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Time and frequency characteristics of the oscillatory level

Lecture



The oscillatory link is an elementary dynamic link of the second order, it has three variable parameters. Therefore, we will pay more attention to its characteristics. Moreover, the oscillating element describes quite complex elements of electromechanical systems and electric drives, for example, such a common element as a DC motor.

The transfer function of the oscillating element -

  Time and frequency characteristics of the oscillatory level

(one)

Where   Time and frequency characteristics of the oscillatory level - gain,   Time and frequency characteristics of the oscillatory level - time constant   Time and frequency characteristics of the oscillatory level - attenuation coefficient.

A distinctive feature of the oscillatory link is that it changes not only its properties, but also its name depending on the magnitude of the attenuation coefficient:

  • if a   Time and frequency characteristics of the oscillatory level - the link is called oscillatory, since its temporal characteristics are oscillatory in nature;

  • if a   Time and frequency characteristics of the oscillatory level - the link is called the inertial (aperiodic) link of the second order, since its temporal characteristics are monotonous, that is, there are no oscillations;

  • if a   Time and frequency characteristics of the oscillatory level - the link is called conservative, since its temporal characteristics have the form of continuous oscillations, they say, the link preserves vibrations.

We obtain the temporal characteristics of the oscillatory level. To do this, we transform its transfer function (1), introducing the notation -

  Time and frequency characteristics of the oscillatory level - attenuation rate,

  Time and frequency characteristics of the oscillatory level - the angular frequency of oscillation

  Time and frequency characteristics of the oscillatory level

(2)

From the Laplace transform tables, we have -

  Time and frequency characteristics of the oscillatory level

Now we can determine the impulse response of the oscillatory link -

  Time and frequency characteristics of the oscillatory level

(3)

An example of the impulse response is shown in Fig. one.

  Time and frequency characteristics of the oscillatory level

Fig. one

Define the transient response of the oscillating element -

  Time and frequency characteristics of the oscillatory level

(four)

An exemplary view of the transient response is shown in Fig. 2

  Time and frequency characteristics of the oscillatory level

Fig. 2

According to fig. 1 and 2, it is easy to judge how the parameters of the oscillatory element affect the temporal characteristics.

We will analyze in more detail the temporal characteristics of the oscillatory level for the case   Time and frequency characteristics of the oscillatory level , that is, we define the temporal characteristics of a conservative link.

The transfer function of the conservative link has the form -

  Time and frequency characteristics of the oscillatory level ,

  Time and frequency characteristics of the oscillatory level - the angular frequency of oscillation

  Time and frequency characteristics of the oscillatory level - attenuation rate.

then the expressions of temporal characteristics (3) and (4) take the following form -

  Time and frequency characteristics of the oscillatory level

(five)

  Time and frequency characteristics of the oscillatory level

(6)

The approximate type of characteristics of a conservative link is shown in fig. 3 and 4.

  Time and frequency characteristics of the oscillatory level

Fig. 3

  Time and frequency characteristics of the oscillatory level

Fig. four

Determine the frequency response of the oscillatory level.

  Time and frequency characteristics of the oscillatory level

(6)

VCHH -

  Time and frequency characteristics of the oscillatory level

(7)

MCH -

  Time and frequency characteristics of the oscillatory level

(eight)

Frequency response -

  Time and frequency characteristics of the oscillatory level

(9)

Phase response -

  Time and frequency characteristics of the oscillatory level

(ten)

Let's build the VCHH and MCH on the same graph, an approximate view of the characteristics is shown in fig. five.

  Time and frequency characteristics of the oscillatory level

Fig. five

The approximate type of APCF is shown in fig. 6

  Time and frequency characteristics of the oscillatory level

Fig. 6

Approximate view of the frequency response and phase response is shown in Fig. 7 and 8, the frequency response function has an extremum (   Time and frequency characteristics of the oscillatory level ) at

  Time and frequency characteristics of the oscillatory level .

  Time and frequency characteristics of the oscillatory level

Fig. 7

  Time and frequency characteristics of the oscillatory level

Fig. eight

Consider the frequency characteristics of a conservative link (   Time and frequency characteristics of the oscillatory level ).

  Time and frequency characteristics of the oscillatory level .

With   Time and frequency characteristics of the oscillatory level characteristics (see fig. 9) have a gap

  Time and frequency characteristics of the oscillatory level .

  Time and frequency characteristics of the oscillatory level

Fig. 9

Determine the phase response of the conservative link -

  Time and frequency characteristics of the oscillatory level

An exemplary view of the phase response is shown in Fig. ten.

  Time and frequency characteristics of the oscillatory level

Fig. ten

We define the logarithmic characteristics of the oscillatory link.

  Time and frequency characteristics of the oscillatory level

(eleven)

We define the asymptotic LAFC of the oscillatory link

  Time and frequency characteristics of the oscillatory level

Slope asymptotes -

  Time and frequency characteristics of the oscillatory level .

The maximum deviation of the asymptotic LAFC from the exact -

  Time and frequency characteristics of the oscillatory level .

The approximate type of LAFC and LPCHH is shown in fig. eleven.

  Time and frequency characteristics of the oscillatory level

Fig. eleven

To obtain the temporal characteristics of the inertial link of the second order (   Time and frequency characteristics of the oscillatory level ) suitable and expressions (3) and (4), obtained above for the oscillatory link. But they can be obtained otherwise.

If a   Time and frequency characteristics of the oscillatory level , you can convert link transfer function -

  Time and frequency characteristics of the oscillatory level

(12)

Where

  Time and frequency characteristics of the oscillatory level .

A link with a transfer function in the form of (12) can be represented in the idea of ​​two aperiodic links, connected in series, as shown in Fig. 12.

  Time and frequency characteristics of the oscillatory level

Fig. 12

The impulse characteristics of these links are of the form -

  Time and frequency characteristics of the oscillatory level .

Then the impulse characteristic of the inertial link of the second order can be obtained using the Laplace transform multiplication of images -

  Time and frequency characteristics of the oscillatory level

(13)

We obtain the transition characteristic by integrating (13) -

  Time and frequency characteristics of the oscillatory level

(14)

The approximate type of temporal characteristics of the inertial (aperiodic) link of the second order is shown in fig. 13.

  Time and frequency characteristics of the oscillatory level

Fig. 13

We obtain the asymptotic LAFC for the second-order inertial link, representing it in the form of two successively connected aperiodic links, (see Fig. 12).

  Time and frequency characteristics of the oscillatory level

In fig. 14 and 15 show LAFC of a second-order inertial unit.

  Time and frequency characteristics of the oscillatory level

Fig. 14

  Time and frequency characteristics of the oscillatory level

Fig. 15

Test questions and tasks

    1. How does the name of an oscillatory link change and depending on what?

    2. Link Transfer Function -

  Time and frequency characteristics of the oscillatory level .

Determine the frequency of oscillation of the temporal characteristics of this link.

Answer :

Oscillation frequency   Time and frequency characteristics of the oscillatory level .

    1. Link Transfer Function -

  Time and frequency characteristics of the oscillatory level .

Determine the attenuation rate of the time characteristics of this link.

Answer :

Attenuation rate   Time and frequency characteristics of the oscillatory level .

    1. Link Transfer Function -

  Time and frequency characteristics of the oscillatory level .

Determine the frequency of oscillation of the temporal characteristics of this link.

Answer :

Oscillation frequency   Time and frequency characteristics of the oscillatory level .

    1. How many quadrants does the AFC of the oscillatory link go through?

Answer :

Two quadrants.

    1. At what frequency does the frequency response have a conservative link, if its transfer function has the form -

  Time and frequency characteristics of the oscillatory level .

Answer :

Frequency response frequency response   Time and frequency characteristics of the oscillatory level .

    1. At what angle does the harmonic signal shift with frequency   Time and frequency characteristics of the oscillatory level dynamic link with transfer function -

  Time and frequency characteristics of the oscillatory level ,

and what is the value of the frequency response at this frequency?

Answer :

The phase angle is   Time and frequency characteristics of the oscillatory level the value of the frequency response -   Time and frequency characteristics of the oscillatory level .


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control