Generating the Closed-Form Second-Order Characteristics of Analog Differential Cells by Symbolic Perturbation

Analog integrated circuit designers always use closed-form design equations in design reasoning. However, most of the time, they have to derive all equations by hand. When high-order effects are of interest, manual derivation would become much harder. Although the art of symbolic circuit analysis has been making steady progress, symbolically generating closed-form equations for high-order effect is still a less well-studied subject. This paper proposes a novel symbolic computation method for generating the second-order characterization of differential cells. It is a perturbation approach that takes the advantage of structural symmetry in differential cells, and is formulated as a systematic computation procedure comprised of several intermediate steps, the computation of each step gives a compact partial expression. Examples are provided to illustrate that closed-form common-mode rejection ratio equations in pole-zero form (unknown before) for two classical operational amplifiers can be generated by applying the symbolic perturbation procedure.