Chapter 6, Problem 6.4E

Interpretation Introduction

(a)

Interpretation:

It is to be shown that an extremum by $y\left(t\right)$ can be exhibited in the step response only if $\frac{1-{\tau }_a/{\tau }_2}{1-{\tau }_a/{\tau }_1}>1$.

Concept introduction:

For a function $f\left(t\right)$, the Laplace transform is given by,

$F\left(s\right)=ℒ\left[f\left(t\right)\right]={\displaystyle {\int }_0^{\infty }f\left(f\right)e^{-st}dt}$   .......... (1)

Here, $F\left(s\right)$ represents the Laplace transform, $s$ is a variable which is complex and independent, $f\left(t\right)$ is any function of time which is being transformed, and $ℒ$ is the operator which is defined by an integral.

$f\left(t\right)$ is calculated by taking inverse Laplace transform of the function $F\left(s\right)$.

PFE is the partial fraction expansion is the method of expanding the denominator of a fraction into simpler terms.

Interpretation Introduction

(b)

Interpretation:

It is to be shown that the overshoot occurs only if ${\tau }_a/{\tau }_1>1$.

Concept introduction:

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.

From the definition of the overshoot, it can be written as:

$\text{OS}=\frac{y\left(t\right)-KM}{KM}$   .......... (2)

Interpretation Introduction

(c)

Interpretation:

It is to be shown that the inverse response occurs only if ${\tau }_a<0$.

Concept introduction:

Inverse response occurs for a dynamic system, when its initial response is in the opposite direction to the final outcome. It generally results when two separate effects occur simultaneously but with different dynamics and directions.

Interpretation Introduction

(d)

Interpretation:

The time at which the extremum in $y\left(t\right)$ exists is to be calculated analytically.

Concept introduction:

For chemical processes, dynamic models consisting ordinary differential equations are derived through unsteady-state conservation laws. These laws generally include mass and energy balances.

The process models generally include algebraic relationships which commence from thermodynamics, transport phenomena, chemical kinetics, and physical properties of the processes.