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Fractional order coupled inductor (FCI) was proposed recently which can enrich existing circuit element system. In this paper, the passivity conditions of FCI are discussed. The study shows that different from the traditional passive coupled inductor, passive fractional order coupled inductor (PFCI) can be non-reciprocal. Traditional integer order coupled inductors are special cases of FCI, and the passivity conditions between them have great differences. Based on passivity conditions of FCI, modified multivariate positive real definition is proposed and multivariate passivity of general FCIs also be proved. Finally, this paper proposes passivity criterions of networks with general FCIs in the complex frequency domain and multivariate domain, respectively.

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