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Are there any means or technologies that can be used to linearly (or close to linear) amplify an audio or LF RF electrical signal that do not require transistors or vacuum tube technologies? (Or equivalent tech such as ICs, photo-transistors, klystrons, etc.)

Specifically linear (or "close enough" as are the usual slightly non-linear simple tube or transistor amp circuits), so disqualifying a two state relay as a possible solution.

Were any electromechanical gadgets tried before Fleming and De Forest?

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Negative-resistance devices such as tunnel diodes can be used as amplifiers/oscillators.

There are also magnetic amplifiers, most often used for electromechanical control.

A relay has gain, but it's highly nonlinear.

Are these the sorts of things you're looking for?

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A generator without permanent magnets can be used as a amplifier. The extra power comes from the power provided by mechanically rotating the shaft. The magnitude of the output signal is then proportional to the shaft rotation speed and the field winding current. Since extra power is being added to the system via the shaft, you can get more power out than you put into the field windings. If you keep the shaft rotating at a fixed speed, the generator can be thought of as a amplifier from the field current to the output.

One problem with this is that these generators usually put out AC. That can be rectified, but will still have some ripple. If you wind the generator with enough poles and spin it fast enough, the AC output frequency can be higher than your highest frequency of interest. In that case, it can be AM demodulated to yield the final amplified signal.

Rigging up something like this to work at audio frequencies will be tricky. However, you can demonstrate the concept without too much trouble at lower frequencies. Connect a LEB (light emitting bulb) to the output of the generator and spin it to get typical line frequency out, like 60 Hz. Now you can control the lightbulb, or a bank of lightbulbs, with much less power than it takes to drive them directly.

The alternator in your car actually works on this principle to keep the rectified output of the alternator at the right voltage. The alternator spins at whatever speed it spins at. The control system modulates the field current so that the alternator output produces the right voltage to keep the battery charged but to not overcharge it. In this case the "amplifier" is being used as a voltage regulator, but the overall concept is the same. Less power is required to run the alternator field windings, which comes from the 12 V DC, than the alternator puts back onto the 12 V DC line. The extra power comes from adding a mechanical load on the engine.

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What would be the effect of using a large number of poles and a commutator (rather than a rectifier or other demodulator)? – supercat Mar 17 '14 at 19:16
@supe: Commutation is a form of rectification in this case. More poles would give less ripple and higher ripple frequency, which would generally help in allowing that noise to be separated from the desired signal. – Olin Lathrop Mar 17 '14 at 19:35
Normally when I think of rectification, I think of something that restricts either voltage or current to a certain polarity. I would expect that the aforementioned generator would be able to handle voltages and currents of either polarity equally well. – supercat Mar 17 '14 at 19:52
@supe: Today most people think of rectification as something you do with diodes, but it didn't used to be that way. The earliest power utility systems were DC. This was produced with synchronous rectification by commutation. That is also how a brushed DC motor running backwards as a generator makes DC instead of the inherently AC produced by magnets moving past coils. There is a wide world out there beyond diodes. – Olin Lathrop Mar 17 '14 at 21:14
Certainly I'm aware that things like rotary converters were used to convert AC to DC before diodes were invented, but here the goal is to end up with an AC waveform. The voltage on any particular coil would represent the original signal ring-modulated with a sinusoid generated by the rotation, but the purpose of the commutator would be to add or subtract combinations of modulated signals so as to reconstruct the original AC signal. Using the term "rectification" to describe a process yielding an AC signal seems unusual. – supercat Mar 17 '14 at 21:37

A pure mechanical audio amplifier called an auxetophone uses an air compressor and a gate that is mechanically modulated. This video shows a recreation of an auxetophone where a solenoid is used to move the gate, but it can be mechanically moved.

In a similar manner, the gate could be replaced with an adjustable power resistor, so you can build a mechanical amplifier that converts the small signal to mechanical movement, which then modulates a much larger power source.

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A transformer can provide gain i.e. a "step-up" transformer can magnify the input voltage by the turns ratio but it isn't a power amplifier. They can be fairly linear.

A tuned circuit (L & C) can also amplify a voltage signal by many times but it is limited to a small band of frequencies close to the resonant point. Beyond that frequency signal output levels fall off.

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Parametric amplifiers can be made with nonlinear capacitors.

In the early days of radio, a device called the coherer was used.. It could be described as a detector with gain. Not linear at all.

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There were various devices in the early history of instrumentation that provided something like gain without actual power gain. For example, the optical lever, the galvanometer, the thermopile.

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