Root Locus Flashcards
How do you construct a root locus diagram?
- Find number of poles
- Find number of zeros
- Find number of branches (asymptotes)
- Find starting point
- Find angles
- Find breakaway/breakin points
- Find where it cross imaginary axis
- Construct graph
How do you find number of poles?
Number of (s) at bottom of transfer function
How do you find number of zeros?
Number of (s) at top of transfer function
How do you find number of branches ( asymptotes) ?
Number of poles - number of zeros
How do you find starting point?
Addition of poles - Addition of zeros / number of branches ( asymptotes)
How do you find the angle?
Formula: (2k+1)180/number of pole - number of zero
Number of branches determines numbers of angles
Always sub from 0,1,2,3 …. etc into k
How do you find breakaway points?
- Convert to closed loop transfer function = G/1 + GH
- Equate denominator to zero by expanding brackets
- Differentiate equation and equal to zero
- Use second degree equation
- Find real point and ignore imaginary
(Breakaway point only necessary when two poles or zeros meet)
How do find where it crosses imaginary?
- Convert to closed loop transfer function = G/1+GH
- Use denominator and open the bracket
- Sub s = JW ( j^2 = -1) (j^3 = -j)
- Separate real from imaginary ( Real there’s no j) (imaginary contains j)
- Equate both real and imaginary equations to zero
- Simplify both equations to get the values for K and Jw
- Use the value of jw
How to draw root locus diagram?
- Label poles with X
- Label zeros with O
- X axis is real and Y axis is imaginary
- Draw asymptotes (branches) with doted line in the directions of the angles
- Draw root locus line with solid line from start point to imaginary axis and it will follow parallel with dotted line
How and when do you find angle of departure?
Angle of departure only necessary when there’s complex poles
Use second degree equation for the polynomial In transfer function to get the complex poles
- Formula = 180 - (Angle 1 + Angle 2) + Angle 3 (angle of zero)
- Connect poles and zeros to create triangles
- Either measure angles with protractor or use SohCahToa
- Substitute angles to equation
How do you find complex pole or zero?
Use second degree equation
On calculator go on menu,
then press A,
then select polynomial,
then select the number that corresponds with first power,
then fill the numbers,
then final answer will plus/minus complex number
How do find where it crosses imaginary? Alternative method
- Covert to CL transfer function
- Make a column of all S’s and powers in descending order
- Write all the coefficients
- Cross multiply the first two rows
- Simplify answer and equal to zero to get value of K
- Make second row an equation and Sub value of K into equation
- Simplify answer and equal to zero to get value of S
How do you calculate K (Gain) when S is given?
- Flip original transfer function
- Sub value of S (breakaway point) into the function
- Calculate the final answer
How to check for value of K for unstable function?
When calculating where it crosses imaginary the value of K is used. This can be done using alternative method.
If K is greater than number calculated then it’s unstable
If K is equal to the number calculated then it’s marginally stable
If K is smaller than number calculated then it’s stable
