Evolution characteristics of a Kerr black hole (BH) are investigated by considering the coexistence of disc accretion with the Blandford–Znajek process (the BZ process) and magnetic coupling of the BH with the surrounding disc (MC process). (i) The rate of extracting energy from the rotating BH in the BZ process and that in MC process are expressed by a unified formula, which is derived by using an improved equivalent circuit. (ii) The mapping relation between the angular coordinate on the BH horizon and the radial coordinate on the disc is given in the context of general relativity and conservation of magnetic flux. (iii) The power and torque in the BZ process are compared with those in MC process in detail. (iv) Evolution characteristics of the BH and energy extracting efficiency are discussed by using the characteristic functions of BH evolution in the corresponding parameter space. (v) Power dissipation on the BH horizon and BH entropy increase are discussed by considering the coexistence of the above energy mechanisms.

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