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This paper presents the adjoint variable method (AVM) for finite-element (FE) analysis of eddy current problems based on complex variables. In the sensitivity analysis based on FE analysis of time-harmonic eddy current fields, the functions for which sensitivity is evaluated are often real-valued, while unknown variables in the FE analysis are complex. When the AVM is applied to such problems, the real-valued functions are differentiated with respect to the complex variables. However, such differentiation cannot be defined because the Cauchy-Riemann equation does not hold. In this paper, the AVM for complex systems is introduced and applied to linear and nonlinear eddy current problems, in the latter of which the harmonic balance method is employed.