# Is the amplification of a resonant filter's self oscillation increased by the gain applied with the filter?

Is the amplification of a resonant filter's self oscillation increased by the gain applied with the filter? The question may not make sense, insofar as I guess that filters are subtractive, so in what sense to they incorporate gain?

But anyway, supposing something increases audible resonance, insofar as e.g. self oscillation is audible: does that mean that the resonsance of the filter has increased?

The technical answer to this question requires quite a bit of semi-advanced engineering maths (at least in the sense that you only really start to see it during an EE or similar university degree). In a digital system you can do what you want, of course, you could write any response (technical term is transfer function) in code. But resonant filters are generally based on the behaviours of second order (or higher) RLC circuits (R = resistor, L = inductor, C = capacitor). These circuits can be arranged in various ways to act as various kinds of filter. A simple RC circuit or RL circuit can be made into a first order (ie single pole in EQ terms) circuit. An RLC circuit has more 'options', and can produce a filter with a steeper roll off (ie second order), or a bandpass, or notch EQ. If you look inside a venerable old eq like a Pultech EQP-1A you'll just see a bunch of inductors and capacitors. That said even in the analog realm eq tech moved on quickly, which I'll address below.

These are as you say passive circuits, so they cannot increase gain overall in any filter response they have. However capacitors and inductors store energy, and in a circuit using both capacitors and inductors there exists a frequency where everything aligns. The phase of the signal is shifted by DIFFERENT amounts in the capacitor and inductor and at some point they overlay perfectly. In an electrical sense this means that the capacitor starts feeding the inductor at the same rate (and visa versa), and in phase. At the output of the filter this manifests as a peak at the resonant frequency.

There is no new energy being created, but this passing back and forth effect can cause the voltage at the output to go above the maximum voltage fed into the circuit for a particular frequency.

However, analog eq tech moved on quickly and started using transistors, op amps, you name it to do the same thing. These are active components. They can be set up in clever ways to do all manner of eq tasks, and can reduce the need for inductors (which are relatively expensive, compared to caps and resistors) by using just caps as the reactive component.

I won't go to deep into the why, and that would also depend on the exact circuit used, but it is in their nature that these components (op amps especially) amplify any input by hundred of times or more by default (limited by the supply voltage in practice) so we use so called negative feedback, taking this output and subtracting it from the input in realtime, to control and stabilise these circuits.

So the filter itself is still subtracting, but it is an active design and in this case energy CAN be added to the system by the power supply. At the resonant frequency the phase of the input and output aligns and the system CAN lock into a state of oscillation, sustained by the active components. A higher resonance (or Q) will produce more oscillation. At higher Q's any tiny background signal (nothing is ever truly 0 in electronics!) will quickly be amplified at the resonant frequency and the self oscillation set up and sustained.

The gain of the filter is less at play, as this comes after the filter stage to make up any loss caused by the filtering. But the resonance knob has a big effect, as it is essentially how much amplification occurs around the resonant frequency. It's very similar to how feedback in a mic/speaker situation works in many ways. Imagine finding the feedback frequency and having an eq in line set to that frequency, you could boost the eq to encourage feedback, or cut it to nullify it. It's not REALLY the same phenomenon but has similarities.

So,

• Passive filters are always subtractive, but can produce peaks at a resonant frequency due to energy bouncing between the reactive components (caps, inductors).
• Active filters still use roughly the same principles, the actual filtering is still subtractive, but now the circuit has an energy supply for it's surrounding components. You can set up conditions where the tiniest background noise present quickly builds up and self oscillates at the resonant frequency.

The mathematics of RLC filters are very well understood and the simpler software eq designs just apply this (the transfer function) to the input signal. So we end up making an eq that mimics what we see IRL. Of course these equations can be modified heavily to get certain desired responses, and are. You can even eradicate resonances from the equations with technologies such as linear phase EQ's, but these use significant processing power as they apply the usual transfer functions but also consider the phase (which conveniently can also be given by the transfer functions) and delay-match everything to achieve 0 phase shift.

I am sure that someone knowing more about digital eq design, or even analog eq design, would say I have vastly over simplified. There are also tonnes of different circuits which self oscillate for subtly different reasons, but this is an overview!

EDIT; to your second question, no the resonance of the filter will always be the same for a given set of settings, it just means you have FOUND the frequency at which it resonates and are feeding that resonance with your input signal.