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Understanding Controller Gain and Phase Shift in Control Systems

This document delves into the fundamental concepts of controller gain in control systems. It outlines the relationship between output changes and input changes through gain, defines ultimate controller gain (Kcu), and discusses its relationship with phase shifts at critical delays. A block diagram illustrates the mathematical relationships governing amplitude ratio and error in system output. Through algebraic manipulation, we derive important principles that inform systems design and performance. The conclusion highlights key insights on the significance of the phase angle and amplitude ratio for system stability and control.

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Understanding Controller Gain and Phase Shift in Control Systems

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  1. Jigsaw Puzzle3rd Team Angela Purkey Jason Powell Mena Aziz Engr 328 4-10-2007

  2. Outline • Controller Gain • Ultimate Controller Gain • Block Diagram • Conclusion

  3. Controller Gain • Controller gain ,Kc , is the change in output divided by the change in input (units) % • You have control to set what the system Gain would be but you can’t manipulate the ultimate controller Gain, Kcu

  4. Ultimate Controller Gain • Kcu = 1/ AR • Kcu obtains a phase shift of (-180°) When the (amplitude ratio intersects with utltimate frequency) • At that instance Kcu reaches a phase angle shift of (-180°) • Amplitude ratio = 1/kcu for • Ex: (0.08 %/(lb/min)) = (1/12 %/(lb/min))

  5. Block Diagram • M(t) = input= A sin (ωt) , A= Amplitude • C(t)= (AR) * A sin(ωt+u), AR= Amplitude Ratio • e(t) = (r(t)- c(t)) • Error = setpoint – system output.

  6. Phase Angle = (-180°) • C(t)= (AR) * A sin (ωt-180°) • sin(wt-180°)= • =sin(ωt)*cos(-180) + cos(ωt)* sin(-180) • =sin(ωt)*(-1) + cos(ωt)*(0) • =-sin(ωt) • C(t)= -(AR) * A sin (ωt-180°) • Evaluated using the dougle angle formula using trig Identity.

  7. Block Diagram Algebra e(t) = (r(t)- c(t))=0-[-(AR)*Asin(ωt)] e(t)=(AR)* Asin(ωt) e(t) m(t) r(t) c(t) Kc System c(t)

  8. Block Diagram Algebra e(t)*Kc=m(t) m(t)=(AR)* Asin(ωt)*kc m(t)=Kc*(AR)*Asin(ωt) But earlier: m(t)=Asin(ωt) e(t) m(t) Kc

  9. Block Diagram Algebra Asin(ωt)= Kc*(AR)*Asin(ωt) Asin(ωt)/Asin(ωt)=(Kc*(AR)*Asin(ωt))/Asin(ωt) 1=Kc*(AR) AR=1/Kc m(t) c(t) System

  10. Conclusion • When the phase angle =-180 , AR=1/Kc • An advantage of knowing this is that we know where AR=1/Kc occurs.

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