Is it possible to use the power from the headphone jack line out to
amplify the signal in order to get the speaker to be louder?
Here's the thing: The power in the headphone jack is in the audio itself. The power in a normal wall outlet is 50/60Hz AC. The power in a headphone jack is also AC and in the frequencies that you want for your audio! There is no additional power contained in a headphone jack other than the audio itself.
So, if you could somehow convert the power in the headphone jack into something useful and convert it back into a form for driving a speaker, and could do it with 100% efficiency then you would end up with exactly what came out of the headphone jack in the first place.
In short: You can't amplify the power without another source of power. But all is not lost!
What you really want is louder audio, and a way to do that is to make your speaker more efficient. Speaker efficiency is usually expressed as "sensitivity" and is specified as XX dB at 1 meter at 1 watt. A normal speaker is 75 to 90 dB (@ 1 meter @ 1 watt). It might not look like it, but 75-90 dB is a huge range! You can simply get or build a speaker with better efficiency.
Another thing that effects efficiency is the speaker enclosure itself, and the type of speaker. Look into horn speakers. There are horn speakers that have sensitivities up to 105 dB (@1m@1w). Although these will likely be too large for your application, it might be interesting anyway.
Driving an 8 ohm speaker with a headphone jack is questionable. Some devices will do this just fine. Some devices might distort but otherwise be fine. And other devices might be permanently damaged. Unfortunately there is no good way to find out without just trying it.
In theory you could make an impedance matching transformer that would show a proper load to the headphone amplifier, but who's output is good enough to drive an 8 ohm speaker. Just keep in mind that a transformer will not increase the power, only convert the form of that power so different devices can play nice together. I have not seen such a transformer for sale, but I also have not looked for one. But in theory you could make one.
Given all the different issues, if it were me I would figure out a way to make a small battery pack with an amplifier in it. Yes, it's bigger than what you want. But it will probably be the best compromise between volume, size, cost, and robustness.
Let's consider the headphones.
- Earbuds. No need for a dedicated amp since earbuds audio quality is terrible anyway.
- In-ear rubber-nipple thingies. I picked a random one off the net: Sennheiser CX300 is 18 ohms and delivers a whooping 118dB for 1V RMS.
- Real headphones. OK, let's pick some not too expensive ones... HD4.20 again 18 ohms and 118dB for 1V RMS.
- Some cheaper ones HD2.10 26 ohms and 110dB/1VRMS
Sooo... if you use the latter ones, listen to an average level of 80dB which is already loud, with peaks around 100dB (if your music is not too compressed), then:
- RMS voltage will be -30dBV, ie 31mV RMS
- Peak voltage around -10...-3 dBV (like half a volt)
- Peak current won't be huge either, like 30mA.
Now, let's look for an opamp.
The "maximum output current" specification is useless here. What we want is an opamp that can output the required current with the least amount of performance degradation. An opamp picked at random may have good distortion figures, but when driving relatively high currents (for an opamp) with low headroom, it's an entirely different story.
For example, TL072 isn't so bad with a 10kOhm load, but if asked to drive any kind of actual current, its performance simply ceases to exist.
The 32 ohm speaker at 3.7V would have about 100mA peak current, which is too much for a headphone.
You will go deaf before the voice coil burns.
The resistor which everyone puts at the output of headphone amps actually degrades performance, but it has a rather important purpose: protect the output stage when the output is shorted.
...and every time you insert a 3.5mm jack into the connector, the output will short during insertion. This is a "feature" of this connector, you gotta live with it. If the connector is mis-inserted, the short can be permanent. You must select an opamp that will survive this, so you can get rid of the resistor for better sound.
(Explanation: headphone impedance varies with frequency like every loudspeaker. With a resistor in series you got a voltage divider. Voltage on headphone thus depends on its impedance at the frequency of interest. So you get bumps and dips in the frequency response).
You mention a discrete amp. With such a low supply voltage, discrete design is very complicated. An opamp would offer much better performance.
For example,
OPA1688 is a nice opamp with high output current and rail to rail output; it is specified to drive headphones. Beware the input is not really rail to rail, it should not go above Vcc-2V, but this should be OK.
If you wanna go real fancy, you can use a nested feedback with OPA1652 as frontent, and a beefy fast opamp as the output driver.
Best Answer
To start with you really need an oscilloscope to look at the range of signal that comes out of the receiver and goes to the head phones. Can you see bursts of signal that correspond to the "beeps" with little or no signal in between?
Secondly you need to conceptually decide how the beeping sounds will be "felt". We have no idea here of the duration of each beep nor what rate the beeps occur. It is highly likely that there cannot be direct conversion of a single beep to a powered on pulse to the vibrator motor. The motor likely has a response time that is more typical for the ring duration of phone which is in the seconds range.
Unless you understand the parameters here it is not time to start building anything. The beep characteristics and the motor response time will fully dictate the type of circuit design that would be required to achieve the vague goal that you have expressed.