# New FDNR topology? Spot the flaw in its behaviour

#### Mark Tillotson

I was trying out various gyrator topologies, emulated and for real and stumbled across one I don't think I've seen before:

It looks like a NIC stacked on top of a NIC, but note the opamp input polarities are inverted from the normal. (The choice of opamp for emulation is arbitrary by the way, I used a NE5532 for the physical realization)

This sort of makes sense as this guarantees high frequency feedback is negative and thus stability. It does emulate and work for real. Changing the capacitors for resistors will destabilize this topology so its only suitable for a FDNR, not a grounded inductor.

But there's another catch.

Although its basically stable it can misbehave, and does so in real life - thought I'd leave this for people to muse over and figure it out.

[I should add that this is not intended to be voltage driven, the real life circuit has a source with an impedance of several kohms]

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#### MarcelvdG

Remarkable that it works at all with that positive DC feedback!

#### tomchr

Paid Member
There's no negative feedback at DC. I'm pretty sure the opamp will lose its mind over that.

Tom

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#### Mark Tillotson

Well its interesting. The temptation is to think that the "input" signal V(node5) is used by U2 to control V(node3) to match it, but that's assuming a zero input impedance. You can also think of U2 acting to drive V(node5) to match V(node3), which is certainly the case at higher frequencies.

But note that the input current controls the difference between V(node5) and V(node4). You can think of current control going down the circuit with voltage control going upwards. That's an oversimplification I'm sure, but it makes some sense of it.

And although the impedance of the capacitors increases indefinitely as you approach DC, this circuit is stable at DC (remove the capacitors and its unstable of course). The criteria for stability are complicated with multiple paths, and subtle anyway (Nyquist).

Clue: The flaw in the circuit shows up as an alternate mode of operation.

#### Mark Tillotson

BTW the flaw can be shown not to exist in the standard form of the FDNR circuit:

#### MarcelvdG

And although the impedance of the capacitors increases indefinitely as you approach DC, this circuit is stable at DC (remove the capacitors and its unstable of course). The criteria for stability are complicated with multiple paths, and subtle anyway (Nyquist).

This is the part that makes no sense to me. For DC, there is no difference between a capacitor and an open branch, so how could it make a difference for the DC bias points whether the capacitors are there? That's assuming it doesn't turn into a (relaxation) oscillator that never settles to a bias point, of course.

Clue: The flaw in the circuit shows up as an alternate mode of operation.

It's clear that there are two stable bias points with one op-amp output at one rail and the other at the other rail, and that the circuit won't work in either of those. The part I don't understand is how it can end up and remain in a state where it does work.

#### Mark Tillotson

Its intriguing - clearly moment to moment it is stable due to negative feedback through the capacitors, but why it converges as a whole over time is a more complicated thing which LTSpice and real components understand but we struggle with!

Nyquist's criterion is a function of the behaviour across frequencies, I don't think you can declare a circuit unstable at DC until you've evaluated the response all the way up till the gain drops below unity. With multiple loops I suspect this principle is unchanged but the "response" becomes a multi-dimensional thing and thus hard to think about.

#### MarcelvdG

You have a point there.

A special case I'm more familiar with, in simulations and calculations that is, is a feedback amplifier with no poles at all. As there are no poles, they can't move into the right half plane, and the circuit remains stable no matter whether you apply positive or negative feedback. Add a capacitor or inductor anywhere and it becomes unstable for positive feedback with a loop gain greater than 1.

Apparently your circuit works the other way around. I guess I better calculate the characteristic polynomial. I can't assume the op-amps to be ideal (nullors), though, as those have no sign and no distinction between positive and negative feedback; when the differential input voltage and input current are 0, nothing changes when you swap the inputs.

#### tomchr

Paid Member
Just a data point: I build an FDNR as shown in Post #5 using an OPA1612. It worked great in the simulator but the opamp lost its mind in reality. It didn't even land at the right DC op point! I put in an LME49860 instead at the circuit worked as designed. Go figure.

All opamps came direct from TI (or via Mouser) and I did try a second OPA1612 just in case. I never figured out why it wouldn't work but I also didn't dedicate much time to figuring out why.

Tom

#### Mark Tillotson

You sometimes need to add a small feedback capacitor between inverting input and output to stabilize opamps, 22pF is a good first guess. Keeping stray capacitance between output and non-inverting input low by layout can help.

#### Mark Tillotson

And the answer is latch-up... Clipping can trigger it for instance:

note the brown/grey input trace which has a large pulse at 4ms to trigger clipping, which then latches the circuit permanently.

And yes positive feedback at low frequency allows this to happen.

If you study the "standard" gyrator form for FDNRs you'll see each inverting input is placed between output and non-inverting input for the same opamp, shielding from any chance of positive feedback I believe. I think that is what makes it workable. However its got possibly a more complex multiple loop structure, so full analysis may prove me wrong!

#### MarcelvdG

I calculated the characteristic polynomial for the case that both op-amps are modelled as voltage-controlled voltage sources with a constant gain and I see that the result can be stable, but it doesn't provide any insight to me yet...

I haven't yet looked at the case where the op-amps integrate (Aol* = omegan*/s).

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#### Elvee

There is no way such a circuit can be DC-stable. A work-around could be the addition of a large cap in series with the +input 10K, + a large resistor to GND to fix the DC potential.
The resulting circuit might work -or not-, but at least it would free it from a hard latchup condition

#### MarcelvdG

Are you mixing up stability and freedom from latch-up or do I misinterpret your post? I would consider a circuit that can either be in a desired bias point or latched up as stable, it just has more stable bias points than intended.

#### Elvee

Take the upper opamp: its DC conditions are undefined, and it should latchup in one state or another, but the lower one may also play a role and result in a multivibrator or chaotic circuit

#### MarcelvdG

That's also what I thought, but Mark's measurements show it can somehow also function as intended (as an FDNR).

#### tomchr

Paid Member
You sometimes need to add a small feedback capacitor between inverting input and output to stabilize opamps, 22pF is a good first guess. Keeping stray capacitance between output and non-inverting input low by layout can help.
Tried that. Didn't work in my case.

Tom

#### Mark Tillotson

Take the upper opamp: its DC conditions are undefined, and it should latchup in one state or another, but the lower one may also play a role and result in a multivibrator or chaotic circuit
node5 has to have external DC bias - for instance I used 3k9 to a DC coupled oscillator. I would imagine different opamps would behave differently at power-up and some might latch up then - I only know NE5532 doesn't tend to with +/-9V rails brought up rapidly.

#### MarcelvdG

I suppose you also have a damping (termination) capacitor from node5 to ground. Is that correct?

#### Mark Tillotson

Doesn't really matter, the Q is limited to significantly less than infinity by the opamps' imperfections anyway, but a cap across the resistor damps the resonance enough to prevent clipping with my low voltage oscillator. The resonance makes it easy to provoke clipping.