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I've recently started to study the root locus concept and I have got this question about the branches. Please refer to this problem here in the link:

http://lpsa.swarthmore.edu/Root_Locus/Example2/Example2.html

In that example, it can be seen that the branches emerging from the open loop poles s=0 and s=-2 when k=0 move along the real axis towards each other as k increases up to the breakaway point. After this, when k is further increased, one of the branches moves downwards and the other moves upwards. It is seen in this case that the branch from the poles s=-2 moves downward and the branch from s=0 moves upward.

Now my question is what is the factor that decides the upward/downward of the branches. Please provide a logical proof.

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  • \$\begingroup\$ 'one of the branches moves downwards and the other moves upwards' - this statement implies that the original branches retain some sort of identity when they merge and then spring forth from the real axis. They don't. They are just roots of an equation. And, in any case, why would it matter? \$\endgroup\$
    – Chu
    Mar 6 '16 at 15:26
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There is no such factor.

The root locus tracks the closed loop pole locations from the open loop poles. As the gain K is increased little by little, it is easy to see how they move towards the breakaway point. Another way to put this is, each time the gain is incremented, the poles move a little and it could be assumed that the new pole location that is very close to the old pole location must be related.

The problem is that when the poles reach the break out point they are real and repeated. When both poles are the same, mathematically they are indistinguishable. Because they are the same, there is no way to know which one will go up and which one will go down because you don't know which pole is which.

Think of it like putting two identical marbles into a black box. When you take them back out, how do you know which one is which?

The computer program arbitrarily chooses which branch goes down and which branch goes up to make the plot look nice. There is no particular logic to it.

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