SIM E-ISSN 2382-672X ISSN 2228-4672 P-ISSN 2382-6711
ASTRALGO Associator
Kron-Roth representations of circuits  
   

2015 Volume 1 1504

$ \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 31 October 2015
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