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Networks#

Network Topology#

Definition: Network Graph

The network graph of an electronic circuit is a directed multigraph which represents the physical structure of the circuit.

Notation

We often denote the number of branches by \(b\) and the number of nodes by \(n\).

Definition: Branch Voltage Vector

Let \(\mathcal{G} = (N, B, s, t)\) be a network graph.

The branch voltage vector is the vector whose components are the voltages across \(\mathcal{G}\)'s branches:

\[ \boldsymbol{v} \overset{\text{def}}{=} \begin{bmatrix}v_1 \\ \vdots \\ v_b\end{bmatrix} \in \mathbb{R}^{b} \]

Definition: Branch Current Vector

Let \(\mathcal{G} = (N, B, s, t)\) be a network graph.

The branch current vector is the vector whose components are the currents flowing along \(\mathcal{G}\)'s branches:

\[ \boldsymbol{i} \overset{\text{def}}{=} \begin{bmatrix}i_1 \\ \vdots \\ i_b\end{bmatrix} \in \mathbb{R}^{b} \]

The network graph must be constructed in a specific way:

Tellegen's Theorem

Let \(\mathcal{G}\) be a network graph.

If \(\boldsymbol{v}(t)\) is the branch voltage vector of \(\mathcal{G}\) at some time \(t\) and \(\boldsymbol{i}(t^{\ast})\) is the branch current vector of \(\mathcal{G}\) at some time \(t^{\ast}\), then \(\boldsymbol{v}(t)\) and \(\boldsymbol{i}(t^{\ast})\) are orthogonal:

\[ \boldsymbol{v}(t) \cdot \boldsymbol{i}(t^{\ast}) = 0 \]
Proof

TODO

Duality#

Definition: Networks

Let \(\boldsymbol{v}\), \(\boldsymbol{i}\) be the branch voltage vector and branch current vector of a network \(\mathcal{N}\) and let \(\boldsymbol{v}^d\), \(\boldsymbol{i}^d\) be the branch voltage vector and branch current vector of a network \(\mathcal{N}^d\).

We say that \(\mathcal{N}\) and \(\mathcal{N}^d\) are dual if there exists some \(R_d \in \mathbb{R}\) such that

\[\boldsymbol{v}^d = R_d \boldsymbol{i} \qquad \boldsymbol{i}^d = \frac{1}{R_d}\boldsymbol{v}.\]

Algorithm: Constructing Duals

We are given a planar network \(\mathcal{N}\) and want to find its dual \(\mathcal{N}^d\) with respect to the duality constant \(R_d\).

  1. Every mesh in \(\mathcal{N}\) corresponds to a node in \(\mathcal{N}^d\).
  • The outside of \(\mathcal{N}\) also corresponds to a node in \(\mathcal{N}^d\).

  1. If a one-port in \(\mathcal{N}\) is on the boundary between two meshes or a mesh and the outside, then its dual is connected to the corresponding dual nodes in \(\mathcal{N}^d\).

  2. Determine the directions for the dual currents and voltages:

  • Assign an identical orientation to each mesh and the outside of \(\mathcal{N}\) (either everything clockwise or everything counter-clockwise).
  • Each current \(i\) in \(\mathcal{N}\) is adjacent to exactly two meshes or is adjacent to a mesh and the outside. The dual \(v^d\) points away from the dual node of the region whose orientation aligns with the direction of \(i\) and points into the dual node the region whose orientation opposes the direction of \(i\).
  • Each voltage \(v\) in \(\mathcal{N}\) is adjacent to exactly two meshes or is adjacent to a mesh and the outside. The dual \(i^d\) points away from the dual node of the region whose orientation aligns with the direction of \(v\) and points into the dual node of the region whose orientation opposes the direction of \(v\).