Simple AM Radio Circuit

What is the purpose of C1 and the tap on L1?

C1 massively improves the tuning quality-factor so that you can more selectively tune to the required radio station with co-channel interference being avoided. As with any crystal radio set, the antenna is always between one-fortieth and one-tenth of a wavelength (to avoid excessive mast height) and, the antenna looks quite capacitive.

In fact a "short" antenna is too capacitive for selective tuning of a single channel so, you add series capacitance (thus lowering the capacitance) to make the tuning more selective.

The lower section of L1 is the tuning inductor that resonates with the now-lowered antenna capacitance.

The L1 tap is an impedance matcher and voltage amplifier for the detector and high-impedance headphones. Without the "step-up" ratio you won't get enough signal through the diode and, you won't be able to hear very much.

C2 makes the whole tuned circuit "tunable" to different channels/stations.

For one thing, C1 helps prevent disaster when your 20m high outdoor antenna falls on a power line.

Apart from that, you can eliminate the capacitor and adjust the tap point on the main coil toward ground, or further up the coil to give strongest signal. You may have to adjust C2 at each tap point to peak the signal.

22 picofarads seems rather small for C1, but may have been optimum for the author's antenna, coil, tap-point, and audio transducer. In the past, the broadcast band was full of signals, so that one signal of many should dominate the audio output. Selecting one-of-many signals may have been an important goal of the designer.

Today, fewer signals compete, so selectivity may be of less concern. These power-less crystal radios give feeble audio if a transmitter site is far, so you might regard sensitivity of more importance - in that case, make C1 larger, and search for the best tapping point.
The goal is to provide a large RF signal to the diode: a large amplitude is more efficiently detected, producing louder audio.

This undefined tapped coil boosts the AM band voltages while suppressing adjacent noise as a high-Q Band-Pass Filter (BPF). The current through the coil boosts the voltage on the larger side. The speaker's load would be a very high-impedance crystal earbud. This 2nd order filter is not as high BW or quality as double-tuned BPF filter with a mutual coupling ferrite slugs between two fixed coils. But it serves 3 purposes in 1 stage as a direct AM BPF, rectifier, ACDC detector with a lower series output impedance at resonance than the load.

Case 1

On the other hand, if the coils have the right values, it is possible to detect both AM/FM voltages and convert FM to AM using the filter's slope. Performance would suffer with this compromise.

Case 2: