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Root Locus Example

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Engineering Funda

The document discusses the steps to plot the root locus of a control system described by the transfer function G(s)=k(s+2)/(s^2+2s+10). It provides the locations of the poles and zeros, and calculates the number of asymptotes, centroid, and angle of asymptotes. It then determines the angle of departure of the root locus and the breakaway points by setting the derivative of the closed loop characteristic equation equal to zero. No calculation of the intersection points of the root locus with the imaginary axis is needed based on the diagram.

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By Prof. Hitesh Dholakiya Root Locus Example Control Engineering E n g i n e e r i n g F u n d a Engineering Funda Android App Control System YT Playlist
Root Locus plot Asymptotes Angle of Departure Break away point Outlines of Session Intersection to imaginary axis E n g i n e e r i n g F u n d a Engineering Funda Android App Control System YT Playlist
Root Locus Plot ? Plot root locus of system given by ? ? = ?(?+?) ??+??+?? ? Step 1 : Find position of poles, zeros and loci ? Position of zeros ? Position of poles ?1 = ?2 ?1 = ?1 + ?3, ?2 = ?1 ? ?3 Imag j4 j3 j2 j1 0 -j1 -j2 -j3 -j4 Real -6 -5 -4 -3 -2 -1 E n g i n e e r i n g F u n d a Engineering Funda Android App Control System YT Playlist
Asymptotes ? Find Number of Asymptotes, Centroid of Asymptotes and Angle of Asymptotes ? Number of Asymptotes ? Centroid of Asymptotes ? Angle of Asymptotes Imag j4 j3 j2 j1 0 -j1 -j2 -j3 -j4 Real -6 -5 -4 -3 -2 -1 Number of Asymptotes = P C Z = 2 C 1 = 1 ? ? = ?(? + ?) ?? + ?? + ?? ?1 = ?2 ?1 = ?1 + ?3, ?2 = ?1 ? ?3 ?? = σ ???? ????? ?? ????? ? σ ???? ????? ?? ????? ? ? ? ?? = ?1 ? 1 ? (2) 2 ? 1 = 0 ? = (2? + 1) ? ? ? 〜 1800 = 1800 ?? = 0 ? =180 E n g i n e e r i n g F u n d a
Angle of Departure ? Angle of Departure Imag j4 j3 j2 j1 0 -j1 -j2 -j3 -j4 Real -6 -5 -4 -3 -2 -1 ?? = 0 ? =180 ?? = 180 ? ? ????? ?? ???? ? ? ????? ?? ????? ?? = 180 ? ??1 ? ??1 = 180 ? 90 ? 71 = 161 ???= 90 ???= 71 E n g i n e e r i n g F u n d a
Break away point ? Characteristics equation ? Find root at ?? ?? = ? Imag j4 j3 j2 j1 0 -j1 -j2 -j3 -j4 Real -6 -5 -4 -3 -2 -1 ?? = 0 ? =180 ???= 90 ???= 71 ÷ 1 + G S H S = 0 ? ? = ?(? + ?) ?? + ?? + ?? ?1 = ?2 ?1 = ?1 + ?3, ?2 = ?1 ? ?3 ÷ 1 + ?(? + ?) ?? + ?? + ?? = 0 ÷ ? = ? (?2 + 2? + 10) (? + 2) ÷ ?? ?? = 0 = ? ? + 2 2? + 2 ? (?2 + 2? + 10) (? + 2)2 ÷ 0 = ?2 + 4? ? 6 ÷ ? = 1.16 ÷ ? = ?5.16 E n g i n e e r i n g F u n d a
Inter section to imaginary axis ? As per Diagram You don¨t need to plot Intersection to imaginary axis E n g i n e e r i n g F u n d a

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Root Locus Example

  • 1. By Prof. Hitesh Dholakiya Root Locus Example Control Engineering E n g i n e e r i n g F u n d a Engineering Funda Android App Control System YT Playlist
  • 2. Root Locus plot Asymptotes Angle of Departure Break away point Outlines of Session Intersection to imaginary axis E n g i n e e r i n g F u n d a Engineering Funda Android App Control System YT Playlist
  • 3. Root Locus Plot ? Plot root locus of system given by ? ? = ?(?+?) ??+??+?? ? Step 1 : Find position of poles, zeros and loci ? Position of zeros ? Position of poles ?1 = ?2 ?1 = ?1 + ?3, ?2 = ?1 ? ?3 Imag j4 j3 j2 j1 0 -j1 -j2 -j3 -j4 Real -6 -5 -4 -3 -2 -1 E n g i n e e r i n g F u n d a Engineering Funda Android App Control System YT Playlist
  • 4. Asymptotes ? Find Number of Asymptotes, Centroid of Asymptotes and Angle of Asymptotes ? Number of Asymptotes ? Centroid of Asymptotes ? Angle of Asymptotes Imag j4 j3 j2 j1 0 -j1 -j2 -j3 -j4 Real -6 -5 -4 -3 -2 -1 Number of Asymptotes = P C Z = 2 C 1 = 1 ? ? = ?(? + ?) ?? + ?? + ?? ?1 = ?2 ?1 = ?1 + ?3, ?2 = ?1 ? ?3 ?? = σ ???? ????? ?? ????? ? σ ???? ????? ?? ????? ? ? ? ?? = ?1 ? 1 ? (2) 2 ? 1 = 0 ? = (2? + 1) ? ? ? 〜 1800 = 1800 ?? = 0 ? =180 E n g i n e e r i n g F u n d a
  • 5. Angle of Departure ? Angle of Departure Imag j4 j3 j2 j1 0 -j1 -j2 -j3 -j4 Real -6 -5 -4 -3 -2 -1 ?? = 0 ? =180 ?? = 180 ? ? ????? ?? ???? ? ? ????? ?? ????? ?? = 180 ? ??1 ? ??1 = 180 ? 90 ? 71 = 161 ???= 90 ???= 71 E n g i n e e r i n g F u n d a
  • 6. Break away point ? Characteristics equation ? Find root at ?? ?? = ? Imag j4 j3 j2 j1 0 -j1 -j2 -j3 -j4 Real -6 -5 -4 -3 -2 -1 ?? = 0 ? =180 ???= 90 ???= 71 ÷ 1 + G S H S = 0 ? ? = ?(? + ?) ?? + ?? + ?? ?1 = ?2 ?1 = ?1 + ?3, ?2 = ?1 ? ?3 ÷ 1 + ?(? + ?) ?? + ?? + ?? = 0 ÷ ? = ? (?2 + 2? + 10) (? + 2) ÷ ?? ?? = 0 = ? ? + 2 2? + 2 ? (?2 + 2? + 10) (? + 2)2 ÷ 0 = ?2 + 4? ? 6 ÷ ? = 1.16 ÷ ? = ?5.16 E n g i n e e r i n g F u n d a
  • 7. Inter section to imaginary axis ? As per Diagram You don¨t need to plot Intersection to imaginary axis E n g i n e e r i n g F u n d a