The use of an eigenstate based equivalent circuit topology is proposed for the analysis and modeling of lossless and lossy bi-periodic scatterers. It can significantly simplify the design of this kind of surfaces, since it reduces the number of elements with respect to other general circuits. It contains at most only two admittances and two transformers depending on one unique transformation ratio. The real parts of these admittances can be assured to be non-negative, an interesting aspect in the modeling of lossy surfaces such as those present in asorbers. Moreover, due to the capability of decomposition into the eigenexcitations of the structure, the circuit provides important physical insight. Different cases of scatterers have been analyzed: symmetric and asymmetric, lossy and lossless. In all these cases, the modeling of the circuit admittances has been successfully achieved with a few RLC elements, positive and frequency independent. In the case of structures with symmetries, the transformation ratio directly reflects the physical orientation of the eigenexcitations of the scatterer. Furthermore, in the case of lossy scatterers but without symmetries, the resulting equivalent circuit reveals that their eigenexcitations are not linear polarizations, but elliptic polarizations whose properties are described by the complex transformation ratio.