Some information is missing: +/- of what? What are the input and output of?
The nearest thing it sounds like, is a bootstrapped amplifier, where the amplifier's supply +/- are fixed relative (may be isolated supplies, or regulated from a higher total supply, using diodes and capacitors to maintain power during peaks when the rails briefly go outside the higher rails), and the output is the sum of amplifier output plus whatever's driving the bootstrap.
This isn't automatically useful, because the bootstrap driver has to supply all the current, and almost all the voltage (i.e., less the bootstrapped supply's range), of the output. Synergy can arise where the amplifiers are different designs, and the features of one are able to fill in the faults of the other.
A sort-of practical example might be, a hybrid class D amplifier, where very low output ripple/noise, and high bandwidth, is desired, but the power bandwidth at high frequencies is only modest.
So, to explain, I need to run through the overall design of such a system. Bear with me...
Specification
For example, maybe 1000W is desired for 0-10kHz, 10W is desired for 10-20kHz, and say << 1mW is desired for >20kHz (assuming a signal bandwidth of 20kHz, so we're not worrying about signal gain up there, this only limits ripple/noise). Let's say the load is 8 ohms, so nominal output is 126Vpk (0-10kHz), 12.6Vpk (10-20kHz), and <126mV (>20kHz). Finally, the damping factor should be more than 10 -- that is, the output impedance should be less than 0.8 ohms, over the whole 0-20kHz range.
Design
We could construct a class D amplifier with Fsw = 100kHz and filter Fc = 15kHz, which attenuates its >126V ripple* by say 30dB (plausible for a 2nd order filter) or say 4.7Vpk, which clearly does not meet the >20kHz limit.
*The amplifier direct output is a square wave, and should be somewhat higher than the maximum output peak -- this is so that it has some control range left during peaks. Also, the peak of its fundamental component is pi/sqrt(
times the square wave peak, so let's say, 150Vpk at 100kHz.
(Real designs, by the way, have higher switching frequency, so the passband is just fine, entirely from the class D amp, no bootstrap required. This may be more of an archaic example -- imagine back in the day when 100kHz was considered fast switching. The filtering requirements also tend to be much more lax, for a number of reasons. Self-contained powered speakers, for example -- assuming the end user doesn't try hacking cables onto it to extend it to more speakers, making a big antenna in the process.)
The filter is also say -1dB around 10kHz (its response is rolling off, either side of the Fc = -3dB point), but we can correct that with a peaking filter in the front end; the bigger problem is the response is dropping off quickly in the 10-20kHz range, so we have neither signal nor power bandwidth in this range. Actually, given that we only need -20dB of the nominal output in this range, we still have plenty of power bandwidth, but we don't have a flat response, and the output impedance is high, and very reactive (due to the filter), so we lose the damping requirement.
(On that note, I wonder how much it actually matters, as far as damping factor into tweeters. Probably, most amps aren't fantastic in the first place, and at 20kHz, it doesn't take much cabling to add enough resistance and inductance to make the amplifier's damping factor essentially irrelevant, anyway.)
So we can add a bootstrap amplifier in series, to correct for these shortcomings. It should be a power op-amp, with fast enough response that it can correct for the drop in output gain, the rise in output impedance, and at least be able to null the remaining switching ripple, if not filter it directly. (That is, we can feed in a complementary switching waveform, tweaking its amplitude and phase shift to null it out. Or if the amp is fast enough, it can do the job by itself with simple negative feedback.)
At 10-20kHz, the 12.6Vpk requirement combined with the damping factor requirement, means it should have at least a square operating curve, i.e., constant voltage up to a current limit of 12.6Vpk / 8Ω = 1.575Apk.
Note this is on top of the first amplifier's peak output, so this series amp must supply at least +/- 17.3A peak! (Evidently this is around a 100W amp, even though we, in principle, only need 10W in its active frequency band. So you can see there are significant sacrifices already.)
That should be, generally the right sort of assumption, anyway. The filter output impedance rises from near-zero at DC (the class D amp basically has an output resistance of Rds(on), chosen very low to reduce conduction losses), to a peak near Fc, then falling above there (shunt capacitor at the output). So, it looks like a resonant tank, and is inductive or capacitive either side of Fc. The impedance peak at Fc, needn't be well defined at all -- in fact if we do nothing, it's limited mainly by inductor losses and Rds(on). And consider what the bootstrap amp is doing: to correct for damping factor, it needs to act as a series negative impedance, and what's that impedance going to do against our resonant circuit? Yeah, sing like a banshee. So we need to dampen this filter with an R+C and/or R||L (maybe going to a higher order filter in the process, to better optimize Zo and ripple together).
And now we get two things: one, the filter's transition band will be softer (so, more attenuation at somewhat lower frequencies, and less at higher), and the damping elements we've added, now suck down a majority of the class D amplifier's output in this band. They're going to get hot as frig! We can mitigate this with a combination of signal shaping (a sharp cutoff above 10kHz say, so the class D amp simply isn't trying to work at all in that range?), and maybe something is possible with recovering power from those resistors (but this tends to be ineffective over a range of operating points; it's also nonlinear, but at least we can account for that with our extra amp).
Anyway, on the assumption that the filter's output impedance is stabilized to a modest impedance (mostly resistive), around the nominal 8 ohms, then the bootstrap needs to behave as approximately -8 ohms, to get the damping factor high again. And in the process, it's delivering up to maybe twice the expected amplitude: once for the 12.6Vpk it needs available in the 10-20kHz band, once again for the signal dropped across the filter's output (which mostly goes into the damping resistors -- so, this amp basically loses 3dB right off the bat, what a shame!).
Interesting optimization opportunity: since the bootstrap amp mostly does work at high frequencies, we might be able to connect it in shunt, instead. That is, we have a damping R+C across the output, but instead of making this between output and ground, we wire it between system output, and "bootstrap" amp output. Now the filter is actually a diplexer, crossing from a beefy class D amp at low frequencies, to a conventional linear (class AB) amp at high frequencies.
So, this still isn't a great motivating example, where a true flying bootstrap amp is useful. So, eh, I guess you can see why they aren't very common -- there are often other ways around it. It seems I didn't pick the best motivating example.
Other examples might include, high voltage circuitry I think? A bootstrap amplifier wired as follower can be useful. It might be controlling complementary current sources (which simultaneously carry its supply current, and their balance sets its voltage between rails). This allows, say, the input leakage of a MOSFET op-amp (< pA), with the voltage range of, like, a tube amp (100s V). Electrometers sometimes do this I think? Obviously, the amp can't respond perfectly quickly, so some careful clamping is needed to protect against ESD, both local to the op-amp (input with respect to its flying rails), and to the total system supply (input to system supply; or input to GND with a GDT or etc.; or flying rails to system rails).
The advantage is, whereas you could simply use a high voltage MOSFET source follower, a single transistor, for such a job -- you lose all the precision. Who knows what Vgs(th) is, and how it changes over time, temp and load current (and even with a CCS load, its current is still changing at AC, so there's a potential nonlinear AC gain error). Besides which, you can't get HV MOSFETs with very low and stable capacitances, so bandwidth and AC linearity will already be a compromise.
(Neat fact: back in the day, there were "electrometer tubes", optimized for very low leakage -- mainly done by using a small, widely placed structure with low operating voltages and temperatures. So the plate currents were ~uA, and transconductance ~µS. You're not going to get much bandwidth that way, but when your problem is not about doing it at AC, but doing it at DC at all, they did the job!)
Tim
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