Electronic – How to implement a feedback to keep the DC gain at zero for this conceptual passive filter

Definition: The feedback factor beta is defined as the portion of the output voltage that is fed back to the differential opamp input (input source grounded). Therefore, regarding beta there is no difference between the non-inverting and inverting configuration. In both cases, we have

$$\beta = \frac{R_{in}}{R_{in} + R_f}$$

Using the general formula for the closed-loop voltage gain \$A_{cl}\$ of an amplifier with feedback

$$A_{cl} = k \cdot \frac{A_{ol}}{1 + A_{ol} \cdot \beta}$$

with open-loop gain \$A_{ol}\$ and the input damping

$$ k = -\frac{R_f}{R_{in} + R_f}$$

we get (for \$A_{ol} \to\infty\$)

$$Acl = -\frac{k}{\beta} = -\frac{R_f}{R_{in}}$$

EDIT: Comment/explanation: The "input damping" factor \$k\$ is defined as "portion of the input voltage arriving at the diff. opamp input" (negative because \$k \cdot V_{in}\$ arrives at the inv. terminal).

I think, I can agree to everything you have explained. That means: Yes, the Barkhausen condition may be applied to steady-state oscillations only. With other words - if the amplitude condition is met (|T(jωo)|=1 or even >1) and the phase condition is NOT met, we can have decaying or even rising oscillations immediately after power switch-on. But - at which frequency? In reality we have the following situation:

The system will always try to oscillate at a frequency where the phase condition is met. This is the only possibility to allow self-excitement of a system with positive feedback. And this frequency very often will be NOT exactly the frequency (wo) where the amplitude condition is fulfilled per design (or "over-fulfilled"). Hence, the system will try to oscillate at another frequency (let`s say w1). If the amplitude condition is still fulfilled at this new frequency w1 - we will have rising oscillation amplitudes (until any limiting mechanism will come into play). However, if the loop gain magnitude will be <1 at the frequency w1 we will observe damped oscillations (or even no oscillations).

Answer to your second question: If Barkhausens condition (amplitude and phase) is fulfilled for one single frequency only, there is no chance that oscillations at other frequencies are build up.

EDIT 1(example)

Here is a very illustrative example which exactly meets the subject of your question:

There is a harmonic oscillator consisting of two active opamp integrators in a closed loop (one is inverting and the other one is non-inverting). Hence, under ideal conditions we have a loop phase of 0 deg over a rather broad frequency range. However, for real opamps the loop phase will cross the 0 deg line at one single frequency fo only. We will observe that the loop gain magnitude will cross the 0 dB line at another frequency f1 which is always larger than fo (f1>fo). This is due to the opamps finite open-loop gain and corresponding phase shift.

That means: The loop gain magnitude at the frequency fo will be > 0 dB and the Barkhausen condition is (over-)fulfilled. The circuit oscillates at a frequency fo (where the phase condition is fulfilled). Note that for all frequencies below fo the loop gain is even larger than at f=fo. Nevertheless, the frequency for steady-state oscillations is f=fo.

Final remark: In particular, this oscillator topology is most suitable for explaining why and at which frequency a linear oscillator will oscillate.

EDIT 2 (Comment to the example cicrcuit):

Perhaps the following considerations may further help to answer your (very intelligent) question. If you try to use one of the available simulation programs you will find that the two-integrator oscillator will NOT continuously oscillate in case of simplified first-order models for the opamps. Why not? Because the loop phase will meet the 0 deg line for infinite frequencies only. For all finite frequencies, the loop phase is very close to 0 deg but not exactly zero (second-order system only). For this reason, after power switch-on we will have slowly decaying oscillations.

To allow continuous oscillations, a simple resistor across the feedback capacitor will turn one of the integrators into a two-pole system - and the loop gain now is of 3rd order with zero loop phase at a finite frequency.