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Frequency response dynamic link

Lecture



The frequency response of a dynamic link is the function of a complex argument.   Frequency response dynamic link obtained by formal substitution   Frequency response dynamic link on   Frequency response dynamic link in terms of transfer function

  Frequency response dynamic link

We obtain the relationship of the frequency response with well-known concepts. To do this, consider the dynamic link with the transfer function   Frequency response dynamic link and signals   Frequency response dynamic link ,   Frequency response dynamic link . Let be   Frequency response dynamic link ,   Frequency response dynamic link - absolutely integrable functions and equal to zero when   Frequency response dynamic link . Then the frequency spectra of these signals (Fourier transform) of these functions can be defined as follows:

  Frequency response dynamic link

  Frequency response dynamic link .

We obtain the ratio of the spectra

  Frequency response dynamic link .

Thus, the frequency response of a dynamic link can be defined as the ratio of the spectrum (Fourier transform) of the output signal to the spectrum of the input signal.

Knowledge of the frequency response of the link allows you to determine the output spectrum on the input

  Frequency response dynamic link .

Consider a dynamic link -

  Frequency response dynamic link

Fig. one

Get the spectrum of the output signal - impulse response

  Frequency response dynamic link .

Then we have

  Frequency response dynamic link ,

that is, the Fourier transform of the impulse response is equal to the frequency response of the dynamic link.

The frequency function characteristic as a function of a complex argument can be represented as follows: -

  Frequency response dynamic link

Where   Frequency response dynamic link - real (real) part   Frequency response dynamic link ,

  Frequency response dynamic link - imaginary part   Frequency response dynamic link ,

  Frequency response dynamic link - module (amplitude)   Frequency response dynamic link ,

  Frequency response dynamic link - phase argument   Frequency response dynamic link .

Amplitude, phase, real and imaginary parts of the frequency response are functions of the frequency, so the frequency response is used and graphically represented as amplitude-phase, real, imaginary, amplitude and phase frequency characteristics.

In the theory of automatic control, the following frequency characteristics of dynamic links are considered and used:

    1. Amplitude-frequency response (AFC) -

  Frequency response dynamic link .

    1. Phase Frequency Response (FRF) -

  Frequency response dynamic link .

    1. Real frequency response (VCHH) -

  Frequency response dynamic link .

    1. Imaginary frequency response (MCH) -

  Frequency response dynamic link .

  1. Amplitude-phase frequency response (AFC), which is defined as a hodograph (trace of the end movement) of the vector   Frequency response dynamic link built on the complex plane when the frequency changes from 0 to   Frequency response dynamic link .

In fig. 2 we will show the frequency characteristics of some dynamic link.

  Frequency response dynamic link

Fig. 2

To clarify the physical meaning of the frequency response, we consider a dynamic link with the transfer function   Frequency response dynamic link and impulse response   Frequency response dynamic link whose input is a harmonic signal   Frequency response dynamic link .

  Frequency response dynamic link

Fig. 3

Recall that the solution of a linear differential equation of a dynamic link, in the framework of the classical method, consists of two components — free and steady-state.

The steady-state component in the case of a harmonic function of time on the right side of the equation is also a harmonic function of time. Therefore, the steady-state signal at the output of the dynamic link can be described by the following expression

  Frequency response dynamic link .

The signal at the output of the link can be determined using the image multiplication theorem

  Frequency response dynamic link

The result is

  Frequency response dynamic link .

For transition to the established mode we suppose   Frequency response dynamic link then we get

  Frequency response dynamic link .

But, on the other hand, we have, by definition, the direct Fourier transform

  Frequency response dynamic link .

therefore

  Frequency response dynamic link .

From here follows a simple algorithm for experimentally determining the frequency response of a linear dynamic link, object or control system for a specific frequency.   Frequency response dynamic link :

  1. Apply a sinusoidal frequency signal to the object input   Frequency response dynamic link and constant amplitude.

  2. Wait until the free component of the transient decays.

  3. Measure the amplitude of the output signal and its phase shift relative to the input signal.

  4. The ratio of the amplitude of the output steady-state signal to the amplitude of the input signal will determine the modulus of the frequency response at a frequency   Frequency response dynamic link .

  5. The phase shift of the output signal relative to the input signal will determine the angle (argument) of the frequency response at a frequency   Frequency response dynamic link .

Using this algorithm for frequencies from zero to infinity, it is possible to experimentally determine the frequency response of a particular device. Functional diagram of the experimental setup for removing frequency characteristics is

  Frequency response dynamic link

Fig. four

At frequency   Frequency response dynamic link on the oscilloscope screen, after the free component is attenuated, the following picture -

  Frequency response dynamic link

Fig. five

Based on fig. 5, a point belonging to the frequency characteristic of the device can be built on the complex plane, and the set of points when the frequency changes from zero to the value when the amplitude of the output steady-state signal becomes negligible will be an amplitude-phase frequency response (AFC). As can be seen from the figure, any necessary frequency response of the device can be constructed from these data.

  Frequency response dynamic link

Fig. 6

For experimental acquisition of frequency characteristics of various objects, specialized devices are used in engineering practice, and recently personal computers equipped with specialized I / O boards and application packages are widely used for such purposes.

Given all the above, it becomes clear and the physical meaning of the frequency response.

It shows how many times the dynamic link (device) operating in steady state changes, the amplitude of the input sinusoid of frequency   Frequency response dynamic link , and at what angle shifts the input sinusoid in phase.

Test questions and tasks

    1. How to determine the frequency response of a dynamic link, if its transfer function is known?

    2. What types of frequency characteristics do you know?

    3. How to determine the amplitude and frequency response argument?

    4. List the main stages of experimental removal of the frequency response of the device.

    5. Explain the physical meaning of the frequency response of the linear dynamic link.

    6. Determine the expression of the frequency response for a given transfer function

  Frequency response dynamic link .

Answer :

  Frequency response dynamic link .

    1. Determine the expression of the frequency response for a given transfer function

  Frequency response dynamic link .

Answer :

  Frequency response dynamic link .

    1. Determine the expression of the amplitude and phase frequency characteristics for the dynamic link with the transfer function -

  Frequency response dynamic link .

Answer :

  Frequency response dynamic link .

  1. To the input of a dynamic link with transfer function

  Frequency response dynamic link ,

constant amplitude harmonic signal with frequency

  Frequency response dynamic link .

At what angle will the output signal be shifted in steady state?

Answer :

  Frequency response dynamic link


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control