I have tried to find schematics on the RFID RC522 board which is used in many tutorials/examples of Arduino connected to RFID RC522. I am planning to use a modified RFID-RC522 board where the antenna is downsized a bit. There are a number of different schematics of that board with different naming on the components. But so far I have not find a schematic exactly like the one below.
Note that if you are using this reader with a 5V Arduino then you will need potential divider circuits on all outputs from the Arduino to cut them down to the 3V3 signals that board needs. Like this:-
Note that your photograph has very poor resolution and is posted upside down so it is impossible to see what your connections are.
Sorry this is not the whole of the problem because if it were then antenna design would be simple and it is not.
What makes you say that? Many people believe many things, it doesn't make them true. An antenna has to resonate at a specific frequency, the one you are trying to send.
To get resonance you need both an inductive and a capacitave value. So an inductance is less that half the story.
Why do you want to do this? Is it for saving space? A smaller antenna will produce a shorter range.
The RC522 works by sending out a signal at 13.56MHz as an excitation frequency to the tag or card. In reply the tag sends out a reply at one quarter of the excitation frequency which is 3.39MHz. So any antenna has to be sensitive to two frequencies. Which makes antenna design a bit tricky if you only want a to use a single coil.
You can make a two coil solution and that would give you a bit of added range.
As I said I can't read your low resolution photograph. But on the normal one I have has eight pins and reads from left to right:-
SDA - SPI data input
SCK - SPI clock input
MOSI - Master Output / slave input
MISO - Master input / slave output
IRQ - Interrupt request - not normally used
GND - The Arduino ground
RST - Rest
3.3V - Supply voltage
I once made a project that used three of these RFID readers connected to a Raspberry Pi. It is in The Mag Pi 47 download it from here for free:- Download issue 47
The schematic is more complex than just a single reader but In think you can get the idea. This is the video of the project:-
Some videos you find on the internet falsely claim you can do read and write operations only on a writable RFID card. That is only if you want to clone a card.
I 'm about to implement an identification system of a moving object. I came across this article of Dr. Rong-Hao Liang (OpenNFCSense API GitHub - howieliang/OpenNFCSense: Open NFCSense Library) which would suite me just fine. The optimal size of the antenna would be about 30mm X 30mm.