R&D SIM C o n t a c t ÷ A b o u t
D e s c r i p t i o n ÷ B o a r d ÷ A u t h o r ÷ I n d e x E-ISSN 2382-672X P-ISSN 2382-6711


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's one page NW manual  

ASTRALGO Science 2015 Volume 1 1504

KRON's one page N/W manual

$ \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 geometry of the Kron-Roth representations of circuits, the main network (n/w) rules are concisely listed. The invariance of input and output powers of a conducting n/w are presented as the discrete Stokes laws.
  Netgeometry [1, 2] Kron-Roth [3, 4] rep 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
$$\begin{CD} @. C_2@>\p_2>>C_1 @.=@. C_1 @>\p_1>>C_0 @.\\ @. @V{\Delta'}VV @VV{Z}V@.@A{Y}AA@AA{\Delta}A\\[-2em] @. \Cdag_2 @<\pdag_2<<\Cdag_1 @.=@. \Cdag_1 @<\pdag_1<<\Cdag_0 @. \end{CD}$$
The research was in part supported by the Estonian Research Council, Grant ETF9038.


[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, 19086 Tallinn, Estonia
  Received 25 May 2015; Revised 31 October 2015
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