Check for correct voltages first.
Then put this little circuit between the ZX and the RAM pack: Now the ZX should run with its internal RAM even if the pack is faulty. If not, there is a serious problem with the pack like short circuit on address lines or so. The Memopaks can do so if the PAL is dead.
The pack now is addressed from 32768 to 49151 and can simply be checked with POKE and PEEK. Test with 255 and 0 and you see which bit(s) will not work.
Once if the ZX is running you can do a lot of checks at the pack.
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16K byte RAM packs normally use 4116 (or equivalent) DRAM chips. These need three supply voltages. See this post for a pin-out.
The +12V is the main supply. The +5V is for the chips input/output circuitry. The -5V supply is the bias supply. Loss of the -5V supply will cause faulty operation and may cause internal damage to the DRAM chips. Hence this is the first voltage to test. If this is okay, test the other voltages.
The Timex Sinclair 16K RAM packs are the same as the Sinclair 16K RAM packs. There are a number of different versions of the PCBs and the circuitry. We only have a ‘full’ schematic of the first version. You may find that you have the issue 3 version (it appears to be slightly more common).
To get a +12V supply and a -5V supply from the nominal +9V from the edge-connector (which is a direct feed from the +9V power connector), a transistor converter/inverter circuit is used. A transistor coupled to a small transformer (referred to tha the ‘coil’) works as an oscillator. The resulting AC is then rectified and smoothed (filtered) to produce the +12V and -5V supplies.
The +5V supply comes from the 7805 voltage regulator in the ZX81 / TS1000 via the edge-connector.