fm-am transmitter/receiver

Yes it is not a perfect FM receiver, but it will work

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I thought the transistor configuration would keep the gain low...

Yes, the receiver is a Regen that oscillates. It also has an AM detector, not an FM detector. If you turn down its positive feedback so that it is almost oscillating then its RF gain is fairly high and if you tune it to one side of the transmitter's frequency then its AM detector can "slope detect" the FM and any AM noises. I think a regen receiver is used in a kid's cheap walkie talkie.

Regen is short for regenerative. This type was used in the early twentieth century to receive AM broadcasts. It's main advantage was simplicity while delivering high gain using as few tubes as possible. This is the first time I have seen someone apply the concept to receiving FM, and my impression is that this particular receiver circuit is supposed to operate well into oscillation, whereas the old AM regen sets were usually tuned to nearly oscillate but not quite (at least for best performance). This design will definitely send out a strong carrier because the regen oscillator is connected directly to an antenna, so this is certain to interfere with other people trying to listen to their FM radios. For that reason alone, I think this is a bad choice for an FM receiver. Add to that the fact that it will not likely receive FM very well, and I think that other alternatives have to be considered before you finish your project.

I recommend that you consider getting an FM receiver IC. Perhaps one of these types would suit:
TDA7012
TDA7088
Si4840
BK1079
Perhaps audioguru can suggest another.

I am not up to date on FM receiver ICs because most are tiny surface mount.
I was given an "FM radio" from The Dollar Store. It came with a battery and headphones that are worth the dollar so the radio circuit with the TDA7088 IC was free. It was severely overloaded with local stations so it could not receive ordinary stations all around me. Its station scanning missed many stations. It was noisy and distorted.

Something like that dollar store receiver is still preferred over the regen receiver. Finding one of those seems like a good idea.

Should I just be able to band pass the signal I want?

Also, I rebuilt the receiver on a breadboard. It's not transmitting a frequency anymore.
For some reason my audio amplifier is getting hot? Not seeing any shorts though.

There are some things about receivers that you need to know. A radio receiver has a difficult job to do. It has to amplify and demodulate your desired signal while at the same time excluding all other signals that may be nearby in frequency.

The sensitivity of a receiver is a measure of its ability to deliver the modulation from a very weak signal that comes in from the antenna. How weak? A good FM receiver should be able to deliver the audio from as little as 1 uV signal (1.33e-14 watts). If we guesstimate that an FM demodulator needs 50mV to operate correctly, then we can estimate the gain necessary would be about 94 dB. This is a huge amount of gain and is almost impossible to realize in a single stage RF amplifier or a single stage AF amplifier. So FM receivers get a bit complicated by the amount of gain needed.

Now, if you have this much gain and nothing else but a demodulator in your receiver, it will be inundated with all of the FM signals in the broadcast band. There will be massive interference because all of the FM channels will be amplified and fed to the demodulator at the same time. You need to filter out the undesired signals while allowing reception of the desired signal. You can't do this at RF because the filter required would be very narrow bandwidth and very high Q. Your signal bandwith is 200KHz, while the RF frequency is, for example, 90 MHz. The resulting filter cannot be built using a simple LC tank circuit because the Q required would be 500 and single stage LC tanks cannot do much better than about 20 or 30. Perhaps one could build an LC ladder filter, but I fear it would be impossible to realilze a Q of 500 practically. If such a filter were practical (using a SAW device or large resonant cavity for example), it would not be helpful because you would need one filter for every channel you wanted to receive, and there are many in the FM band. So, we have to find another way to filter.

In the past, filtering had traditionally been done using a superheterodyne architecture. In a superheterodyne receiver, the desired channel is translated (moved in frequency) to a fixed intermediate frequency using a mixer and variable frequency local oscillator. At the intermediate frequency (IF) it is much more practical to filter the channel with a fixed bandwidth IF filter. Also, because the frequency is much lower it is also much more practical to apply a lot of gain without suffering instability (undesired oscillation of an amplifier). The demodulator is also optimized to function at the IF frequency, a much easier task than an RF demodulator.

Modern receivers don't use superheterodyne as much any more because there is a better way. That way is called "direct conversion" and it works very similarly to superheterodyne except that the IF is actually set to zero. In other words, the incoming signal is converted to zero frequency instead of an IF. This has only become practical in the last twenty years or so thanks to integration of direct conversion (sometimes also called "zero IF") receivers inside integrated circuits. The technical difficulties in making a zero IF receiver function correctly in an IC were immense, and included problems such as "how do you demodulate FM when direct conversion folds the sidebands on top of each other at baseband" and "how do you prevent re-radiation of the on-channel LO". Well, these are topics for a different day. Suffice to say that converting to baseband rather than an IF means that you can put most of your gain at baseband (which is easy) and you can put all of your filtering inside a DSP or hardware low-pass filter rather than an RF or IF bandpass filter. This is easier too, and can all be done on-chip, unlike the IF filter of a superheterodyne set.

So nowadays we have fully integrated receivers-on-a-chip. But they are complex and include an RF amplifier (the LNA), two down converting mixers fed by a synthesized variable frequency local oscillator, two multi-stage AF amplifiers, two sets of low pass channel filters and two sets of demodulators or A/D converters.

I mention all of this to help you realize that it is not trivial to receive FM in a useful and practical receiver and if you want to build one that works reasonably adequately, you should probably find a receiver chip and learn how to make that work. Alternatively, you should find a scrap portable FM radio because even the poorest portable radio that you can probably find for free in a scrap box at a garage sale, for example, will work well enough.

Would this work?

How mission critical are the capacitor values?

Also is there a DIP version of this?

This TDA7021T looks like a reasonable choice and I see no reason why it would not work, if you build it correctly.
The capacitor on pin 1 forms a low pass AF filter, which doesn't appear to be too critical, but I wouldn't vary the value more than 50%.
On pin 2, you should include a 10K resistor to Vp in parallel with the capacitor to disable internal muting. I think the cap value on that pin is not very critical in this case. On pin 4, stick with the 10nF cap. On pin 5 you have your LO tank circuit so these are critical components. The L and C must resonate to the 88 to 108 MHz range and be tunable with a variable cap.
The cap on pin 6 appears to be simple bypass at IF so value is not super critical but should probably not vary more than a decade from 100nF. The caps on pin 7 and 8 are somewhat sensitive to value and you should match their values for these. The cap from pin 10 to pin 11 is a carefully chosen value, so stick to it. The cap feeding into pin 12 is DC blocking and not critical, but make it a ceramic type. The pin 13 cap should match the one used on pin 12.
Pin 15 should stick with the value they have.

Make sure you review the data sheet for this part. You can find one here: http://pdf.datasheetcatalog.com/datasheet/philips/TDA7021T.pdf

Many of these FM receiver ics have become obsolete so you may have some trouble finding this part. It is not likely you will find any FM receiver ICs that are DIP versions. Surface mount is preferred for RF applications, so DIP simply isn't popular. This part can be dead-bugged fairly easily, I think, on a piece of double sided pcb material.

I'd at least like to understand what is wrong with my assembly,

I wish there were more info available on line describing the theory of operation of your simple receiver. I've only found one so far that tried to explain it, and that was still not good enough to fully understand.

Could it be that my antenna is poor

Receivers like this are very tolerant of many kinds of antenna. Your wire is fine, but the length should ideally be twice as long. However, if you don't hear any stations with this antenna, a longer one will make only a tiny difference and will not be noticeably better, so don't worry about the antenna.

So, to understand what is going on, I built your receiver as you have on your schematic. I have not included the LM386 or processor, just the two-transistor rf receiver. I'm using two 2N3904 transistors for this. I'm just getting a feel for its operation now, but I can report that it does indeed receive local FM radio stations. There are a number of problems that reinforce my opinion that this is not a good receiver.

The first major problem is that the frequency stability is hopelessly bad. Any movement of my body near the antenna changes the frequency a lot. Any time I touch the variable capacitor with my tuning tool the frequency varies. Any change to the AF output load changes the frequency a lot. On top of all this, the frequency drifts slowly away from the channel I am trying to hear.

The second problem is that linearity of the audio is fair to poor and is highly dependent on fine tuning the rf frequency to find a sweet spot with lower distortion. Since the frequency moves around so easily, this is a major headache.

Sensitivity and selectivity are actually useable as is. The regenerative action of the circuit multiplies the Q of the tank circuit so that we only receive one broadcaster at a time. Sensitivity is not great and this is with a 15 inch wire antenna lying on the bench but is adequate for local stations.

I'll report some more later.

The amount of AF output I get is about 1 mV, roughly. That is a very low level and I think you need a gain of more than 20 following it to work well. When you use higher gain audio stages, you can run into problems with AC hum if your grounding is not good. My AF amplifier has a gain of around 300 and I'm fighting the AC hum and losing to some degree.

I've played around with this receiver a little bit. The way it works is that the oscillation frequency of the receiver is pulled, sympathetically, by the incoming RF signal. This is also referred to as injection locking. If you tune the tank circuit so that the off-air signal is off to one side of the LC circuit's resonant frequency, you are basically causing the amplitude of the oscillation to go up and down in sympathy with the FM modulation on the received signal. This is slope detection. Since the oscillator is operating non-linearly, it functions also to down convert (or AM detect) the envelope of the oscillation and this baseband AC is coupled out via the output capacitor.

This circuit is awfully crude and, I think, only exists as an experimental curiousity. If your goal is to have a functioning data receiver, this circuit ain't gonna do it.

I have attached a picture of my receiver board. I also have a 2 minute sound file with the audio output from my receiver. PM if you want a copy of that and I can email it.

The lm386 does have good amplification when it works.

Sorry to hear about your impending loss of sanity. The first thing I want to mention is that the 22K ohm resistor is critical to limiting the current to the receiver and in fact is the only thing that really does so. The circuit will not oscillate without it, I think, and if you had no resistor there for a while, I'm surprised your transistors are still ok. They should not warm up at all.

Perhaps you would improve the circuit reliability if you built it on a piece of circuit board such as my example. This insures that connections won't give you problems and is quite strong, allowing you to clip test leads to the parts without breaking connections.

There isn't very many parts so construction of this thing shouldn't be too much of a challenge to get right.

Some amplifiers (not sure if LM386 is one of these) will break into oscillation when gain is high because of parasitic capacitance resulting from using a plug-in prototyping board rather than pcb or dead-bug assembly. When oscillating, an amplifier often draws quite a lot of current, making it run hot, and because the oscillation may well be above 20KHz, you may not even hear that it is doing this. This kind of thing may be intermittent. You should be able to spot this problem by viewing the output on the scope.

An ideal FM demodulator outputs noise with a triangular frequency response. By this I mean to say that its noise output level increases with audio frequency. So, an FM demodulator used for audio is always immediately followed by a "de-emphasis" circuit. This circuit is nothing more than a simple low pass filter. Usually, we use a series resistor followed by a shunt capacitor. For broadcast FM receivers, the values should provide a 75 uS time constant (multiply R x C = 75 uS). The use of de-emphasis obviously affects the desired audio so to compensate for this de-emphasis filter, the audio transmitter in FM always has a "pre-emphasis" filter to compensate for the presence of the de-emphasis filter in the receivers. Its up to you to decide if you want to use such filters with your data transmitter and receiver as they may not be necessary, but without the de-emphasis you have to live with some additional treble hiss in your output.