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This page basically already exists here. And it's probably more straightforward. And it's bilingual (German / English).

So there's few chips on Atari's Asteroids (and some similar boards) that you can't get new. Or at all. Or at sane prices.
Except for the DACs. The AD561 is hard to get, only pops up every so often and when it does, it does so at eyewatering prices. And they stopped producing them long ago.

So I set out to find a replacement, that's still being made and get it to work in an Asteroids (after determining its DACs were both shot).
And the winner is: AD7533JNZ. At 16 Eur they aren't exactly penny articles, but still better than what the AD561 goes for on the Bay.

Now keep in mind I'm self-taught and only studied what interested me or what I needed to get stuff working.
So I failed at getting a bipolar output from the AD7533JNZ which means the picture is gonna sit in a corner of the screen and be rotated 180 degrees as well.
So I use a TL084 (you should use TL082 as only two of the op-amps are needed, but that's what I had) to shift and amplify the picture.

Here is the schematic. Note the capacitor between pins 2/3 and 15, it's pretty important (but its capacity isn't, I encourage you to try out different ones - my prototype actually uses a 0.1µ, I guess bigger is better until they start getting too slow). Without it, the lines will be terribly squiggly.

asteroids_dac_adapter.png

And btw. if you are my Physics teacher in 9th grade or when it was that we had to draw schematics - if a line on a schematic goes from one component to another and just barely fails to touch the pin there, it counts as a connection! Hey I wanna see you draw a perfect schematic under time pressure! Also I'm lazy.

The 561 is the same orientation as the 7533 and what's also not clear is that the output from the 7533 doesn't go to the Op-Amp, but back into the board. The Op-Amp is unrelated to the DAC (more or less). The Op-Amp provides centering, axis flipping as well as a bit of amplification - and the Op-Amp portion of the above schematics expect you to replace both DACs with a 7533, otherwise you'll have to only connect one axis to the Op-Amp. What sucks about me and my diagrams is I imply more common sense than I had like 10 years ago.

Preparation:
First you have to remove the original DACs and put sockets in their place if there aren't any yet. You'll also have to remove the X- and Y-Axis test clips on the top edge of the PCB - these actually serve as jumper wires, so when they're removed, the signals won't reach the monitor anymore and we can intercept them, manipulate them and send these signals to the monitor.
Also you need a +15V source which unfortunately isn't among the pins of the original 561, so you'll need to solder a wire once you're done constructing that contraption up there - twice (one for X, one for Y).
Take note that of the Test clips you removed, the upper contact next to the board edge connects to the monitor (where the Op-Amps output connects) while the lower one carries the signal output from the board's op-amp, so that's what goes to the negative input of the Op-Amp.
Also center the centering pots before testing.

Errata:
It looks like on the Op-Amp input instead of a 1k resistor I use 2k2 and I use them on both channels (one is missing on these schematics). And on the feedback it appears I have 3k3 on both X and Y. Probably the image was pre-scaled for 4:3 by the other op-amps on the board already.
 
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