single impulse in the form d( t) or d( t – t 0) functions under zero initial conditions. Its dimension is equal to the ratio of the dimension of the reaction to the product of the dimension of the impact on the time, so it can have the dimensions s -1 , Oms -1 , Cms -1 .
The impulse function d( t) can be considered as a derivative of the unit step function d( t) = d 1(t)/dt. Respectively, impulse response is always the time derivative of the transient response: k(t) = h(0 +)d( t) + dh(t)/dt. This relationship is used to determine the impulse response. For example, if for some chain h(t) = 0,7e –100t, then k(t) = 0.7d( t) – 70e –100 t. The transient response can be determined by the classical or operator method for calculating transients.
between temporary and frequency characteristics chain there is a connection. Knowing the operator transfer function, one can find an image of the chain reaction: Y(s) = W(s)X(s), i.e. the transfer function contains complete information about the properties of the circuit as a signal transmission system from its input to the output under zero initial conditions. In this case, the nature of the impact and response correspond to those for which the transfer function is determined.
Transmission function for linear circuits does not depend on the type of input action, so it can be obtained from the transient response. Thus, under the action of a unit step function 1( t) transfer function, taking into account the fact that 1( t) = 1/s, is equal to
W(s) = L [h(t)] / L = L [h(t)] / (1/s), where L [f(t)] - notation direct conversion Laplace over a function f(t). The step response can be determined in terms of the transfer function using the inverse Laplace transform, i.e. h(t) = L –1 [W(s)(1/s)], where L –1 [F(s)] - notation for the inverse Laplace transform over a function F(s). Thus, the transient response h(t) is a function whose image is W(s) /s.
Under the action of the unit impulse function d( t) Transmission function W(s) = L [k(t)] / L = L [k(t)] / 1 = L [k(t)]. So the impulse response of the circuit is k(t) is the original transfer function. According to the known operator function of the circuit, using the inverse Laplace transform, one can determine the impulse response: k(t) W(s). This means that the impulse response of the circuit uniquely determines the frequency response of the circuit and vice versa, since
W(j w) = W(s)s = j w. Since the transient response of the circuit can be found from the known impulse response (and vice versa), the latter is also uniquely determined by the frequency response of the circuit.
Example 8 Calculate the transient and impulse characteristics of the circuit (Fig. 15) for the input current and output voltage for the given parameters of the elements: R= 50 Ohm, L 1 = L 2 = L= 125 mH,
FROM= 80 uF.
Rice. fifteen
Solution. We apply the classical calculation method. Characteristic equation Z in = R + pL +
+ 1 / (PC) = 0 for the given parameters of the elements has complex conjugate roots: p 1,2 =
= – d j w A 2 = - 100 j 200, which determines the oscillatory nature of the transient process. In this case, the laws of change of currents and voltages and their derivatives in general view are written like this:
y(t) = (M cosw A 2 t+ N sinw A 2 t)e– d t + y vyn; dy(t) / dt =
=[(–M d+ N w A 2) cos w A 2 t – (M w A 2 + N d) sinw A 2 t]e– d t + dy ex / dt, where w A 2 - frequency of free oscillations; y vyn - forced component of the transient process.
First we find a solution for u C(t) and i C(t) = C du C(t) / dt, using the above equations, and then using the Kirchhoff equations, we determine the necessary voltages, currents and, accordingly, transient and impulse responses.
To determine the constants of integration, the initial and forced values of these functions are necessary. Their initial values are known: u C(0 +) = 0 (from the definition h(t) and k(t)), because i C(t) = i L(t) = i(t), then i C(0 +) = i L(0 +) = 0. We determine the forced values from the equation compiled according to the second Kirchhoff law for t 0 + : u 1 = R i(t) + (L 1 + L 2) i(t) / dt + u C(t), u 1 = 1(t) = 1 = const,
from here u C() = u C vyn = 1, i C() = i C vyn = i() = 0.
Compose equations for determining the constants of integration M, N:
u C(0 +) = M + u C ex (0 +), i C(0 +) = FROM(–M d+ N w A 2) + i C vyn (0 +); or: 0 = M + 1; 0 = –M 100 + N 200; from here: M = –1, N= -0.5. The obtained values allow us to write solutions u C(t) and i C(t) = i(t): u C(t) = [–cos200 t-0.5sin200 t)e –100t+ 1] B, i C(t) = i(t) = e –100 t] = 0,02
sin200 t)e –100 t A. According to Kirchhoff's second law,
u 2 (t) = u C(t) + u L 2 (t), u L 2 (t) = u L(t) = ldi(t) / dt= (0.5cos200 t– 0.25sin200 t) e –100t B. Then u 2 (t) =
=(–0.5cos200 t– 0.75sin200 t) e –100t+ 1 = [–0.901sin(200 t + 33,69) e –100t+1]B.
Let's check the correctness of the obtained result by initial value: one side, u 2 (0 +) = -0.901 sin (33.69) + 1 = 0.5 and on the other hand, u 2 (0 +) = u C (0 +) + u L(0 +) = 0 + 0.5 - the values are the same.
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).
Unit 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) 
.
A remarkable feature of linear systems - the validity of the superposition principle - opens a direct path to the systematic solution of problems about the passage of various signals through such systems. The dynamic representation method (see Chapter 1) allows representing signals as sums of elementary impulses. If it is possible in one way or another to find the reaction at the output that occurs under the influence of an elementary impulse at the input, then the final step in solving the problem will be the summation of such reactions.
The planned path of analysis is based on the temporal representation of the properties of signals and systems. Equally applicable, and sometimes much more convenient, is analysis in the frequency domain, when the signals are given by series or Fourier integrals. 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 the 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 exist if 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 impulse signal of an 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 take into account that the integral is the limit value of the sum, so the linear operator T, based on the principle of superposition, can be brought under the integral sign. Further, the operator T "acts" only on quantities that depend on the current time t, but not on the integration variable 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 impulse of duration T. The impulse occurs at t = 0 and has an amplitude
Determine the output response of this system when a step signal is applied to the input
When applying the Duhamel integral formula (8.8), note that the output signal will look different depending on whether or not the current value exceeds the duration of the impulse response. When we have
If then for , the function vanishes, so
The found output reaction is displayed as a piecewise line graph.
Generalization to the multidimensional case.
So far, it has been assumed that both the input and output signals are one-dimensional. In more general case systems with inputs and outputs should introduce partial impulse responses, each of which displays the signal at the output when a delta function is applied to the input.
The set of functions forms an impulse response matrix
The Duhamel integral formula in the multidimensional case takes the form
where - -dimensional vector; - -dimensional vector.
Condition of physical realizability.
Whatever the specific form of the impulse response of a physically feasible system, the most important principle must always be fulfilled: the output signal corresponding to the impulse input cannot occur until the moment the impulse appears at the input.
This implies a very simple constraint on the form of admissible impulse responses:
This condition is satisfied, for example, by the impulse response 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 value of time:
Formula (8.13) has a clear physical meaning: a linear stationary system, processing the input signal, carries out the weighted summation of all its instantaneous values that existed "in the past" at - The role of the weight function is played by the impulse response of the system. It is fundamentally important that a physically realizable system is under no circumstances able to operate with "future" values of the input signal.
A physically realizable system must also be stable. This means that its impulse response must satisfy the absolute integrability condition
Transition characteristic.
Let the signal represented by the Heaviside function act at the input of the linear stationary system.
output reaction
called the transient response of the system. Since the system is stationary, the transient response is invariant under time shift:
The previously stated considerations about the physical feasibility of the system are completely transferred to the case when the system is excited not by the delta function, but by a single jump. Therefore, the transient response of a physically realizable system is nonzero only at while at t There is a close connection between the impulse and transient responses. 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 the derivation of relation (8.8), we obtain another form of the Duhamel integral:
Frequency transfer coefficient.
In the mathematical study of systems, of particular interest are such input signals that, being transformed by the system, remain unchanged in form. If there is equality
then is an eigenfunction of the system operator T, and the number X, generally complex, is its eigenvalue.
Let us show that the complex signal for any value of frequency is an eigenfunction of a linear stationary operator. To do this, we use the Duhamel integral of the form (8.9) and calculate
This shows 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 transfer 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 responses, or in the frequency domain by setting the frequency transfer 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 linear system, which has inputs and outputs, can be described by a matrix of frequency transfer coefficients
Between the matrices there is a connection law 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 gain (AFC) is an even, and the phase angle (PFC) is an odd function of frequency.
It is much more difficult to answer the question of what the frequency transfer coefficient should be in order for the physical realizability conditions (8.12) and (8.14) to be satisfied. Let us 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 exists
Consider a specific example illustrating 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 transfer coefficient is given by the system of equalities:
Yes, based on expression (8.20), the impulse response of such a filter
The symmetry of the graph of this function with respect to the point t = 0 indicates the unrealizability of an ideal low-pass filter. However, this conclusion follows directly from the Paley-Wiener criterion. Indeed, the integral (8.27) diverges for any frequency response that vanishes on some finite segment of the frequency axis.
Despite the unrealizability of an ideal LPF, this model is successfully used for an approximate description of the properties frequency filters, assuming that the function contains a phase factor that depends linearly on the frequency:
It is easy to check that here the impulse response
The parameter, equal in absolute value to the slope of the PFC, determines the time delay of the maximum of the function h(t). It's clear that this model the more accurately reflects the properties of the implemented system, the greater the value
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 impact 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:
, i.e. actually 
.
Generalizing the formulas, we obtain a connection 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 .
Because the 
is an image of the derivative of the transient response, then 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 with the help of 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 that 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 on the input:
the chain is connected: with a change in the energy state ... (+0),. Uc(-0) = Uc(+0). 3. Transitional characteristic electrical chains is: Response to a unit step...
Study chains second order. Search for input and output characteristics
Coursework >> Communication and communication3. Transitional and impulse characteristics chains Laplace image transitional characteristics has a look. For getting transitional characteristics in ... A., Zolotnitsky V. M., Chernyshev E. P. Fundamentals of the theory electrical chains.-SPb.: Lan, 2004. 2. Dyakonov V. P. MATLAB ...
Basic provisions of the theory transitional processes
Abstract >> PhysicsLaplace; - temporary, using transitional and impulse characteristics; - frequency, based on ... the classical method of analysis transitional fluctuations in electrical chains transitional processes in electrical chains described by equations...
Impulse (weight) response or impulse function
chains
- this is its generalized characteristic, which is a time function, numerically equal to the response of the circuit to a single impulse action at its input under zero initial conditions (Fig. 13.14); in other words, this is the response of a circuit free of initial energy storage to the Diran delta function 
at her entrance.
|
|
Function 
can be determined by calculating the transition 
or transmission 
circuit function.
Function calculation 
using the transition function of the circuit. Let under the input action 
the reaction of a linear electrical circuit is 
. Then, due to the linearity of the circuit, with an input action equal to the derivative 
, the reaction of the chain will be equal to the derivative 
.
As noted, when 
, 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 a 
, then the impulse function has the form
.
(13.9)
Therefore, the dimension of the impulse response is equal to the dimension of the transient response divided by time.
Function calculation 
using the transfer function of the circuit. According to expression (13.6), when acting on the input of the function 
, the response of the function will be the transition function 
type:
.
On the other hand, it is known that the image of the derivative of a function with respect to time 
, 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. Let us find the impulse function of the 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 plot 
the chain is shown in fig. 13.15.
conclusions
impulse response 
introduced for the same two reasons as the transient response 
.
1. Single impulse action 
- intermittent and therefore rather heavy external influence for any system or circuit. Therefore, it is important to know the reaction of the system or chain under such an impact, i.e. impulse response 
.
2. With the help of some modification of the Duhamel integral, knowing 
calculate the response of the system or circuit to any external perturbation (see further subsections 13.4, 13.5).
4. Overlay integral (duhamel).
Let an arbitrary passive two-terminal network (Fig. 13.16, a) is connected to a source continuously changing from the moment 
voltage 
(Fig. 13.16, b).
|
|
Need to find the current 
(or voltage) in any branch of the two-terminal network after the key is closed.
We will solve the problem in two stages. First, we find the desired value by turning on the two-terminal network for a single voltage jump, which is given by a single step function 
.
It is known that the reaction of the chain to a single jump is step response (function)
.
For example, for 
– circuits transient function for current 
(see clause 2.1), for 
– circuit voltage transient function 
.
At the second stage, continuously changing voltage 
replace by 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 stresses 
, shifted relative to each other by time intervals 
and having a plus sign for the increasing and minus sign for the falling branch of the given voltage curve.
The component of the desired current at the moment 
from direct voltage 
is equal to:
.
The component of the desired current from an elementary voltage jump 
included at the moment of time 
is equal to:
.
Here the argument of the transition function is the time 
, since the elementary voltage jump 
starts to work for a while 
later than the closing of the key, or, in other words, since the time interval between the moment 
the beginning of the action of this jump and the time 
equals 
.
Elementary power surge
,
where 
is the scale factor.
Therefore, the desired component of the current
Elementary power surges are switched on in the time interval from 
until the moment 
, for which the desired current is determined. Therefore, summing the current components 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 applied voltage
(13.11)
called overlay integral (superposition) or Duhamel integral (the first form of writing this integral).
Similarly, the problem is solved when connecting the circuit and the current source. According to this integral, the reaction of the chain, in general, 
at some point 
after the start of exposure 
determined by all that part of the impact that took place before the point in time 
.
By changing variables and integrating by parts, one can obtain other forms of writing the Duhamel integral, equivalent to expression (13.11):
The choice of the form for writing the Duhamel integral is determined by the convenience of calculation. For example, if 
is expressed by an exponential function, the 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 in front of the integral vanishes.
Arbitrary impact 
can also be represented as a sum of sequentially connected pulses, as shown in Fig. 13.17.
|
|
For an infinitesimal pulse duration 
we obtain formulas for the Duhamel integral similar to (13.13) and (13.14).
The same formulas can be obtained from relations (13.13) and (13.14) by replacing a with the derivative function 
impulse function 
.
Conclusion.
Thus, based on the formulas of the Duhamel integral (13.11) - (13.16) and the time characteristics of the circuit 
and 
time functions of circuit responses can be determined 
on arbitrary influences 
.
