Optimum Passive Differentiators

A general, nth order, the transfer function (TF) is derived, whose time-domain response approximates optimally that of an ideal differentiator, optimality criterion chosen being the maximization of the first n derivatives of the ramp response at t = 0⁺. It is shown that transformerless, passive, unbalanced realizability is ensured for n < 3, but for n > 3, the TF is unstable. For n = 3, the TF is not realizable, however, near optimum results can be obtained by perturbation of the pole locations. Optimum TFs are also derived for the additional constraint of inductorless realizability. It is shown that TFs for n ≥ 2 are not realizable. For all n, however, near optimum results can be achieved by small perturbations of the pole locations; this is illustrated in this chapter for n = 2. Network realizations, for a variety of cases, are also given.

Appendix

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].