Memory elements (known as mem-elements) include memristor, memcapacitor and meminductor. They are considered the keys to developing the new generation of intelligent and neuromorphic devices. However, the complicated nano-scale manufacturing technique makes them difficult to produce, impeding the research and commercialization activities of mem-element devices. The emerging independent/universal emulators is considered one of the most promising candidates for mem-elements research. In this paper, for the completeness of circuit theory, the mathematical representations, classification, concept of unfolding with the extended formulation of three mem-elements are described, respectively. Then, a newly proposed floating and grounded memory interface circuits with wide frequency range are presented, together with the mathematical model and fingerprints. We also discuss the main affecting factors of frequency and demonstrate that the frequency of memristor is as high as 900kHz, and the highest frequency of memcapacitor also exceeds 500kHz. Finally, we explore a potential application based on the proposed interface circuit, i.e., a simple chaotic circuit. Also, the models, strange attractors, and Lyapunov exponents of three chaotic systems with mem-element are obtained, respectively. There is no doubt that the good agreement among theoretical analysis, simulation and experimental results verifies the practicability and flexibility of these interface circuits.