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 ...

 

[rezeew]

Yes, you are correct. To find the numerator of a transfer function, we need to use the Laplace transform to convert the system from the time domain to the frequency domain. The Laplace transform of a system's output divided by its input gives us the transfer function. In this case, the poles of the transfer function are located at -2±3j, which can be expressed as (s+2)2+32. Therefore, the transfer function can be written as 3/((s+2)2+32). This means that the numerator of the transfer function is 3. This is important because it tells us the gain or amplification of the system at different frequencies. In your case, you are using the transfer function to perform a reverse Laplace transform and the value of the numerator is crucial in this process.