One of the major bottlenecks of the traditional current transformer energy harvester (CTEH) is the instable output induced by the wide-range variations of the current in transmission lines. In this work, a novel CTEH capable of generating a stable output is demonstrated by using a core fabricated with saturable magnetic material. The stable output of the CTEH is enabled by the constant voltage-time product in its saturable characteristic. The proposed CTEH is implemented with a resistive load representing the load of electronic devices. When the current in the primary side of the CTEH's increases from 1 to 1000 A, the maximum power on the load can reach about 0.5 W, demonstrating the feasibility of using the CTEH with the saturable magnetic core as a quasi-stable power supply