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This circuit is a radio receiver.
The above picture is quoted from here and here. My question is in the diagram.
It looks like a transformer symbol.
Is it even essential? Can't we use a transformer and just use the inductor (highlighted in blue in the picture) as our antenna?
We can just use an op-amp as amplifier instead of a transformer to amplify voltage, right? It seems the author of the video does not use a transformer, right?
Consider what is attached to the other side of the transformer.
If the rod by itself ("loopstick antenna") is sensitive enough for the receiver chip, you might not have an aerial connected at all. (This does seem to be pretty standard for AM radios I've seen.) In that case, it does nothing, and can probably be left out.
If you do use an aerial, its impedance will be extremely high (essentially the reactance of its self-capacitance), so it must be reduced via transformer or other matching network to suit the receiver's input resistance. Presumably, the aerial side has more turns, giving it a higher impedance (inductance of both sides, or turns counts, are not given on the screenshot, so this is just a guess).
Transformers have other features as well, though they aren't used here: for example, galvanic isolation, or to create a double-tuned circuit (only one side is resonant here). So that leaves just the above explanation.
There may be other requirements specific to the IC, which the datasheet or related documentation can explain (granted, perhaps not very well: in which case, asking about that separately might be a good idea).
One thing to know is that transformers do not amplify. The term amplifier is usually applied to a circuit or device where there is the capability of a net power gain (see this discussion), the product of the output voltage and current can be greater than the product of the input voltage and current. In a transformer the output power is always less than the input power (in an ideal transformer they would be the same, but real ones have losses).
The 'transformer' in the schematic is a loopstick antenna. It is two coils of wire on a ferrite rod. The reason it is wired as a transformer is for impedance matching. Sometimes you will see it done with a single coil that is tapped near one end to create an auto-transformer. Matching the antenna to the RF amplifier input impedance can also be done with a capacitive divider, or an LC network.
Video clearly talks about LC tank circuit, inductors and capacitors, and shows the formula for resonant frequency.
$$f_R = \frac {1} {2 \pi \sqrt {LC}}$$
This is the circuit from the referenced data sheet. It does show a transformer used, so it is clear where you got your misunderstanding, but it is there mainly as an inductor, not as a transformer.
It's also clear why the manufacturer used it (because they are easier to obtain than making an inductor). But the selected transformer characteristics are not in the data sheet (aside from inductance), so it is completely understandable why the youtuber went the way they did.
This is the circuit, the youtuber implemented. Notice the transformer has been simplified to an inductor symbol. In the video, he then has to search for an appropriate trimmer capacitor. This is closer to the classic AM radio, with wire wrapped around a ferrite rod to form the inductor L and a parallel plate air capacitor connected to a dial form a capacitor C to tune to frequencies in the AM frequency band. His trimmer capacitor will have less range, but that is not the point of his video.
From Wikipedia.
As used (by the youtuber), the copper wrapped iron rod, would be better referred to as an inductor or choke.
Any transformer/inductor AND capacitor connected in parallel form a resonant frequency, tank circuit or tunable circuit to select a specific frequency via the resonant frequency formula, but proper selection of L and C, will select a frequency in the AM radio band from 540 kHz to 1600 kHz.
Transformers are used with RF for many tasks.
Filtering, impedance adapting, oscillators ...
At RF high frequencies, the op-amp is not really the best choice for amplifying, even if some can do it ...
