I started off with a schematic found in an article on the website of German c't magazine.
This RFID tag finder is based on a Clapp oscillator which generates a 13.56 MHz sine wave. The inductor of the oscillator is formed by a loop antenna, which I constructed by winding 5 turns of insulated wire on a spool of fishing line that I had lying around. When the circuit is powered on, the loop antenna radiates a electromagnetic field. When a RFID tag is placed near the loop, a current is induced in the tag's antenna, which will draw power from the oscillator, decreasing the amplitude of the sine wave.
The original schematic detects the presence of a tag by monitoring the supply current of the oscillator, which develops a voltage drop across a current sensing resistor. However, this voltage drop is quite small, and is also dependent on hands or metal objects in the vicinity of the loop antenna.
I modified the circuit to measure the amplitude of the sine wave directly, eliminating the current sensing resistor and adding a envelope detector circuit, consisting of a diode, a capacitor and a resistor. The resulting voltage drops significantly when a tag is present, but doesn't change as much when hands or metal objects are placed on the antenna.
A voltage comparator switches on a LED when the voltage drops under a certain reference level, which is adjustable by means of a potentiometer. The level is adjusted as to light the LED only when a tag is present.
The result is a detector which is insensitive to hands or metal objects, but can detect RFID tags when placed within 3 or 4 cm distance from the loop antenna.