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You most likely have a bad transistor or two or three or four or five or six in the battery charge circuit. The HP97 service manual with schematics is available on the HPMUSEUM CDROM set.
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I'm new to the HP97 myself. There is a complete service manual on the MOHPC CD-ROMs and also a user's manual for either the 97 or the 67, which works almost identically. As I recall from opening my 97, the only large resistor is the one that limits the charging current. There is also another resistor in series that is shorted out when the power switch is on, to allow the AC adapter to provide current for both charging and running the calculator. I just examined my machine, there is a 4.7 ohm 2W wirewound resistor next to the large chip (1820-1751) visible by just removing the bottom of the unit. Then on the keyboard PCB (visible when you unplug the circuit boards and remove the middle assembly), there are two large resistors. One is an 8.2 ohm 2W wirewound which I found is the one shorted out when the power switch is "on". There is also another 4.7 ohm 2W wirewound but it measures less than 1/2 ohm, it must be across a coil. I'll look at my MOHPC CDs later.
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The HP97 (and all other Topcats AFAIK) includes a circuit to protect the machine from overvoltage if the battery pack goes open-circuit
This works by detecting if the input voltage is higher than Vss (the +ve supply to the chips) and if so, it uses a transistor and resistor to load the supply (the HP charger is designed to have a high internal resistance so this works)
until the input voltage is less than Vss. The problems start if the PSU converter circuit doesn't run. Then Vss is a little less than the input voltage (1 diode-drop IIRC), so this protection circuit tries to pull the input voltage down to 0V. Needless to say it gets rather hot trying (I've had Topcats draw over 1A from my bench supply when then happens).
Check the oscillator transistor in the power converter circuit (the one connected to the little torroidal transformer on the logic board. It often fails. A 2N3904 -- but not just any NPN trasnsitor -- works here. As others have
menmtioned the service manual for the 97 (including schematics and PCB layouts) is on the MoHPC CD-ROMs.
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thank you for your helpful contributions,
the interior of my hp-97 is looking like this:
http://www.hpmuseum.org/97intern.jpg (large resistor top right)
obviously it was operated with the hp-power-adapter 3va, 8v, which i received with the calc. this might be not suitable for this hp-97 (input voltage is not shown on it)
martin