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ASTRALGO Science

ASTRALGO Science is an international multidisciplinary peer reviewed scholarly & scientific research journal. The main aim is dissemination and publishing of the professional scientific knowledge and information. The ASTRALGO Science Editorial Board members are internationally recognized science experts as well as experienced conference organizers and editors of several scientific journals & proceedings. Standard & open access options.
BIG BANG Speed Test: Kron-Roth representations of circuits  

ASTRALGO Science 2015 Volume 1 1504

Note on Kron-Roth representations of circuits

 
$ \DeclareMathOperator{\IM}{Im} \DeclareMathOperator{\Ker}{Ker} \newcommand{\dag}{\dagger} \newcommand{\p}{\partial } \newcommand{\pdag}{\partial^{\dag}} \newcommand{\Adag}{A^{\dag}} \newcommand{\Cdag}{C^{\dag}} \newcommand{\ee}{\varepsilon} \newcommand{\be}{\beta} \renewcommand{\bra}{\langle} \renewcommand{\ket}{\rangle} $
  Following the Kron-Roth representations of circuits, the main network rules are concisely listed. The invariance of input and output powers of a conducting network are presented as the discrete Stokes laws.
 
  Netgeometry [1, 2] Kron-Roth [3, 4] Transform
Differential $\p:=(\p_1,\p_2)$ $\Adag,\, C$ $\p_1 \mapsto \Adag, \quad \p_2 \mapsto C$
Composition $C_2 \stackrel{\p_2}{\longrightarrow} C_1 \stackrel{\p_1}{\longrightarrow} C_0$ $C'_2 \stackrel{C}{\longrightarrow} C'_1 \stackrel{\Adag}{\longrightarrow} C'_0$ Mesh $\mapsto$ Branch $\mapsto$ Node
Orthogonality $\p_1\p_2=0$ $\Adag C=0$ $\IM{\p_2}=\Ker{\p_1}$
Laplace 1 $\Delta\hspace{.25em}:=\p_1 Y\pdag_1$ $Y':=\Adag Y A$ $\Delta\hspace{.3em}\mapsto Y'\quad$ Admittance
$\hspace{2.75em}$ 2 $\Delta':=\pdag_2 Z \p_2$ $Z'\,:=\Cdag Z C$ $\Delta'\mapsto Z'\quad\hspace{.2em}$ Impedance
Kirchhoff CL $\p_1 i= \beta\hspace{2em}=\p_1\beta'$ $\Adag J= I'\hspace{.75em}=\Adag I$ $i\hspace{.5em}\,\,\mapsto J,\hspace{.5em} \beta\mapsto I',\hspace{.5em} \beta'\mapsto I$
$\hspace{3.25em}$ VL $Z i \hspace{.25em}= V\hspace{1.5em}:=\varepsilon-\pdag_1\phi$ $ZJ\hspace{.5em}=V\hspace{.5em}:=e+AE'$ $\varepsilon\hspace{.5em}\,=e,\hspace{.5em}\, \phi\hspace{.25em}\mapsto -E'$
  $Z(\beta'+\p_2\mu)\,=\varepsilon-\pdag_1\phi$ $Z(I+Ci')\,=e+A E'$ $\mu\hspace{.25em}\,\mapsto i'$
  $Z(\beta'+\mu')\hspace{.75em}\,=\varepsilon-\phi'=V$ $Z(I+i)\hspace{1em}\,=e+E=V$ $\mu'\,\mapsto i,\hspace{.5em} \phi'\mapsto -E$
Corollary $\pdag_2V=\pdag_2\varepsilon=:\varepsilon'$ $\Cdag V\hspace{.125em}=\Cdag e=:e'$ $\varepsilon'\hspace{.25em}=e'$
Poisson 1 $\Delta \phi\hspace{.4em}= \p_1 (Y\varepsilon-\beta')$ $Y'E'=\Adag(I-Ye)$  
$\hspace{2.75em}$ 2 $\Delta'\mu\hspace{.2em}=\pdag_2 (\varepsilon-Z \beta')$ $Z'i'\hspace{.625em}=\Cdag(e-ZI)$  
Problem 1 $1.\hspace{.5em}\, Zi\hspace{.5em}=V$ $1.\hspace{.5em} ZJ\hspace{.5em}=V$  
  $2.\hspace{.5em} \p_1i\hspace{.5em}=\beta$ $2.\hspace{.5em} \Adag J\hspace{.125em}=I'$  
  $3.\hspace{.5em} \pdag_2V=\varepsilon'$ $3.\hspace{.5em} \Cdag V=e'$  
$\hspace{3em}$ 2 $1.\hspace{.5em} Z(\beta'+\mu')=\varepsilon-\phi'$ $1.\hspace{.5em} Z(I+i)=e+E$  
  $2.\hspace{.5em} \p_1\mu'=0$ $2.\hspace{.5em} \Adag i\hspace{.4em}=0$  
  $3.\hspace{.5em} \pdag_2\phi'=0$ $3.\hspace{.5em} \Cdag E = 0$  
Power OUT $\bra\be'|\phi'\ket=\bra\be|\phi\ket$ $\bra I|E\ket\hspace{.25em}=\bra I'|E'\ket$ $\bra\beta'|\pdag_1\phi\ket=\bra\p_1 \beta'|\phi\ket\hspace{.75em}$ Stokes
$\hspace{2.25em}$ IN $\bra\mu'|\varepsilon\ket\hspace{.5em}=\bra\mu|\varepsilon'\ket$ $\bra i|e\ket\hspace{.75em}=\bra i'|e'\ket$ $\bra\p_2\mu|\varepsilon\ket\hspace{.5em}=\bra\mu|\pdag_2\varepsilon\ket\hspace{1.125em}$ Stokes
$$\begin{CD} @. C_2@>\p_2>>C_1 @.=@. C_1 @>\p_1>>C_0 @.\\ @. @V{\Delta'}VV @VV{Z}V@.@A{Y}AA@AA{\Delta}A\\[-2em] @. \Cdag_2 @<\pdag_2<<\Cdag_1 @.=@. \Cdag_1 @<\pdag_1<<\Cdag_0 @. \end{CD}$$

References

[1] Paal E and Umbleja M 2014 J. Phys.: Conf. Series 532 012022
[2] Paal E and Umbleja M 2015 ASTRALGO Sci. 1 1501; ibid 1502
[3] Kron G 1939 Tensor Analysis of Networks (New York: John Wiley & Sons); 1965 (London: Macdonald & Co)
[4] Roth J P 1955 Proc. Nat. Acad. Sci. USA 41 518
  E Paal, Tallinn University of Technology, Estonia
  Received 25 May 2015; Revised 30 October 2015, 30 November 2015
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