Tehnîkaya Locusê Rêgîn a Serastina | Wêneya Locusê Rêgîn

Taqnîka locus rootyê di cîhanek kontrolan de hûn hatin paqij kirin ji sînor 1948 derbasdar Evans. Her sistem fîzikî bi funksiyon transfeirîn wek forma bêtirin nîşan didin

Hûn dikarin poles û zeros bigihin ji G(s). Cîh û zeros ku ne ku bin navkirin di navbera stabîlî, stabîlîya berhevkirin, pêşkeftina kevankir û analîza çewtiyê. Dema sistem bi xebitîya bikar bîne inductance û capacitance seretî yên diger li ser sistema têne, li vir rojek pole û zeros dide. Di taqnîka locus rootyê di cîhanek kontrolan de hûn ênin da ku piştgiriya cîhên, hilberên wê û malperên yekbûyî. Malperên wan bi kar werin da ku şopandinê ya performansa sistema bigihin.
Hûn bi xwe bibin da ku dema ku min taqnîka locus rootyê paqij bikam, pir tiştên çend an jînên lêgerînê ya vê taqnîka ji sînora din bistînin. Çend an jînên taqnîka locus rootyê li jêr nivîsraw.

Anjamanên Taqnîka Locus Root

1. Taqnîka locus rootyê di cîhanek kontrolan de herî ezbere ye ji sînora din.

2. Bi alaka locus rootyê hûn dikarin performansa hêla sistemê bigihin.

3. Locus rootyê rêzikariya bikarhatin da ku parametrên nîşan bidin.

Li navbera taqnîka locus rootyê van terma li ser her duvên vê gotara bikar bînin.

1. Equation Characteristic Li Ser Taqnîka Locus Rootyê : 1 + G(s)H(s) = 0 namek equation characteristic e. Ji bo hilberên equation characteristic û hilberên dk/ds equals to zero, hûn dikarin break away points bigihin.

2. Break away Points : Di dema ku du root loci ji pole destpêk û li ser virgulek din biceribin, hûn dikarin çapkerin bigihin da ku li vir çapkerin wê li ser virgulên din biceribin. An jî breakaway points di dema ku roots of the characteristic equation 1 + G(s)H(s) = 0 occur. Nîrvana K herî mezin e di navbera breakaway points. Breakaway points dibane real, imaginary an complex.

3. Break in Point : Şertê break in be di plotê de bike wek hevdî sa:Root locus divê be di navbera du zeros adjacent de li ser x-axis.

4. Navenda Gravity : Ji seddeye centroid û bi nîşan didin da ku jî navendê ku asymptotes destpêk. Mathematik, bi nîrvana poles û zeros di transfer function de hêviyê nîşan didin. Navenda gravity herî real e û bi sigmaA nîşan didin.
   
   Ji ber ku, N ji nîrvana poles û M ji nîrvana zeros.

5. Asymptotes of Root Loci : Asymptote ji navenda gravity an centroid destpêk û li ser infinity at definite some angle. Asymptotes direction dike root locus ji dema ku wê li ser break away points biceribin.

6. Angle of Asymptotes : Asymptotes ji angle ên ji x-axis û ji formula wek hevdîsa bike:
   
   Ji ber ku, p = 0, 1, 2 ……. (N-M-1)
   N ji nîrvana poles total
   M ji nîrvana zeros total.

7. Angle of Arrival or Departure : Hûn dikarin angle of departure bike da ku ji system complex poles hene. Angle of departure bi nîrvana 180-{(sum of angles to a complex pole from the other poles)-(sum of angle to a complex pole from the zeros)} bike.

8. Intersection of Root Locus with the Imaginary Axis : Ji bo nîşandan navendê intersection root locus bi imaginary axis, hûn dikarin Routh Hurwitz criterion bike. Yekam, hûn dikarin auxiliary equation bike û value of K corresponding dike value of the point of intersection bike.

9. Gain Margin : Hûn define gain margin bi kêmîn da ku design value of the gain factor dikare bi multiply bike pêro system unstable bike. Mathematically it is given by the formula
   

10. Phase Margin : Phase margin bi formula wek hevdîsa bike:
    

11. Symmetry of Root Locus : Root locus symmetric about the x axis or the real axis.

How to determine the value of K at any point on the root loci? Now there are two ways of determining the value of K, each way is described below.

1. Magnitude Criteria : At any points on the root locus we can apply magnitude criteria as,
   
   Using this formula we can calculate the value of K at any desired point.

2. Using Root Locus Plot : The value of K at any s on the root locus is given by
   

Root Locus Plot

Ev taqnîka locus rootyê di cîhanek kontrolan de hîn nîşan didin da ku stabîlîya sistema taybetandîn bide. Ji bo nîşandan stabîlîya sistema bi taqnîka locus rootyê hûn dikarin range of values of K bike da ku performansa hêla sistema taybetandîn bide û operasyon stable bike.
Ji bo nîşandan root locus hûn dikarin bêtirin results bike. Results li jêr nivîsraw:

1. Region where root locus exists : After plotting all the poles and zeros on the plane, we can easily find out the region of existence of the root locus by using one simple rule which is written below,
   Only that segment will be considered in making root locus if the total number of poles and zeros at the right hand side of the segment is odd.

2. How to calculate the number of separate root loci ? : A number of separate root loci are equal to the total number of roots if number of roots are greater than the number of poles otherwise number of separate root loci is equal to the total number of poles if number of roots are greater than the number of zeros.

Procedure to Plot Root Locus

Bi hemî vê points hûn dikarin root locus plot bikerin ji bo her cîha sistema. Ji bo nîşandan procedure of making a root locus.

1. Find out all the roots and poles from the open loop transfer function and then plot them on the complex plane.

2. All the root loci starts from the poles where k = 0 and terminates at the zeros where K tends to infinity. The number of branches terminating at infinity equals to the difference between the number of poles & number of zeros of G(s)H(s).

3. Find the region of existence of the root loci from the method described above after finding the values of M and N.

4. Calculate break away points and break in points if any.

5. Plot the asymptotes and centroid point on the complex plane for the root loci by calculating the slope of the asymptotes.

6. Now calculate angle of departure and the intersection of root loci with imaginary axis.

7. Now determine the value of K by using any one method that I have described above.

   By following above procedure you can easily draw the root locus plot for any open loop transfer function.

8. Calculate the gain margin.

9. Calculate the phase margin.

10. You can easily comment on the stability of the system by using Routh Array.

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