A current limiting and averaging circuit for driving a peripheral core circuit with a lower limit current value in response to a supply signal, includes a current limiting module, an energy storage module, and a converter module. The current limiting module provides a limited supply signal whose current value is smaller than or equal to an upper limit value according to the supply signal. The energy storage module stores a storage signal according to the limited supply signal when the upper limit value is higher than the lower limit current value and provides a discharge signal according to the storage signal when the upper limit value is lower than the lower limit current value. The converter module provides a driving signal for driving the peripheral core circuit in response to the limited supply signal or the limited supply signal and the discharge signal.