Constant phase traces on a Nyquist plot: how?

Question

I know how to trace closed-loop transfer function constant magnitude curves on a Nyquist plot.

I have not yet been able to find a geometric construct to trace closed-loop transfer function constant phase curves. I know these curves are perpendicular to the constant magnitude curves (why?).

Given M a point on the open-loop transfer function, O the origin, and A the point (-1, j0), arg(OM) - arg(AM) = constant should do it. Alas, it leads me nowhere.

Could you give me some tips?

Thank you very much in advance.

Mark · Accepted Answer

I don't have this at home, but I do have the equations for M and N circles for inverse Nyquist diagrams M being circles of constant magnitude or gain, and N being circles of constant phase:

u/y = X + 1 + jY M = | u / y |

1/M = SQRT [ (X+1)^2 + Y^2 ] 1/M^2 = (X+1)^2 + Y^2 Circle with centre at X=-1, Y=0, and radius r = 1/M

Lines of constant phase for the INVERSE Nyquist diagram are given by: alpha = /_ u / y = -tan^-1 ( Y / (X+1))

Hope this helps. I'll see if I can find the normal (not inverse) stuff tomorrow.

I used to have a classical control theory by Atkinson with this in. Loaned it to a colleague.

Fred Stevens · Answer

Yes, how many excellent books have we lost by lending them to colleagues!?

Constant phase traces on a Nyquist plot: how?

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