Finding Transfer Function & Step Response with Matlab

steveb

Joined Jul 3, 2008
2,436
I tried to find the roots and this was my result:
Is it correct?

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:

\( H(s)={{A_o\beta s}\over{s^2+\beta s + \omega_0^2}} \)

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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steveb

Joined Jul 3, 2008
2,436
So for:


I have

Ao = C1R2
B = k+1
wo = 1

Is this correct?
No, you would need to convert the formula to have a factor of one in front of the s^2 term in the denominator. That is, divide every term in the numerator and denominator by R1R2C1C2. The Ao and B will then easily be seen as the coefficients for s^0 and s^1 respectively. The next part is to find Ao which leads to a cumbersome formula, but it's easy to identify it and write it out.
 

steveb

Joined Jul 3, 2008
2,436
I'm still a bit lost. I divided both the numerator and denominator by R1R2C1C2 and got the following:


But how will I find Ao, B, and wo?
Well, B and wo are easy because you can directly read them off the formulas.

You are on the right track. Ao is a little more tricky but to too bad if you equate the coefficients directly.
 
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