Affine Two-Ports#
Theorem: Strictly Linear Decomposition
Let \(\mathcal{F}\) be an affine two-port.
If \(\mathcal{F}\) has an explicit representation
\[ \begin{bmatrix}p_1 \\ p_2\end{bmatrix} = \boldsymbol{M}\begin{bmatrix}q_1 \\ q_2\end{bmatrix} + \begin{bmatrix}P_1 \\ P_2\end{bmatrix}, \]
then it is equivalent to a strictly linear two-port \(\mathcal{F}'\) whose corresponding representation matrix is \(\boldsymbol{M}\) and which has independent time-invariant ideal sources attached to it in the following configuration:
- If \(p_k\) is the current flowing into the \(j\)-th port of \(\mathcal{F}\), then a DC source is attached at the \(j\)-th port so that the current flowing into the \(j\)-th port of \(\mathcal{F}'\) is \(i_j - P_k\).
- If \(p_k\) is the voltage across the \(j\)-th port of \(\mathcal{F}\), then a voltage source is attached at the \(j\)-th port so that the voltage across the \(j\)-th port of \(\mathcal{F}'\) is \(v_j - P_k\).
Proof
TODO