Root Locus Examples - Michigan State University



Root Locus Examples

(graphed using MatLab)

|MatLab Program: |[pic] |

|num = [1] | |

|poles = [0;-1+j;-1-j;-3] | |

|den = poly(poles) | |

|rlocus(num,den) | |

|Notes: | |

|n=4 poles, m=0 zeros | |

|asymptotes at ±45,±135 | |

|asymptote cg @ | |

|(-3-1-1+0)/4 = -1.25 | |

|Angle of Departure, θd | |

|([pic]+ arctan(0.5) + 90 + 135) = 180 | |

|[pic] = -71.6 deg | |

|MatLab Program: |[pic] |

|num = [1] | |

|poles=[0;-1+4*j;-1-4*j;-2] | |

|den = poly(poles) | |

|rlocus(num,den) | |

|Notes: | |

|n = 4 poles, m = 0 zeros | |

|asymptotes at ±45,±135 | |

|asymptote cg @ | |

|(-2-1-1+0)/4 = -1.0 | |

|Angle of Departure, [pic] | |

|([pic]+ arctan(4) + (180-arctan(4)) + 90) = 180 | |

|[pic] = -90 deg | |

|MatLab Program: |[pic] |

|zeros = [-1+3*j;-1-3*j] | |

|num = poly(zeros) | |

|poles = [0;-1+2*j;-1-2*j;-2] | |

|den = poly(poles) | |

|rlocus(num,den) | |

|Notes: | |

|n = 4 poles, m = 2 zeros | |

|asymptotes at ±90 degrees | |

|asymptote cg @ | |

|[(-2-1-1+0)-(-1-1)]/2 = -1.0 | |

|Angle of Departure, [pic] | |

|[([pic]+ arctan(2)+(180-arctan(2))+ 90) - | |

|(-90+90) ] = 180 => [pic] = -90 deg | |

|Angle of Arrival, [pic] | |

|[(90 + arctan(3)+(180-arctan(3))+ 90) - | |

|(-([pic]+90) ] = 180 => [pic] = 90 deg | |

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