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 impulse area 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 
.
Summarizing the formulas, we obtain the relationship between the operator transfer function of the circuit and the operator transient and impulse characteristics of the circuit:
Thus, knowing one of the characteristics of the chain, you can determine any others.
Let's make the identical 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 characteristics 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 response 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.
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 caused by this impulse action 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 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, ...
18. Reaction of linear circuits to 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 turns 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 an ideal 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 circuit's response to an arbitrary action can be determined. The transient and impulse characteristics are interconnected as well as the corresponding influences are interconnected. 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 response 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) 
.
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 expression (11.20), the voltage across the inductance
where 
.
We carry out scaling according to expression (13.5) and 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:
,
(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 input of the circuit 
, 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 
- spasmodic, 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), you can determine the response of a system or chain to any form of external influences.
To judge the capabilities of electrical devices that receive and transmit input influences, resort to the study of their transient and impulse characteristics.
Transient response h(t) of a linear circuit that does not contain independent sources is numerically equal to the response of the circuit to the effect of a single current or voltage jump in the form of a unit step function 1 ( t) or 1 ( t – t 0) with zero initial conditions (Fig. 14). The dimension of the transient characteristic 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 response of the circuit to the action of a single impulse in the form d ( t) or d ( t – t 0) functions with zero initial conditions. Its dimension is equal to the ratio of the dimension of the reaction to the product of the dimension of impact on time, therefore it can have dimensions with –1, Oms –1, Cms –1.
Impulse function d ( t) can be regarded as the derivative of the unit step function d ( t) = d 1(t)/dt... Accordingly, the 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.
There is a relationship between the timing and frequency characteristics of a circuit. Knowing the operator transfer function, you 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 system for transmitting signals from its input to the output at zero initial conditions. In this case, the nature of the impact and reaction correspond to those for which the transfer function is determined.
The transfer function for linear circuits does not depend on the type of input action, therefore it can be obtained from the transient response. So, when acting at the input of a unit step function 1 ( t) transfer function taking into account that 1 ( t) = 1/s, is equal to
W(s) = L [h(t)] / L = L [h(t)] / (1/s), where L [f(t)] - notation for the direct Laplace transform over the function f(t). The transient response can be defined 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 of the inverse Laplace transform over the function F(s). Thus, the transient response h(t) is a function whose image is equal to W(s) /s.
When a single impulse function d ( t) Transmission function W(s) = L [k(t)] / L = L [k(t)] / 1 = L [k(t)]. Thus, the impulse response of the circuit k(t) is the original transfer function. By the known operator function of the chain using the inverse Laplace transform, you 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 known impulse response can be used to find the transient response of the circuit (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,
WITH= 80 μF.
Rice. 15
Solution. Let's use 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 transition process. In this case, the laws of change of currents and voltages and their derivatives in general form are written as follows:
y(t) = (M cosw A 2 t+ N sinw A 2 t)e- d t + y vy; 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 out / dt, where w A 2 - frequency of free vibrations; y forced - a forced component of the transition process.
First, we will 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 required voltages, currents and, accordingly, the transient and impulse characteristics.
To determine the constants of integration, the initial and forced values of these functions are required. 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. The forced values are determined from the equation composed according to the second Kirchhoff's law for t 0 + : u 1 = R i(t) + (L 1 + L 2) i(t) / dt + u C(t), u 1 = 1(t) = 1 = сonst,
from here u C() = u C vyn = 1, i C() = i C out = i() = 0.
Let us compose equations to determine the integration constants M, N:
u C(0 +) = M + u C out (0 +), i C(0 +) = WITH(–M d + N w A 2) + i C out (0 +); or: 0 = M + 1; 0 = –M 100 + N 200; from here: M = –1, N= –0.5. The obtained values allow you to write solutions u C(t) and i C(t) = i(t): u C(t) = [–Сos200 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= (0,5сos200 t- 0.25sin200 t) e –100t B. Then u 2 (t) =
= (- 0.5sos200 t- 0.75sin200 t) e –100t+ 1 = [–0.901sin (200 t + 33,69) e –100t+ 1] B.
Let's check the correctness of the result obtained by the initial value: on the one hand, 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.
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 is called the ratio of the chain reaction to a step action to the magnitude of this action at zero initial conditions.
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 effect 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 voltage response across the capacitance 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 moment 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 impulse area 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.
Summarizing the formulas, we obtain the relationship between the operator transfer function of the circuit and the operator transient and impulse characteristics of the circuit:
Thus, knowing one of the characteristics of the chain, you can determine any others.
Let's make the identical 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 characteristics 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 response 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.
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 caused by this impulse action 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:
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)
