# Towards 2D Electronic Circuits in the Spatial Domain

Nicolas Ratier

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Nicolas Ratier. Towards 2D Electronic Circuits in the Spatial Domain. 13th WSEAS International Conference on CIRCUITS, Jan 2009, Besançon, France. pp.212-218. hal-00957616

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# Towards 2D electronic circuits in the spatial domain

NICOLAS RATIER

FEMTO-ST Institut

Time-Frequency department

32 avenue de l'Observatoire

25044 Besançon Cedex

FRANCE

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 that periodic networks of resistances (PNR) 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 present how to construct an electronic circuit of a linear PDE up to the fourth order. The discrete solution at each voltage node of the center of each cell composing this circuit converges towards the solution of the PDE *g*(*x*, *y*), with its boundary conditions, for a given input signal *f*(*x*, *y*) when the number of periodic cells increases.

*Key-Words:* Periodic networks of resistances, 2D circuits, Partial differential equations

## 1 Introduction

Electronic circuits are modelled by ODE (ordinary differential equations) of a time variable *t* with its associated initial conditions [1]. We will show, in this paper, that Periodic Networks of Resistances (Called PNR) can be used to define a kind of 2-D networks in the spatial domain. These networks are then defined by PDE (partial differential equations) of two space variables (*x*, *y*) with their associated boundary conditions. Table 1 summarizes the equivalence between regular networks and periodic networks of resistances.

<table><caption>Table 1: Equivalence between networks and PNR.</caption><thead><tr><th>Networks</th><th>PNR</th></tr></thead><tbody><tr><td>Time domain</td><td>Spatial domain</td></tr><tr><td>1D Variable <i>t</i></td><td>2D Variables (<i>x</i>, <i>y</i>)</td></tr><tr><td>ODE</td><td>PDE</td></tr><tr><td>Initial conditions</td><td>Boundary conditions</td></tr></tbody></table>

Our method is based on the design of electronic circuits which “computed” locally the finite difference approximation of a partial differential equation. These circuits must have the properties of being possibly assembled then periodically recopied. The whole periodic network is then composed of the numerous repeated identical cells, except at the boundaries where

the cell takes into account the boundary conditions. So we can say that PNR “solve” or “is described” by a PDE.

The 14+1 partial differential operators up to order 4 are summarized in table 2. It is sufficient to study 9 of them (at the left side of the table), all the other ones can be deduced by symmetry.

<figure><img src="image_2.png" /><figcaption>Table 2: Partial differential operators up to order 4.</figcaption></figure>

The following sections propose a study of eight unit circuits which “code” these 8 finite difference (FD) approximations. The name of the sections “Molecule $\alpha\beta$” refer to the FD approximation of the derivative $\frac{\partial^{\alpha+\beta}}{\partial x^{\alpha} \partial y^{\beta}}$ with an error proportional to the step size of order 2. The calculation of the harmonic
## 4 Molecule 10

The molecule of the first derivative is represented by (7). This molecule is not symmetrical and thus cannot be represented by a symmetrical network.

$$ \frac{\partial v}{\partial x} \approx \frac{1}{2h} \{ \begin{smallmatrix} -1 \\ 0 \\ 1 \end{smallmatrix} \} \quad (7) $$

To implement the molecule, we must construct an asymmetrical circuit. The proposed one uses the potential $-V_{i,j}$. It is obtained by a linear VCVS $e_0$ of value -1. (VCVS stands for Voltage-Controlled Voltage Source). The circuit is drawn on Fig. 3.

<figure><img src="image_3.png" /><figcaption>Figure 3: Network of molecule 10.</figcaption></figure>

The resistances $\rho_1$ connect the potentials $V_i$ and $V_{i\pm 1}$ and the resistance $\rho_2$ connect the potentials $V_i$ and $-V_{i+1}$. Applying KCL at node $V_i$ and after some algebraic manipulations, one can obtain (8).

$$ \frac{1}{2h} \left[ V_i \left( \frac{1}{r_0} + \frac{2}{\rho_1} + \frac{1}{\rho_2} \right) - V_{i-1} \left( \frac{1}{\rho_1} \right) + V_{i+1} \left( \frac{1}{\rho_2} - \frac{1}{\rho_1} \right) \right] = \frac{1}{2h} i_0 \quad (8) $$

If we choose $\rho_1 = 2h\rho_0$, $\rho_2 = 2h\rho_0/2$, and $r_0 = -2h\rho_0/4$, then the potential at node $V_i$ estimates the first derivative of the function $v(x)$ at a given point (Eq. (9)).

$$ \frac{\partial v}{\partial x} = \rho_0 i_0 \quad (9) $$

The resistances of the cells can be taken as $r_1 = r_3 = \rho_1/2$ and $r_2 = r_4 = \rho_2/2$.

## 5 Molecule 30

The molecule of the third derivative is represented by (10). Like every odd order derivative, the third molecule is not symmetrical and thus, it cannot be represented by a symmetrical network. Moreover, it expands over two steps at each side.

$$ \frac{\partial^3 v}{\partial x^3} \approx \frac{1}{2h^3} \begin{smallmatrix} -1 \\ 2 \\ 0 \\ -2 \\ 1 \end{smallmatrix} \quad (10) $$

The proposed circuit is represented on Fig. 4. The circuit is a combination of the method used in molecule 40 to obtain the value of the potential, and those used in molecule 10 to get an asymmetrical value in the molecule. The pathes 1 and 2 code the outer values of the molecule $\{-1, 1\}$, while the pathes 3 and 4 code its inner values $\{2, -2\}$. Note that the minus sign is inverted between these outer and inner values, this is translated into a direction change in the network (compare pathes 2 and 4).

<figure><img src="image_4.png" /><figcaption>Figure 4: Network of molecule 30.</figcaption></figure>

The resistances $\rho_1$ connect the potentials $V_i$ and $V_{i\pm 2}$ and the resistance $\rho_2$ connect the potentials $V_i$ and $-V_{i+2}$. In a similar way, the resistances $\rho_3$ connect the potentials $V_i$ and $V_{i\pm 2}$ and the resistance $\rho_2$ connect the potentials $V_i$ and $-V_{i+1}$. Applying KCL at node $V_i$ and after some algebraic manipulations, one can obtain (11).

$$ \frac{1}{2h^3} \left[ V_i \left( \frac{1}{r_0} + \frac{2}{\rho_1} + \frac{1}{\rho_2} + \frac{2}{\rho_3} + \frac{1}{\rho_4} \right) - V_{i-2} \left( \frac{1}{\rho_1} \right) + V_{i+2} \left( \frac{1}{\rho_2} - \frac{1}{\rho_1} \right) + V_{i-1} \left( \frac{1}{\rho_4} - \frac{1}{\rho_3} \right) - V_{i+1} \left( \frac{1}{\rho_3} \right) \right] = \frac{1}{2h^3} i_0 \quad (11) $$

If we choose $\rho_1 = 2h^3\rho_0$, $\rho_2 = 2h^3\rho_0/2$, $\rho_3 = 2h^3\rho_0/2$, $\rho_4 = 2h^3\rho_0/4$ and $r_0 = -2h^3\rho_0/12$, then the potential at node $V_i$ estimates the third derivative of the function $v(x)$ at a given point (Eq. (12)).

$$ \frac{\partial^3 v}{\partial x^3} = \rho_0 i_0 \quad (12) $$

The resistances of the cells can be taken as $r_{\{1,3,7,9\}} = \rho_1/4$, $r_{\{4,6,10,12\}} = \rho_2/4$, $r_{\{2,8\}} = \rho_3/2$ and $r_{\{5,11\}} = \rho_4/2$.

## 6 Molecule 11

The molecule 11 derivative is represented by (13). This molecule is the tensor product of two asymmetrical molecules (the molecules 10 and 01) and is there-
braic manipulations, one can obtain (20).

$$
\begin{aligned}
& \frac{1}{2h^3} \left[ V_{i,j} \left( \frac{1}{\rho_0} + \frac{2}{\rho_1} + \frac{1}{\rho_2} \right) \right. \\
& \qquad \left. + \frac{2}{\rho_3} + \frac{1}{\rho_4} + \frac{2}{\rho_5} + \frac{1}{\rho_6} \right) \\
& + V_{i,j-1} \left( \frac{1}{\rho_2} - \frac{1}{\rho_1} \right) - V_{i,j+1} \left( \frac{1}{\rho_1} \right) \\
& + V_{i-1,j+1} \left( \frac{1}{\rho_4} - \frac{1}{\rho_3} \right) - V_{i+1,j-1} \left( \frac{1}{\rho_3} \right) \\
& + V_{i+1,j+1} \left( \frac{1}{\rho_6} - \frac{1}{\rho_5} \right) - V_{i-1,j-1} \left( \frac{1}{\rho_5} \right) \right] \\
& = \frac{1}{2h^3} i_0
\end{aligned}
\qquad (20)
$$

If we choose $r_0 = -2h^3/12$ and the sets of $\rho_\alpha$ so that,

<table><tr><td>ρ<sub>1</sub> = 2h<sup>3</sup>ρ<sub>0</sub>/2</td><td>ρ<sub>2</sub> = 2h<sup>3</sup>ρ<sub>0</sub>/4</td></tr><tr><td>ρ<sub>3</sub> = 2h<sup>3</sup>ρ<sub>0</sub></td><td>ρ<sub>4</sub> = 2h<sup>3</sup>ρ<sub>0</sub>/2</td></tr><tr><td>ρ<sub>5</sub> = 2h<sup>3</sup>ρ<sub>0</sub></td><td>ρ<sub>6</sub> = 2h<sup>3</sup>ρ<sub>0</sub>/2</td></tr></table>

the potential at node $V_i$ estimates the 2–1 partial derivative of the function $v(x)$ at a given point (Eq. (21)).

$$ \frac{\partial^3 v}{\partial x^2 \partial y} = \rho_0 i_0 \qquad (21) $$

The actual resistances are represented by segments between two dots. The 6 resistances along the paths 1 and 2 can be taken as $r_{\langle 1 \rangle, \langle 2 \rangle} = \rho_1/2$, while the resistances along the paths 3, 4, 5 and 6 can be taken as $r_{\langle 1 \rangle, \langle 2 \rangle, \langle 3 \rangle, \langle 4 \rangle, \langle 5 \rangle, \langle 6 \rangle} = \rho_2/3$. The value of the actual resistances along the paths define, by periodicity, the resistances in the elementary cell.

## 9 Molecule 31

The molecule 21 derivative is represented by (22). This molecule is the tensor product of two asymmetrical molecule (the molecules 30 and 01) and is therefore symmetrical.

$$ \frac{\partial^4 v}{\partial x^3 \partial y} \approx \frac{1}{4h^4} \left\{ \begin{array}{cccc} \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.3, every node/.style={scale=0.3}] \node[circle, fill, inner sep=1pt] (0) at (0,0) {}; \node[circle, fill, inner sep=1pt] (1) at (1,0) {}; \node[circle, fill, inner sep=1pt] (2) at (2,0) {}; \node[circle, fill, inner sep=1pt] (3) at (3,0) {}; \draw[->] (0) -- (1); \draw[->] (0) -- (2); \draw[->] (0) -- (3); \node[above] at (0,0) {0}; \node[above] at (1,0) {0}; \node[above] at (2,0) {0}; \node[left] at (0,0) {1}; \node[left] at (1,0) {2}; \node[left] at (2,0) {3}; \end{tikzpicture} & \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.3, every node/.style={scale=0.3}] \node[circle, fill, inner sep=1pt] (0) at (0,0) {}; \node[circle, fill, inner sep=1pt] (1) at (1,0) {}; \node[circle, fill, inner sep=1pt] (2) at (2,0) {}; \node[circle, fill, inner sep=1pt] (3) at (3,0) {}; \draw[->] (0) -- (1); \draw[->] (0) -- (2); \draw[->] (0) -- (3); \node[above] at (0,0) {0}; \node[above] at (1,0) {0}; \node[above] at (2,0) {0}; \node[left] at (0,0) {1}; \node[left] at (1,0) {2}; \node[left] at (2,0) {3}; \node[right] at (3,0) {0}; \node[right] at (3,0) {1}; \end{tikzpicture} & \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.3, every node/.style={scale=0.3}] \node[circle, fill, inner sep=1pt] (0) at (0,0) {}; \node[circle, fill, inner sep=1pt] (1) at (1,0) {}; \node[circle, fill, inner sep=1pt] (2) at (2,0) {}; \node[circle, fill, inner sep=1pt] (3) at (3,0) {}; \draw[->] (0) -- (1); \draw[->] (0) -- (2); \draw[->] (0) -- (3); \node[above] at (0,0) {0}; \node[above] at (1,0) {0}; \node[above] at (2,0) {0}; \node[right] at (3,0) {0}; \node[right] at (3,0) {1}; \end{tikzpicture} & \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.3, every node/.style={scale=0.3}] \node[circle, fill, inner sep=1pt] (0) at (0,0) {}; \node[circle, fill, inner sep=1pt] (1) at (1,0) {}; \node[circle, fill, inner sep=1pt] (2) at (2,0) {}; \node[circle, fill, inner sep=1pt] (3) at (3,0) {}; \draw[->] (0) -- (1); \draw[->] (0) -- (2); \draw[->] (0) -- (3); \node[above] at (0,0) {0}; \node[above] at (1,0) {0}; \node[above] at (2,0) {0}; \node[right] at (3,0) {0}; \node[right] at (3,0) {1}; \end{tikzpicture} \end{array} \right\} (22) $$

The molecule is separated into two submolecules around the center point $(i, j)$. The first submolecule is the inner one $(-2, 2, -2, 2)$, it can be defined under a circuit form from the circuit of the molecule 11 by inverting and scaling the resistance $\rho_\alpha$. The circuit of

the second molecule, the outer one $(1, -1, 1, -1)$ is drawn on Fig 9. As indicated in the network, $\rho_1$ correspond to the resistances between the potentials $V_{i,j}$ and $V_{i \pm 2, j \pm 1}$ and $\rho_2$ are the resistances between the potentials $V_{i \mp 2, j \mp 1}$.

<figure><img src="image_5.png" /><figcaption>Figure 9: Network of outer submolecule of molecule 31.</figcaption></figure>

The circuit of the molecule 31 is the “schematic sum” of the network of Fig. 9 and the network of molecule 11 (with resistances renamed $\rho'_1$ and $\rho'_2$). A “schematic sum” means that the two circuits must be superposed with separated dots in the cell border, and that these two circuits have the same current source $i_0$. Applying KCL at node $V_i$ and after some algebraic manipulations, one can obtain (23).

$$ \frac{1}{4h^4} \left[ V_{i,j} \left( \frac{2}{\rho_1} + \frac{2}{\rho_2} \right) + V_{i,j} \left( \frac{2}{\rho'_1} + \frac{2}{\rho'_2} \right) \right. \\
- \frac{1}{\rho_1} V_{i+2,j+1} - \frac{1}{\rho_2} V_{i-2,j+1} \\
- \frac{1}{\rho_1} V_{i-2,j-1} - \frac{1}{\rho_2} V_{i+2,j-1} \\
- \frac{1}{\rho'_1} V_{i+1,j+1} - \frac{1}{\rho'_2} V_{i-1,j+1} \\
\left. - \frac{1}{\rho'_1} V_{i-1,j-1} - \frac{1}{\rho'_2} V_{i+1,j-1} \right] = \frac{1}{4h^4} i_0 \qquad (23) $$

If we choose $\rho_1 = -4h^4\rho_0$, $\rho_2 = 4h^4\rho_0$ and the corresponding resistances $\rho'_1$ and $\rho'_2$ of the molecule 11 so that $\rho'_1 = 4h^2\rho_0/2$ and $\rho'_2 = -4h^2\rho_0/2$, the potential at node $V_i$ estimates the 3–1 partial derivative of the function $v(x)$ at a given point (Eq. (24)).

$$ \frac{\partial^4 v}{\partial x^3 \partial y} = \rho_0 i_0 \qquad (24) $$

The actual resistances of Fig. 9 are represented by segments between two dots. The 4 resistances along the paths 1 and 2 can be taken as $r_{\langle 1 \rangle, \langle 2 \rangle} = \rho_1/4$. The value of the actual resistances along the paths define, by periodicity, the resistances in the elementary cell.
## 10 Simulation and comments

We have developed a program which generate the Spice network corresponding to the chosen finite difference discretisation scheme from a (linear) partial differential equation. The generated file contained the N or $N^2$ cells of the domain, and the cells at the boundaries to take account of the boundary conditions. All the circuits defined in the previous sections have been independently checked by simulating the PDE:

$$ \frac{\partial^{\alpha+\beta} v}{\partial x^{\alpha} \partial y^{\beta}} = \alpha! \beta!, $$

with $\alpha + \beta \le 4$ on the domain $\Omega = [0, 1] \times [0, 1]$, with the right boundary conditions. The Spice solution for the potential $V_{i,j}$ has been compared with the PDE solutions $v(x, y) = x^{\alpha} y^{\beta}$ for different mesh sizes. On these academic examples, there is no significant difference between the solutions.

The case $\alpha = \beta = 0$ corresponds to the Molecule 00, which is the trivial molecule $\textcircled{1}$. The unit circuit which “codes” the equation $V_i = \rho_0 i_0$ is given in Fig 10. This case is important, it allows to manage the terms of EDP which involves the function itself (or derivative of order 0 of a function to justify the terminology Molecule 00).

<figure><img src="image_6.png" /><figcaption>Figure 10: Network of molecule 00.</figcaption></figure>

This paper considers the problem of designing the electronic linear circuit of a (linear) PDE. Reference [2] considers the reverse problem, i.e. to build the (linear) PDE of an electronic linear circuit (The method is based on an extension of an homogenization modelling theory initially developed to study composite materials).

Unfortunately, the two theories operate on circuits with very different topologies. Thus, the common networks they can treat simply reduce to one circuit. Indeed, The periodicity of the electronic network implies that each node $n$ located on the boundary of the unit cell has its counterpart $n'$ on the opposite side. The method used in [2] imposes that each such node couple $(n, n')$ is linked by at least a crossing path. All the networks developed here lead to unit cells which have not generally this property, the exception is the circuit of molecule 20 (strictly, the molecule 40 can be analysed too by taking 2 adjacent cells).

## 11 Conclusion

We have proposed a concept of periodic network of resistances (PNR) to define a kind of 2D electronic circuit in the spatial domain. We adopt an electronician point of view, i.e. designing the electronic circuits and simulating them. The matrix equations of our periodic circuit are the same that the corresponding set of finite difference equation to resolve a given PDE. However, the accuracy and the stability of the numerical schemes have to be carefully studied.

By doing a “schematic sum” of some elementary networks defined in the previous sections, it is possible to build the 2D electronics circuit which is a solution of a given linear partial differential equation, for example the plate equation:

$$ \frac{\partial^4 u}{\partial x^4} + 2 \frac{\partial^4 u}{\partial x^2 \partial y^2} + \frac{\partial^4 u}{\partial y^4} = \frac{P}{D}. $$

Moreover, generalization to non constant coefficient PDE or PDE with an arbitrary function in the right hand side is faireble by the same method. By using multiplier, even nonlinear PDE could be manageable. All these generalizations suppose implicitly that the finite difference equations relative to the periodic network are mathematically valid.

An example of application of PNR could be the real time control of a system, with a PDE control law implemented under the form of an analog integrated circuit instead of a numerical one. It is difficult to qualify the gain in speed, that may be of a factor 1000.

Of course, a practical implementation of the method described in this paper under the form of integrated circuits is challenging, some of the numerous problems are the integration of the many NIC (Negative Impedance Convertor) to simulated the negative resistances: How to choose the magnitude of the elements ? Should the scaling operate on the resistances or on the current sources ? What is the effect of the scattering of the component values (some finite difference schemes are very sensitive to it, but finite difference schemes deriving from physical equations have “generally” a good behavior).

## References:

[1] L.O. Chua, A.D. Desoer and S.K. Kuh, Linear and Nonlinear Circuits, “McGraw-Hill Series in Electrical Engineering”, 1987.

[2] M. Lenczner, “Homogenization of linear spatially periodic electronic circuits”, Networks and Heterogeneous Media, 1 (2006), no. 3, pp. 467–494.