Recently, metamaterial absorbers (MAs) with a multi-layered anisotropic substrate have received significant attention due to their huge potential for application in major engineering fields like aircraft stealthing, electromagnetic sensing, and materials processing, etc. However, the working mechanism of this type of structural materials has not been well-understood yet, as the classical equivalent circuit model was only proposed to describe the conventional overall isotropic metal-substrate MAs. In this paper, for the first time, a generalized equivalent circuit model that considering the anisotropy of the multi-layered substrate is constructed, based on new findings about the unique distribution of the induced current inside the MA with a carbon fiber reinforced polymer (CFRP) composite substrate - a typical multi-layered anisotropic laminate. The effectiveness of the generalized analytical model is validated by predicting the structure-performance relationship of the CFRPsubstrate MA, which is in excellent agreement with numerical simulation results based on Maxwell's equations. Experimental cases have also been conducted to demonstrate the strong power of this model in inverse design of several tunable MAs. Through the above research, the scope of the equivalent circuit modelling has been greatly broadened, which can help to design a series of MAs with more extreme performance in future.