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Root locus plot of LC Tank Oscillator

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promach

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1) Why "As gain (GmRp) increases, closed loop poles move into right half S-plane" ? http://lpsa.swarthmore.edu/Root_Locus/Example1/Example1.html does not have such right half plane route. Why can't the root locus travel along the vertical imaginary axis ?

2) From https://www.ee.usyd.edu.au/tutorials_online/matlab/rlocus/rlocus.html , why "The root locus of an (open-loop) transfer function is a plot of the locations (locus) of all possible closed loop poles" ? Could anyone give further maths explanation on this statement ?

3) For https://en.wikipedia.org/wiki/Closed-loop_pole , why "For negative feedback systems, the closed-loop poles move along the root-locus from the open-loop poles to the open-loop zeroes as the gain is increased" ?

Screenshot from 2017-12-31 20-20-00.png

Screenshot from 2017-12-31 20-20-12.png

Screenshots extracted from http://inf-server.inf.uth.gr/courses/CE433/tutorials/VCO.pdf#page=11
 

CataM

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Why can't the root locus travel along the vertical imaginary axis ?
You would know if you read the 10 rules of drawing a Root Locus. Actually, by the time you get to the 8th rule, you already have answered your question.

The root locus of an (open-loop) transfer function is a plot of the locations (locus) of all possible closed loop poles" ? Could anyone give further maths explanation on this statement ?
That is a common question. In all major control systems books, they call the Open Loop transfer function to what is actually called in electronics the Loop Gain. Loop Gain = Open Loop gain * Feedback factor.

Math is simple: Root locus simply plots the roots of A(s)*H(s)=-1 or equivalently, 1+A(s)*H(s)=0, which are the closed loop poles i.e. the zeros of the characteristic polynomial.
I repeat again. Open Loop transfer function is NOT A(s), but it is what is called in electronics the loop gain i.e. A(s)*H(s).

A simple conclusion one could draw from this is that the Root Locus can be applied ONLY FOR NEGATIVE FEEDBACK SYSTEMS.

"For negative feedback systems, the closed-loop poles move along the root-locus from the open-loop poles to the open-loop zeroes as the gain is increased" ?
Poles tend to repel the branches of the Root Locus and zeros tend to attract them... It is a conclusion from the rules.
 

promach

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Poles tend to repel the branches of the Root Locus and zeros tend to attract them
@CataM

There are a lot of root locus rules online. However, there is not much on maths proof/explanation about the quote statement above.

Could anyone elaborate in further detail ?
 
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CataM

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why is P-Z = number of asymptotes ?
Because poles that do not have a zero to pair with, they find their zero in the infinite, so a asymptote leads them there.
 

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@CataM

What about "Poles tend to repel the branches of the Root Locus and zeros tend to attract them " ?
 

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Why as gain increases, closed-loop poles move into right half plane ?
 

CataM

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Why as gain increases, closed-loop poles move into right half plane ?
That is what you find after drawing the Root Locus. If we would have known before drawing this, the Root Locus would have been useless.
 

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