What will be the Gain Margin (GM) and the Phase Margin (PM) of a closed loop

TF T(s) = 500000/(s^{2} + 700s + 250000) ?

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DFCCIL Executive S&T 2018 Official Paper

Option 1 : ∞, 35

CT 1: Current Affairs (Government Policies and Schemes)

54560

10 Questions
10 Marks
10 Mins

__Concept:__

For a negative unity feedback system, if CLTF is given, the Open-loop transfer function is evaluated as:

OLTF \( = \frac{G}{{1 + GH - G}}\)

The phase margin is defined as:

\(PM=180^\circ - ϕ \)

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:__

\( TF= \frac{{500000}}{{{s^2} + 700s + 250000}}\)

Hence OLTF will be:

\(GH = \frac{{500000}}{{{s^2} + 700s + 250000 - 500000}}\)

\( = \frac{{500000}}{{{s^2} + 700s - 250000}}\)

Replacing s with jω:

\(G(jω )H(jω ) = \frac{{500000}}{{ - {ω ^2} + 700jω - 250000}}\)

For ωgc:

\(\left| {\frac{{500000}}{{ - {ω ^2} + 700jω - 250000}}} \right|=1\)

\(\frac{{500000}}{{\sqrt {\left( { - {ω ^2} - {{250000}}} \right)^2 + {{\left( {700ω } \right)}^2}} }}=1\)

\( \frac{{{{\left( {500000} \right)}^2}}}{{{{\left( { - {ω ^2} - 250000} \right)}^2} + {{\left( {700ω } \right)}^2}}}=1\)

\(\frac{{25 × {{10}^{10}}}}{{{ω ^4} + 500000{ω ^2} + 625 × {{10}^8} + 490000{ω ^2}}}=1\)

\(\frac{{25 × {{10}^{10}}}}{{{ω ^4} + 990000{ω ^2} + 625 × {{10}^8}}}=1\)

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) \(=\frac{{500000}}{{ - 412667 + j282324}}\)

ϕ = -180o + tan-1\([\frac{{282324}}{{412667}}]\) = ( -180o + 34.37) degrees

PM = \(180^\circ \) + ϕ

= 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 = ∞