射影空间中的电路理论与齐次电路模型
Circuit theory in projective space and homogeneous circuit models
摘要：
本文提出了一个通用的框架，线性电路分析的基础上基本方面的射影几何。我们使用一个灵活的方法，其中没有先验分配的电气性质的电路分支是必要的。这样的分配最终仅通过设置某些模型参数来完成，以避免需要在电压源和电流源之间进行区分的方式，并且另外使得可以摆脱对阻抗的电压或电流控制假设。这铺平了道路，为一个完全通用的$m$维减少任何电路定义的$m$两端，非耦合线性元件，相反，在一个步骤或另一个施加某些限制，允许的设备上的最经典的方法。约简具有$$\begin{pmatrix} AP \\ BQ \end{pmatrix} u = \begin{pmatrix} AQ \\ -BP \end{pmatrix} \bar{s}.$$的形式。这里，$A$和$B$捕获图形拓扑，而$P$、$Q$、$\bar{s}$包括所有电路元件的同构描述;未知的$u$是（比方说）“种子”变量的$m$维向量，从该向量获得电流和电压作为$i=Pu -Q\bar{s}$、$v=Qu + P\bar{s}$。计算实现是直接的。这些模型允许非退化配置的多齐次基尔霍夫多项式方面的一般特征，并在这个方向上，我们提出了一些独立的兴趣，涉及矩阵树定理的结果。我们的方法可以很容易地与经典的方法相结合，只使用同质的描述某些分支，产生部分同质的模型。我们还指出了如何容纳受控源和耦合设备在均匀的框架。几个例子说明了结果。
英文摘要：
This paper presents a general framework for linear circuit analysis based on elementary aspects of projective geometry. We use a flexible approach in which no a priori assignment of an electrical nature to the circuit branches is necessary. Such an assignment is eventually done just by setting certain model parameters, in a way which avoids the need for a distinction between voltage and current sources and, additionally, makes it possible to get rid of voltage- or current-control assumptions on the impedances. This paves the way for a completely general $m$-dimensional reduction of any circuit defined by $m$ two-terminal, uncoupled linear elements, contrary to most classical methods which at one step or another impose certain restrictions on the allowed devices. The reduction has the form $$\begin{pmatrix} AP \\ BQ \end{pmatrix} u = \begin{pmatrix} AQ \\ -BP \end{pmatrix} \bar{s}.$$ Here, $A$ and $B$ capture the graph topology, whereas $P$, $Q$, $\bar{s}$ comprise homogeneous descriptions of all the circuit elements; the unknown $u$ is an $m$-dimensional vector of (say) ``seed'' variables from which currents and voltages are obtained as $i=Pu -Q\bar{s}$, $v=Qu + P\bar{s}$. Computational implementations are straightforward. These models allow for a general characterization of non-degenerate configurations in terms of the multihomogeneous Kirchhoff polynomial, and in this direction we present some results of independent interest involving the matrix-tree theorem. Our approach can be easily combined with classical methods by using homogeneous descriptions only for certain branches, yielding partially homogeneous models. We also indicate how to accommodate controlled sources and coupled devices in the homogeneous framework. Several examples illustrate the results.
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