band passtransfer function
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.
If you wish to redraw the circuit without the op-amp, try something like this:
simulate this circuit – Schematic created using CircuitLab
See the difference?
You are trying to create a detailed model of a sloppy process. For one thing, the gain-bandwidth product is a minimum spec, or sometimes even just "typical". The actual gain-bandwidth can be quite different for any one part, and I wouldn't count on the phase margin either. The raw DC gain can also vary a great deal, as can other parameters.
Getting this detailed is useless for circuit design because the point is to have the circuit work with the minimum guaranteed parameters, but keep working for higher ones. If you are looking at this level of detail around what a 741 can do, then you are using the wrong opamp.
Best 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}\$