Root Locus

• Root locus


  The locus of the roots of the characteristic equation by varying system gain K from zero to infinity
• Characteristic equation of closed loop control system

  1. 1 + G(s)H(s) = 0

  2. G(s)H(s) = K N(s)/D(s)

  3. D(s) + KN(s) = 0
• K = 0


  Closed loop poles are equal to open loop poles
• K = ∞


  Closed loop poles are equal to open loop zeros
• Root locus branches start at open loop poles and end at open loop zeros
• Angle condition


  The point at which the angle of the open loop transfer function is an odd multiple of 180°
• Magnitude condition


  The point (which satisfied the angle condition) at which the magnitude of the open loop transfer function is one
• ∠G(s)H(s) = tan^-1(0/-1) = (2n + 1)π
• |G(s)H(s)| = 1
• Root locus


  A graphical representation in s-domain that is symmetrical about the real axis

  
• Open loop poles and zeros exist in the s-domain having the values either as real or as complex conjugate pairs
• Constructing the root locus

  1. Locate the open loop poles and zeros in the 's' plane

  2. Find the number of root locus branches

  3. Identify and draw the real axis root locus branches

  4. Find the centroid and the angle of asymptotes

  5. Find the intersection points of root locus branches with an imaginary axis

  6. Find Break-away and Break-in points

  7. Find the angle of departure and the angle of arrival

  
• Steps to find break-away and break-in points


  1. Write P(s) in terms of K from the characteristic equation

  2. Differentiate P(s) with respect to s and make it equal to zero

  3. Substitute these values of s in the above equation

  4. The values of K for which the P(s) value is positive are the break points
• Steps to find angle of departure and angle of arrival

  1. Calculate the angle of departure and the angle of arrival at complex conjugate open loop poles and complex conjugate open loop zeros respectively

  2. The formula for the angle of departure is φd = -φ

  3. The formula for the angle of arrival is φa = +φ
• Root locus


  • Number of root locus branches is equal to the number of poles of the open loop transfer function
  • Two root locus branches intersect the imaginary axis
  • One break-away point on the real axis root locus branch between the poles

  
• From root locus diagrams, we can know the range of K values for different types of damping

  
• Adding open loop poles

  Some root locus branches move towards right half of s-plane, decreasing damping ratio, increasing damped frequency, decreasing delay time, rise time and peak time, but affecting system stability

  
• Adding open loop zeros

  Some root locus branches move towards left half of s-plane, increasing damping ratio, decreasing damped frequency, increasing delay time, rise time and peak time, improving system stability

  
• Based on requirements, open loop poles or zeros can be added to the transfer function