In this talk we will first recall the classical Kawashima-Shinzuta (KS) condition on the hyperbolic system with partial relaxation.Second,we show a result on the spectral analysis of the damped Euler-Maxwell system which actually violates the KS condition and admits the regularity-loss property.Third,we present a recent result on the large time behavior of solutions to the two-fluid Euler-Maxwell system,which is a joint work with Qingqing Liu and Changjiang Zhu.In that work we give an analytical proof of solutions tending time-asymptotically to diffusion waves in the context of two-fluid plasma with collisions.