Explanation Given: P ( s ) = 137.2 × 10 − 6 s 2 + 0.0224 s + 196 × 10 − 6 e − 39 s And unity feedback system i.e. for unity feedback system H ( s ) = 1 Concept Used: Since we know that closed loop transfer function is as below: T ( s ) = G ( s ) 1 + G ( s ) H ( s ) use this to obtain characteristic equation i.e. 1 + G ( s ) H ( s ) , which will be the denominator of closed loop transfer function. Calculation: G ( s ) = K 137.2 × 10 − 6 s 2 + 0.0224 s + 196 × 10 − 6 ( 1 − 39 2 s ) ( 1 + 39 2 s ) = K 137.2 × 10 − 6 s 2 + 0.0224 s + 196 × 10 − 6 [ 2 − 39 s 2 + 39 s ] For unity feedback system, solve for 1 + G ( s ) H ( s ) . 1 + G ( s ) H ( s ) = 1 + K 137.2 × 10 − 6 s 2 + 0.0224 s + 196 × 10 − 6 [ 2 − 39 s 2 + 39 s ] × 1 = 0 = ( s 2 + 0.0224 s + 196 × 10 − 6 ) ( 2 + 39 s ) + K 137.2 × 10 − 6 ( 2 − 39 s ) = 0 = 2 s 2 + 39 s 3 + 0.0448 s + 0.8736 s 2 + 392 × 10 − 6 + 7488 × 10 − 6 s + 2744 K × 10 − 6 − 5350.8 K s × 10 − 6 = 39 s 3 + 2.8736 s 2 + 0.007488 s + 0.0448 s − 0.005358 K s + 392 × 10 − 6 + 2774 K × 10 − 6 = 39 s 3 + 2.8736 s 2 + ( 0.052288 − 0.005358 K ) s + ( 0.000392 + 0

7th Edition

ISBN: 9781118170519

Author: Norman S. Nise

Publisher: WILEY

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Chapter 6, Problem 70P

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The range ofK for the system defined by given specification will be stable: -

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