**
Plenary Lecture**

Towards 2D Electronic Circuits in the Spatial Domain

**Associate Professor Nicolas Ratier**

Graduate engineering school in Mechanics and Microtechnics (ENSMM)

FEMTO-ST laboratory

Besancon, France

E-mail: nicolas.ratier@femto-st.fr

**Abstract:** Electronic circuits are, by
nature, functions of one independent variable in the time domain. They
operate (in the sense of differential operators) from an input signal e(t)
to an output signal s(t). In this sense, we can speak of the electronic
circuits of an ODE (ordinary differential equation) with its associated
initial condition.

We show, in this talk, that periodic circuits can be used to define a kind
of 2D circuits in the spatial domain. These circuits operate as a PDE
(partial differential equations) from an input signal f(x,y) to an output
signal g(x,y). The two independent variables are now the coordinate axes x
and y. As to the initial conditions, they are replaced by boundary
conditions.

We first present how to construct an electronic circuit of a linear PDE and
how the boundary conditions are transposed to electronic elements. The
discrete solution at each voltage node of this circuit converges towards the
solution of the PDE g(x,y) for a given input signal f(x,y) when the number
of periodic cells increases.

Next, we consider the reverse problem, i.e. to build the PDE of an
electronic linear circuit. Electric conditions imposed at the boundary are
converted into Dirichlet or Neumann conditions. The solution of this problem
is based on an extension of an homogenization modelling method initially
developed by theoretical mechanicians to study composite materials.

**Brief Biography of the Speaker:**

Nicolas Ratier was born in Paris in 1965. He is graduated in electronics at
Paris XI University, Orsay. In 1993, he received the Ph. D. degree in
microelectronics from Toulouse III University for his work on the simulation
of micromechanical capacitive pressure sensors. He is associated professor
at ENSMM (graduate engineering school in Mechanics and Microtechnics) and at
FEMTO-ST laboratory, both at Besancon, France. From 1994 to 2006, his
primary research activities were the simulation of ultrastable quartz
crystal oscillators. Most of these researches have been completed in the
framework of contracts executed for the french space agency (CNES). Since
2007, his current research concerns the study of spatially distributed
periodic electronic circuits to control array of MEMS.