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ESE Electronics 2014 Paper 2: Official Paper

Option 3 : Phase margin decreases

CT 3: Building Materials

2962

10 Questions
20 Marks
12 Mins

__Concept:__

**Phase Margin**

It indicates the additional phase angle required to add to the system at ω_{gc }to drive the system to the verge of stability.

\(PM = 180^\circ + {\left. {\angle GH\left( {j\omega } \right)} \right|_{\omega = {\omega _{gc}}}}\)

**Gain Crossover frequency (ω _{gc})**

The frequency at which magnitude equal to 1 in linear or 0 in dB.

__Analysis:__

The transfer function of the proportional controller will be of the form

TF = K_{p}

Let the transfer function of the proportional controller be

\(G\left( s \right) = \frac{{{K_p}}}{{s + 1}}\)

Now introducing the delay element the new transfer function of the system is

\({G_1}\left( s \right) = \frac{{{K_p}{e^{ - s{\tau _d}}}}}{{s + 1}}\)

ϕ = 0° - tan^{-1}(ω)

PM = 180° - tan^{-1}(ω_{gc})

\({\phi _1} = - \omega {\tau _d}\frac{{180^\circ }}{\pi } - {\tan ^{ - 1}}\left( \omega \right)\)

\(P{M_1} = 180^\circ - {\omega _{gc}}{\tau _d}\frac{{180^\circ }}{\pi } - {\tan ^{ - 1}}\left( {{\omega _{gc}}} \right)\)

Phase angle **ϕ _{1} > ϕ**. So, the PM will decrease and then stability also decreases.

**NOTE: ** If the transportation delay/lag is introduced then the system stability will decrease