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18. Find the sensitivity of T[s] with respect to a, when s² + (1 + a)s +2 s3 + 2as² + 4s + 10a asD[s]-a(2s²+10)N[s] T[s] = N[s] D[s]

18. Find the sensitivity of T[s] with respect to a, when s² + (1 + a)s +2 s3 + 2as² + 4s + 10a asD[s]-a(2s²+10)N[s] T[s] = N[s] D[s]

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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Title: Sensitivity Analysis of Transfer Functions --- **Problem 18: Sensitivity of T(s) with respect to parameter a** We are tasked with finding the sensitivity of the transfer function T(s) with respect to the parameter a, given by: \[ T(s) = \frac{N(s)}{D(s)} = \frac{s^2 + (1+a)s + 2}{s^3 + 2as^2 + 4s + 10a} \] **Options for the Sensitivity Function:** (a) \(\frac{asD[s] - a(2s^2 + 10)N[s]}{N[s]D[s]}\) (b) \(\frac{sD[s] - (2s^2 + 10)N[s]}{N[s]D[s]}\) (c) \(\frac{aD[s] - aN[s]}{N[s]D[s]}\) (d) \(\frac{sD[s] - sN[s]}{N[s]D[s]}\) --- **Explanation:** When analyzing the sensitivity of a transfer function, we determine how the function \( T(s) \) reacts to changes in system parameters—in this case, the parameter \( a \). The correct expression will accurately represent how variations in \( a \) influence \( T(s) \), using the derivative of both numerator and denominator with respect to \( a \). This is a typical task involved in control systems engineering, aimed at understanding system stability and performance changes, forming a critical part of robust control design.
Transcribed Image Text:Title: Sensitivity Analysis of Transfer Functions --- **Problem 18: Sensitivity of T(s) with respect to parameter a** We are tasked with finding the sensitivity of the transfer function T(s) with respect to the parameter a, given by: \[ T(s) = \frac{N(s)}{D(s)} = \frac{s^2 + (1+a)s + 2}{s^3 + 2as^2 + 4s + 10a} \] **Options for the Sensitivity Function:** (a) \(\frac{asD[s] - a(2s^2 + 10)N[s]}{N[s]D[s]}\) (b) \(\frac{sD[s] - (2s^2 + 10)N[s]}{N[s]D[s]}\) (c) \(\frac{aD[s] - aN[s]}{N[s]D[s]}\) (d) \(\frac{sD[s] - sN[s]}{N[s]D[s]}\) --- **Explanation:** When analyzing the sensitivity of a transfer function, we determine how the function \( T(s) \) reacts to changes in system parameters—in this case, the parameter \( a \). The correct expression will accurately represent how variations in \( a \) influence \( T(s) \), using the derivative of both numerator and denominator with respect to \( a \). This is a typical task involved in control systems engineering, aimed at understanding system stability and performance changes, forming a critical part of robust control design.
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