Input impedance of twin-t notch filter

[miketan3904]

Hi guys! I want to derive the equation for the filter shown in the picture but I do not know how to start. I do know that the imput impedance is Vin/Iin but how should I write the equation to get the expression for the input impedance. I would appreciate if someone can give me some hints. Thanks.

[RoGeorge]

Handbook Of Operational Amplifier Active RC Networks
https://www.ti.com/lit/an/sboa093a/sboa093a.pdf

[T3sl4co1l]

In general, write down the currents into each node (there are 6 nodes plus ground).  The currents are determined by the voltages on the two nodes each component connects between, and the transfer function of that component: I = V/R for the resistors and I = \$j \omega\C V$ for the capacitors.  The op-amp and its feedback resistors can be treated as an ideal gain function, simplifying a few nodes (a more accurate model will treat it as an integrator, which in turn simplifies the area into a single pole transfer function).  Finally, solve the matrix through whatever means you like -- you can do substitution and elimination by hand, or en masse via Gauss-Jordan elimination or whatever, or better yet (and more accurately), just cram it all into a CAS (computer algebra system) and crank out the result.

If that's too advanced, suggested reading is on op-amp circuit solutions, what assumptions and relations are used to do it; and on the basics of linear algebra, i.e. it's just a system of equations, and the rules for manipulating matrices and vectors are the same algebraic rules (hence linear algebra); that breaks things down to high school algebra levels.

Tim

[Benta]

The generalized names of this kind of circuit analysis are "Kirchhoff's current law" and "Kirchhoff's voltage law". Which one to use is dependent on circuit topology and to some extent personal taste.
Basically you set up equations for each current loop or each voltage node and solve.
It's not as hard as it sounds, but can be tedious sometimes.

You need to use the LaPlace equivalents for the reactive parts (capacitors, inductors).
Z: sL and 1/sC
Y: sC and 1/sL
where Z is impedance and Y is admittance.

In this case, I'd recommend setting up the voltage node equations, as TeslaCoil also suggests.

[RoGeorge]

Looking closer at the original schematic, R4 and R5 form a DC positive reaction loop.  That will either latch up or latch down the op-amp in DC, or at least will act like a hysteresis.

Is that intentional?

[Benta]

Looking closer at the original schematic, R4 and R5 form a DC positive reaction loop.  That will either latch up or latch down the op-amp in DC, or at least will act like a hysteresis.

Is that intentional?

Well seen, George. I hadn't looked at the filter topology itself. As it's shown, it will not work. Seems to be an attempt to simplify the original two amp twin-t notch filter. Nice try, but no cigar.

[palpurul]

You can try to simulate the design in spice and measure the input impedance. I know this is not what you're looking for, but you can try to "guess" an analytical solution that kind of "fits" the simulation results based on the values that you use in the simulation. You can even change the values to see how the input impedance graph "moves" which can give you a hint.

Just throwing ideas around, I am sorry If I confused you