Circuit Theory/Active Filters/Example90

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Active filter ... find transfer function

Start with writing a node equation:

i_1 -i_2 - i_3 -i_0 = 0
\frac{V_{in}- V_{node}}{R_1} - V_{node}sC_1 - \frac{V_{node}-V_o}{R_3} - \frac{V_{node}}{R_2} = 0

But the last expression for io could have also been written using C2. And we need to eliminate Vnode. So our second equation is associated with the feedback resistor:

\frac{V_{node}}{R_2} - (0-V_osC_2) = 0

Now these two equations can be solved for Vo/Vin:

solve([(vin-vnode)/r1 - vnode*s*c1 - (vnode-vout)/r3 - vnode/r2, vnode/r2 - (0-vout*s*c2)],[vout,vin])

The transfer function is going to be the ratio of Vout/Vin so:

vout := -vnode/(c2*r2*s)
vin := (vnode*(r1 + c2*r1*r2*s + c2*r1*r3*s + c2*r2*r3*s + c1*c2*r1*r2*r3*s^2))/(c2*r2*r3*s);
vout/vin
-r3/(r1 + c2*r1*r2*s + c2*r1*r3*s + c2*r2*r3*s + c1*c2*r1*r2*r3*s^2)

Yields this:

-\frac{R_3}{R_1 + (C_2R_1R_2 + C_2R_1R_3 + C_2R_2R_3)s  + C_1C_2R_1R_2R_3s^2}

Which looks like a low pass filter (inverted because of the op amp).