# 特勒根定理

## 定律

${\displaystyle \sum _{k=1}^{b}U_{k}I_{k}=0.}$

${\displaystyle \sum _{k=1}^{b}U_{k1}I_{k2}=0.}$
${\displaystyle \sum _{k=1}^{b}U_{k2}I_{k1}=0.}$

## 定义

${\displaystyle a_{ij}={\begin{cases}1,&{\text{如果支路}}j{\text{与节点}}i{\text{相关联，且支路方向背离节点}}\\-1,&{\text{如果支路}}j{\text{与节点}}i{\text{相关联，且支路方向指向节点}}\\0,&{\text{如果支路}}j{\text{与节点}}i{\text{无关联}}\end{cases}}}$

• 基尔霍夫电流定律：
${\displaystyle \mathbf {A} \mathbf {I} =\mathbf {0} }$
• 基尔霍夫电压定律：
${\displaystyle \mathbf {U} =\mathbf {A^{T}} \mathbf {U} }$

${\displaystyle u_{k}}$表示各节点相对${\displaystyle P_{0}}$电势差.

## 证明

{\displaystyle {\begin{aligned}\mathbf {U^{T}} \mathbf {I} =\mathbf {(A^{T}U)^{T}} \mathbf {I} =\mathbf {(U^{T}A)} \mathbf {I} =\mathbf {U^{T}AI} =\mathbf {0} \end{aligned}}}

${\displaystyle \sum _{k=1}^{b}U_{k}I_{k}=\mathbf {U^{T}} \mathbf {I} =0}$

## 应用

${\displaystyle \sum _{j=1}^{n_{P}}W_{j}{\frac {\operatorname {d} Z_{j}}{\operatorname {d} t}}=\sum _{k=1}^{n_{f}}W_{k}f_{k}+\sum _{j=1}^{n_{P}}w_{j}p_{j}+\sum _{j=1}^{n_{t}}w_{j}t_{j},\quad j=1,\dots ,n_{p}+n_{t}}$

## 参考资料

1. ^ Tellegen's Theorem and Electrical Networks by Paul Penfield, Jr., Robert Spence, and Simon Duinker, The MIT Press, Cambridge, MA, 1970

• 《电路理论基础》，C.A. Desoer and E.S. Kuh, McGraw-Hill, New York, 1969
• 《特勒根定理和热力学不等式》, G.F. Oster and C.A. Desoer, J. Theor. Biol 32 (1971), 219–241
• "Network Methods in Models of Production", Donald Watson, Networks, 10 (1980), 1–15