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This paper deals with the study and analysis of several rational approximations to approach the behavior of arbitrary-order differentiators and integrators in the frequency domain. From the Riemann–Liouville, Grünwald–Letnikov and Caputo basic definitions of arbitrary-order calculus until the reviewed approximation methods, each of them is coded in a Maple 18 environment and their behaviors are compared. For each approximation method, an application example is explained in detail. The advantages and disadvantages of each approximation method are discussed. Afterwards, two model order reduction methods are applied to each rational approximation and assist a posteriori during the synthesis process using analog electronic design or reconfigurable hardware. Examples for each reduction method are discussed, showing the drawbacks and benefits. To wrap up, this survey is very useful for beginners to get started quickly and learn arbitrary-order calculus and then to select and tune the best approximation method for a specific application in the frequency domain. Once the approximation method is selected and the rational transfer function is generated, the order can be reduced by applying a model order reduction method, with the target of facilitating the electronic synthesis.

A synthesis methodology to search on all realisations of CMOS voltage followers (VFs) and current followers (CFs), is introduced by manipulation of generic cells. Nullators are used to model the ideal behaviour of generic VFs, and norators for generic CFs. Norators and nullators are added until to form joined nullator-norator pairs, which are further biased and synthesized by transistors. Finally, two generated topologies performing good follower behaviour are selected to synthesize the CCII-and an oscillator, for which SPICE simulation results are given.