To check my understanding:
1. Plain RC passive filters, however you cascade them, can never have particularly shrap fall-offs at the pass band edge?
For example, for a 2nd order filter, the middle term of the transfer function denominator (the \$\omega^1\$ term) has a minimum coefficient of 2 in the limiting case when R2/R1 --> infty, i.e. the second RC loads the first RC minimally. And such a quadratic (of the form x^2 + 2x + 1, scaled to the given cutoff frequency) is a repeated root (in the denominator: repeated pole), always real.
IIRC, the equal-value RCRC lowpass gives a middle coefficient of 7. So it's pretty de-Q-ed.
You only get oscillating transient (impulse or step) response for complex poles, i.e. [analytical] response of the form \$t^n e^{\frac{-t}{\tau}} e^{j \omega t}\$; the cases directly correspond.
Mind, with clever application of zeroes, an RC filter can still have (voltage) gain slightly over 1, but it's still very gradual with respect to frequency; you can't make a sharp peak. (You can make a sharp valley (notch), though.)
2. LC filters can quite easily achieve sharp fall-offs and huge Q values?
I wouldn't say "huge", but commercially available parts fall in the 10-200 range, and you can make your own in the thousands. More than some thousands -- I mean, thousands you're looking at quite an expensive construction to begin with, but beyond that you pretty much have to look at mechanical or acoustical solutions.
Anyway, with clever band planning, such high selectivity isn't usually a problem. It's more a matter of frequency stability, and the shortcut of having a single pre-made slam-dunk component, that makes crystal/ceramic filters so valuable.
That's somewhat in historical context, for example VHF and low UHF radios were just fine with LC tuning and vacuum tube amplifiers throughout the 1930s to 60s. Maybe not great; even SW is a bit of a challenge without crystals (and a synthesizer for that matter) if you're working SSB. But they managed, and some manufacturers made some very finely tuned equipment, temperature compensated, mechanically compensated, the works. You'd see crystals mostly for precision work, for narrow-band or special purpose radios, and most of all of course, television colorburst.
And, whatever telecom used at the time; I don't have a clue what the market was like -- and it might not be very obvious even for those in the know, just because everything went through Western Electric back then, Bell was very vertically integrated -- but I at least imagine they used a lot of crystals as well. The advantage there being, telecom bands were strictly defined (these are bands on the line, not in air!), so mixing a voice band with a selected sequence of crystal tones, and some nice sharp LC filters to clinch the band edges, was an effective way to multiplex (FDM) many voice calls on a single trunk line (and, ultimately where T1 lines came from, once those bands were reused for digital purposes).
With microwave comms being pervasive today, the use of narrow sub-bands, digital; crystals are pretty much mandatory.
Or, there are actually some "silicon oscillators", that I'm not clear on whether they're just very well tuned RC oscillators, or LC or wave something or other, but they offer specs comparable to crystals. I think crystals are still the cheaper option overall, so still dominant.
3. Because real inductors are pretty non-ideal, it is favoured to use op amps to make virtual inductors ("gyrators") using combinations of op amps, resistors and caps? And with real inductors often having pretty wide tolerances, 10% or such, what is supposed to be say a narrow bandwidth filter with say a 2dB permissible passband ripple, ends up with a passband where there are 15dB ripples present if the L and C values aren't accurate to multiple significant figures (I found an online simulator page which showed this difference when you told it to give solutions with exact coponent values vs solutions rounded to the nearest commonly available value for components).
Yes, but not quite as stated. Yes, to the extent that inductors are non-ideal
where?: at low frequencies where the part size is massive, or at high impedances where inductance is hard to construct at all.
Op-amps aren't good at high Q, as it doesn't take much phase shift or gain error to push that complex pole pair from the left half-plane (sharp resonance) to the right (unstable, an oscillator). Not to say it can't be done, but it is tricky, and a frequency-mixing scheme becomes desirable.
Tolerance isn't much of an issue, in that, if the filter spec can't be met by off-the-shelf or mildly-custom parts, and the system can't be re-designed to, or to use other methods, then it needs to be tuned. Which goes even for precisely machined e.g. cavity filters for example, for multi-user antenna installations, cellular transceivers, etc., which regularly use high-order diplexing filters to separate rx/tx, or multiple customers (of the antenna/tower installation), tuned with a whole mess of screws perturbing the cavities.
4. Is there a way to make these "gyrators" with single transistor setups rather than two-input op amps? I understand gyrators need feedback paths, in some cases transistors can do this?
5. Can R, C and transistors be used to make a fairly sharp fall-off bandpass then?
Consider the Bode plot of the element: at low frequency (down to DC), there is some "DCR" due to finite amp gain (i.e., 1/gm). At mid frequencies, Z ~ F and we have inductance. At high frequencies, the amp itself rolls off, or stray C takes over, and it drops off again, and there is some R at the peak. There may be a flat region at HF due to limited amp speed/gain, or circuit resistances, before C takes over; depends.
Well, simply given an impedance function with one zero and one pole, Z = (1 + s/w_z) / (1 + s/w_p), for w_z < w_p, the slope of that midband transition region, where it's inductive, is slightly less than proportional, and by how much, depends on how close those corner bends in the curve are.
You can think of the Bode plot as a perfect sharp bend, plus this curvature function, where the nonideality of what would otherwise be a sharp bend (from flat to +20 dB/dec) is superimposed on the ideal function.
The closer the two corners are together, the more of their nonideality overlaps.
And this is a simple rational function, so the nonideality is simply inverse with distance from the corner. We're all doing proportions here.
So, if you want a Q of 10, you need the corner and shelf frequencies at least a factor of 10 below and above the range of interest. A total range of 100x is pretty reasonable; an emitter follower can even manage that.
Or if you want a Q of 1000, you need a 10^6 range of pure slope, to begin approaching that number in the middle of it. That's pretty serious, but a good op-amp probably can manage.
You can imagine, any little gain/phase error will tweak that response, and sure you could easily tweak in some negative resistance intentionally, but now you're artificially peaking a network and the response isn't just the component values (including synthesized inductors) and coupling factors, but it depends on gain factors too, and probably device properties, and the whole thing starts getting just as messy as you had (in part) meant to avoid by using active circuitry.
(There are of course many reasons to use active filters, and many reasons not to. One of them is that, for the cascaded tuned amplifier topology, coupling (feedback) between stages is ideally zero, so the pole pair of each individual stage can be designed and tested independently. In an LC filter, couplings are perfectly symmetrical, and adjusting any one component value changes the whole response, necessitating changes almost everywhere else.)
This may be food for thought: for my 20m radio project a while ago, I opted for an active filter to clean up the final AF output. I used a simple follower:
6V6 at these bias conditions, is a close model to the 5702 submini pentode I used (in turn similar to 6AK5, etc.). SPICE models of little-used parts can be dubious, but the operating point checks out in this case, and the model matches measurements.
This is just a plain old Sallen-Key unity-gain prototype, set up for a bandpass response. It's maximally bandpassed, in the sense that the slopes are ideally 40dB/dec; it's 2nd order and the amplifier is doing its job, peaking response slightly.
The -47dB-ish shelf at HF matches the feed-forward path, R2+R3 into R9 || Rk, where Rk is about 1/gm, and gm is about 4mS or 1/250Ω here.
Is there any other type of circuit than a true filter which can let one get an output (either the analog signal itself, or something heavily distorted but giving proportionate results for peak values when put through a rectifier...), roughly proportional to signal strength, if and only if a signal at some specific frequency (+/- a tolerance bandwith around it, incase the signal's frequency is a little bit off) is present?
Thanks
Welcome to the wild world of hetrodyning, and mixing in general.
I expect the reference here:
Ive just resurrected a.circuit I designed in 1979. It uses 8 10% capacitors, one 10% resistor, and 8 analogue switches.
The key performance points are Q>1000, flat passband, and initial rolloff is >20000dB/decade. Stopband can be 20-40 dB down. And yes, despite typing this on a tablet, those are the right number of zeros 
is a synchronous type, omitting
a few components (oscillator or clock input, amplifiers or buffers), and caveats or assumptions (aliasing filters?).
Switched capacitor filters work on two principles:
1. A capacitor toggled between an input and an output port, conveys a slug of charge per clock pulse, proportional to the voltage between ports; doing this at Fclk means a current flow of (V1 - V2) * Fclk * C, in other words Fclk * C is an equivalent resistance.
2. Signals alias around Fclk. There will also be charge injection due to imperfectly balanced switches, so use exactly at Fclk (or harmonics) can be dubious, or challenging, or limited in noise floor; but in any case the effect is a synchronous modulator and detector all in one.
We can implement what is apparently a very narrow bandpass filter, by using simply a very low cutoff lowpass filter, and synchronously modulating it up around a center frequency. Downsides include aliasing, poor skirts, etc. (Of course, using a higher-order filter, and a transfer function rather than a self-impedance -- using an input and an output mixer -- can afford a much squarer passband.)
Spotting modulators in the wild is a handy trick. Consider the familiar ZVS oscillator circuit:
The supply inductor (47-200uH) acts with the load (seemingly an LC resonant tank) with respect to its center frequency. That is, when lightly loaded, the amplitude (and proportionally, the cycle-averaged voltage at the center tap) can bounce and oscillate; and when a load is attached, AC current is drawn, which is reflected as DC current draw -- the AC current draw is synchronously detected by the alternating transistors, drawing supply current, as it of course and necessarily must.
The conservation of energy isn't a surprise, but what's neat about this perspective is, if we modulate the input voltage, or load current, and if we have a bandpass filter circuit attached to the output, or lowpass to the input, the frequency response of that filter is visible on the other side, due to reciprocity across the modulator.
Put still another way: the LC tank resonates at Fo with impedance Zo, but it manifests to the supply port as an equivalent RC circuit, where C is twice the 0.68uF (plus whatever other equivalent is attached). Thus we expect the amplitude/envelope to bounce at a rate determined by this equivalent C, and the series inductor; and dampened by reflected equivalent load resistance.
It is a tricky matter of course, because as a self-oscillating modulator, center frequency is always the peak frequency it latched onto, and that frequency will be skewed by whatever impedance is attached to it. If we use an external oscillator, the equivalence is much more direct.
The general lesson (and by "lesson", I hardly mean to teach the entirety of the subject; for my part, it took years to develop a full appreciation of all the implications, what it means, how it works, and a familiarity with network theory; at best, more just to hint at where to go, and get a flavor for what's going on here), is that reciprocity is a double-edged sword. At the same time it makes filters more complicated (every value influences every other), but it also makes clever mixing strategies possible.
Tim
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