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Elementary (typical) dynamic links

Lecture



Any linear ACS can be represented as a transfer function in the form of a Bode.

  Elementary (typical) dynamic links ,(one)

Where   Elementary (typical) dynamic links can be either real or complex conjugate. Consider separately each case.

Valid Zeros and Poles

We transform the factors from (1) by entering the notation

  Elementary (typical) dynamic links ,

as a result, we have factors of the following form -

  Elementary (typical) dynamic links (2)

Complex conjugate zeros and poles

In this case, we have the roots of the form -

  Elementary (typical) dynamic links ,

and their corresponding factors

  Elementary (typical) dynamic links .

We introduce the notation -

  Elementary (typical) dynamic links ,

we obtain the factors of the following form

  Elementary (typical) dynamic links , (3)

in the numerator and denominator of the transfer function.

Then (1) with regard to (2) and (3) can be written in the following form

as a result, we have factors of the following form -

  Elementary (typical) dynamic links . (2)

Complex conjugate zeros and poles

In this case, we have the roots of the form -

  Elementary (typical) dynamic links ,

and their corresponding factors

  Elementary (typical) dynamic links .

We introduce the notation -

  Elementary (typical) dynamic links ,

we obtain the factors of the following form

  Elementary (typical) dynamic links , (3)

in the numerator and denominator of the transfer function.

Then (1) with regard to (2) and (3) can be written in the following form

  Elementary (typical) dynamic links ,(four)

Where

  Elementary (typical) dynamic links .

From (4) it follows, taking into account the rule of equivalent transformation of structural schemes, that a linear ACS can be represented as a serial connection of elementary dynamic links of the 1st and 2nd order with transfer functions of the following form

  Elementary (typical) dynamic links .(five)

In addition, the transfer function of the ACS can be represented in the form of Hevisite -

  Elementary (typical) dynamic links .

From which it follows that the ACS can be represented as parallel-connected links with transfer functions of the form (5). In addition, the transfer functions of the 1st and 2nd order describe many functional components of the control systems.

Such dynamic links are called elementary or typical links, the study of their properties and characteristics gives a lot in the synthesis and analysis of real and complex systems.

Typical links include the following dynamic links:

    1. Inertialess (scaling, proportional) link

  Elementary (typical) dynamic links .

    1. Differentiator link

  Elementary (typical) dynamic links .

    1. Integrating link

  Elementary (typical) dynamic links .

    1. Aperiodic link

  Elementary (typical) dynamic links .

    1. Oscillatory link

  Elementary (typical) dynamic links .

    1. Boost links

  Elementary (typical) dynamic links .

Comment

The following links are not elementary in the full sense of the word, but they are often classified as typical due to their wide distribution.

    1. Real differentiating link

  Elementary (typical) dynamic links .

    1. Integral with delay

  Elementary (typical) dynamic links .

  1. Proportional-integral link

  Elementary (typical) dynamic links .

The characteristics (time and frequency) of typical links can be obtained analytically from their transfer functions, while it is convenient to use a summary diagram showing the relationship of mathematical models of dynamic links.

  Elementary (typical) dynamic links

Fig. one

Instant link

Transmission function

  Elementary (typical) dynamic links .

Time characteristics

  Elementary (typical) dynamic links ,

  Elementary (typical) dynamic links .

  Elementary (typical) dynamic links

Frequency response

  Elementary (typical) dynamic links ,

  Elementary (typical) dynamic links .

  Elementary (typical) dynamic links

  Elementary (typical) dynamic links

  Elementary (typical) dynamic links

Differentiator link

Transmission function

  Elementary (typical) dynamic links .

Time characteristics

  Elementary (typical) dynamic links ,

  Elementary (typical) dynamic links .

Frequency response

  Elementary (typical) dynamic links ,

  Elementary (typical) dynamic links .

  Elementary (typical) dynamic links

  Elementary (typical) dynamic links

  Elementary (typical) dynamic links

Integrating link

Transmission function

  Elementary (typical) dynamic links .

Time characteristics

  Elementary (typical) dynamic links ,

  Elementary (typical) dynamic links .

  Elementary (typical) dynamic links

Frequency response

  Elementary (typical) dynamic links ,

  Elementary (typical) dynamic links .

  Elementary (typical) dynamic links

  Elementary (typical) dynamic links

  Elementary (typical) dynamic links

Test questions and tasks

  1. Give the definition of a typical dynamic link.

  2. Why do typical dynamic links study in such detail?

  3. List the dynamic links that are classified as typical (elementary).

  4. How to determine the impulse response of a dynamic link from the transfer function?

  5. How to determine the transient characteristic of a dynamic link by transfer function?

  6. How to determine the frequency response of a dynamic link from the transfer function?

  7. What type of link shifts the harmonic signal of any frequency at an angle   Elementary (typical) dynamic links towards the delay?

  8. What type of link shifts the harmonic signal of any frequency at an angle   Elementary (typical) dynamic links in the direction of advance?

  9. What type of link does not change the phase of the harmonic signal of any frequency?


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control