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Logarithmic frequency characteristics of automatic control systems

Lecture



Aperiodic link

Transmission function -

  Logarithmic frequency characteristics of automatic control systems .

Frequency response -

  Logarithmic frequency characteristics of automatic control systems ,

Frequency response and phase response

  Logarithmic frequency characteristics of automatic control systems .

Logarithmic characteristics

  Logarithmic frequency characteristics of automatic control systems

In this case, at the frequency -

  Logarithmic frequency characteristics of automatic control systems

we have

  Logarithmic frequency characteristics of automatic control systems .

Consider two characteristic ranges for the aperiodic link:

  Logarithmic frequency characteristics of automatic control systems

(one)

  Logarithmic frequency characteristics of automatic control systems

(2)

  Logarithmic frequency characteristics of automatic control systems ,

  Logarithmic frequency characteristics of automatic control systems .

Expressions (1) and (2) are the equations of straight lines - asymptotes, to which LAFH aspire away from the point of their conjugation.   Logarithmic frequency characteristics of automatic control systems . As we will see later, in the synthesis and analysis of systems, it is more convenient to use not exact, but asymptotic characteristics.

  Logarithmic frequency characteristics of automatic control systems

As we saw when working with the simplest typical links, frequency characteristics can be obtained from the transfer function. In more complex cases, when solving the problems of synthesis and analysis of the ACS, there is a need to obtain the characteristics of the ACS based on the known characteristics of the links included in the ACS.

The most frequently used case is when the links in the ACS are switched on sequentially, as shown in Fig. one.

  Logarithmic frequency characteristics of automatic control systems

Fig. one

In accordance with the rules of equivalent transformations, the transfer function of the entire ACS will have the form -

  Logarithmic frequency characteristics of automatic control systems .

Get the frequency response of the ACS

  Logarithmic frequency characteristics of automatic control systems

Consequently,

AChH SAU -

  Logarithmic frequency characteristics of automatic control systems

(3)

AFF SAU

  Logarithmic frequency characteristics of automatic control systems

(four)

We obtain, using expressions (3) and (4), the logarithmic characteristics of the ACS:

LAFH -

  Logarithmic frequency characteristics of automatic control systems

(five)

LFCH -

  Logarithmic frequency characteristics of automatic control systems

(6)

Thus, the logarithmic frequency characteristics of the ACS can be defined as the sum of the logarithmic frequency characteristics of the series-connected components of the ACS units. The logarithmic scale and the use of asymptotes allows summation graphically.

TAU also uses the properties of logarithmic frequency characteristics of dynamic links, the transfer functions of which are reciprocal -

  Logarithmic frequency characteristics of automatic control systems .

Let the frequency characteristics of the link   Logarithmic frequency characteristics of automatic control systems known:

Frequency response -

  Logarithmic frequency characteristics of automatic control systems ,

LAFH -

  Logarithmic frequency characteristics of automatic control systems ,

LFCH -

  Logarithmic frequency characteristics of automatic control systems .

Then the frequency characteristics of the link   Logarithmic frequency characteristics of automatic control systems have the form:

Frequency response -

  Logarithmic frequency characteristics of automatic control systems ,

LAFH -

  Logarithmic frequency characteristics of automatic control systems ,

LFCH -

  Logarithmic frequency characteristics of automatic control systems .

Thus, LAFH and LFCH of the inverse dynamic links are located symmetrically with respect to the frequency axis, which is confirmed by the previously obtained LAFH and LFCH of the differentiating and integrating links.

Example

For SAU was determined transfer function. LACHH SAU should be determined.

  Logarithmic frequency characteristics of automatic control systems .

Decision

Imagine the ACS in the form of serially connected dynamic links.

  Logarithmic frequency characteristics of automatic control systems

We obtain asymptotic LAFC for each aperiodic link

  Logarithmic frequency characteristics of automatic control systems

Using the properties of LAFC of reciprocal links, we obtain the asymptotic LPPH of forcing links   Logarithmic frequency characteristics of automatic control systems .

  Logarithmic frequency characteristics of automatic control systems

We obtain the asymptotic LAFC of SAU by performing the graphical summation of LAFC of links

  Logarithmic frequency characteristics of automatic control systems .

The task is greatly simplified by the fact that asymptotic graphs of links have sections with integer slope.

  Logarithmic frequency characteristics of automatic control systems

We obtain LAFC and LFCH typical units using the above.

Real differentiating link

Transmission function

  Logarithmic frequency characteristics of automatic control systems .

Imagine a link in the following form

  Logarithmic frequency characteristics of automatic control systems

Then LAFH and LFCHH have the form -

  Logarithmic frequency characteristics of automatic control systems ,

  Logarithmic frequency characteristics of automatic control systems .

  Logarithmic frequency characteristics of automatic control systems

Integrating Link with Lag

Transmission function

  Logarithmic frequency characteristics of automatic control systems .

Imagine a link in the following form

  Logarithmic frequency characteristics of automatic control systems

Then LAFH and LFCHH have the form -

  Logarithmic frequency characteristics of automatic control systems ,

  Logarithmic frequency characteristics of automatic control systems .

  Logarithmic frequency characteristics of automatic control systems

Proportional-integral link

Transmission function

  Logarithmic frequency characteristics of automatic control systems .

Imagine a link in the following form

  Logarithmic frequency characteristics of automatic control systems

Then LAFH and LFCHH have the form -

  Logarithmic frequency characteristics of automatic control systems ,

  Logarithmic frequency characteristics of automatic control systems .

  Logarithmic frequency characteristics of automatic control systems

Test questions and tasks

    1. How can be used to obtain the frequency characteristics of the system that the system can be represented in the form of parallel-connected typical dynamic links?

    2. How do LAFC and LPCh of dynamic links, the transfer functions of which are reciprocal?

    3. Which consecutively type dynamic links should be split into a really differentiating link to get its asymptotic LAFH and LFCH?

    4. Which consecutively connected dynamic links should be broken into an integrating link with delay in order to get its asymptotic LAFH and LFCH?

    5. Which serially-connected typical links should be split proportionally to the integrating link in order to get its asymptotic LAFC and LFCh?

    6. Link Transfer Function -

  Logarithmic frequency characteristics of automatic control systems ,

At what frequency LFCHH will matter   Logarithmic frequency characteristics of automatic control systems .

Answer :

At frequency   Logarithmic frequency characteristics of automatic control systems .

    1. Link Transfer Function -

  Logarithmic frequency characteristics of automatic control systems ,

As with frequency   Logarithmic frequency characteristics of automatic control systems Will the exact and asymptolic LAFCH of this link differ?

Answer :

Asymptotic LAFC will be less accurate on   Logarithmic frequency characteristics of automatic control systems .

    1. The transfer function of the object has the form -

  Logarithmic frequency characteristics of automatic control systems ,

Build an asymptotic LAFC object?

Answer :

  Logarithmic frequency characteristics of automatic control systems


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control