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A linear transformation approach to the design of a family of passive second-order digital filters is presented. Starting from the definition of the power dissipation in a two-port network, a power dissipation expression for a linear transformed two-port network is derived. A passive recursive second-order structure can be obtained by linear transformation of a passive gyrator circuit. By trying all feasible Voltage, Current, Power, Voltage -current, Current-voltage transformations, twenty-one new passive second-order digital filter structures were obtained. Under finite arithmetic operations, this family of digital filters is free from zero-input and forced response oscillations by applying magnitude truncation at delays' inputs.