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If you have used Boss pedals long enough, you probably have noticed a strange inconsistency in how external power supplies are used with various Boss pedals.
When Boss introduced their compact pedal line around 1978, they designed an unregulated 12-volt DC external power supply to power the pedals called the ACA adapter. Components inside the pedal would drop the operating voltage to 9 volts DC. As the Boss pedal line grew, so did the power draw of most of their pedals. It was decided that a new power adapter was needed and it was redesigned as a 9-volt DC unregulated power supply. In these newer pedals, the voltage-regulating components were removed. This new power adapter is known as the PSA adapter. These two power adapter configurations coexisted until late 1997 when Boss decided to make their entire pedal line PSA adapter compatible. To add to the confusion, they also revamped the ACA power supply to be 9 volts. It's a big problem for people who own a pre-1997 Boss pedal because they cannot power their pedals with the new ACA power supply.
Just to confuse matters even more, an ACA type pedal will usually work with a 9-volt PSA type adaptor if it is used in a daisy chain configuration with some other "normal" 9-volt pedals. This is because the ACA pedals use a shared ground that bypasses the ACA power circuit.
Older ACA pedal (Boss DF-2)
Newer PSA pedal (Boss LS-2)
So, check the sticker on the bottom of your Boss pedal. If it says PSA, you can use just about any 9-volt DC unregulated power supply. If it says ACA, find out if it was produced before or after August 1997. If was produced before 1997, you more than likely need to use an ACA power adapter unless you're running your pedals in a daisy-chain configuration.
Of course, a standard 9-volt battery maybe used in any Boss compact pedal and it should work as expected as the battery circuit circumvents the voltage dropping components.