  A reconfigurable intelligent surface (RIS) is a planar structure that is
engineered to dynamically control the electromagnetic waves. In wireless
communications, RISs have recently emerged as a promising technology for
realizing programmable and reconfigurable wireless propagation environments
through nearly passive signal transformations. With the aid of RISs, a wireless
environment becomes part of the network design parameters that are subject to
optimization. In this tutorial paper, we focus our attention on communication
models for RISs. First, we review the communication models that are most often
employed in wireless communications and networks for analyzing and optimizing
RISs, and elaborate on their advantages and limitations. Then, we concentrate
on models for RISs that are based on inhomogeneous sheets of surface impedance,
and offer a step-by-step tutorial on formulating electromagnetically-consistent
analytical models for optimizing the surface impedance. The differences between
local and global designs are discussed and analytically formulated in terms of
surface power efficiency and reradiated power flux through the Poynting vector.
Finally, with the aid of numerical results, we discuss how approximate global
designs can be realized by using locally passive RISs with zero electrical
resistance (i.e., inhomogeneous reactance boundaries with no local power
amplification), even for large angles of reflection and at high power
efficiency.
