Impulse response of a circuit example. Transient and impulse characteristics of linear circuits. Basic provisions of the theory of transition processes

⁄ and .

Operator images of the circuit reaction to external

tion of impact. Expressing

through camera images of external

impacts

Ai

; we get

0 operator images of transient and impulse nature

stick have a particularly simple form:

Thus, the impulse response of the circuit

This is a function

Circuit reaction - terminal voltage

at the output of the circuit is equal to the voltage at terminals 1-1": u 2 |t 0

u 1| t 0

1 B. Over time

As the voltage across the inductance decreases, tending to zero at t

∞. Accordingly

1 and tends to zero

Duhamel integral.

Knowing the response of the circuit to a single disturbing influence, i.e. transient conductivity function and/or transient voltage function, you can find the response of the circuit to an influence of an arbitrary shape. The method, the calculation method using the Duhamel integral, is based on the principle of superposition.

When using the Duhamel integral to separate the variable over which the integration is performed and the variable that determines the moment of time at which the current in the circuit is determined, the first is usually denoted as , and the second as t.

Let at the moment of time to the circuit with zero initial conditions (passive two-terminal network PD in Fig. 1) a source with a voltage of arbitrary shape is connected. To find the current in the circuit, we replace the original curve with a step one (see Fig. 2), after which, taking into account that the circuit is linear, we sum up the currents from the initial voltage jump and all voltage steps up to moment t, which come into effect with a time delay.

At time t, the component of the total current determined by the initial voltage surge is equal to .

At the moment of time there is a voltage surge , which, taking into account the time interval from the beginning of the jump to the time point of interest t, will determine the current component.

The total current at time t is obviously equal to the sum of all current components from individual voltage surges, taking into account , i.e.

Replacing the finite time increment interval with an infinitesimal one, i.e. passing from the sum to the integral, we write

.

(1)

Relationship (1) is called Duhamel integral.

It should be noted that voltage can also be determined using the Duhamel integral. In this case, instead of transition conductivity, (1) will include a transition voltage function.

Calculation sequence using
Duhamel integral

As an example of using the Duhamel integral, we determine the current in the circuit in Fig. 3, calculated in the previous lecture using the inclusion formula.

Initial data for calculation: , , .

1. Transient conductivity

.

18. Transfer function.

The relation of the influence operator to its own operator is called the transfer function or transfer function in operator form.

A link described by an equation or equations in a symbolic or operator form can be characterized by two transfer functions: a transfer function for the input value u; and the transfer function for the input quantity f.

And

Using transfer functions, the equation is written as . This equation is a conditional, more compact form of writing the original equation.

Along with the transfer function in operator form, the transfer function in the form of Laplace images is widely used.

Transfer functions in the form of Laplace images and operator form coincide up to notation. The transfer function in the form, Laplace images can be obtained from the transfer function in operator form, if the substitution p=s is made in the latter. In the general case, this follows from the fact that differentiation of the original - symbolic multiplication of the original by p - under zero initial conditions corresponds to multiplication of the image by a complex number s.

The similarity between transfer functions in the form of the Laplace image and in the operator form is purely external, and it occurs only in the case of stationary links (systems), i.e. only under zero initial conditions.

Let's consider a simple RLC (series) circuit, its transfer function W(p)=U OUT /U IN

Fourier integral.

Function f(x), defined on the entire number line is called periodic, if there is a number such that for any value X equality holds . Number T called period of the function.

Let us note some properties of this function:

1) Sum, difference, product and quotient of periodic functions of period T is a periodic function of period T.

2) If the function f(x) period T, then the function f(ax)has a period.

3) If f(x) - periodic function of period T, then any two integrals of this function, taken over intervals of length T(in this case the integral exists), i.e. for any a And b equality is true .

Trigonometric series. Fourier series

If f(x) is expanded on a segment into a uniformly convergent trigonometric series: (1)

Then this expansion is unique and the coefficients are determined by the formulas:

Where n=1,2, . . .

Trigonometric series (1) of the type considered with coefficients is called trigonometric Fourier series.

Complex form of the Fourier series

The expression is called the complex form of the Fourier series of the function f(x), if defined by equality

, Where

The transition from the Fourier series in complex form to the series in real form and back is carried out using the formulas:

(n=1,2, . . .)

The Fourier integral of a function f(x) is an integral of the form:

, Where .

Frequency functions.

If you apply to the input of a system with a transfer function W(p) harmonic signal

then after the transition process is completed, harmonic oscillations will be established at the output

with the same frequency, but different amplitude and phase, depending on the frequency of the disturbing influence. From them one can judge the dynamic properties of the system. Dependencies connecting the amplitude and phase of the output signal with the frequency of the input signal are called frequency characteristics(CH). Analysis of the frequency response of a system in order to study its dynamic properties is called frequency analysis.

Let's substitute expressions for u(t) And y(t) into the dynamics equation

(aоp n + a 1 pn - 1 + a 2 p n - 2 + ... + a n)y = (bоp m + b 1 p m-1 + ... + b m)u.

Let's take into account that

pnu = pnU m ejwt = U m (jw)nejwt = (jw)nu.

Similar relationships can be written for the left side of the equation. We get:

By analogy with the transfer function, we can write:

W(j), equal to the ratio of the output signal to the input signal when the input signal changes according to the harmonic law, is called frequency transfer function. It is easy to see that it can be obtained by simply replacing p by j in the expression W(p).

W(j) is a complex function, therefore:

where P() - real frequency response (RFC); Q() - imaginary frequency response (ICH); A() - amplitude frequency response (AFC): () - phase frequency response (PFC). The frequency response gives the ratio of the amplitudes of the output and input signals, the phase response gives the phase shift of the output quantity relative to the input:

;

If W(j) is represented as a vector on the complex plane, then when changing from 0 to + its end will draw a curve called vector hodograph W(j), or amplitude-phase frequency response (APFC)(Fig. 48).

The AFC branch when changing from - to 0 can be obtained by mirroring this curve relative to the real axis.

TAU is widely used logarithmic frequency characteristics (LFC)(Fig.49): logarithmic amplitude frequency response (LAFC) L() and logarithmic phase frequency response (LPFC) ().

They are obtained by taking the logarithm of the transfer function:

LFC is obtained from the first term, which is multiplied by 20 for scaling reasons, and not the natural logarithm is used, but the decimal one, that is, L() = 20lgA(). The value of L() is plotted along the ordinate axis in decibels.

A change in signal level by 10 dB corresponds to a change in its power by a factor of 10. Since the power of the harmonic signal P is proportional to the square of its amplitude A, a change in the signal by 10 times corresponds to a change in its level by 20 dB, since

log(P 2 /P 1) = log(A 2 2 /A 1 2) = 20log(A 2 /A 1).

The abscissa axis shows the frequency w on a logarithmic scale. That is, unit intervals along the abscissa axis correspond to a change in w by a factor of 10. This interval is called decade. Since log(0) = -, the ordinate axis is drawn arbitrarily.

The LPFC obtained from the second term differs from the phase response only in the scale along the axis. The value () is plotted along the ordinate axis in degrees or radians. For elementary links it does not go beyond: - +.

Frequency characteristics are comprehensive characteristics of the system. Knowing the frequency response of the system, you can restore its transfer function and determine its parameters.

Feedback.

It is generally accepted that a link is covered by feedback if its output signal is fed to the input through some other link. Moreover, if the feedback signal is subtracted from the input action (), then the feedback is called negative. If the feedback signal is added to the input action (), then the feedback is called positive.

The transfer function of a closed circuit with negative feedback - the link covered by negative feedback - is equal to the forward circuit transfer function divided by one plus the open circuit transfer function

The closed-loop transfer function with positive feedback is equal to the forward-loop transfer function divided by one minus the open-loop transfer function

22. 23. Quadrupoles.

When analyzing electrical circuits in problems of studying the relationship between variables (currents, voltages, powers, etc.) of two branches of the circuit, the theory of four-terminal networks is widely used.

Quadrupole- This is a part of a circuit of any configuration that has two pairs of terminals (hence its name), usually called input and output.

Examples of a four-terminal network are a transformer, amplifier, potentiometer, power line and other electrical devices in which two pairs of poles can be distinguished.

In general, quadripoles can be divided into active, whose structure includes energy sources, and passive, branches of which do not contain energy sources.

To write the equations of a four-terminal network, we select in an arbitrary circuit a branch with a single energy source and any other branch with some resistance (see Fig. 1, a).

In accordance with the principle of compensation, we replace the original resistance with a source with voltage (see Fig. 1,b). Then, based on the superposition method for the circuit in Fig. 1b can be written

Equations (3) and (4) are the basic equations of the quadripole; they are also called quadripole equations in A-form (see Table 1). Generally speaking, there are six forms of writing the equations of a passive quadripole. Indeed, a four-terminal network is characterized by two voltages and and two currents and. Any two quantities can be expressed in terms of the others. Since the number of combinations of four by two is six, then six forms of writing the equations of a passive quadripole are possible, which are given in Table. 1. Positive directions of currents for various forms of writing equations are shown in Fig. 2. Note that the choice of one or another form of equations is determined by the area and type of problem being solved.

Table 1. Forms of writing the equations of a passive quadripole

Form	Equations	Connection with the coefficients of the basic equations
A-shape	; ;
Y-shape	; ;	; ; ; ;
Z-shape	; ;	; ; ; ;
H-shape	; ;	; ; ; ;
G-shape	; ;	; ; ; ;
B-shape	; .	; ; ; .

Characteristic impedance and coefficient
propagation of a symmetrical quadripole

In telecommunications, the operating mode of a symmetrical four-port network is widely used, in which its input resistance is equal to the load resistance, i.e.

.

This resistance is designated as and called characteristic resistance symmetrical four-port network, and the operating mode of the four-port network, for which it is true

,

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
.

Impulse is a function without any time support. With differential equations is used to obtain the natural response of the system. Its natural response is a reaction to the initial state. A forced response of a system is a response to input, neglecting its initial formation.

Since the impulse function does not have any time support, it is possible to describe any initial state arising from the corresponding weighted quantity, which is equal to the mass of the body produced by the speed. Any arbitrary input variable can be described as a sum of weighted pulses. As a result, for a linear system it is described as the sum of the “natural” responses to the states represented by the quantities under consideration. This is what the integral explains.

When the impulse response of a system is calculated, a natural response is essentially produced. If the sum or integral of a convolution is examined, it is basically solving this input to a series of states, and then the initially formed response to those states. Practically for the impulse function, we can give an example of a punch in boxing, which lasts very little, and after that there will be no next one. Mathematically, it is only present at the starting point of a realistic system, having a high (infinite) amplitude at that point, and then fades away continuously.

The impulse function is defined as follows: F(X)=∞∞ x=0=00, where the response is a characteristic of the system. The function in question is actually a region of a rectangular pulse at x=0, the width of which is assumed to be zero. At x=0 the height of h and its width 1/h is the actual start. Now, if the width becomes negligible, that is, almost tends to zero, this makes the corresponding height h tend to infinity. This defines the function as infinitely high.

Design response

The impulse response is as follows: Whenever an input signal is assigned to the system (block) or processor, it modifies or processes it to give the desired output warning depending on the transfer function. System response helps determine the fundamentals, design, and response for any sound. The delta function is a generalized function that can be defined as the limit of a class of specified sequences. If we take a pulse signal, then it goes without saying that it is the spectrum of direct current in the frequency domain. This means that all harmonics (ranging from frequency to +infinity) contribute to the signal in question. The spectrum of the frequency response indicates that this system provides this order of amplification or attenuation of this frequency or suppresses these oscillating components. Phase refers to the shift provided for different harmonics of the frequency.

Thus, the impulse characteristics of a signal indicate that it contains the entire frequency range, and is therefore used for system testing. Because if any other notification method is used, it will not have all the necessary designed parts, hence the response will remain unknown.

Device response to external factors

When processing an alert, the impulse response is its output when it is represented by a brief input signal called an impulse. More generally, it is the reaction of any dynamic system in response to some external changes. In both cases, the impulse response describes a function of time (or perhaps as some other independent variable that parameterizes dynamic behavior). It has infinite amplitude only at t=0 and zero everywhere, and, as the name suggests, its impulse i, e acts for a short period.

When applied, any system has a transfer function from input to output, which describes it as a filter that affects the phase and the above quantity in the frequency range. This frequency response using impulse methods is measured or calculated digitally. In all cases, the dynamic system and its characteristic can be real physical objects or mathematical equations describing such elements.

Mathematical description of pulses

Since the function under consideration contains all frequencies, the criteria and description determine the response of the linear time invariant structure for all quantities. Mathematically, how momentum is described depends on whether the system is modeled in discrete or continuous time. It can be modeled as the Dirac delta function for continuous-time systems or as the Kronecker value for a discontinuous design. The first represents the limiting case of a pulse that was very short in time, maintaining its area or integral (thus producing an infinitely high peak). Although this is not possible in any real system, it is a useful idealization. In Fourier analysis theory, such a pulse contains equal parts of all possible excitation frequencies, making it a convenient test probe.

Any system in the large class known as linear time invariant (LTI) is completely described by an impulse response. That is, for any input, the output can be calculated in terms of the input and the immediate concept of the quantity in question. The impulse description of the linear transformation is an image of the Dirac delta function under the transformation, analogous to the fundamental solution of the partial differential operator.

Features of pulse designs

It is usually easier to analyze systems using transfer impulse responses rather than responses. The quantity under consideration is the Laplace transform. A scientist's improvement in a system's output can be determined by multiplying the transfer function by this input action in the complex plane, also known as the frequency domain. The inverse Laplace transform of this result will give the output in the time domain.

Determining the output directly in the time domain requires convolving the input with the impulse response. When the transfer function and Laplace transform of the input are known. A mathematical operation that applies to two elements and implements a third may be more complex. Some prefer the alternative of multiplying two functions in the frequency domain.

Real Application of Impulse Response

In practical systems, it is impossible to create a perfect pulse for input data for testing. Therefore, a short signal is sometimes used as an approximation of the magnitude. Provided that the pulse is short enough compared to the response, the result will be close to the true, theoretical one. However, in many systems, an entry with a very short, strong pulse can drive the design into a nonlinear mode. So instead it is driven by a pseudo-random sequence. Thus, the impulse response is calculated from the input and output signals. The response, viewed as a Green's function, can be thought of as an "influence" - how the entry point affects the output.

Characteristics of pulse devices

Speakers is an application that demonstrates the very idea (the development of impulse response testing in the 1970s). Loudspeakers suffer from phase inaccuracy, a defect, as opposed to other measured properties such as frequency response. This underdeveloped criterion is caused by (slightly) delayed oscillations/octaves, which are mainly the result of passive cross-passes (especially higher order filters). But also caused by resonance, internal volume or vibration of body panels. The response is the finite impulse response. Its measurement provided a tool for use in reducing resonances through the use of improved materials for the cones and cabinets, as well as changes in the crossover of the speakers. The need to limit amplitude to maintain system linearity has led to the use of inputs such as pseudo-random sequences of maximum length, and to the assistance of computer processing to obtain the remaining information and data.

Electronic change

Impulse response analysis is a fundamental aspect of radar, ultrasound imaging, and many areas of digital signal processing. An interesting example would be broadband Internet connections. DSL services use adaptive equalization techniques to help compensate for signal distortion and interference introduced by the copper telephone lines used to deliver the service. They are based on outdated circuits, the impulse response of which leaves much to be desired. This has been replaced by modernized coverage for the use of the Internet, television and other devices. These advanced designs have the potential to improve quality, especially since today's world is all about internet connections.

Control systems

In control theory, impulse response represents the response of a system to a Dirac delta input. This is useful when analyzing dynamic structures. The Laplace transform of the delta function is equal to one. Therefore, the impulse response is equivalent to the inverse Laplace transform of the system transfer function and filter.

Acoustic and sound applications

Here, impulse responses allow the sound characteristics of a location, such as a concert hall, to be recorded. Various packages are available containing location-specific alerts, from small rooms to large concert halls. These impulse responses can then be used in convolution reverberation applications to allow the acoustic characteristics of a specific location to be applied to the target sound. That is, in fact, analysis occurs, separation of various alerts and acoustics through a filter. The impulse response in this case can give the user a choice.

Financial component

In modern macroeconomic modeling, impulse response functions are used to describe how it responds over time to exogenous quantities, which scientific researchers usually call shocks. And are often simulated in the context of vector autoregression. Impulses that are often considered exogenous from a macroeconomic perspective include changes in government spending, tax rates and other fiscal policy parameters, changes in the monetary base or other capital and credit policy parameters, changes in productivity or other technological parameters; transformation in preferences, such as degree of impatience. Impulse response functions describe the response of endogenous macroeconomic variables such as output, consumption, investment and employment during a shock and at subsequent points in time.

More specifically about impulse

Essentially, current and impulse response are interrelated. Because each signal can be modeled as a series. This occurs due to the presence of certain variables and electricity or a generator. If the system is both linear and time-dependent, the response of the instrument to each response can be calculated using the reflexes of the quantity in question.