# Circuit for Inverse Chebyshev or Elliptic LPF

I know I came across a 2 op-amp, 2 capacitor circuit that can be used for a single section of an Inverse Chebyshev (a.k.a. "Chebyshev Type II) or an Elliptic (Cauer) filter. It has a pair of zeros on the $j\omega$-axis at $\pm j\omega_z$, and with resonant frequency $\omega_0<\omega_z$ and the transfer function is:

$$H(s) = \frac{1 + \left(\tfrac{s}{\omega_z}\right)^2 }{1+\tfrac{1}{Q}\tfrac{s}{\omega_0}+\left(\tfrac{s}{\omega_0}\right)^2}$$

I know how to brute force derive a circuit with a pair of integrators and the canonical form we learned in Linear Circuits class 4 decades ago. Such as this:

I just thought I saw a more elegant circuit, with one less op-amp and a couple fewer resistors, that pretty much guaranteed that the zeros lie on the $j\omega$-axis and at a higher frequency than the resonant frequency $\omega_0$. Anybody know how to save a couple of parts with this? Is there a single op-amp, two-capacitor, 4-resistor circuit that can do this?

A sorta Sallen-Key with zeros.

• I'm so ashamed of myself. 4 decades ago, I would know this. But I can't remember this now. Aug 2, 2018 at 7:55

There are several one-opamp circuits with Chebyshev II or Cauer behaviour, for example:

• double-T-feedback circuits with positive (fixed) gain,
• Boctor-filter (based on Multi-feedback topology),
• Scultety-structure
• GIC-based structures

(Hint: Google for Boctor and Scultety)

For example, see here:

http://www.schematica.com/active_filter_resources/a_list_of_active_filter_circuit_topologies.html

• my goodness, what a useful link! i had never heard of schematica.com before. Aug 2, 2018 at 17:23
• looks to me that the Boctor might be the one. (i don't want more storage elements than the order of the filter, so the double T is less preferred.) Aug 2, 2018 at 17:25
• Yes - another point: Double-T is very sensitive to component tolerances (exact matching required). In this respect, GIC structures are best - however with two opamps.
– LvW
Aug 3, 2018 at 7:31

I'm wondering, will this do it?

simulate this circuit – Schematic created using CircuitLab

• doing the node-voltage method now. what an interesting trip down memory lane. Aug 2, 2018 at 8:19
• Why not simulate it? Aug 2, 2018 at 8:19
• i wanna transfer function. Aug 2, 2018 at 8:20
• Well you can try this circuit or that circuit and derive the TF for each and examine the TF to see if it matches or you can just mess around with a sim until you get the TF looking about right then derive it. It's your time you'll be spending. Aug 2, 2018 at 8:21
• This opamp circuit only has one zero, so it can't realize the transfer function in the question. Aug 2, 2018 at 12:30