Can someone please tell me the equation or the reason how the transfer function
G(s) = 1/(s+2) is converted to
G(jω) = 2 - jω / ω^2 + 4
G(s) = 1/(s+2) is converted to
G(jω) = 2 - jω / ω^2 + 4
Yes - in post#1 the brackets are suppressed.I would think the correct form is
\(G(j\omega)=\frac{(2-j\omega)}{(\omega^2+4)}\)
What pole-zero cancellation? Isn't this simply a case of rationalization of the denominator?
\(G(j\omega)=1s^{-1}\frac{(2s^{-1}-j\omega)}{(\omega^2+4s^{-2})}=1s^{-1}\frac{(2s^{-1}-j\omega)}{(2s^{-1}+j\omega)(2s^{-1}-j\omega)}\)I would think the correct form is
\(G(j\omega)=\frac{(2-j\omega)}{(\omega^2+4)}\)
What pole-zero cancellation?
Yes - sometimes it is confusing.Here, s is 'seconds', which is why I think it was a stupid convention to adopt s as the variable for the Laplace transform.
Yes I see your point. I guess one could then arbitrarily state that any transfer function can be regarded as an irreducible function multiplied by an infinite array of unique matched pole-zero pairs.Yes - in post#1 the brackets are suppressed.
The original expression has no zero at all.
However, the final expression has one zero at s=2 and a pole at s=2.
The final expression is not a function of s, but of jω. It has a zero at jw=2r/s and two poles at jw=±2r/s. If left as a function of s, it would still have two poles, one at +2r/s and one at -2r/s.Yes - in post#1 the brackets are suppressed.
The original expression has no zero at all.
However, the final expression has one zero at s=2 and a pole at s=2.
OK - from the mathematical point of view this is correct.The final expression is not a function of s, but of jω. It has a zero at jw=2r/s and two poles at jw=±2r/s. If left as a function of s, it would still have two poles, one at +2r/s and one at -2r/s.
No argument on that score - someone should tell the professors skilled in the art to desist from discussing the concept with their students.To t_n_k:
I tend to reserve the idea of pole-zero cancellation as a design "tool" for making improvements in system stability.
I strongly do NOT recommend this method. Neither does this "design tool" improve stability nor has it any other advantages. You never can match both frequencies and a mismatched pole-zero pair has severe disadvantages in the time domain. Sometimes a pole or a zero cannot be avoided - and in this case one can try to make a compensation. However, that`s another situation.
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