Finding the Numerator of a Transfer Function

[Steve Collins]

I am attempting the question shown in the attachment.

It can be seen that the poles are located at -2 ± 3j which expressed in terms of s is (s + 2)² + 3².

This is the denominator, but how is the numerator of the transfer function found?

Edit:

Looking at the Laplace look-up table I would want the numerator to be 3 giving:

3/((s + 2)² + 3²) so that i could use e^-atcosωt to perform the reverse Laplace transform in part b.

Is this correct?

 

[rude man]

There are no zeros in your plot, ergo there is no numerator other than a constant. The constant cannot be determined from the plot (unless it's contained in those funny numbers within the white part of the plot. I have never seen a plot like that before.) You can assume it's 3 but any other real constant is OK also. That should be obvious since L^-1{cF(s)} → cf(t), c a constant.

My table says L^-1{1/[(s+a)² + b²]} → (1/b)e^-atsin(bt).

 

[Steve Collins]

There are no zeros in your plot, ergo there is no numerator other than a constant. The constant cannot be determined from the plot (unless it's contained in those funny numbers within the white part of the plot. I have never seen a plot like that before.) You can assume it's 3 but any other real constant is OK also. That should be obvious since L^-1{cF(s)} → cf(t), c a constant.

My table says L^-1{1/[(s+a)² + b²]} → (1/b)e^-atsin(bt).

Yes you are correct, I have misread the table and copied the entry from the line above.

 

[rude man]

BTW you could also have done the inversion by partial fraction expansion, if you're comfy with manipulating complex numbers just a reminder probably ...