The time characteristics of circuits include transient and impulse responses.

Consider a linear electrical circuit that does not contain independent sources of current and voltage.

Let the external influence on the circuit be the switch-on function (unit jump) x (t) = 1 (t - t 0).

Transient response h (t - t 0) of a linear circuit that does not contain independent energy sources is the ratio of the reaction of this circuit to the effect of a single current or voltage jump

The dimension of the transient characteristic is equal to the ratio of the dimension of the response to the dimension of the external influence, therefore the transient characteristic can have the dimension of resistance, conductivity, or be a dimensionless quantity.

Let the external influence on the chain have the form of the -function

x (t) = d (t - t 0).

Impulse response g (t - t 0) a linear chain that does not contain independent energy sources is called the reaction of the chain to an action in the form of an-function with zero initial conditions /

The dimension of the impulse response is equal to the ratio of the dimension of the response of the circuit to the product of the dimension of the external action and time.

Like the complex frequency and operator characteristics of a circuit, the transient and impulse characteristics establish a connection between the external influence on the circuit and its response, however, unlike the first characteristics, the argument of the latter is time t rather than angular w or complex p frequency. Since the characteristics of the circuit, the argument of which is time, are called temporal, and the characteristics, the argument of which is the frequency (including the complex one), are called frequency, the transient and impulse characteristics refer to the temporal characteristics of the circuit.

Each operator characteristic of the circuit H k n (p) can be associated with the transient and impulse characteristics.

(9.75)

At t 0 = 0 operator images of transient and impulse responses have a simple form

Expressions (9.75), (9.76) establish the relationship between the frequency and time characteristics of the circuit. Knowing, for example, the impulse response, we can use the direct Laplace transform to find the corresponding operator characteristic of the circuit

and from the known operator characteristic H k n (p) using the inverse Laplace transform, determine the impulse response of the circuit

Using expressions (9.75) and the differentiation theorem (9.36), it is easy to establish a connection between the transient and impulse characteristics

If at t = t 0 the function h (t - t 0) changes abruptly, then the impulse response of the circuit is related to it by the following relation

(9.78)

Expression (9.78) is known as the generalized derivative formula. The first term in this expression is the derivative of the transient response at t> t 0, and the second term contains the product of the d-function and the value of the transient response at the point t = t 0.

If the function h 1 (t - t 0) does not undergo a discontinuity at t = t 0, that is, the value of the transient characteristic at the point t = t 0 is equal to zero, then the expression for the generalized derivative coincides with the expression for the ordinary derivative., The impulse response circuit is equal to the first derivative of the transient response with respect to time

(9.77)

To determine the transient (impulse) characteristics of a linear circuit, two main methods are used.

1) It is necessary to consider the transient processes that take place in a given circuit when exposed to a current or voltage in the form of a switch-on function or a-function. This can be done using classical or operator transient analysis methods.

2) In practice, to find the temporal characteristics of linear circuits, it is convenient to use a path based on the use of relations that establish a relationship between frequency and time characteristics. Determination of temporal characteristics in this case begins with drawing up an operator circuit equivalent circuit for zero initial conditions. Further, using this scheme, the operator characteristic H k n (p) is found corresponding to the given pair: external influence on the chain x n (t) is the reaction of the chain y k (t). Knowing the operator characteristic of the circuit and applying relations (6.109) or (6.110), the sought time characteristics are determined.

It should be noted that when qualitatively considering the reaction of a linear circuit to the effect of a single current or voltage pulse, the transient process in the circuit is divided into two stages. At the first stage (at tÎ] t 0-, t 0+ [) the circuit is under the influence of a single impulse, imparting a certain energy to the circuit. In this case, the currents of inductors and capacitance voltages change abruptly to a value corresponding to the energy supplied to the circuit, while the laws of commutation are violated. At the second stage (at t ³ t 0+) the action of the external influence applied to the circuit has ended (while the corresponding energy sources are turned off, that is, they are represented by internal resistances), and free processes arise in the circuit due to the energy stored in the reactive elements at the first stage of the transient process. Consequently, the impulse response characterizes free processes in the circuit under consideration.

1. TASK

The circuit of the investigated circuit [Fig. 1] No. 22, in accordance with the option of assignment 22 - 13 - 5 - 4. Parameters of the circuit elements: L = 2 mH, R = 2 kOhm, C = 0.5 nF.

The external influence is given by the function:, where a is calculated by the formula (1) and is equal to.

Figure 1. Wiring diagram of a given circuit

It is necessary to determine:

a) an expression for the primary parameters of a given two-port network as a function of frequency;

b) the complex voltage transmission coefficient of the four-port network in the no-load mode at the terminals;

c) amplitude-frequency and phase-frequency characteristics of the voltage transmission coefficient;

d) the operator's voltage transmission coefficient of the four-port network in the no-load mode at the terminals;

e) the transient response of the circuit;

e) impulse response of the circuit;

g) the response of the circuit to a given input action when the load is disconnected.

2. CALCULATION PART

.1 Determination of the primary parameters of a four-port network

To determine the Z - parameters of the four-terminal network, we will compose the equations of electrical equilibrium of the circuit by the method of loop currents using a complex circuit equivalent circuit [Fig. 2]:

Figure 2. Complex equivalent circuit of a given electrical circuit

Choosing the direction of traversing the contours, as indicated in [Fig. 2], and considering that

we write down the contour equations of the circuit:

Substitute the values ​​and into the resulting equations:

(2)

The resulting equations (2) contain only currents and voltages at the input and output terminals of a four-port network and can be converted to the standard form of writing the basic equations of a four-port network in the form Z:

(3)

Transforming equations (2) to the form (3), we get:

Comparing the obtained equations with equations (3), we obtain:

quadripole voltage idle amplitude

2.2 Determination of the voltage transmission coefficientin idle mode at the output

We find the complex voltage transfer coefficient from terminals to terminals in no-load mode () at the output using the values ​​obtained in paragraph 2.1 expressions for primary parameters:

2.3 Determination of amplitude-frequencyand phase-frequencyvoltage transmission coefficient characteristics

Consider the resulting expression for as the ratio of two complex numbers, find an expression for the frequency response and phase response.

The frequency response will look like:

From formula (4) it follows that the phase-frequency characteristic will have the form:

Where, rad / s is found from the equation

Frequency response and phase response graphs are shown on the next page. [fig.3, fig.4]

Figure 3. Frequency response

Figure 4. Phase response

Limit values ​​and at to control calculations, it is useful to determine without resorting to calculation formulas:

· Given that the resistance of the inductance at constant current is zero, and the resistance of the capacitance is infinitely large, in the circuit [see. fig. 1], you can break the branch containing the capacitance and replace the inductance with a jumper. In the resulting circuit and, because the input voltage is in phase with the voltage at the terminals;

· At an infinitely high frequency, the branch containing the inductance can be broken, because the inductance resistance tends to infinity. Despite the fact that the resistance of the capacitance tends to zero, it cannot be replaced with a jumper, since the voltage across the capacitance is a response. In the resulting circuit [see. Fig. 5], for,, the input current is ahead of the input voltage in phase, and the output voltage coincides in phase with the input voltage, therefore .

Figure 5. Electrical diagram of a given circuit at.

2.4 Determination of the operating voltage transmission ratioquadripole in idle mode on the clamps

The operator circuit of the equivalent circuit in appearance does not differ from the complex equivalent circuit [Fig. 2], since the analysis of the electrical circuit is carried out under zero initial conditions. In this case, to obtain the operator voltage transmission coefficient, it is sufficient to replace the operator in the expression for the complex transmission coefficient:

We transform the last expression so that the coefficients at the highest powers in the numerator and denominator are equal to one:

The function has two complex conjugate poles:; and one real zero: .

Figure 6. Pole-zero function diagram

The pole-zero diagram of the function is shown in Fig. 6. Transient processes in the circuit have an oscillatory damping character.

2.5 Definition of transientand impulsecircuit characteristics

The operator expression allows you to get images of the transient and impulse responses. It is convenient to determine the transient response using the relationship between the Laplace image of the transient response and the operator transmission coefficient:

(5)

The impulse response of the circuit can be obtained from the ratios:

(6)

(7)

Using formulas (5) and (6), we write the expressions for the images of the impulse and transient characteristics:

We transform the images of the transient and impulse responses to a form convenient for determining the originals of the time characteristics using the Laplace transform tables:

(8)

(9)

Thus, all images are reduced to the following operator functions, the originals of which are given in the Laplace transform tables:

(12)

Considering that for this considered case , , , we find the values ​​of the constants for expression (11) and the values ​​of the constants for expression (12).

For expression (11):

And for expression (12):

Substituting the obtained values ​​into expressions (11) and (12), we get:

After transformations, we get the final expressions for the temporal characteristics:

The transient process in this circuit ends after switching for the time , where - is defined as the reciprocal of the absolute minimum value of the real part of the pole. Because , then the decay time is (6 - 10) μs. Accordingly, we choose the interval for calculating the numerical values ​​of the time characteristics ... Transient and impulse response graphs are shown in Figs. 7 and 8.

For a qualitative explanation of the type of transient and impulse characteristics of the circuit to the input terminals, an independent voltage source. The transient response of the circuit numerically coincides with the voltage at the output terminals when a single voltage jump is applied to the circuit at zero initial conditions. At the initial moment of time after switching, the voltage across the capacitor is zero, since, according to the laws of commutation, at a finite value of the jump amplitude, the voltage across the capacitance cannot change abruptly. Hence, that is. When the voltage at the input can be considered constant and equal to 1V, that is. Accordingly, only direct currents can flow in the circuit, therefore the capacitance can be replaced by an open, and the inductance by a jumper, therefore, in the circuit converted in this way, that is. The transition from the initial state to the steady state occurs in an oscillatory mode, which is explained by the process of periodic exchange of energy between inductance and capacitance. Damping of oscillations occurs due to energy losses on resistance R.

Figure 7. Transient response.

Figure 8. Impulse response.

The impulse response of the circuit numerically coincides with the output voltage when a single voltage pulse is applied to the input ... During the action of a single pulse, the capacitance is charged to its maximum value, and the voltage across the capacitance becomes equal to

.

When the voltage source can be replaced with a short-circuited jumper, and a damped oscillatory process of energy exchange between the inductance and the capacitance occurs in the circuit. At the initial stage, the capacitance is discharged, the capacitance current gradually decreases to 0, and the inductance current increases to its maximum value at. Then the inductance current, gradually decreasing, recharges the capacitor in the opposite direction, etc. When, due to the dissipation of energy in the resistance, all currents and voltages of the circuit tend to zero. Thus, the oscillatory nature of the voltage across the capacitance damping over time explains the form of the impulse response, and and .

The correctness of the impulse response calculation is confirmed qualitatively by the fact that the graph in Fig. 8 passes through 0 at those times when the graph in Fig. 7 has local extremes, and the maxima coincide in time with the inflection points of the graph. And also the correctness of the calculations is confirmed by the fact that the graphs and, in accordance with formula (7), coincide. To check the correctness of finding the transient characteristic of the circuit, we will find this characteristic when a single voltage jump is applied to the circuit using the classical method:

Let us find independent initial conditions ():

Let us find the dependent initial conditions ():

To do this, turn to Fig. 9, which shows a circuit diagram at a time, then we get:

Figure 9. Circuit diagram at time

Let's find the forced component of the response:

To do this, refer to Fig. 10, which shows the circuit diagram after switching. Then we get that

Figure 10. Circuit diagram for.

Let's compose a differential equation:

To do this, we first write down the equation of the balance of currents in the node according to the first Kirchhoff's law and write down some equations based on the second Kirchhoff's laws:

Using the component equations, we transform the first equation:

Let us express all unknown voltages in terms of:

Now, differentiating and transforming, we obtain a differential equation of the second order:

Substitute the known constants and get:

5. Let's write down the characteristic equation and find its roots:
to zero. The time constant and the quasi-period of the oscillation of the temporal characteristics coincide with the results obtained from the analysis of the operator gain; The frequency response of the circuit under consideration is close to the frequency response of an ideal low-pass filter with a cutoff frequency .

List of used literature

1. Popov V.P. Fundamentals of circuit theory: Textbook for universities - 4th ed., Rev. - M .: Higher. shk., 2003 .-- 575s.: ill.

Korn, G., Korn, T., A Handbook of Mathematics for Engineers and High School Students. Moscow: Nauka, 1973, 832 p.

Earlier, we considered frequency characteristics, and time characteristics describe the behavior of a circuit in time for a given input action. There are only two such characteristics: transient and impulse.

Transient response

The transient response - h (t) - is the ratio of the circuit's response to an input step action to the magnitude of this action, provided that there were no currents or voltages in the circuit before it.

The stepwise action has a schedule:

1 (t) - single step action.

Sometimes a step function is used that does not start at time "0":

To calculate the transient response, a constant EMF (if the input action is voltage) or a constant current source (if the input action is a current) is connected to a given circuit and the transient current or voltage specified as a response is calculated. After that, divide the result by the value of the source.

Example: find h (t) for u c with an input action in the form of voltage.

Example: solve the same problem with an input action in the form of a current

Impulse response

The impulse response - g (t) - is the ratio of the circuit's response to the input action in the form of a delta function to the area of ​​this action, provided that there were no currents or voltages in the circuit before connecting the action.

d (t) - delta function, delta impulse, unit impulse, Dirac impulse, Dirac function. This is the function:

It is extremely inconvenient to calculate g (t) by the classical method, but since d (t) is formally a derivative, it can be found from the relation g (t) = h (0) d (t) + dh (t) / dt.

To experimentally determine these characteristics, one has to act approximately, that is, it is impossible to create the exact required effect.

A sequence of pulses similar to rectangular falls at the input:

t f - the duration of the leading edge (the rise time of the input signal);

t and - pulse duration;

Certain requirements are imposed on these impulses:

a) for the transient response:

The T pause should be so large that by the time the next pulse arrives, the transient process from the end of the previous pulse is practically over;

T and should be so large that the transient process caused by the appearance of the impulse also practically had time to end;

T f should be as small as possible (so that for t cp the state of the circuit does not practically change);

X m should be, on the one hand, so large that the reaction of the chain could be registered with the available equipment, and on the other hand, so small that the studied chain retains its properties. If all this is so, register the graph of the chain reaction and change the scale along the ordinate axis by X m times (X m = 5B, divide the ordinates by 5).

b) for the impulse response:

t pauses - the requirements are the same and for X m - the same, there are no requirements for t f (because even the pulse duration t f itself should be so short that the state of the circuit does not practically change. If all this is so, the reaction is recorded and the scale is changed along the ordinate by the input pulse area.

Results according to the classical method

The main advantage is the physical clarity of all the quantities used, which makes it possible to check the course of the solution from the point of view of the physical meaning. In simple chains, it is very easy to get the answer.

Disadvantages: as the complexity of the problem increases, the complexity of the solution increases rapidly, especially at the stage of calculating the initial conditions. It is not convenient to solve all problems by the classical method (practically no one is looking for g (t), and everyone has problems when calculating problems with special contours and special sections).

Before switching,.

Therefore, according to the laws of commutation, u c1 (0) = 0 and u c2 (0) = 0, but from the diagram it can be seen that immediately after the key is closed: E = u c1 (0) + u c2 (0).

In such problems, one has to apply a special procedure for finding the initial conditions.

These disadvantages can be overcome in the operator method.

Linear circuits

Test number 3

Self-test questions

1. List the main properties of the probability density of a random variable.

2. How are the probability density and the characteristic function of a random variable related?

3. List the basic laws of distribution of a random variable.

4. What is the physical meaning of the dispersion of an ergodic random process?

5. Give some examples of linear and non-linear, stationary and non-stationary systems.

1. A random process is called:

a. Any random change in some physical quantity over time;

b. A set of functions of time, subject to some common statistical regularity;

c. A set of random numbers obeying some statistical regularity common to them;

d. A collection of random functions of time.

2. The stationarity of a random process means that throughout the entire period of time:

a. The mathematical expectation and variance are unchanged, and the autocorrelation function depends only on the difference in time values t 1 and t 2 ;

b. The mathematical expectation and variance are unchanged, and the autocorrelation function depends only on the times of the beginning and end of the process;

c. The mathematical expectation is unchanged, and the variance depends only on the difference in time values t 1 and t 2 ;

d. The variance is unchanged, and the mathematical expectation depends only on the start and end times of the process.

3. Ergodic process means that the parameters of a random process can be determined by:

a. Several end-to-end implementations;

b. One final implementation;

c One endless realization;

d. Several endless realizations.

4. The power spectral density of an ergodic process is:

a. Truncated realization spectral density limit divided by time T;

b. Spectral density of the final realization with duration T divided by time T;

c. Truncated realization spectral density limit;

d. Spectral density of the final realization with duration T.

5. The Wiener - Khinchin theorem is a relation between:

a. Energy spectrum and mathematical expectation of a random process;

b. Energy spectrum and variance of a random process;

c. Correlation function and variance of a random process;

d. Energy spectrum and correlation function of a random process.

The electrical circuit converts the signals arriving at its input. Therefore, in the most general case, the mathematical model of the circuit can be specified in the form of a relationship between the input action S in (t) and output response S out (t) :

S out (t) = TS in (t),

where T- chain operator.

Based on the fundamental properties of the operator, one can draw a conclusion about the most essential properties of the chains.

1. If the chain operator T does not depend on the amplitude of the impact, then the chain is called linear. For such a circuit, the principle of superposition is valid, reflecting the independence of the action of several input actions:

T = TS in1 (t) + TS in2 (t) +… + TS inxn (t).

Obviously, with linear transformation of signals in the response spectrum, there are no oscillations with frequencies different from the frequencies of the exposure spectrum.

The class of linear circuits is formed by both passive circuits, consisting of resistors, capacitors, inductors, and active circuits, including transistors, lamps, etc. But in any combination of these elements, their parameters should not depend on the amplitude of the impact.

2. If a shift of the input signal in time leads to the same shift of the output signal, i.e.

S out (t t 0) = TS in (t t 0),

then the chain is called stationary. The stationarity property does not apply to circuits containing elements with time-variable parameters (inductors, capacitors, etc.).

The temporal characteristics of the electrical circuit are transient h (l) and impulse k (t) specifications. Time characteristic electrical circuit is called the response of the circuit to a typical action at zero initial conditions.

Transient response an electrical circuit is the response (reaction) of a circuit to a unit function under zero initial conditions (Fig.13.7, a, b), those. if the input value is / (/) = 1 (/), then the output value will be /? (/) = NS(1 ).

Since the impact begins at the moment of time / = 0, then the response /? (/) = 0 at / in). In this case, the transient response

will be written as h (t- t) or L (/ - t) - 1 (r-t).

The transient response has several varieties (Table 13.1).

Impact type	Reaction type	Transient response
Single voltage surge	Voltage	^?/(0 U (G)
Single surge current	Voltage	2(0 TO,( 0

If the action is specified in the form of a single voltage surge and the response is also voltage, then the transient response turns out to be dimensionless and is the transmission coefficient Kts (1) by voltage. If the output quantity is current, then the transient characteristic has the dimension of conductivity, is numerically equal to this current "and is the transient conductance ?(1 ). Similarly, when exposed to a current surge and a voltage response, the transient response is the transient resistance 1(1). If, in this case, the output quantity is current, then the transient characteristic is dimensionless and is the transmission coefficient K / (g) by current.

There are two ways to determine the transient response - calculated and experimental. To determine the transient response by calculation, it is necessary: ​​to determine the response of the circuit to a constant impact using the classical method; the received response is divided by the magnitude of the constant action and thereby determine the transient response. In the experimental determination of the transient response, it is necessary: ​​to apply a constant voltage to the input of the circuit at the time t = 0 and to take the oscillogram of the circuit response; the obtained values ​​are normalized relative to the input voltage - this is the transient response.

Consider the example of the simplest circuit (Fig. 13.8) the calculation of transient characteristics. For a given chain in Ch. 12 it was found that the reaction of a chain to a constant impact is determined by the expressions:

Dividing "c (G) and / (/) by the effect?, We obtain the transient characteristics, respectively, for the voltage across the capacitance and for the current in the circuit:

Transient response graphs are shown in Fig. 13.9, a, b.

To obtain the transient voltage response across the resistance, the current transient response should be multiplied by / - (Figure 13.9, c):

Impulse response (weight function) is the response of the chain to the delta function with zero initial conditions (Fig.13.10, a - v):

If the delta function is mixed relative to zero by m, then the reaction of the chain will also be shifted by the same amount (Fig. 13.10, d); in this case, the impulse response is written in the form / s (/ - t) or ls (/ - t)? 1 (/ -t).

The impulse response describes a free process in the circuit, since the influence of the form 5 (/) exists at the moment / = 0, and for T * 0 the delta function is equal to zero.

Since the delta function is the first derivative of the unit function, then between /; (/) and to (I) there is the following relationship:

With zero initial conditions

Physically, both terms in expression (13.3) reflect two stages of the transient process in the electric circuit when it is exposed to a voltage (current) pulse in the form of a delta function: the first stage is the accumulation of some final energy (electric field in capacitors C or magnetic field in inductances?) the duration of the impulse (Dg -> 0); the second stage is the dissipation of this energy in the circuit after the end of the pulse.

From expression (13.3) it follows that the impulse response is equal to the transient response divided by a second. By calculation, the impulse response is calculated from the transient response. So, for the previously given circuit (see Fig. 13.8), impulse responses in accordance with expression (13.3) will have the form:

Impulse response graphs are shown in Fig. 13.11, a-c.

To determine the impulse response experimentally, it is necessary to apply, for example, a rectangular pulse with a duration of

... At the output of the circuit - the curve of the transient process, which is then normalized relative to the area of ​​the input process. The normalized oscillogram of the response of a linear electrical circuit will be the impulse response.