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Measure battery voltage with a meter: at least about 12.4 volts with the engine off, and about 13.5 to 14.5 volts while running. Do not disconnect a battery cable with the engine running, because the voltage spike can damage the alternator diodes. If running voltage stays near 12 volts, replace the alternator with one that matches the car's wiring rather than only the swapped engine.
The owner has a Honda Accord with a 1998 Honda Prelude VTEC engine and a non-VTEC alternator from a 1994 Accord, and the battery is not charging. One reply suggested disconnecting the negative battery cable while running, but later replies warned that this can spike voltage and damage the rectifier. Experts advised checking open-circuit and running voltage and using the alternator that matches the vehicle wiring.