Cyclone IV reconfiguration

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do Cyclone IV FPGAs support software programmable reconfiguration, or any other form of the run-time reconfiguration?

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Yes, all FPGAs support reconfiguration. Whether or not your particular board design can handle it is another question.

FPGAs can be configured using their JTAG connection, or from an external serial EEPROM, or from an external Flash device.

Did you have a particular development board in mind that you had a specific question about?

Cheers,

Dave