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The roots of the closed-loop characteristic equation define the system characteristic responses

o Introduced by W. R. Evans in 1948

o Graphical method, in which movement of poles in the s-plane is sketched when some parameter is varied The path taken by the roots of the characteristic equation when open loop gain K is varied from 0 to ∞ are called root loci

o Direct Root Locus = 0 < k < ∞

o Inverse Root Locus = - ∞ < k < 0

o The roots of the closed-loop characteristic equation define the system characteristic responses

o Their location in the complex s-plane lead to prediction of the characteristics of the time domain responses in terms of:

o damping ratio ζ,

o natural frequency, wn

o damping constant ζ, first-order modes

o Consider how these roots change as the loop gain is varied from 0 to∞

o Symmetrical about real axis

o RL branch starts from OL poles and terminates at OL zeroes

o No. of RL branches = No. of poles of OLTF

o Centroid is common intersection point of all the asymptotes on the real axis

o Asymptotes are straight lines which are parallel to RL going to ∞ and meet the RL at ∞

o No. of asymptotes = No. of branches going to ∞

o At Break Away point , the RL breaks from real axis to enter into the complex plane

o At BI point, the RL enters the real axis from the complex plane

o Locate the OL poles & zeros in the plot Find the branches on the real axis

o Find angle of asymptotes & centroid

o Φa= ±180º(2q+1) / (n-m)

o ζa = (Σpoles - Σzeroes) / (n-m) Find BA and BI points

o Find Angle Of departure (AOD) and Angle Of Arrival (AOA)

o AOD = 180º- (sum of angles of vectors to the complex pole from all other poles) + (Sum of angles of vectors to the complex pole from all zero)

o AOA = 180º- (sum of angles of vectors to the complex zero from all other zeros) + (sum of angles of vectors to the complex zero from poles)

o Find the point of intersection of RL with the imaginary axis.

**Step 1:** Write the characteristic equation as

1+ *F*(*s*)= 0

**Step 2: **Rewrite preceding equation into the form of poles and zeros as follows

**Step 3:**

Locate the poles and zeros with specific symbols, the root locus begins at the open-loop poles and ends at the open loop zeros as K increases from 0 to infinity

If open-loop system has n-m zeros at infinity, there will be n-m branches of the root locus approaching the n-m zeros at infinity

**Step 4:**

The root locus on the real axis lies in a section of the real axis to the left of an odd number of real poles and zeros

**Step 5:**

The number of separate loci is equal to the number of open-loop poles

**Step 6:**

The root loci must be continuous and symmetrical with respect to the horizontal real axis

**Step 7:**

The loci proceed to zeros at infinity along asymptotes centered at centroid and with angles

**Step 8:**

The actual point at which the root locus crosses the imaginary axis is readily evaluated by using Routh‗s criterion

**Step 9:**

Determine the breakaway point d (usually on the real axis)

**Step 10:**

Plot the root locus that satisfy the phase criterion

**Step 11:**

Determine the parameter value *K1* at a specific root using the magnitude criterion

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