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Friday, 22 June 2018

Topological Approach to Symbolic Pole-Zero Extraction Incorporating Design Knowledge


This paper addresses the problem of automatic analytical pole-zero extraction for multi-stage operational amplifiers with frequency compensation. Traditional methods mainly rely on numerical reference to derive approximate pole-zero expressions without incorporating any design knowledge. Such methods suffer from bad interpretability of the auto-generated results. This paper takes a topological approach and attempts to advocate that certain form of design knowledge can be incorporated in the symbolic term selection process for a pole-zero generation. The generation engine selects the dominant terms by a formal inspection of the token patterns that are correlated to gain factors and compensation elements. Since the gain factors and compensation elements of an opamp are pertinent to the topological details of a circuit, the proposed pole-zero extraction method is closer to design conception than other numerical reference-based methods. Consequently, the generated pole/zero results are better interpretable. Application to a class of multi-stage operational amplifiers with a variety of compensation structures demonstrates that the proposed method is effective and can match human-derived results.

Index Terms—

analog design automation, graph-pair decision diagram (GPDD), pole and zero (PZ), an operational amplifier (opamp), symbolic analysis.


In the design of analog integrated circuit (IC) cells frequency compensations in the forms of feedback and/or feedforward paths are commonly used. A variety of compensation strategies have been studied in many multi-stage operational amplifier (opamp) design works. To ensure that the closed-loop circuit has sufficient stability margins, the pole-zero (PZ) properties of the transfer function must be examined before proceeding to size and biasing. For the traditional one- or two-stage opamps, the pole-zero analysis is relatively simpler and can be carried out by manual analysis. When multi-stage designs are considered, compensation strategy becomes more intricate and analysis gets more involved. Technology advancement drives device feature size to be much smaller and supply voltages to be further lower. Designers are thus forced to consider multi-stage designs to realize opamps with much higher gain and better signal swing. Recently, research contributions to multi-stage opamp design are quickly rising. For opamps involving more than two stages, manual transfer function analysis becomes less wieldy. Designers have to manually derive all terms by solving nodal equations or using signal-flow graph-based analysis. Then designers use their design knowledge to screen the long list of terms and prune those less significant terms. This process of term selection highly relies on the human knowledge on the analog design practice, which is nontrivial for the beginners. However, deriving closed-form pole-zero (PZ) expressions is still considered a necessary analytical step in the contemporary design practice. This is because several crucial design steps like compensation, stability, and other frequency performance related reasoning must be based on the closedform PZ expressions. A key step toward PZ derivation is to generate a simplified transfer function that captures the circuit frequency response. Therefore, the main work for symbolic PZ generation is to construct a simplified symbolic transfer functions in which only the dominant coefficient terms are retained. A basic requirement on the usefulness of automatically generated PZ expressions is that such closed-form results should be readable or interpretable, so that based on such results designer can proceed to reason on pole-zero relocation, cancelling, and resolving conflicts, etc. Many frequency compensation related design techniques are discussed in the survey paper  In order to facilitate design space exploration and speed up the analytical design process, a tool capable of automatic pole-zero generation would be of great help. To this date, numerous methods have been proposed for the problem of automatic pole-zero extraction. However, most of them cannot successfully serve the design need due to missing of certain critical ingredients that are commonly involved in human reasoning.

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