You may have double checked this by now, but I thought I would make a recommendation here. It can be very cumbersome to deal with the individual component values, as you can see in your equations above. Hence, it's a good rule to put your transfer function in a standard form when you have a second order system like this. Basically, the form you have is the classic bandpass response and the general formula for such systems is as follows:I tried to find the roots and this was my result:
Is it correct?
\( H(s)={{A_o\beta s}\over{s^2+\beta s + \omega_0^2}} \)
The parameters are as follows:
\(\omega_o\) center angular frequency in rad/s
\(\beta\) measure of angular frequency bandwidth in rad/s
\( A_o\) Gain factor (or value at center frequency)
Once you have this form, you can express the system poles in terms of center frequency and bandwidth. The formulas will be simpler and it will be easier for others to verify your work.
Note that other forms of the transfer function can be used with damping factor or Q-factor relations in place of \(\beta\).
EDIT: removed plot because it had error in it. Note that if \( \beta\) is much less than \(\omega_o\) then the bandwidth \( \beta\) is approximately the full width of the frequency band with magnitude greater than 0.707 times the maximum amplitude.
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