Chapter 5, Problem 5.15E

Interpretation Introduction

(a)

Interpretation:

An expression for the measured temperature of an electrically heated process as a function of time is to be derived.

Concept introduction:

For the second order transfer function of the form,

$G\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\frac{K}{{\tau }^2{s}^2+2\zeta \tau s+1}$   ..... (1)

Where, $K,\tau ,\text{ and }\zeta$ are the system parameters, response $\left(y\left(t\right)\right)$ for the step input of magnitude $M$ for underdamped system $\left(0\le \zeta <1\right)$ is given by the expression:

$y\left(t\right)=KM\left\{1-e^{-\zeta t/\tau }\left[\cos \left(\frac{\sqrt{1-{\zeta }^2}}{\tau }t\right)+\frac{\zeta }{\sqrt{1-{\zeta }^2}}\sin \left(\frac{\sqrt{1-{\zeta }^2}}{\tau }t\right)\right]\right\}$   ..... (2)

Interpretation Introduction

(b)

Interpretation:

The maximum observed temperature of the process and the time of its occurrence are to be determined.

Concept introduction:

For the second order transfer function of the form,

$G\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\frac{K}{{\tau }^2{s}^2+2\zeta \tau s+1}$   ..... (1)

Where, $K,\tau ,\text{ and }\zeta$ are the system parameters, response $\left(y\left(t\right)\right)$ for the step input of magnitude $M$ for underdamped system $\left(0\le \zeta <1\right)$ is given by the expression:

$y\left(t\right)=KM\left\{1-e^{-\zeta t/\tau }\left[\cos \left(\frac{\sqrt{1-{\zeta }^2}}{\tau }t\right)+\frac{\zeta }{\sqrt{1-{\zeta }^2}}\sin \left(\frac{\sqrt{1-{\zeta }^2}}{\tau }t\right)\right]\right\}$   ..... (2)

The maximum value of the response of a system to achieve its peak from the desired response of the given system is known as overshoot. It exceeds its final steady-state value. The formula to calculate overshoot (OS) for an underdamped system is:

$\text{OS}=\exp \left(-\pi \zeta /\sqrt{1-{\zeta }^2}\right)$   ..... (4)