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Frequency and time characteristics of linear circuits. Time characteristics of linear circuits Time characteristics of linear circuits

The time characteristic of a circuit is a function of time, the values of which are numerically determined by the reaction of the circuit to a typical action. The response of a circuit to a given typical action depends only on the circuit diagram and the parameters of its elements and, therefore, can serve as its characteristic. Time characteristics are determined for linear circuits, not containing independent energy sources, and under zero initial conditions. Temporal characteristics depend on the type of the specified typical impact. Due With this divides them into two groups: transient and impulse time characteristics.

transient response, or transient function, is determined by the response of the circuit to the impact of a single step function. It has several varieties (Table 14.1).

If the action is given in the form of a single voltage jump and the reaction is also voltage, then the transient response is dimensionless, numerically equal to the voltage at the output of the circuit and is called the transient function or transfer coefficient KU(t) by voltage. If the output value is current, then the transient characteristic has the dimension of conductivity, is numerically equal to this current and is called transient conductivity Y(t). Similarly, when exposed in the form of current and reaction in the form of voltage, the transient function has the dimension of resistance and is called the transition resistance Z(t). If, in this case, the output value is current, then the transient characteristic is dimensionless and is called the transient function or transfer coefficient K I (t) no current.

AT general case any type of transient response is denoted by h(t). The transient characteristics are easily determined by calculating the response of the circuit to a single step action, i.e., by calculating the transient process when the circuit is turned on for constant pressure 1 V or on D.C. 1 A.

Example 14.2.

Find temporary crossings about these characteristics of a simple rC-circuit (Fig. 14.9, a), if in about the effects are stresses.

1. To determine the transient characteristics, we calculate the transient process when a voltage is applied to the input of the circuit u(t) - 1 (t). This corresponds to the inclusion of the circuit at the moment t=0 to the source of constant e. d.s. e 0 \u003d 1 AT(Fig. 14.9.6). Wherein:

a) the current in the circuit is determined by the expression

so the transient conductivity is

b) voltage on the capacitance

so the voltage transition function

Pulse the characteristic, or impulse transient function, is determined by the response of the circuit to the action of the δ(t)-function. Like the transient response, it has several varieties, determined by the type of impact and response - voltage or current. In general, the impulse response is denoted by a(t).

Let's establish a connection between the impulse response and the transient response of a linear circuit. To do this, we first determine the response of the circuit to an impulse action of short duration t И =Δt, presenting it as a superposition of two step functions:

In accordance with the principle of superposition, the reaction of the circuit to such an impact is determined using the transient characteristics:

For small Δt, we can write

where S and =U m Δƒ is the area of the impulse.

At Δt 0 and U m the resulting expression describes the response of the chain to the δ(t)-function, t . e, defines the impulse response of the circuit:

With this in mind, the response of a linear circuit to a pulsed action of short duration can be found as the product impulse function to the impulse area:

This equality underlies the experimental definition of the impulse function. It is the more accurate, the shorter the pulse duration.

Thus, the impulse response is the derivative of transient response:

It is taken into account here that h(t)δ(t)=h(0)δ(t), and multiplication h(t) on l(t) is equivalent to saying that the value of the function h(t) at t<0 равно нулю.

By integrating the resulting expressions, it is easy to verify that

Equalities (14.17) and (14.19) are a consequence of equalities (14.14) and (14.15). Since impulse responses have the dimension of the corresponding transient response divided by time. To calculate the impulse response, you can use the expression (14.19), i.e., calculate it using the transient response.

Example 14.3.

Find the impulse response of a simple rC circuit (see Fig. 14.9, a). Solution.

Using the expressions for the transient responses obtained in Example 14.2, with the help about using expression (14.19) we find the impulse responses;

The time characteristics of typical links are given in table. 14.2.

Timing is usually calculated in the following order:

the points of application of the external influence and its type (current or voltage) are determined, as well as the output value of interest - the reaction of the circuit (current or voltage in some part of it); the required time characteristic is calculated as the response of the circuit to the corresponding typical action: 1(t) or δ(t),

The expressions (5.17), (5.18) given in the previous paragraph for the gains can be interpreted as transfer functions of a linear active two-terminal network. The nature of these functions is determined by the frequency properties of the parameters Y.

Having written in the form of functions, we come to the concept of the transfer function of a linear active four-terminal network. The generally dimensionless complex function is an exhaustive characteristic of a quadripole in the frequency domain. It is determined in the stationary mode with harmonic excitation of the quadripole.

It is often convenient to represent the transfer function in the form

The module is sometimes called the amplitude-frequency characteristic (AFC) of the quadripole. The argument is called the phase-frequency characteristic (PFC) of the quadripole.

Another exhaustive characteristic of a quadripole is its impulse response, which is used to describe a circuit in the time domain.

For active linear circuits, as well as for passive circuits, the impulse response of the circuit means the response, the reaction of the circuit to an impact that has the form of a single impulse (delta function). The connection between is easy to establish using the Fourier integral.

If a single EMF pulse (delta function) with a spectral density equal to unity for all frequencies acts at the input of a quadripole, then the spectral density of the output voltage is simply . The response to a single pulse, i.e., the impulse response of the circuit, is easily determined using the inverse Fourier transform applied to the transfer function:

In this case, it should be taken into account that before the right side of this equality there is a factor 1 with the dimension of the area of the delta function. In a particular case, when a b-voltage impulse is meant, this dimension will be [volt x second].

Accordingly, the function is the Fourier transform of the impulse response:

In this case, before the integral, we mean the factor one with the dimension [volt x second]^-1.

In the future, the impulse response will be denoted by the function , which can be understood not only as voltage, but also as any other electrical quantity that is a response to the impact in the form of a delta function.

As with the representation of signals on the plane of the complex frequency (see § 2.14), in the theory of circuits, the concept of the transfer function is widely used, considered as the Laplace transform of the function 8

Unit functions and their properties An important place in the theory of linear circuits is occupied by the study of the reaction of these circuits to idealized external influences, described by the so-called unit functions. A single step function (Heaviside function) is a function: The graph of the function 1(t-t 0) has the form of a step or a jump, the height of which is 1. We will call a jump of this type a single one.

Unit functions and their properties Due to the fact that the product of any limited time function f(t) by 1(t-t 0) is zero at t

Unit functions and their properties If at t=t 0 a source of harmonic current or voltage is included in the circuit, then the external effect on the circuit can be represented as: If the external effect on the circuit at time t=t 0 changes abruptly from one fixed value X 1 to another X 2, then

Unit functions and their properties

Unit functions and their properties Consider a rectangular pulse with duration and height 1/ t (Fig.). Obviously, the area of this momentum is equal to 1 and does not depend on t. With a decrease in the duration of the pulse, its height increases, and at t → 0 it tends to infinity, but the area remains equal to 1. A pulse of infinitesimal duration, infinitely high height, whose area is equal to 1, will be called a single pulse. The function that determines the unit impulse is denoted (t-t 0) and is called the δ-function or the Dirac function.

Unit functions and their properties Using the δ-function, one can extract the values of the function f(t) at arbitrary times t 0. This feature of the δ function is usually called the filtering property. For t 0 =0, operator images of unit functions have a particularly simple form:

Transient and impulse responses of linear circuits on the impact of a single surge of current or voltage. The dimension of the transient response is equal to the ratio of the dimension of the response to the dimension of the external action, so the transient response can have the dimensions of resistance, conductivity, or be a dimensionless quantity.

Transient and impulse responses of linear circuits The impulse response h(t-t 0) of a linear circuit that does not contain independent energy sources is the ratio of the response of this circuit to the action of an infinitely short impulse of infinitely high height and a finite area to the area of this impulse under zero initial conditions: numerically equal to the response of the circuit to the impact of a single impulse. 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 influence and the time.

Transient and impulse responses of linear circuits Like the complex frequency and operator responses of a circuit, the transient and impulse responses establish a relationship between the external influence on the circuit and its response, however, unlike the complex frequency and operator responses, the argument of the transient and impulse responses is the time t, not the angular ω or complex p frequency. Since the characteristics of the circuit, the argument of which is time, are called temporary, and the argument of which is frequency (including complex) - frequency characteristics, the transient and impulse characteristics refer to the temporal characteristics of the circuit.

Transient and impulse response of linear circuits Thus, the impulse response of the circuit hkv(t) is a function whose image, according to Laplace, is the operator characteristic of the circuit Hkv(p), and the transient response of the circuit gkv(t) is a function whose operator image is is equal to Hkv(p)/p.

Determination of the chain response to an arbitrary external impact External impact on the chain is represented as a linear combination of elementary components of the same type: and the chain response to such an impact is found as a linear combination of partial responses to the impact of each of the elementary components of the external impact separately: external influences, elementary (trial) influences in the form of a harmonic function of time, a single jump and a single impulse are most widely used.

Determining the response of a circuit to an arbitrary external action by its transient response Let us consider an arbitrary linear electric circuit that does not contain independent energy sources, the transient response g(t) of which is known. Let the external influence on the circuit be given as an arbitrary function x=x(t) equal to zero at t

Determination of the response of a circuit to an arbitrary external action by its transient response The function x(t) can be approximately represented as a sum of non-unit jumps or, what is the same, as a linear combination of unit jumps, shifted relative to each other by: In accordance with the definition of the transient response the reaction of the circuit to the impact of a non-unit jump applied at the time t= k is equal to the product of the jump height and the transient response of the circuit g(t-k). Therefore, the reaction of the circuit to the impact, represented by the sum of non-unit jumps (6. 114), is equal to the sum of the products of the heights of the jumps and the corresponding transient characteristics:

Determining the response of a circuit to an arbitrary external action by its transient response It is obvious that the accuracy of representing the input action as a sum of non-unit jumps, as well as the accuracy of representing the reaction of the circuit, increases with decreasing time step. As → 0, summation is replaced by integration: The expression is known as the Duhamel integral (overlay integral). Using this expression, you can find the exact value of the circuit response to a given impact x=x(t) at any time t after switching. Integration in is carried out on the interval t 0

Determining the reaction of a circuit to an arbitrary external action by its transient response Using the Duhamel integral, one can determine the reaction of a circuit to a given action even in the case when the external action on the circuit is described by a piecewise continuous function, i.e., a function that has a finite number of finite discontinuities . In this case, the integration interval must be divided into several intervals in accordance with the continuity intervals of the function x=x(t) and the reaction of the circuit to the final jumps of the function x=x(t) at the break points should be taken into account.

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

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

Since the action begins at the time / = 0, then the response /?(/) = 0 at /s). At the same time, the transient response

will be written in the form h(t- m) or A(/-m) - 1(r-m).

The transient response has several varieties (Table 13.1).

Type of impact	Type of reaction	Step response
Single voltage spike	Voltage	^?/(0 U (G)
single current surge	Voltage	2(0 TO,( 0

If the action is given in the form of a single voltage jump and the reaction is also a voltage, then the transient response turns out to be dimensionless and is the transfer coefficient Kts(1) by voltage. If the output value is the current, then the transient characteristic has the dimension of conductivity, is numerically equal to this current "and is the transient conductivity ?(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 value is current, then the transient characteristic is dimensionless and is the transfer 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: by the classical method to determine the response of the circuit to a constant impact; divide the received response by the value of the constant action and thereby determine the transient response. When experimentally determining the transient response, it is necessary: to apply a constant voltage to the input of the circuit at the time / = 0 and take an oscillogram of the circuit response; normalize the obtained values with respect to the input voltage - this is the transient response.

Consider, using the example of a simple circuit (Fig. 13.8), the calculation of transient responses. For this circuit in Ch. 12, it was found that the response of the circuit to a constant impact is determined by the expressions:

Dividing "s (G) and / (/) by the impact?, we obtain the transient characteristics, respectively, for the voltage on the capacitance and for the current in the circuit:

Graphs of transient characteristics are shown in fig. 13.9, a, b.

To obtain a transient voltage response across the resistance, multiply the current transient response by /- (Fig. 13.9, c):

Impulse response (function of weight) is the response of the circuit to the delta function under zero initial conditions (Fig. 13.10, a - in):

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

The impulse response describes a free process in the circuit, since the action of the form S(/) exists at the time / = 0, and for Г*0 the delta function is equal to zero.

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

Under zero initial conditions

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

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

Graphs of impulse responses are presented 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

. At the output of the circuit - the curve of the transition process, which is then normalized with respect to the area of the input process. The normalized oscillogram of the reaction of a linear electrical circuit will be the impulse response.

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COURSE WORK

Time and frequency characteristics of linear electrical circuits

Initial data

Scheme of the circuit under study:

Element parameters value:

External influence:

u 1 (t)=(1+e - bt) 1 (t) (B)

As a result of execution term paper need to find:

1. Expression for the primary parameters of a given quadripole as a function of frequency.

2. Find an expression for the complex voltage transfer coefficient K 21 (j w) quadripole in idle mode on terminals 2 - 2".

3. Amplitude-frequency K 21 (j w) and phase-frequency Ф 21 (j w

4. Operator voltage transfer coefficient K 21 (p) of a four-terminal network in idle mode on terminals 2-2 ".

5. Transient response h(t), impulse response g(t).

6. Response u 2 (t) to a given input action in the form u 1 (t)=(1+e - bt) 1 (t) (B)

1. DefineYparameters for a given quadripole

I1=Y11*U1+Y12*U2

I2=Y21*U1+Y22*U2

To make it easier to find Y22, let's find A11 and A12 and express Y22 in terms of them.

Experience 1. XX on clips 2-2 "

Let's make the change 1/jwС=Z1, R=Z2, jwL=Z3, R=Z4

Let's produce a circuit equivalent circuit

Z11=(Z4*Z2)/(Z2+Z3+Z4)

Z33=(Z2*Z3)/(Z2+Z3+Z4)

U2=(U1*Z11)/(Z11+Z33+Z1)

Experience 2: short circuit on clamps 2-2 "

By the method of loop currents, we will make equations.

a) I1 (Z1+Z2) - I2*Z2=U1

b) I2 (Z2+Z3) - I1*Z2=0

From equation b) we express I1 and substitute it into equation a).

I1=I2 (1+Z3/Z2)*(Z1+Z2) - I2*Z2=U1

A12=Z1+Z3+(Z1*Z3)/Z2

Hence we get that

Experience 2: short circuit on clamps 2-2 "

Let's make an equation using the method of loop currents:

I1*(Z1+Z2) - I2*Z2=U1

I2 (Z2+Z3) - I1*Z2=0

We express I2 from the second equation and substitute it into the first:

We express I1 from the second equation and substitute it into the first:

For mutual quadripole Y12=Y21

Matrix A of the parameters of the considered quadripole

2 . Find the complex voltage transfer coefficientTo 21 (jw ) quadripole in idle mode on terminals 2-2 ".

Complex voltage transfer coefficient K 21 (j w) is determined by the relation:

You can find it from the system of standard basic equations for Y parameters:

I1=Y11*U1+Y12*U2

I2=Y21*U1+Y22*U2

So, according to the condition for idling I2=0, we can write

We get the expression:

K 21 (j w)=-Y21/Y22

Let's replace Z1=1/(j*w*C), Z2=1/R, Z3=1/(j*w*C), Z4=R, we get an expression for the complex voltage transfer coefficient K 21 (j w) in idle mode on clamps 2-2"

Let's find the complex voltage transfer coefficient K 21 (j w) quadripole in idle mode on terminals 2-2 "in numerical form by substituting the values of the parameters:

Let us find the amplitude-frequency K 21 (j w) and phase-frequency Ф 21 (j w) characteristics of the voltage transfer coefficient.

Let us write an expression for K 21 (j w) in numerical form:

Let's find the calculation formula for the phase-frequency Ф 21 (j w) characteristics of the voltage transfer coefficient as arctg of the imaginary part to the real one.

As a result, we get:

Let us write an expression for the phase-frequency Ф 21 (j w) characteristics of the voltage transfer coefficient in numerical form:

Resonance frequency w0=7*10 5 rad/s

Let's build frequency response graphs (Appendix 1) and phase response (Appendix 2)

3. Find the operator voltage transfer coefficientK 21 x (p) quadripole in idle mode on terminals 2-2 "

operator voltage pulse circuit

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

Let us write the expression for the operator voltage transfer coefficient К21х(р) in numerical form:

Find the value of the argument р n , at which M(p)=0, i.e. poles of the function K21x(p).

Let us find the values of the argument p k for which N(p)=0, i.e. zeros of the function K21x(p).

Let's make a pole-zero diagram:

Such a pole-zero diagram testifies to the oscillatory damped nature of transient processes.

This pole-zero diagram contains two poles and one zero

4. Timing calculation

Let us find the transient g(t) and impulse h(t) characteristics of the circuit.

The operator expression K21 (p) allows you to get an image of the transient and impulse responses

g(t)hK21 (p)/p h(t)hK21 (p)

Let's transform the image of transient and impulse responses to the form:

Let us now define the transient characteristic g(t).

Thus, the image is reduced to the following operator function, the original of which is in the table:

Thus, we find the transition characteristic:

Let's find the impulse response:

Thus, the image is reduced to the following operator function, the original of which is in the table:

Hence we have

Let's calculate a series of g(t) and h(t) values for t=0h10 (µs). And we will build graphs of transient (Appendix 3) and impulse (Appendix 4) characteristics.

For a qualitative explanation of the type of transient and impulse responses of the circuit, we connect an independent voltage source e (t) = u1 (t) to the input terminals 1-1 ". The transient response of the circuit numerically coincides with the voltage at the output terminals 2-2" when exposed to a single circuit voltage jump e(t)=1 (t) (V) at zero initial conditions. At the initial moment of time after switching, the voltage on the capacitance is equal to zero, because according to the laws of switching, at a finite value of the amplitude of the input jump, the voltage across the capacitance cannot change. Therefore, looking at our chain, we see that u2 (0)=0 i.e. g(0)=0. Over time, at t, tending to infinity, only direct currents will flow through the circuit, which means that the capacitor can be replaced by a break, and the coil by a short-circuited section, and looking at our circuit, it can be seen that u2 (t) = 0.

The impulse response of the circuit numerically coincides with the output voltage when a single voltage pulse e(t) = 1d(t) V is applied to the input. During the action of a single pulse, the input voltage is applied to the inductance, the current in the inductor increases abruptly from zero to 1/L, and the voltage across the capacitance does not change and is zero. At t>=0, the voltage source can be replaced by a short-circuited jumper, and a damped oscillatory process of energy exchange between inductance and capacitance occurs in the circuit. At the initial stage, the inductance current gradually decreases to zero, charging the capacitance to the maximum voltage value. In the future, the capacitance is discharged, and the inductance current gradually increases, but in opposite direction, reaching the largest negative value at Uc=0. When t tends to infinity, all currents and voltages in 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, with h(?) equal to 0

6. Calculation of the response to a given input action

Using the superposition theorem, the impact can be represented as partial impacts.

U 1 (t) \u003d U 1 1 + U 1 2 \u003d 1 (t) + e - bt 1 (t)

The response U 2 1 (t) coincides with the transient response

The operator response U 2 2 (t) to the second partial action is equal to the product of the operator chain transfer coefficient and the image of the Laplace exponent:

Find the original U22 (p) according to the Laplace table:

Define a, w, b, K:

Finally, we get the original response:

Calculate a series of values and build a graph (Appendix 5)

Conclusion

In the course of the work, the frequency time characteristics of the circuit were calculated. Expressions are found for the response of the circuit to harmonic action, as well as the main parameters of the circuit.

The complex conjugate poles of the voltage operator coefficient indicate the damped nature of transient processes in the circuit.

Bibliography

1. Popov V.P. Fundamentals of the theory of circuits: A textbook for universities - 4th ed., Revised, M. Vyssh. school, 2003. - 575 p.: ill.

2. Biryukov V.N., Popov V.P., Sementsov V.I. Collection of problems on the theory of circuits / ed. V.P. Popov. M.: Higher. school: 2009, 269 p.

3. Korn G., Korn T., Handbook of mathematics for engineers and university students. M.: Nauka, 2003, 831 p.

4. Biryukov V.N., Dedulin K.A., Methodological guide No. 1321. Guidelines for the implementation of course work on the course Fundamentals of the theory of circuits, Taganrog, 1993, 40 p.

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