IOE Control System Note BEL BCT BEX Steps for root locus

 IOE Control System Note | Plot of root locus & loci | Design lead lag compensator | RH criteria | Bode plot | | By IOE All Subject Notes | |

 Root Locus analysis

👉 Root locus is a graphical method in which roots of characteristic equation are plotted in plane for different values of gain. The locus of the roots of the  characteristic equation when gain is varied from zero to infinity is called root locus. 

Steps to sketch root locus in graph 

Write OLTF:

⇛ we know 

OLTF for open loop transfer function is :G(s)H(s)

 CLTF for close loop transfer function is :G(s)/(1+G(s)H(s))

Step1: Plot the open loop poles and zeros.:

⇛ compare denominator with zero to get Poles: S = ?....

⇛ compare numerator with zero to get Zeros: S= ?,...

Step2: Determine the root loci on real axis.

⇛ Any point on the real axis is a part of root locus if and only if number of poles and zeros to its right is odd.(positive axis on graph  (σ) तिर बाट हेरेर odd number of Poles वा zeros भेटेको points सम्म Dark गर्ने )

Therefore root locus lies between [ ? to ? ]  and [ ? to ? ]

Step3: Number of root loci 

⇛We let, 

N = number of separate root loci  

P = number of finite poles

Z = number of finite zeros

Rules:

N = P ; if  P > Z

N = Z ; if Z > P

N = P = Z ; if poles = zeros

We have to find number of root loci N = ?

Step4: Centroid of Asymptotes(σA)

⇛ The plot of intersection of asymptotes with real axis is called centroid of asymptotes and given by

 

(indicate with dot in graph paper because we are going to draw angles at this point)

 

 Step 5: Angle of asymptotes 

 ⇛ The branches of root locus tend to infinity along a set of straight line called asymptotes . These asymptotes making an angle with real axis.

 

Where K = 0,1,2,3. . . . (always include 0 in K values)

K को value  मा zero (0) सधै include गर्ने तर K को value always (P - Z) वटा लिने |

asymptotes angle with real axis is given by formula:

 (K को different value ले माथिको formula बाट phi को different value आएको हुन्छ त्यो सबै angles लाइ protector को मदत ले σA को point मा Draw गर्ने )

 Step6: Calculation of break away or break in point

⇛ The break away point corresponds to a point in the s-plane where multiple roots of
characteristic equation occurs. The break away or break in point can be determined from the roots of 

Write characteristic equation i.e. [  1 +G(s)H(s)  ] solve this and get the value of K in terms of S. Then substitute in above equation you will get a quadratic equation and after this use calculator to solve quadratic equation and get Different values of S = ? 

👉Dark गरेको axis  line  मा यी सबै points  हरु पर्छ कि पर्दैन conform  गर्ने 

• पर्ने points = either break in or break away

• नपर्ने points लाइ neglect  गर्ने

👉 Another method to check either points are break in | break away or not : put the values of k in the expression of K 

•  positive  र real  आयो भने = either break in or break away

• negative  वा complex  आयो भने = neglect  गर्ने

 Rules: एउटा S को value  २ वटा poles  वा zero  को बिचमा

 

 Mark break in or break away points on the graph 

Step7: Determine the point where the root cross the imaginary axis.

⇛Write ch. equation and write Routh array table

⇛ Form auxiliary equation and solve to get value of S = ? (imaginary point) plot these in graph

Final step: Draw a curve passing through break away points and parallel to root locus line and also 

 If complex poles are present :👾 we need to draw curve for it

·         Complex pole आयो भने angle of departure निकालनुपर्छा .

·         Complex zero आयो भने angle of arrival निकालनुपर्छा .

For angle of departure:

For angle of arrival:

• ·        Negative angle आयो भने Reference line भन्दा तल बनाउने
• ·        Positive angle आयो भने Reference line भन्दा माथि बनाउने

 

Upper angle = - Lower angle

 

For drawing curve at complex pole : 

Complex pole ले के भन्छ मलाई angle of departure संगै मलाई समातेर अनि angle of asymptotes लाई  parallel हुने गरि जाऊ। 

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