Transfer Function Of Multiple Feedback Bandpass Filter

I am trying to figure out the transfer function for this bandpass filter (images below). I am sorry for badly drawn circuit. The operational amplifier is an ideal op amp.

I'd like to know if I correctly analyzed the circuit. I don't think so because when I try to simplify the transfer function into bandpass filter transfer function, something seems to be off.

1 Answer

By nodal analysis, using $$\\large\Sigma\$$(currents away from node) = 0:

Let $$\\small V\$$ be the voltage at the $$\\small R_1,\: R_2,\: C_3,\: C_4\$$ node, then:

$$\\large\frac{V-V_E}{R_1}+\frac{V}{R_2}+\frac{V}{Z_3}+\frac{V-V_S}{Z_4}=0\$$

And, at the summing junction node:

$$\\large\frac{-V}{Z_3}+\frac{-V_S}{R_5}=0\$$

Eliminating $$\\small V\$$:

$$\\large\frac{V_S}{V_E}=\frac{-R_2Z_4R_5}{R_1R_2(Z_3+Z_4+R_5)+(R_1+R_2)Z_3Z_4}\$$

• Thanks Chu for the answer. On your nodal analysis, You used +V/R2. Is it because R2 is directly branched by the ground? (the current runs from low voltage to high voltage) – user167987 Nov 25 '17 at 14:45
• ALWAYS use the rule: sum of currents AWAY from each node =0. This helps to avoid errors due to getting the signs wrong. In each numerator term, the node voltage under consideration will appear first, once again minimising sources of error. – Chu Nov 25 '17 at 18:22
• You have an $R_3$ in your answer but there's no $R_3$ in the schematic – Andrew Aug 20 at 4:08
• @Andrew Typo ... $Z_3$. Thanks – Chu Aug 22 at 21:54
• user167987...after inserting Z3=1/sC3 and Z4=1/sC4 into the equation you should rewrite/simplify the formula with the goal to have the form: sT1/(1+sT2+s²T²). with T=R*C. This is the classical bandpass form for identifying bandwidth and midfrequency. – LvW Aug 23 at 9:37