Passivity and its Relation to Stability in LRTI $N$-Ports

Among the properties of Linear Resistive Time Invariant (LRTI) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex>-port networks, passivity plays a crucial role. In fact, passivity has a strong relation to stability of the corresponding Linear Dynamic Time Invariant (LDTI) circuit, where a positive dynamic element, i.e. a capacitor or an inductor, is connected to each port. This relation is less intuitive than expected and requires a more formal and in depth approach than that normally found in circuit theory textbooks. Passivity is also very important to characterize AC <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex>-ports, even if, in this case, it takes a somewhat different meaning. Passivity in AC is important in applications, especially those concerning power grids. Reciprocity of these elements will also be considered.