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How to setup negative -10 V for an op-amp to be designed on a PCB? I read that -10 V can be obtained from a power supply by connecting positive to ground and negative of the supply to the node where we want -10 V.

How can we do it if we have only a single +10 V source and need to generate -10 V and +10 V for discrete op-amp operation? Is it possible to obtain without using additional components?

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It's not possible without additional components. The only way is with some kind of switching supply, either inductor based or capacitive charge pump.

There are packaged DC-DC converters (modules) which can be used in various ways, however 12V and 15V are more common nominal input and fixed output voltages.

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