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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.
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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 |
$Z i \hspace{.25em}= V\hspace{1.5em}:=\varepsilon-\pdag_1\phi$ |
$ZJ\hspace{.5em}=V\hspace{.5em}:=e+AE'$ |
$\varepsilon\hspace{.5em}\,=e,\hspace{.5em}\, \phi\hspace{.25em}\mapsto -E'$ |
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$Z(\beta'+\p_2\mu)\,=\varepsilon-\pdag_1\phi$ |
$Z(I+Ci')\,=e+A E'$ |
$\mu\hspace{.25em}\,\mapsto i'$ |
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$Z(\beta'+\mu')\hspace{.75em}\,=\varepsilon-\phi'=V$ |
$Z(I+i)\hspace{1em}\,=e+E=V$ |
$\mu'\,\mapsto i,\hspace{.5em} \phi'\mapsto -E$ |
| Corollary |
$\pdag_2V=\pdag_2\varepsilon=:\varepsilon'$ |
$\Cdag V\hspace{.125em}=\Cdag e=:e'$ |
$\varepsilon'\hspace{.25em}=e'$ |
| Poisson 1 |
$\Delta \phi\hspace{.4em}= \p_1 (Y\varepsilon-\beta')$ |
$Y'E'=\Adag(I-Ye)$ |
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| $\hspace{2.75em}$ 2 |
$\Delta'\mu\hspace{.2em}=\pdag_2 (\varepsilon-Z \beta')$ |
$Z'i'\hspace{.625em}=\Cdag(e-ZI)$ |
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| Problem 1 |
$1.\hspace{.5em}\, Zi\hspace{.5em}=V$ |
$1.\hspace{.5em} ZJ\hspace{.5em}=V$ |
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$2.\hspace{.5em} \p_1i\hspace{.5em}=\beta$ |
$2.\hspace{.5em} \Adag J\hspace{.125em}=I'$ |
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$3.\hspace{.5em} \pdag_2V=\varepsilon'$ |
$3.\hspace{.5em} \Cdag V=e'$ |
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| $\hspace{3em}$ 2 |
$1.\hspace{.5em} Z(\beta'+\mu')=\varepsilon-\phi'$ |
$1.\hspace{.5em} Z(I+i)=e+E$ |
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$2.\hspace{.5em} \p_1\mu'=0$ |
$2.\hspace{.5em} \Adag i\hspace{.4em}=0$ |
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$3.\hspace{.5em} \pdag_2\phi'=0$ |
$3.\hspace{.5em} \Cdag E = 0$ |
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| 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 |
| $$\begin{CD}
@. C_2@>\p_2>>C_1 @.=@. C_1 @>\p_1>>C_0 @.\\
@. @V{\Delta'}VV @VV{Z}V@.@A{Y}AA@AA{\Delta}A\\[-2em]
@. \Cdag_2 @<\pdag_2<<\Cdag_1 @.=@. \Cdag_1 @<\pdag_1<<\Cdag_0 @.
\end{CD}$$
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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 |
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E Paal, Tallinn University of Technology, Estonia |
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Received 25 May 2015; Revised 31 October 2015 |
