Question
Download Solution PDFWhat will be the Gain Margin (GM) and the Phase Margin (PM) of a closed loop
TF T(s) = 500000/(s2 + 700s + 250000) ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
For a negative unity feedback system, if CLTF is given, the Open-loop transfer function is evaluated as:
OLTF
The phase margin is defined as:
where ϕ is the phase of the OLTF at ωgc
ωgc (Gain crossover frequency) = frequency at which system gain is unity.
If the polar plot is not crossing the negative real axis, then GM = ∞
For a negative unity feedback system, the phase at ω = 0 is 0 degrees.
And phase at ω = ∞ is:
(p-z) × 90°
p = number of open-loop poles and z = no. of open-loop zeros.
Analysis:
Hence OLTF will be:
Replacing s with jω:
For ωgc:
25× 1010 = ω4 + 990000ω2 + 625× 108
On solving the above, we get:
gc = 403.32 rad/sec
Now we will calculate the phase of the system at ωgc
GH(jω)
ϕ = -180o + tan-1
PM =
= 180o + (- 180o + 34.37o) = 34.37o
If we draw the polar plot of the above system, then we get to see that its polar plot will not cut the negative real axis, in that case:
GM = ∞
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