I've seen some different forms for the equation of a transfer function and wonder which one is most correct and why? Does it depend on type of filter etc.?
Let's take an example and its transfer function: $$H(w)=\frac{V_o(w)}{V_i(w)}=\frac{R_2jwL}{R_1(R_2+jwL)+R_2jwL}$$
With some algebra you can get it on two different forms:
\$H_1(w)=\bigl(\frac{R_2}{R_1+R_2}\bigr)\frac{jw}{jw+\bigl(\frac{R_1R_2}{L(R_1+R_2}\bigl)}\$ where \$\frac{R_1R_2}{L(R_1+R_2)}\$ is the pole (and \$jw\$ is a zero?).
\$H_2(w)=\bigl(\frac{L}{R_1}\bigr)\frac{jw}{1+j\Biggl(\frac{w}{\bigl(\frac{R_1R_2}{L(R_1+R_2)}\bigl)}\Biggr)}\$ where \$\frac{R_1R_2}{L(R_1+R_2)}\$ is the pole (and \$jw\$ is a zero?).
\$H_1(0)=0\$, \$H_2(0)=0\$, \$H_1(\infty)=\frac{R_2}{R_1+R_2}\$ and \$H_2(\infty)=\frac{R_2}{R_1+R_2}\$.
Questions:
They are clearly the same (if I've calculated it all right), but which form is most correct and standard to use? Does it differ from type of filter, i.e low vs. high pass etc.? It seems easier to use \$H_1\$ to read of the constant \$\frac{R_2}{R_1+R_2}\$ as \$w\$ approach \$\infty\$, although I've seen the form av \$H_2\$ more often and it seems standard two try to get a "\$1+...\$" in the denominator?
Thank you!