Pulse and transient characteristics of rc circuits. Transfer function and impulse response of a circuit. Transient and impulse characteristics

Academy of Russia

Department of Physics

Lecture

Transient and impulse characteristics of electrical circuits

Eagle 2009

Educational and educational goals:

Explain to students the essence of the transient and impulse characteristics of electrical circuits, show the connection between the characteristics, pay attention to the use of the characteristics under consideration for the analysis and synthesis of electrical circuits, and aim at high-quality preparation for practical training.

Distribution of lecture time

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

Study questions:

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

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

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

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

Conclusion……………………………………………………5 min.

1. Transient characteristics of electrical circuits

The transient response of a circuit (like the pulse response) refers to the temporary characteristics of the circuit, i.e., it expresses a certain transient process under predetermined influences and initial conditions.

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

Transient response of the circuit is the ratio of the reaction of a chain to a stepwise impact to the magnitude of this impact at zero initial conditions.

By definition,

If the transient response of the circuit is known (or can be calculated), then from the formula you can find the reaction of this circuit to a stepwise effect at zero NL

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

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

Hence .

From here the operator transition characteristic of the circuit is found using the operator transfer function.

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

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

Example: determine the transient response for the response of voltage to capacitors in series

Here the reaction to a stepwise effect of magnitude

where does the transition characteristic come from:

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

2. Duhamel integrals

The transient response is often used to find the response of a circuit to a complex stimulus. Let us establish these relations.

Let us agree that the impact

Set impact

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

Stepped impact with differential

An infinitesimal step effect with a difference

According to the reaction superposition principle

Usually in the last formula

3. Pulse characteristics of electrical circuits

Impulse response of the circuit is called the ratio of the reaction of a circuit to a pulsed action to the area of ​​this action under zero initial conditions.

By definition,

where is the circuit’s response to impulse action;

– impact pulse area.

Using the known impulse response of the circuit, one can find the response of the circuit to a given impact: .

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

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

.

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

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

By definition:

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

.

Hence, .

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

, i.e. actually .

Generalizing the formulas, we obtain a connection 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 circuit, you can determine any others.

Let us carry out the identical transformation of equality by adding to the middle part.

Then we will have .

Since is an image of the derivative of the transition characteristic, then the original equality can be rewritten as:

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

If, then.

The inverse relationship between these characteristics has the form:

.

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

If the powers 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 characteristics for voltages and in the series circuit shown in Figure 4.

Let's define:

Using the correspondence table, let's move on to the original:

.

The graph of this function is shown in Figure 5.

Rice. 5

Transfer function:

According to the correspondence table we have:

.

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

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

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 following condition must always be satisfied:

4. Convolution (overlay) integrals

Let us consider the procedure for determining the response of a linear electrical circuit to a complex influence 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, let us imagine the impact as a sum of rectangular pulses of infinitesimal duration, one of which, corresponding to the moment in time, is shown in Figure 7. This pulse is characterized by duration and height.

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

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

Using the principle of superposition, the total reaction of a circuit can be defined as the sum of an infinitely large number of infinitesimal components caused by a sequence of infinitely small area pulses preceding the instant in time.

Thus:

.

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

.

The resulting relation 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:

.

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

chain is connected: with a change in energy state... (+0),. Uc(-0) = Uc(+0). 3. Transitional characteristic electrical chains is: Response to a single step...

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 pulse action at its input under zero initial conditions (Fig. 13.14); in other words, it is the response of a circuit free of initial energy reserve to the Diran delta function
at its entrance.

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

Function calculation
using the circuit's transient function. Let at the input influence
the reaction of a linear electric 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
, and if
, then the chain reaction will be
, i.e. impulse function

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

. (13.8)

If
, then the impulse function has the form

. (13.9)

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

Function calculation
using the circuit transfer function. According to expression (13.6), when acting on the function input
, 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
chain is equal to the inverse Laplace transform of its transfer
functions.

Example. Let us find the pulse function of the circuit whose equivalent circuits 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 circuit is shown in Fig. 13.15.

Conclusions

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

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

2. Using some modification of the Duhamel integral, we can, knowing
calculate the response of a system or circuit to any external disturbance (see further paragraphs 13.4, 13.5).

4. Imposition integral (duhamel).

Let an arbitrary passive two-terminal network (Fig. 13.16, A) is connected to a source that continuously changes 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 switch is closed.

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

It is known that the reaction of a circuit to a single jump is step response (function)
.

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

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

Component of the desired current at the moment from constant voltage
is equal to:

.

Component of the desired current from an elementary voltage surge
, switched on at the moment of time is equal to:

.

Here the argument of the transition function is time
, since an elementary voltage surge
takes effect temporarily 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 moment of time equals
.

Elementary power surge

,

Where
– scale factor.

Therefore, the required current component

Elementary voltage surges are included in the time interval from
until the moment , for which the required current is determined. Therefore, summing up the current components from all jumps, moving to the limit at
, and taking into account the current component from the initial voltage surge
, we get:

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

(13.11)

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

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

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

The choice of the form of writing the Duhamel integral is determined by the convenience of calculation. For example, in case
is expressed by an exponential function, formula (13.13) or (13.14) turns out to be convenient, which is due to the ease of differentiation of the exponential function.

At
or
It is convenient to use a form of notation in which the term before the integral vanishes.

Voluntary influence
can also be presented as a sum of sequentially connected pulses, as shown in Fig. 13.17.

For infinitesimal pulse durations
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), replacing them 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
to voluntary influences
.

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 you to represent signals in the form of sums of elementary pulses. If, in one way or another, it is possible to find the reaction at the output that arises under the influence of an elementary impulse at the input, then the final stage of solving the problem will be the summation of such reactions.

The intended 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 signals are specified by Fourier series or 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 system’s response to an input signal. This means that the function h(t) satisfies the equation

Since the system is stationary, a similar equation will exist if the input action is shifted in time by the 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 approximates the response of a system to an input pulse 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 own oscillations.

Duhamel integral.

Knowing the impulse response of a linear stationary system, one can formally solve any problem about 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 output reaction corresponding to it

Now let us 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 included under the sign of the integral. 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. Relationship (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 further stages of the solution are reduced to completely formalized operations.

Example 8.4. Some linear stationary system, the internal structure of which is unimportant, has an impulse response that is a rectangular video pulse of duration T. The pulse 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), you should pay attention to the fact 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 at the function vanishes, therefore

The found output reaction 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 the 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 a matrix of impulse responses

The Duhamel integral formula in the multidimensional case 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 satisfied: the output signal corresponding to the impulse input action cannot arise until the moment the impulse appears at the input.

This leads to a very simple restriction on the type of permissible impulse characteristics:

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 value of time:

Formula (8.13) has a clear physical meaning: a linear stationary system, processing the signal arriving at the input, carries out the operation of a weighted summation of all its instantaneous values ​​that existed “in the past” at - The role of the weighting function is played by the impulse response of the system. It is fundamentally important that a physically implemented system is under no circumstances capable of operating with “future” values ​​of the input signal.

A physically realizable system must, in addition, be stable. This means that its impulse response must satisfy the condition of absolute integrability

Transition characteristic.

Let a signal represented by the Heaviside function act at the input of a linear stationary system.

Output reaction

is usually called the transient characteristic of the system. Since the system is stationary, the transient response is invariant with respect to the time shift:

The previously stated considerations 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 is different from zero only at while at t There is a close connection between the impulse and transient characteristics. Indeed, since then based on (8.5)

The differentiation operator and the linear stationary operator T can change places, so

Using the dynamic representation formula (1.4) and proceeding in the same way as when deriving relation (8.8), we obtain another form of the Duhamel integral:

Frequency transmission coefficient.

In the mathematical study of systems, of particular interest are those 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, in the general case complex, is its eigenvalue.

Let us show that a complex signal at any frequency value 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 a 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 related to each other by the Fourier transform. Therefore, always, knowing the function, you can determine the impulse response

We have come to the most important point 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 ease of calculations.

In conclusion, we note that the frequency properties of a linear system having inputs and outputs can be described by a matrix of frequency transfer coefficients

There is a connection law between the 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 is received 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 transmission coefficient (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 transmission coefficient should be in order for the conditions of physical realizability (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

Let's consider a specific example illustrating the properties of the frequency transfer coefficient 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 given 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 the impracticability of an ideal low-pass filter. However, this conclusion directly follows from the Paley-Wiener criterion. Indeed, integral (8.27) diverges for any frequency response that vanishes at some finite segment of the frequency axis.

Despite the impracticability of an ideal low-pass filter, this model is successfully used to approximately describe the properties of frequency filters, assuming that the function contains a phase multiplier that linearly depends on frequency:

As is easy to check, here is the impulse response

The parameter, equal in magnitude to the slope coefficient of the phase response, determines the time delay of the maximum of the function h(t). It is clear that this model reflects the properties of the implemented system more accurately, the larger the value

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

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

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

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

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

From this expression it follows that the spectral function of the characteristic is equal to the complex transmission coefficient of the circuit, i.e. or, using the direct Fourier transform, we write:

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

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

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

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

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

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

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

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

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

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

This characteristic is presented in Fig. 5.10b. It corresponds to the oscillogram of free oscillations in the circuit, which we considered in paragraph 1.5.

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