Cascaded Op-Amp circuit analysis

Question

Given this circuit I’m asked to determine the transfer function from \$V_i\$ to \$V_o\$, but there are no answers given and my result disagrees with my earlier DC analysis.

I think think the first “stage” is just a voltage follower. By superposition, it’s easy to show that \$V_o’ = V_i\$ (though I'm unsure whether I'm allowed to split \$V_i\$ like that):

^{simulate this circuit – Schematic created using CircuitLab}

For the second stage

^{simulate this circuit}

Since \$A_2\$ is ideal, there is no current through \$C_2 \parallel R_5\$ so \$0V\$ drop across it, and the inverting port follows the non-inverting one, thus there is \$-V_i\$ across \$R_3\$, and by Ohm's law \$i = -\frac{V_i}{R_3}\$ running through it.

After which I did KVL (red arrows): $$ V_o + i(R_4 \parallel C_3) + V_i = 0\\ V_o -\frac{V_i}{R_3}(R_4 \parallel C_3) + V_i = 0\\ V_o + V_i[1 - \frac{1}{R_3}(R_4 \parallel C_3)] = 0\\ V_o = - V_i[1 - \frac{1}{R_3}(R_4 \parallel C_3)]\\ \frac{V_o}{V_i} = H(s) = \frac{1}{R_3}(R_4 \parallel C_3) - 1\\ H(s) = \frac{1}{R_3} \frac{\frac{R_4}{sC_3}}{R_4 + \frac{1}{sC_3}} - 1 = \frac{1}{R_3} \frac{R_4}{sR_4C_3 + 1} - 1\\ H(s) = \frac{R_4 - sR_3R_4C_3 - R_3}{sR_3R_4C_3 + R_3}\\ H(0) = \frac{R_4 - R_3}{R_3}\\ $$

However, I've found earlier through DC analysis (\$R_1, R_2, (C_2 \parallel R_5)\$ are short, \$C_3, C_1\$ are open) that the DC-gain should be \$1 + \frac{R_4}{R_3}\$.

I'm not even sure where to look to find my mistake.

Answer 1

The first line doing the KVL was wrong, leading to the wrong result. It's supposed to be $$ V_o + i(R_4 \parallel C_3) - V_i = 0 $$

Answer 2

If opamps are ideal, R1, R2, R5 play no role.

A1 is a buffer indeed. Note that due to virtual ground, no current flows through R6, and thus none through R7. So vin is buffered to vout.

A2 is a non-inverting stage, with gain G2 = 1 + R4/R3 = 1 + 27/9 = 4.

So, the TF(0) = 1 * 4. Which agrees with what you found before.