Power Circuit (Atari Lynx II)
The power circuit of the Atari Lynx II is a ground switched circuit. Unlike most consoles where the positive voltage rail of the system is normally switched on when turned on, the Lynx II is different and it disconnects the battery/DC ground from the rest of the system until turned on.

The entire circuit is quite complicated and a full breakdown of how it worked is best explained in this YouTube video.
Forcing Power On Without Game
Some models of the Lynx II have a switch between pin 33 (5V) and pin 31 (ROM detect) of the cartridge slot.

When a game is inserted the cartridge shorts pin 31 to 33 and provides power to the U6 hex inverter chip on the power on circuit, which in turn controls Q12 enabling power on.
To bypass this part of the circuit (just the U6 hex inverter and requirement for a game inserted), short pins 31 and 33 of the cartridge slot.

Your console should now power on without a game inserted and show INSERT GAME on the screen if everything else was working fine.
Bypassing Entire Power Up Circuit
Before you get into diagnosing a faulty power circuit, faulty ribbon or many other parts that constitute getting the power circuit to work, you can do a very quick test to power up the console without needing any ribbons, screen, or even a working power circuit.
This test helps quickly identify if the console is working with the exception of the power up circuit or components.
Get a bench power supply and set it to 5V. Connect the ground lead to the shield or one of the solder pads where the shield used to be.
This is important is the shield ground is after the Q12 switching transistor which bypasses the power up circuit.

The shield is on the green trace side, while the battery springs and DC jack are on the blue side which requires a fully working power on circuit and ribbons.
Now attach your red lead (making sure your bench is set to 5V, not 9V!), to the battery positive spring.
This spring goes through the 9V rail which gets lowered to 5V by the power on circuit. By setting it to 5V in the first place and bypassing the ground, we do not need anything to work in the power circuit to power on the console, we are simply providing a regulated 5V directly to the system.
Once power is applied the console should boot up fully, it should show INSERT GAME on the screen, or load the game if one is inserted, and audio should play.
Power Draws From Bypass
If you have just a speaker connected, no LCD, and bypass the power circuit as shown above, a generally working console should draw around 200mA.
If you connect a screen it will jump up to 350-400mA, and anything above that in general is likely a component failing, shorted or faulty.
If you have excess current, use a thermal camera or IPA to detect where the short/fault is.
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