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Purpose: To improve the ability of energy harvester, a Helmholtz–Duffing oscillator with nonlinear capacitance is considered, which has an asymmetric characteristic. In this paper, we investigates the effect of potential function asymmetries on the dynamic behaviors under harmonic excitation. Method: The electromechanical coupling governing equations of Helmholtz–Duffing energy harvester is established. Then its dynamic response in the region of the primary resonance is sought by the harmonic balance method. Results: The performances of the presented asymmetric nonlinear primary resonance energy harvester and the linear stiffness system are compared. Results clearly demonstrate an improved bandwidth over a case which does not involve an asymmetric potential energy function. Conclusion: It is found that multiple-valued steady-state solutions of frequency-response equation can exist, indicating that the jump phenomenon occurs. Besides, the amplitude–frequency response curves have double-jumping phenomena by the bend, first to the left and then to the right. The influence of the stiffness parameters on the shape of the frequency-response curves is also discussed. It is observed that different shapes of frequency-response curves can exist. The approximate analytical outcomes are in good agreement with the numerical results.