I feel dumb for asking this but i have a question. I have a step-down transformer which steps 220v down to 12V which is further step down to barely 3V. I want to understand how Arduino will use 3V to display 220v on my LCD. Does it make sense?

What Arduino do you have? Is it a 5V or is it a 3.3V board? How is the 12V stepped down to 3V? It is 3V DC, right?

Used a step-down transformer with a rectifier to being it down to 12V DC

We can't help if you will not answer questions.

I feel dumb for asking this but i have a question. I have a step-down transformer which steps 220v down to 12V which is further step down to barely 3V. I want to understand how Arduino will use 3V to display 220v on my LCD. Does it make sense?

A transformer’s output voltage and input voltage are in a fixed ratio. If you know the ratio and the output voltage you know the input voltage.

You didn’t explain how the 12V is stepped down to 3V though.

Do you want to measure the 220V (AC I assume) or do you just want to know if the power is on or off?

Considerations -

An actual transformer is AC-in and AC-out and the output voltage (secondary) is proportional to the input voltage (primary) so it CAN be used to measure power line voltage.

You'd need to rectify and filter to DC with a diode and capacitor (the Arduino can be damaged by negative voltages) and you'd need a voltage divider (2 resistors) to knock the voltage down to 3V or less.

Note that the peak of a sine wave is 1.4 times RMS so a 12V transformer will give you about 16VDC after rectification & filtering, allowing for the ~0.6V drop across the rectifier. Also, a "12V" transformer will usually put-out more than it's rated voltage under no-load or under light-load. So you could end-up with 20VDC or so and you'd have to measure and "calibrate" your readings (in software).

A power supply puts-out DC and most are regulated so the output is NOT proportional to the input. It can be used to detect the presence of 220V but it can't measure it. (A transformer is part of a power supply.)

It is a 5V board and the voltage is being reduced to approx 3V by voltage divider circuit. I’m interested to know how the ADC will work in the current scenario.

Transformer 3 burns out analog pin 3... No Arduino pin can tolerate any negative voltage lower than -0.3V. Feed AC to the pin and its zapped instantly.

You've missed ground and Vcc connections to the Arduino in that circuit, BTW.

It is a 5V board and the voltage is being reduced to approx 3V by voltage divider circuit. I'm interested to know how the ADC will work in the current scenario.

OK, assuming 0 to 220 VAC now is equal to 0 to 3 VDC. This after all your transformers and rectification and also assumes filtered DC. Using a 5.0 Volt referenced ADC like an Arduino Uno for example which has a 10 Bit ADC (2^10) tells me 0 to 5 Volts is measured as 0 to 1023 bits or 1024 quantization levels. That's a resolution of 4.882 mV per step. Three Volts would be 614.5 bits.

The simple way would be to use a MAP Function:
https://www.arduino.cc/reference/en/language/functions/math/map/

That or do the math in your code.

Again, this all assumed you have exactly 0 to 220 VAC = 0 to 3.0 VDC and all the change is a linear curve.

Ron

And assuming that Vcc (default Aref) is exactly 5.00V.