In this section, we examine the nature of

*b* _{ o } and substantiate the assertions made in Section 5 that higher the value of

*b* _{ o }, better is the approximation. For the first-order case, the TF and the corresponding ramp response are, respectively,

$$ H_{1} (s) = \frac{{b_{o} s}}{{s + b_{o} }} $$

(22.49)

and

$$ v_{1} (t) = (1 - {\text{e}}^{{ - b_{o} t}} )u(t) $$

(22.50)

Clearly, higher the *b* _{ o }, closer is *v* _{1}(*t*) to *u*(*t*) which is the ideal ramp response. Maximum value of *b* _{ o } can be unity in *H* _{1}(*s*) [F-G conditions of realizability of *H* _{1}(*s*)].

For the second-order case, the TF and the ramp response are, respectively

$$ H_{2} (s) = \frac{{s^{2} + b_{o} s}}{{s^{2} + b_{o} s + b_{o} }} $$

(22.51)

and

$$ \nu_{2} (t) = \left[ {1 - \frac{{{\text{e}}^{{ - b_{o} t/2}} }}{{(4 - b_{o} )^{1/2} }}\cos \left( {\omega_{o} t - \tan^{ - 1} \frac{{b_{o} }}{{2\omega_{o} }}} \right)} \right]u(t), $$

(22.52)

where

$$ \omega_{o} = \left( {b_{o} - b_{o}^{2} /4} \right)^{1/2} $$

(22.53)

As \( \left| {\cos \left( {\omega_{o} t - \tan^{ - 1} \frac{{b_{o} }}{{2\omega_{o} }}} \right)} \right| \le 1, \) smaller the value \( \frac{{{\text{e}}^{{ - b_{o} t/2}} }}{{(4 - b_{o} )^{1/2} }}, \) closer is *v* _{2}(*t*) to *u*(*t*). Increase of *b* _{ o } decreases \( {\text{e}}^{{ - b_{o} t/2}} \) faster (i.e. exponentially) than (4 − *b* _{ o })^{1/2}. Thus higher the *b* _{ o }, smaller is the value of \( \frac{{{\text{e}}^{{ - b_{o} t/2}} }}{{(4 - b_{o} )^{1/2} }} \) and consequently closer is the *v* _{2}(*t*) to the ideal value *u*(*t*).

For the general case, a semi-rigorous argument can be forwarded as follows. As

$$ H_{n} (s) = \frac{{s^{n} + b_{n - 1} s^{n - 1} + \cdots + b_{2} s^{2} + b_{o} s}}{{s^{n} + b_{n - 1} s^{n - 1} + \cdots + b_{2} s^{2} + b_{1} s + b_{o} }} $$

(22.54)

and

$$ V_{n} (s) = \frac{1}{s}\left[ {\frac{{s^{n - 1} + b_{n - 1} s^{n - 2} + \cdots + b_{3} s^{2} + b_{2} s + b_{o} }}{{s^{n} + b_{n - 1} s^{n - 1} + \cdots + b_{2} s^{2} + b_{1} s + b_{o} }}} \right] $$

(22.55)

$$ \underline{\underline{\Delta }} \frac{1}{s}G(s) $$

(22.56)

where

$$ G(s) = \frac{{s^{n - 1} + b_{n - 1} s^{n - 2} + \cdots + b_{3} s^{2} + b_{2} s + b_{o} }}{{s^{n} + b_{n - 1} s^{n - 1} + \cdots + b_{2} s^{2} + b_{1} s + b_{o} }} $$

(22.57a)

$$ = \frac{{1 + (b_{2} /b_{o} )s + (b_{3} /b_{o} )s^{2} + \cdots }}{{1 + (b_{1} /b_{o} )s + (b_{2} /b_{o} )s^{2} + \cdots }} $$

(22.57b)

Equation

22.56 shows that

*v* _{ n }(

*t*) =

*L* ^{−1} *V* _{ n }(

*s*) can be interpreted as the unit step response of the low-pass function

*G*(

*s*). Equation

22.55, together with the initial and final value theorems of Laplace transforms shows that

*v* _{ n }(

*t*) rises from a value zero at

*t* = 0 to unity at

*t* = ∞. To enable us make

*v* _{ n }(

*t*) achieve unity value in as short a time as possible, we must choose

*b* _{0} such that the rise time

*τ* _{ r }, of

*v* _{ n }(

*t*) is as small as possible. Using Elmore’s formula [

2], with the assumption that the plot of

*v* _{ n }(

*t*) is monotonic, (whereby Elmore’s formula can be applied), we get

$$ \tau_{r} = \frac{1}{{b_{o} }}\left\{ {2\pi \left[ {\left( {b_{1}^{2} - b_{2}^{2} } \right) - 2b_{o} \left( {b_{2} - b_{3} } \right)} \right]} \right\}^{{\frac{1}{2}}} $$

(22.58)

*τ* _{ r }, decreases monotonically with the increase of

*b* _{ o }. Thus

*b* _{ o } should be as large as possible. The assumption of

*v* _{ n }(

*t*) being monotonic has implications as mentioned in the Appendix of [

1].