Engineering Why does my integrator pole disappear when I simplify this?

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The discussion centers on the confusion surrounding the disappearance of the integrator pole when applying algebraic simplification in two attempts. In the first attempt, the integrator pole at s=0 is lost, while in the second attempt, it remains intact. The user realizes that a mistake in their algebraic manipulation led to this discrepancy, particularly involving the expression R_ρC_ρ + 1/s. A follow-up question addresses whether the timing of equating the denominator to zero affects the presence of poles, emphasizing the importance of maintaining standard terms in 's' throughout the simplification process. The overall takeaway is the need for careful algebraic handling to avoid losing critical poles in system analysis.
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Homework Statement
I have a 2nd order low-pass filter (expected to be driven by a current source) that I know has an integrator pole, a LHP pole and a LHP zero.

I need to find the location of these which I am doing by trying to find the impedance of this circuit.
Relevant Equations
n/a
I have tried two attempts at this and the strange this is - depending on where and how I apply my algebraic simplification (multiplying by s/s), I get a different answer. In attempt 1, I lose the integrator s=0 pole some how but in attempt 2, it's all fine.

Attempt 1

1714300951872.png


Attempt 2
1714300980690.png


PS: I have not completed this, my question is purely regarding why does the integrator pole dissapear.

So, why does the integrator pole in attempt 1 disappear but not in attempt 2?? I am really confused!
 
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1714309976049.png

BLUE BOX should be
R_\rho C_\rho +1/s
which makes attempts 1 and 2 have same result.
 
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anuttarasammyak said:
View attachment 344193
BLUE BOX should be
R_\rho C_\rho +1/s
which makes attempts 1 and 2 have same result.
Oops. Can't believe I did that even though I reviewed my work 3 times! I was going crazy!
Thank you.
 
anuttarasammyak said:
View attachment 344193
BLUE BOX should be
R_\rho C_\rho +1/s
which makes attempts 1 and 2 have same result.
A follow up question - does it matter when I equate the denominator to 0 to get the poles. For example, if I replaced the blue box with R_\rho C_\rho +1/s, there would a 1/s term at the top, yet the bottom would be unchanged. If I just left the 1/s term on top and equated the bottom to zero at this stage of the work, I would still lose my s = 0 pole.

Do I have to ensure that the entire function is in standard terms of 's' and no quotients before equating to zero?
 
The result of attempt 2 would be written as
\frac{A}{s}+\frac{B}{s+c}
where
c=R_\rho^{-1}(C_\rho^{-1}+C_2^{-1})
You can get constants A and B by calculation. You find it sum of simple pole functions. You do not have to do this reduction in applying residue theorem. The result of attempt 2 is well enough to do it. But be cafeful in your formula so that numerator does not diverge at poles.
 
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