Have I obtained the correct simplified transfer function expression?
I am trying to obtain a transfer function for an electrical circuit. However, the response of the TF that I obtain doesn't correlate to the response of the circuit itself when I simulate it.
Therefore, I have obviously done something wrong in my derivation. I have spent a good couples of days and plenty of scrap papers trying to see where I have missed something, but I keep on arriving at the same answer.
So, in order save myself the headache of repeatedly doing the same thing only to obtain the same outcome, I've decided to put my dilemma to the community!
Below is my working out from the mid-way point to the end result and if anyone can point out what I have missed, I would really appreciate it.
So the starting expression is:
$$ I_2(s)((\frac{C_2Ls^2+C_2(R_1+R_2)s+1}{C_2Ls^2+C_2R_1s})(\frac{C_1Ls^2+C_1R_1s+1}{C_1s}))-I_2(s)(Ls+R_1)=E_i(s)\tag{1} $$
$$ I_2(s)((\frac{C_2Ls^2+C_2(R_1+R_2)s+1}{C_2Ls^2+C_2R_1s})(\frac{C_1Ls^2+C_1R_1s+1}{C_1s})-(Ls+R_1))=E_i(s) $$
$$ I_2(s)(\frac{(C_2Ls^2+C_2(R_1+R_2)s+1)({C_1Ls^2+C_1R_1s+1})}{(C_2Ls^2+C_2R_1s)(C_1s)})-(Ls+R_1))=E_i(s) $$
$$ I_2(s)(\frac{(C_2Ls^2+C_2(R_1+R_2)s+1)(C_1Ls^2+C_1R_1s+1)-(C_1C_2Ls^3+C_1C_2R_1s^2)(Ls+R_1)}{(C_1C_2Ls^3+C_1C_2R_1s^2)})\tag{2}$$
Expanding the numerator for the positive term
$$ (C_2Ls^2+C_2(R_1+R_2)s+1)(C_1Ls^2+C_1R_1s+1)$$
$$ (C_2Ls^2)(C_1Ls^2) + (C_2Ls^2)(C_1R_1s) + (C_2Ls^2)(1) + (C_2(R_1+R_2)s)(C_1Ls^2) + (C_2(R_1+R_2)s)(C_1R_1s)+(C_2(R_1+R_2)s)(1)) + (1)(C_1Ls^2) + (1)(C_1R_1s) + (1)(1) $$
$$ (C_1C_2L^2s^4) + (C_1C_2R_1Ls^3) + (C_2Ls^2) + (C_1C_2L(R_1+R_2)s^3) + (C_1C_2R_1(R_1+R_2)s^2) + C_2(R_1+R_2)s + (C_1Ls^2) + (C_1R_1s) + 1 $$
$$ C_1C_2L^2s^4 + C_1C_2R_1Ls^3 + C_2Ls^2 + C_1C_2L(R_1+R_2)s^3 + C_1C_2R_1(R_1+R_2)s^2 + C_2(R_1+R_2)s + C_1Ls^2 + C_1R_1s + 1 $$
$$ C_1C_2L^2s^4 + C_1C_2R_1Ls^3 + C_2Ls^2 + C_1C_2R_1Ls^3 + C_1C_2R_2Ls^3 + C_1C_2R_1^2s^2 + C_1C_2R_1R_2s^2 + C_2(R_1+R_2)s + C_1Ls^2 + C_1R_1s + 1 $$
$$ C_1C_2L^2s^4 + 2C_1C_2R_1Ls^3 + C_2Ls^2 + C_1C_2R_2Ls^3 + C_1C_2R_1^2s^2 + C_1C_2R_1R_2s^2 + C_2(R_1+R_2)s + C_1Ls^2 + C_1R_1s + 1 \tag{3}$$
Expanding the numerator for the negative term
$$ -(C_1C_2Ls^3+C_1C_2R_1s^2)(Ls+R_1)$$
$$ -((C_1C_2Ls^3)(Ls)+(C_1C_2Ls^3)(R_1)+(C_1C_2R_1s^2)(Ls)+(C_1C_2R_1s^2)(R_1))$$
$$ -(C_1C_2L^2s^4 + C_1C_2R_1Ls^3 + C_1C_2R_1Ls^3 + C_1C_2R_1^2s^2)\tag{4}$$
Subtracting $(4)$ from $(3)$
$$ C_1C_2L^2s^4 + 2C_1C_2R_1Ls^3 + C_2Ls^2 + C_1C_2R_2Ls^3 + C_1C_2R_1^2s^2 + C_1C_2R_1R_2s^2 + C_2(R_1+R_2)s + C_1Ls^2 + C_1R_1s + 1 -C_1C_2L^2s^4 – C_1C_2R_1Ls^3 – C_1C_2R_1Ls^3 – C_1C_2R_1^2s^2 $$
$$ C_1C_2L^2s^4 + 2C_1C_2R_1Ls^3 + C_2Ls^2 + C_1C_2R_2Ls^3 + C_1C_2R_1^2s^2 + C_1C_2R_1R_2s^2 + C_2(R_1+R_2)s + C_1Ls^2 + C_1R_1s + 1 – C_1C_2L^2s^4 – 2C_1C_2R_1Ls^3 – C_1C_2R_1^2s^2 $$
$$ C_2Ls^2 + C_1C_2R_2Ls^3 + C_1C_2R_1R_2s^2 + C_2(R_1+R_2)s + C_1Ls^2 + C_1R_1s + 1 $$
Combining like terms
$$ C_1C_2R_2Ls^3 + (C_1(C_2R_1R_2 + L)+ C_2L)s^2 + (C_2(R_1+R_2) + C_1R_1)s + 1 $$
Therefore, I get
$$ I_2(s)\frac{C_1C_2R_2Ls^3 + (C_1(C_2R_1R_2 + L)+ C_2L)s^2 + (C_2(R_1+R_2) + C_1R_1)s + 1}{C_1C_2Ls^3+C_1C_2R_1s^2} = E_i(s)$$
And I arrive at this final transfer function each and every time:
$$ \frac{I_2(s)}{E_i(s)}=\frac{C_1C_2Ls^3+C_1C_2R_1s^2}{C_1C_2R_2Ls^3 + (C_1(C_2R_1R_2 + L)+ C_2L)s^2 + (C_2(R_1+R_2) + C_1R_1)s + 1} \tag{5}$$
Therefore, if anyone has noted any step(s) that I might have missed, I would really appreciate it, if you could please point it out 🙂
Best Answer
I can confirm that my derivation was right. The issue that I had wasn't anything to do with my algebra, but my circuit analysis.
The final transfer function that I was actually after was $\dfrac{E_o(s)}{E_i(s)}$ and not $\dfrac{I_2(s)}{E_i(s)}$ which is what I had and that is why the response of the TF didn't correlate to the response of the circuit.