How to experimentally measure the time characteristics of linear circuits. Timing characteristics of the circuit. Transfer function in operator form
The timing characteristics of circuits include transient and impulse characteristics.
Let us consider a linear electrical circuit that does not contain independent sources of current and voltage.
Let the external influence on the circuit be the switching function (single jump) x(t) = 1(t - t 0).
Step response h(t - t 0) of a linear circuit that does not contain independent energy sources is called the ratio of the reaction of this circuit to the influence of a single current or voltage jump
The dimension of the transient characteristic is equal to the ratio of the response dimension 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 circuit have the form of an -function
x(t) = d(t - t 0).
Impulse response g (t - t 0) linear chain that does not contain independent energy sources is called the reaction of the chain to an impact in the form of an -function at zero initial conditions/
The dimension of the impulse response is equal to the ratio of the dimension of the circuit response to the product of the dimension of the external influence 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 reaction, however, unlike the first characteristics, the argument of the latter is time t, not angular w or complex p frequency. Since the characteristics of a circuit whose argument is time are called temporary, and the characteristics whose argument is frequency (including complex) are called frequency characteristics, then 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 transient and impulse characteristics.
(9.75)
At t0 = 0 operator images of the transient and impulse characteristics have a simple form
Expressions (9.75), (9.76) establish a connection between the frequency and time characteristics of the circuit. Knowing, for example, the impulse response, you 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 transition 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 represents the derivative of the transition characteristic at t > t 0, and the second term contains the product of the d-function and the value of the transition characteristic at the point t=t0.
If the function h 1 (t - t 0) does not undergo a discontinuity at t = t 0, i.e., the value of the transient response 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., impulse response circuit is equal to the first derivative of the transient response with respect to time
(9.77)
To determine the transient (pulse) characteristics of a linear circuit, two main methods are used.
1) It is necessary to consider the transient processes that occur in a given circuit when it is exposed to current or voltage in the form of a switching function or -function. This can be done using classical or operator methods of transient analysis.
2) In practice, to find the time characteristics of linear circuits, it is convenient to use a path based on the use of relationships that establish a connection between frequency and time characteristics. Determining the timing characteristics in this case begins with drawing up an operator circuit equivalent circuit for zero initial conditions. Next, using this scheme, find the operator characteristic H k n (p) corresponding to a given pair: external influence on the circuit x n (t) - reaction of the circuit y k (t). Knowing the operator characteristic of the circuit and applying relations (6.109) or (6.110), the required time characteristics are determined.
It should be noted that when qualitatively considering the response of a linear circuit to the action of a single current or voltage pulse, the transient process in the circuit is divided into two stages. At the first stage (with tО] t 0- , t 0+ [) the circuit is under the influence of a single impulse, which imparts a certain energy to the circuit. In this case, the inductor currents and capacitor voltages change abruptly to a value corresponding to the energy entering the circuit, and the laws of commutation are violated. At the second stage (with t ³ t 0+) the action of the external influence applied to the circuit has ended (the corresponding energy sources are turned off, i.e., represented by internal resistances), and free processes arise in the circuit, occurring due to the energy stored in the reactive elements at the first stage of the transition process. Consequently, the impulse response characterizes free processes in the circuit under consideration.
The expressions (5.17), (5.18) given in the previous paragraph for the gain factors can be interpreted as transfer functions of a linear active two-port network. The nature of these functions is determined by the frequency properties of the Y parameters.
Having written it in the form of functions, we come to the concept of the transfer function of a linear active two-port network. Dimensionless in the general case, the complex function is an exhaustive characteristic of a four-port network in the frequency domain. It is determined in a stationary mode with harmonic excitation of a four-terminal network.
It is often convenient to represent the transfer function in the form
The module is sometimes called the amplitude-frequency response (AFC) of a quadripole network. The argument is called the phase-frequency response (PFC) of a quadripole network.
Another comprehensive characteristic of a quadripole is its impulse response, which is used to describe the circuit in the time domain.
For active linear circuits, as well as for passive ones, the impulse response of a circuit means the response, the reaction of the circuit to an impact in the form of a single impulse (delta function). The connection between is easy to establish using the Fourier integral.
If a single pulse (delta function) of an emf with a spectral density equal to unity for all frequencies is applied at the input of a four-port network, then the spectral density of the output voltage is simply equal to . The response to a single impulse, i.e. the impulse response of the circuit, is easily determined using the inverse Fourier transform applied to the transfer function:
It is necessary to take into account that in front of the right side of this equality there is a factor of 1 with the dimension of the area of the delta function. In the particular case, when we mean a b-voltage pulse, 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 a factor of one with the dimension [volt x second]^-1.
In what follows, we will denote the impulse response as a function, by which we can mean not only voltage, but also any other electrical quantity that is a response to an impact in the form of a delta function.
As with the representation of signals on the complex frequency plane (see § 2.14), in circuit theory the concept of a transfer function 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.
Unit step function (using the Heaviside function) called function
Graph of function 1(7 - (0) has the form of a step or jump, the height of which is equal to one (Fig. 6.16, A). We will call a jump of this type single. At t 0 = Q for a unit step function the notation 1(0) is used (Fig. 6.16, b).
Due to the fact that the product of any limited function of time f(t) pa 1 (t - t 0) equals zero at t and equals /(0 at t > t 0:
Heaviside function l(f - t 0) convenient to use for analytical representation of various external influences
Rice. 6.16.
When connecting a circuit to a source of constant current or voltage, an external influence on the circuit
Where to - switching moment.
This type of external influence is called non-unit jump. Using the Heaviside function, expression (6.95) can be represented as
If at t= ?о a source of harmonic current or voltage is included in the circuit
then the external influence on the circuit can be represented as
If an external influence on the circuit at the moment of time t = t§ changes abruptly from one fixed value X ( to another X 2, That
External influence on the circuit, having the form of a rectangular pulse with a height X and duration t u(Fig. 6.17, A), can be represented as the difference of two identical jumps
shifted in time by? and (Fig. 6.17, b, V):
Rice. 6.17.
Rice. 6.18.
Consider a rectangular pulse of duration At and height X/At(Fig. 6.18, A). Obviously, the area of this pulse is equal to unity and does not depend on At. As the pulse duration decreases, its height increases, and when At-*? 0 it tends to infinity, but the area of the pulse remains equal to unity. A pulse of infinitesimal duration, infinitely large height, the area of which is equal to unity, will be called single impulse.
The function that defines a single pulse is designated 5 (t - to) and is called a 5-function or "Dirac function". Thus,
At? 0 = 0 for the 5-function the designation 5 is used (t). When constructing time diagrams of function b (t - to) And 8(t) we will depict it as a vertical arrow with a 00 icon near the tip (Fig. 6.18, b, c).
To establish the connection between the 5-function and the unit step function, we use expression (6.96). Believing X = 1 /At and rushing At to zero, we get
Thus, The 8-function is the derivative of the unit step function, and the unit step function - integral of the 8-function.
A rigorous justification for operations on unit functions, including the operation of differentiation of a unit step function, is given in the theory of generalized functions. For a qualitative justification of such operations, functions 1(7: - / 0) and 6(7 - t 0) It is convenient to represent them as limit values of some simpler functions for which the corresponding operations are definite. Consider, for example, the function.gDG) (Fig. 6.19, A), satisfying the conditions
Derivative of a function X(t) by time (Fig. 6.19, b) has the form of a rectangular pulse with a duration At and height 1/D t:
At At -*? 0 function X(t) degenerates into a unit step function, and the function dx ( (t)/dt- into b-function:
whence it follows that 
Rice. 6.19.
When performing various operations on unit functions, the commutation moment tQ It is convenient to divide it into three different points: ? 0 _ is the moment of time immediately preceding the switching, ? 0 is the actual commutation moment and? ()+ - the moment of time immediately following commutation. Taking this into account, from condition (6.98) we can obtain
In general
Product of an arbitrary limited time function /(?) by 8(? - ? 0)
Conditions (6.103) are also satisfied by the product f(t 0)6(t- ?o)> therefore,
From expressions (6.102) and (6.104) it follows that the integral of the product of an arbitrary limited function /(?) by 6(1: - tg) is equal to either the value of this function at t = to(if point to belongs to the integration interval), or zero (if the point? 0 does not belong to the integration interval):
Thus, using the 5-function, you can extract the values of the function/(?) at arbitrary times? 0 - This feature of the 8-function is usually called filtering property.
To determine the response of linear electrical circuits to an external influence in the form of a single jump or a single impulse, it is necessary to find images of unit Laplace functions. Using the considered properties of unit functions, we obtain
At t 0 = 0 operator representations of unit functions have a particularly simple form:
- For a more rigorous definition of the 5-function, see, for example, the work.
Previously, we considered frequency characteristics, and time characteristics describe the behavior of a circuit over time for a given input action. There are only two such characteristics: transient and impulse.
Step 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 before it there were no currents or voltages in the circuit.
The graph has a stepwise effect:
1(t) - single step effect.
Sometimes a step function is used that does not start at moment “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 current) is connected to a given circuit and the transient current or voltage specified as a reaction is calculated. After this, divide the result by the source value.
Example: find h(t) for u c with input action in the form of voltage.
Example: solve the same problem with input action in the form of current
Impulse response
The impulse response - g(t) - is the ratio of the circuit's response to an input influence in the form of a delta function to the area of this influence, provided that before connecting the influence there were no currents or voltages in the circuit.
d(t) - delta function, delta impulse, unit impulse, Dirac impulse, Dirac function. This is the function:
It is extremely inconvenient to calculate g(t) using 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 ones fall at the input:
t f - duration of the leading edge (rise time of the input signal);
t and - pulse duration;
These impulses have certain requirements:
a) for the transient response:
T pause should be so large that by the time the next pulse arrives, the transition process from the end of the previous pulse is practically over;
T should be so large that the transient process caused by the occurrence of a pulse also practically has time to end;
T f should be as small as possible (so that during t cf the state of the circuit practically does not change);
X m should, on the one hand, be so large that using the existing equipment it would be possible to register the reaction of the chain, and on the other hand, it should be so small that the chain under study retains its properties. If all this is true, record the circuit reaction graph and change the scale along the ordinate axis by X m times (X m = 5V, divide the ordinate by 5).
b) for impulse response:
t pause - the requirements are the same for X m - the same, there are no requirements for t f (because even the pulse duration t f itself must be so short that the state of the circuit practically does not change. If all this is so, record the reaction and change the scale along the ordinate axis by the area of the input pulse.
Results using the classical method
The main advantage is the physical clarity of all quantities used, which allows you to check the progress of the solution from the point of view of physical meaning. In simple circuits it is possible to get the answer very easily.
Disadvantages: as the complexity of the problem increases, the complexity of the solution quickly increases, especially at the stage of calculating the initial conditions. Not all problems are convenient to solve using the classical method (almost no one looks for g(t), and everyone has problems when calculating problems with special contours and special sections).
Before switching, .
Consequently, according to the commutation laws, u c1 (0) = 0 and u c2 (0) = 0, but from the diagram it is clear that immediately after closing the key: E= u c1 (0)+u c2 (0).
In such problems it is necessary to use a special procedure for searching for initial conditions.
These shortcomings can be overcome in the operator method.
The time characteristic of a circuit is a function of time, the values of which are numerically determined by the response of the circuit to a typical impact. The reaction of a circuit to a given typical impact depends only on the circuit diagram and the parameters of its elements and, therefore, can serve as its characteristic. Temporal characteristics are determined for linear circuits that do not contain independent energy sources and under zero initial conditions. Temporary characteristics depend on the type of specified typical impact. In connection With This divides them into two groups: transient and impulse time characteristics.
Transition characteristic or transition function, is determined by the response of the circuit to the influence 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 characteristic turns out to be 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 quantity is current, then the transition characteristic has the dimension of conductivity, is numerically equal to this current and is called transition conductivity Y(t). Similarly, when acting in the form of a current and reacting in the form of a voltage, the transition function has the dimension of resistance and is called the transition resistance Z(t). If the output quantity is current, then the transition characteristic is dimensionless and is called the transition function or transfer coefficient K I (t) no current
In general, a transition characteristic of any type is denoted by h(t). Transient characteristics are easily determined by calculating the circuit's response to a single step action, i.e., calculating the transient process when the circuit is switched on to a constant voltage of 1 V or a constant current of 1 A.
Example 14.2.
Find temporary crossings O These characteristics of a simple rC circuit (Fig. 14.9, a), if in O 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 switching on of the circuit at the moment t=0 to a source of constant e. d.s. e 0 =1 IN(Fig. 14.9,6). In this case:
a) the current in the circuit is determined by the expression
therefore the transition conductivity is
b) voltage across the capacitance
therefore the voltage transition function
Pulse the characteristic, or impulse transient function, is determined by the response of the circuit to the influence of the δ(t) function. Like the transient characteristic, it has several varieties, determined by the type of impact and reaction - voltage or current. In general, the impulse response is denoted by a(t).
Let us 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 a short pulse action t И =Δt, representing it by superimposing two step functions:
In accordance with the superposition principle, the response of the circuit to such an impact is determined using transient characteristics:
For small Δt we can write
Where S and =U m Δƒ- impulse area.
At Δt 0 and Um the resulting expression describes the reaction of the chain to the δ(t)-function, t . e, determines the impulse response of the circuit:
Taking this into account, the response of a linear circuit to a pulse of short duration can be found as the product of the pulse function and the pulse area:
This equality underlies the experimental determination of the impulse function. The shorter the pulse duration, the more accurate it is.
Thus, the impulse response is the derivative of the step 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 indicating 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 characteristics have the dimension of the corresponding transient response divided by time. To calculate the impulse response, you can use expression (14.19), i.e., calculate it using the transient response.
Example 14.3.
Find the impulse characteristics of a simple rC circuit (see Fig. 14.9, a). Solution.
Using the expressions for the transient characteristics obtained in Example 14.2, using O Using expression (14.19) we find the impulse characteristics;
The timing characteristics of typical links are given in Table. 14.2.
Calculation of timing characteristics is usually carried out in the following order:
the points of application of the external influence and its type (current or voltage), as well as the output quantity of interest - the reaction of the circuit (current or voltage in some section of it) are determined; the required time characteristic is calculated as the response of the circuit to the corresponding typical impact: 1(t) or δ(t),