Finding the Numerator of a Transfer Function

• Steve Collins
In summary, the question is asking about finding the numerator of a transfer function given the denominator (s + 2)2 + 32 and how it can be determined using the Laplace look-up table. The conversation also mentions the option of using e-atcosωt to perform the reverse Laplace transform and confirms that there are no zeros in the plot. The conclusion is that the constant in the numerator cannot be determined from the plot and can be assumed to be any real constant. The table shows that the inverse Laplace transform of 1/[(s+a)2 + b2] is (1/b)e-atsin(bt).
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?

Attachments

• pole zero config.pdf
110.2 KB · Views: 246
Last edited:
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).

rude man said:
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.

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

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.

1. What is the numerator of a transfer function?

The numerator of a transfer function is the polynomial expression in the numerator of the transfer function equation. It represents the output of the system in response to the input signal.

2. How is the numerator of a transfer function related to the system's dynamics?

The numerator of a transfer function is directly related to the system's dynamics as it represents the system's response to the input. It can provide information about the amplitude and frequency characteristics of the system.

3. Can the numerator of a transfer function be zero?

Yes, the numerator of a transfer function can be zero. This means that there is no output response to the input signal, which indicates that the system is not sensitive to that particular input.

4. How can the numerator of a transfer function be simplified?

The numerator of a transfer function can be simplified by factoring out common terms from the polynomial expression. This can help to reduce the complexity of the transfer function and make it easier to analyze and manipulate.

5. Is the numerator of a transfer function affected by changes in the system's parameters?

Yes, changes in the system's parameters, such as gains or time delays, can affect the numerator of a transfer function. This is because these parameters directly impact the system's response to the input signal, which is represented by the numerator.

Replies
1
Views
855
Replies
5
Views
2K
Replies
1
Views
391
Replies
2
Views
1K
Replies
1
Views
2K
Replies
16
Views
1K
Replies
3
Views
2K
Replies
23
Views
5K
Replies
2
Views
1K
Replies
12
Views
2K