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This paper investigates the circuit synthesis problem for a certainpositive-real bicubic (third-order) admittance with a simple pole at the origin(s = 0) to be realizable as a one-port series-parallel damper-spring-inertercircuit consisting of at most six elements, where the results can be directlyapplied to the design and physical realization of inerter-based controlsystems. Necessary and sufficient conditions for such a specific bicubicadmittance to be realizable by a one-port passive series-parallel mechanicalcircuit containing at most six elements are derived, and a group of mechanicalcircuit configurations covering the whole set of realizability conditions arepresented together with element value expressions. The conditions and elementvalue expressions are related to the admittance coefficients and the roots ofcertain algebraic equations. The circuit synthesis results of this paper areillustrated by several numerical examples including the control system designof a train suspension system. Any realization circuit in this paper containsmuch fewer passive elements than the ten-element realization circuit by thewell-known Bott-Duffin circuit synthesis approach. The investigations of thispaper can contribute to the theory of circuit synthesis and many other relatedfields.

Series-Parallel Mechanical Circuit Synthesis of a Positive-Real Third-Order Admittance Using at Most Six Passive Elements for Inerter-Based Control