2005 RISP International Workshop on Nonlinear
Circuit and Signal Processing (NCSP'05)
Hawaii, USA, Mar. 4-6, 2005.

Analysis of Two-Dimensional Conductive Plates Based on CNNs
Masayoshi Oda∗ , Zonghuang Yang∗∗ , Yoshifumi Nishio∗ , Akio Ushida∗∗∗∗Tokushima University, Japan
Email: {masayoshi, nishio}@ee.tokushima-u.ac.jp∗∗
Nichia Corporation, Japan∗∗∗
Tokushima Bunri University, Japan

1. Introduction

discretized model shown in Fig. 1 [10]. The discrete steps areRecently, there have been many papers on the analysis oftransmission lines, because they are becoming more important for designing VLSI chips as the operation speed of thechips becomes higher. On the other hand, a device simulation is also very important to design Integrated Circuits (ICs)and to understand the qualitative behavior. The methods foranalyzing conductive plates have been proposed. For semiconductor devices, the circuit equations are described by partial differential equations and are usually solved by the finite
element techniques. The linearization of the semiconductor
equations are efficient solution for their transient and steadystate [1-2]. As applications for the resistance calculations,two basic methods have been proposed. One of them is basedon a boundary element method [3], and the other is a finite element method [4]. These methods are really time-consuming,
because, to obtain exact solutions, the device must be divided
into many small sections.
Since the middle of last decade, Cellular Neural Networks(CNNs) have been noted in the fields of circuits and systems[5-6]. The CNN is an array of fundamental elements, called
cell. Neighbor cells are coupled each other directly, while
distant cells are connected with propagation effects. Oneparadigm of CNN applications has been proposed as an implementation of solving Partial Differential Equations (PDEs)
and systems of Ordinary Differential Equations (ODEs) [7-8].
Therefore, CNN has been adopted to simulate various nonlinear spatio-temporal phenomena such as traveling waves, autowaves, spiral waves and so on [9].
In this study, we propose an analysis method of potential
propagation in conductive plates based on two-layer CNN.
The computer simulated results show that the two-layer CNN
can simulate the potential propagation in conductive plates
for various sets of circuit parameters.

j-1

jvi-1, jj+1i-1
LRvi, j-1

LRLvi, jL
RR vi, j+1iii, jC

Gvi+1, ji+1

Figure 1: Discretized model.
equidistance in both directions. Hence, the circuit parameters
R and L are the same value in both directions.
We assume the model as the lumped constant circuit,hence, the state equations of the discretized model are described by the following ordinary differential equations.
dvi,jdt
dii,jdt

G1vi,j + ii,jCCR1= − ii,j + (vi−1,j + vi+1,jLL+vi,j−1 + vi,j+1 − 4vi,j ).
= −After the normalization of time:1τ=√t,LC
we havedvi,jdτ

2. Two–Dimensional Circuits
We consider a conductive plate. In order to analyze thepotential propagation of the conductive plate, we discretize
the plate spatially and propose two-dimensional circuits as- 447 -dii,jdτ

r
= −Gr

Lvi,j +Cr

(1)

(2)

Lii,jCrCC= −Rii,j +(vi−1,j + vi+1,jL
L+vi,j−1 + vi,j+1 − 4vi,j ).

(3)

3. Two-Layer Cellular Neural Networks

4. Simulation Results

Cellular Neural Network has given rise to wide interestsin theoretical researches for various generalizations and theirapplications in the areas like as image processing, patternrecognition, motion detection, and computer vision as solving
various types of nonlinear differential equations. In our previous researches, we have reported various applications using
two-layer CNNs. In the two-layer CNN, the connection between the first layer and the second layer is given by the Ctemplate.
The state equations and output equations of the two-layer
CNN are described as follows:[State Equations]

Between the two equations (3) and (4), we can see remarkable similarity. Both of them are systems of locally interconnected ordinary differential equations. By comparing them, it
is easy to design the cloning templates of the two-layer CNN
describing the two-dimensional circuits as follows:X
dvx1ij (t)= −vx1ij (t) +A1 (i, j; k, l)vy1kl (t)dt
C(k,l)∈Nr (i,j)
X+
B1 (i, j; k, l)vu1kl (t)C(k,l)∈Nr (i,j)

X+C1 (i, j; k, l)vy2kl (t) + I1

C(k,l)∈Nr (i,j)

0A1 = 000q
1−G0LC



0 q00


L0 , C1 = 0C000

B1 = 0,0




0 ,0

I1 = 0.
(7)

00q
A2 = 0 1 − R C
L00q





CqL0q C0 , C2 =  L
00
I2 = 0.

0−4qCL

0q C,L
0CX
dvx2ij (t)L= −vx2ij (t) +A2 (i, j; k, l)vy2kl (t)B2 = 0,dt
X C(k,l)∈Nr (i,j)(8)+B2 (i, j; k, l)vu2kl (t)As illustrated examples, we simulated the two-impulse reC(k,l)∈Nr (i,j)
X
sponses on the uniform plate for two different circuit param+
C2 (i, j; k, l)vy1kl (t) + I2 .
eters. One is damped oscillation case, and the other is idealC(k,l)∈Nr (i,j)
lossless case for the same initial condition. The initial state(4) is shown in Fig. 2, where i and j mean discretized spaces,[Output Equations]and v means the voltage. Namely, the plate is analyzed by thetwo-layer CNN with 100×100 cells.
1vy1ij (t) = (|vx1ij (t) + 1| − |vx1ij (t) − 1|)2(5)1vy2ij (t) = (|vx2ij (t) + 1| − |vx2ij (t) − 1|).
2v

The cloning templates connecting cells are given as follows:
Ai−1,j−1 Ai−1,j Ai−1.j+1
Ai,jAi,j+1 Am =  Ai,j−1
Ai+1,j−1 Ai+1,j Ai+1,j+1
Bi−1,j−1 Bi−1,j Bi−1.j+1Bi,jBi,j+1 
Bm =  Bi,j−1(6)Bi+1,j−1 Bi+1,j Bi+1,j+1Ci−1,j−1 Ci−1,j Ci−1.j+1Ci,jCi,j+1 
Cm =  Ci,j−1Ci+1,j−1 Ci+1,j Ci+1,j+1Im

10.5
0-0.5
-10102030
40
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i60
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80
90
100 0102030

405060

70

80

90100

jFigure 2: Initial state.

=cwhere m = 1, 2 and c is a constant. Am , Bm , and Cm 4.1. Damped Oscillation
are feedback templates, control templates, and coupling temFirst, we simulate the potential propagation of the plate forplates, respectively. These templates define the behavior of
the case of damped oscillation. We used the following temthe two-layer CNNs with input and initial states.

- 448 -plates:
0
A1 = 00
0 000.9 0 , C1 = 00 00

B1 = 00
100(9)

I1 = 0.


00000.1
A2 = 0 0.999 0 , C2 = 0.1 −0.400000.1
B2 = 0simulate the potential propagation in conductive plates forvarious sets of circuit parameters.
Our future research is how to modify the cloning templates
when some obstacles are put on the conductive plates.

00 ,0References00.1 ,0I2 = 0.

(10)Transient responses are shown in Fig. 3. We observe thatthe potential propagation is damped down as time goes. Notethat the scale of the voltage is quite small for (c).
4.2. Ideal Lossless Oscillation
Next, we consider the ideal lossless plate. The inductance
L and the capacitance C of the plate are the same values as
the damped case, but the resistance R and the conductance Gare set to be zero, respectively. The cloning templates for thiscase are given as follows:0 0 00 0 0
A1 = 0 1 0 , C1 = 0 10 0 ,0 0 00 0 0(11)
B1 = 0,I1 = 0.


0A2 = 00
B2 = 0,010


000.10 , C2 = 0.1 −0.4
000.1[2] M. S. Obrecht, M. I. Elmasry and E. L. Heasell,“TRASIM: Compact and Efficient Two-DimensionalTransient Simulator for Arbitrary Planner Semiconductor Devices,” IEEE Trans. on Computer-Aided Design,
vol. 14, no. 4, pp. 447-458,1995.
[3] Z. Wang and Q. Wu, “A Two-Dimensional Resistance
Simulator Using the Boundary Element Method,” IEEETrans. on Computer-Aided Design, vol. 11, no. 4, pp.
497-504, 1992.
[4] T. Mitsuhashi and K. Yoshida, “A Resistance Calculation Algorithm and its Application to Circuit Extraction,” IEEE Trans. on Computer-Aided Design, vol. 6,
no. 3, pp. 337-345, 1987.
[5] L. O. Chua and L. Yang, “Cellular Neural Networks:Theory,” IEEE Trans. on Circuits Syst., vol. 35, pp.
1257-1272, 1988.
[6] L. O. Chua and L. Yang, “Cellular Neural Networks:Application,” IEEE Trans. on Circuits Syst., vol. 35, pp.
1273-1290, 1988.
[7] T. Roska, L. O. Chua, D. Wolf, T. Kozek, R. Tetzlaff
and F. Puffer, “Simulating Nonlinear Waves and Partial
Differential Equations via CNN–Part I,” IEEE Trans. onCircuits Syst., vol. 42, no. 10, pp 807-815, Oct. 1995.

00.1 ,0I2 = 0.

(12)
The transient responses are shown in Fig. 4. We can observe that the oscillation does not damped down. Note thatthe scales of the voltages are the same for (a), (b) and (c).
5. Conclusions[1] M. S. Obrecht, “A New Stable Method for Linearizationof Discretized Basic Semiconductor Equation,” Solid
State Electron., vol. 36, no. 4. pp. 643-648, 1993.

[8] T. Kozek, L. O. Chua, T. Roska, D. Wolf, R. Tetzlaff, F.
Puffer and K. Lotz, “Simulating Nonlinear Waves andPartial Differential Equations via CNN–Part II,” IEEE
Trans. on Circuits Syst., vol. 42, no. 10, pp 816-820,
Oct. 1995.
[9] Z. Yang, Y. Nishio and A. Ushida, “Generation ofVarious Types of Spatio-Temporal Phenomena in TwoLayer Cellular Neural Networks,” IEICE Trans. on Fundamentals, vol. E87-A, no. 4, pp. 864-871, 2004.

In this study, it has been shown that the two-dimensional
conductive plates can be solved by the framework of the two- [10] A. Ushida, Y. Tanji and Y. Nishio, “Analysis of Twolayer CNNs. After discretizing the plate into two-dimensional
dimensional Circuits Based on Multi-conductor Theocircuits, the ODEs describing the circuit was related with therem,” IEICE Tech. Report, no. NLP97-16, pp. 25-29,two-layer CNN equations. By using the relation, the cloning
1997.
templates of the two-layer CNN were obtained. The computer simulated results showed that the two-layer CNN could- 449 -v

v

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Figure 3: Potential propagation (damped oscillation).

Figure 4: Potential propagation (ideal lossless oscillation).

- 450 -