Graphical Analysis of the filter A transfer function

In this section, further evidence that a factor of $\sqrt{f_0}$ has been neglected will be presented. Graphs obtained using the simulation package Mathematica will be shown which affirm the conjecture that equation (2.2) has a factor missing.

 

Figure 8.23:   Plots of H(s) and H_A(s)

The top plot in figure 8.23 shows plots of the transfer function as a cascade of pole zero pairs, H_A(s), given by equation (2.1),

$$H_A(s)=\prod_{n=1}^8{\frac{1}{\sqrt{7}}\cdot\frac{s+\sqrt{7}\alpha_n}{s+\alpha_ n}}$$

where $\alpha_{n+1}=7\alpha_n$ and $\alpha_8=2\pi \cdot 6.72 rad/s$.

The bottom plot of the same figure is a plot of the function H(s) which H_A(s) purports to approximate. H(s) is given by equation (2.2)

$$H_{A}(s)\approx \frac{ \sqrt{2 \pi f_0}}{s^{\frac{1}{2}}} \q... ...ere} f_{0}=\frac{\alpha_{1}}{2 \pi \sqrt[4]{7}}\approx 5\mu Hz.$$

If it is actually the case that

$$H_A(s)\approx\sqrt{\frac{2\pi}{s}}$$

then it would be expected that a log-log plot of each would result in curves with gradients of $-\frac{1}{2}$ where

$$H(s)\approx \sqrt{f_0}H_A(s)\approx 0.00224\cdot H(s).$$

Referring again to the top plot figure 8.23, it can be seen that H_A(s) approximates $s^{-\frac{1}{2}}$ behaviour as it has approximately the same gradient as the bottom plot. But H_A(s) is displaced upwards from H(s). By considering the vertical intercepts it may be seen, for example, that

so $H(0.1)\approx \sqrt{f_0} H_A(0.1)$. This suggests that a factor of $\sqrt{f_0}$ is required in order that $H(s)\approx H_A(s)$, as before.