Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
74 views
in Electronics by (115k points)
closed by

A plant transfer function is given as,

\({\rm{G}}\left( {\rm{s}} \right) = \left( {{{\rm{K}}_{\rm{p}}} + \frac{{{{\rm{K}}_{\rm{I}}}}}{{\rm{s}}}} \right)\frac{1}{{{\rm{s}}\left( {{\rm{s}} + 2} \right)}}\)

When the plant operates in a unity feedback configuration the condition for the stability of the closed-loop system is
1. \({{\rm{K}}_{\rm{P}}} > \frac{{{{\rm{K}}_{\rm{I}}}}}{2} > 0\)
2. \(2{{\rm{K}}_{\rm{I}}}{\rm{\;}} > {\rm{\;}}{{\rm{K}}_{\rm{P}}}{\rm{\;}} > {\rm{\;}}0\)
3. \(2{{\rm{K}}_{\rm{I}}}{\rm{\;}} < {\rm{\;}}{{\rm{K}}_{\rm{P}}}\)
4. \(2{{\rm{K}}_{\rm{I}}}{\rm{\;}} > {\rm{\;}}{{\rm{K}}_{\rm{P}}}\)

1 Answer

0 votes
by (152k points)
selected by
 
Best answer
Correct Answer - Option 1 : \({{\rm{K}}_{\rm{P}}} > \frac{{{{\rm{K}}_{\rm{I}}}}}{2} > 0\)

Concept:

According to RH stability criteria there are two necessary conditions:

First is there should not be any sign changes in the first column of the tabulation and Second Is there should not be any missing coefficients in the equation 1 + G(s) H(s).

The RH criteria can be solved as follows:

\({s^3} - - - {\rm{\;\;\;}}{{\rm{a}}_0} \searrow {\rm{\;\;\;\;\;}} \swarrow {{\rm{a}}_2}{\rm{\;\;\;\;\;\;}}{{\rm{a}}_4}\)

\({s^2} - - - {\rm{\;\;\;}}{{\rm{a}}_1}{\rm{\;}} \nearrow {\rm{\;\;\;}} \nwarrow {{\rm{a}}_3}{\rm{\;\;\;\;\;\;\;}}0\)

\({s^1} - - - {\rm{\;\;\;}}\frac{{{{\rm{a}}_1}{{\rm{a}}_2} - {{\rm{a}}_0}{{\rm{a}}_3}}}{{{{\rm{a}}_1}}}{\rm{\;}} = {\rm{A\;\;\;\;\;\;\;\;\;\;}}0{\rm{\;}}\)

\({s^0} - - - {\rm{\;\;\;}}\frac{{{\rm{A}}{{\rm{a}}_3}}}{{\rm{A}}} = {{\rm{a}}_3}\)

Calculation:

Given the open-loop transfer function

\({\rm{G}}\left( {\rm{s}} \right) = \frac{{\left( {{\rm{s}}{{\rm{K}}_{\rm{P}}} + {{\rm{K}}_{\rm{I}}}} \right)}}{{{{\rm{s}}^2}{\rm{\;}}\left( {{\rm{s}} + 2} \right)}}\)

The characteristic equation of the above transfer function will be

⇒ q(s) = s3 + 2s2 + sKP + KI = 0

From RH-criteria

\(\begin{array}{*{20}{c}} {{s^3}}\\ {{s^2}}\\ {{s^1}}\\ {{s^0}} \end{array}\left| {\begin{array}{*{20}{c}} 1&2\\ K_P&{K_I}\\ {\frac{2K_P-K_I}{K_P}}&0\\ K_I&{} \end{array}} \right.\)

From the Routh criterion, the condition for stability is

⇒ 2KP > KI and KI > 0

∴ \({{\rm{K}}_{\rm{P}}} > \frac{{{{\rm{K}}_{\rm{I}}}}}{2} > 0\) 

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

Categories

...