
Research Article


Analysis of Capacitance Networks


J.H. Asad


ABSTRACT

This study showed that infinite two dimensional (i.e.,
2D) complex networks consisting of identical capacitors each with capacitance
1farad can be analyzed using basic concepts of physics rather than using
complicated principles. In this study the equivalent capacitance between
adjacent nodes of a square infinite network consisting of identical capacitors
each of 1farad capacitance is determined. The method is applied also
to other networks (i.e., triangular, honeycomb and kagome networks).





INTRODUCTION
Infinite resistive network problems have served as excellent vehicles for helping
physics students and electrical engineering recognize and appreciate the power
and superposition and symmetry in the analysis of electrical networks. These
problems have been studied and illustrated well using random walk approach (Jeng,
2000), superposition and symmetry approach (Venezian,
1994; Atkinson and Steenwijk, 1999; Asad
et al., 2005c) and finally using the socalled Lattice Green’s Function
approach (Cserti, 2000; Asad et
al., 2004, 2005a, b, c,
2006) . A special case of this class of problems involves
the calculation of the effective resistance between two adjacent nodes of an
infinite uniform twodimensional (2D) resistive lattice.
In particular, the effective resistance between two adjacent nodes of the 2D
Liebman resistive mesh (i.e., infinite 2D square resistive lattice) was calculated
by Aitchison (1964) and the result was found to be (1/2)R.
Bartis (1967) calculated the resistance between adjacent
nodes for three other infinite 2D resistive lattices, the triangular, Honeycomb
and kagome lattices. He found that the effective resistances to be (1/3)R, (2/3)R
and (1/2)R, respectively. Aitchison (1964) and Bartis
(1967) showed how to analyze infinite 2D resistive networks using the undergraduate
level (i.e., Ohm’s and Kirchhoff’s Laws).
Now is it possible to analyze complex infinite networks using undergraduate
principles and laws?
This note is written to answer the above question. So, the aim of this note
is to show that infinite 2D networks consisting of identical capacitors each
of capacitance 1farad can also be treated and given to the undergraduate level.
The method used in this short note is the same as that presented in Bartis
(1967). So, here we introduce the problem with its complete solution as
a counter example.
RESULTS AND DISCUSSION
The problem here is the determination of the equivalent capacitance between
adjacent nodes of a square infinite network consisting of identical capacitors
each of 1farad capacitance. To do this, let us concentrate at a given
element C near the center of Fig. 1 and assume the charge
passing through in two cases: First, consider a potential difference of
m Volt exists between one end of C and the other four sides of the network
at infinity; Second, consider a potential difference of m Volt is established
between infinity and the other end of C.

Fig. 1: 
Infinite square network consisting of identical capacitors
each of capacitance 1farad 

Fig. 2: 
Infinite triangular network consisting of identical capacitors
each of capacitance 1farad 
It is clear from the symmetry of the situation that in both cases listed above
a quarter of the charge passes through C. Now superimpose these two cases, choosing
m such that the potential difference across C is 2Volt. Thus it is easy to
see that half of the charge then flows through C. From Ohm’s law, the charge
in C is 2coulomb. Therefore, it follows that the total charge flowing in the
network is 4coulomb and as a result its effective capacitance between adjacent
nodes is 2farad. The same result was recently obtained by Asad
et al. (2004) using LGF method and Asad et
al. (2006) using the superposition of charge distribution method.
The above analysis can also be applied to other networks (i.e., triangular,
honeycomb and kagome networks) but with some minor changes (Fig.
24). Now, assuming all the elements of Fig.
24 have a capacitance of 1farad, one can finds that
the equivalent capacitance between adjacent nodes for the earlier mentioned
networks is equal to 3, 3/2 and 2 farad, respectively.

Fig. 3: 
Infinite honeycomb network consisting of identical capacitors
each of capacitance 1farad 

Fig. 4: 
Infinite honeycomb network consisting of identical capacitors
each of capacitance 1farad 
CONCLUSION
It is important to notice that the present approach used to analyze the
Capacitance network used the concepts of symmetry and the superposition.
These two ideas are certainly not beyond the undergraduates. So, we can
analyze the capacitance network for the undergraduate students. Finally,
the content of this short note is helpful for electric circuit design.
In this study, infinite complex 2D networks of identical capacitors has
been analyzed using basic undergraduate principles. The results obtained
in this study are very similar to those results obtained using complex
methods (i.e., LGF method and superposition of charge distribution method
which is based on complex mathematics.

REFERENCES 
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