Electrical Elements#

Definition: Electrical Element

An \(n\)-terminal electrical element is an abstract mathematical model of a lumped electronic (sub)circuit with \(n\) accessible terminals.

Definition: Inputs and Outputs

Some of the \(n\) terminals may be called inputs and others may be called outputs.

Definition: Current Vector

The current vector of an \(n\)-terminal network at time \(t\) is defined as the vector whose components are the currents flowing into the \(n\) terminals at time \(t\):

\[ \begin{bmatrix} i_1(t) \\ \vdots \\ i_n(t) \end{bmatrix} \]

Notation

\[ \boldsymbol{i}(t) \qquad \mathbf{i}(t) \]

Definition: Potential Vector

The potential vector of an \(n\)-terminal network at time \(t\) is defined as the vector whose components are the electrostatic potentials at the \(n\) terminals at time \(t\):

\[ \begin{bmatrix} \varphi_1(t) \\ \vdots \\ \varphi_n(t) \end{bmatrix} \]

Definition: Voltage Vector

The voltage vector of an \(n\)-terminal network at time \(t\) is defined as the vector whose components are the voltages between the terminals and some reference point at time \(t\):

\[ \begin{bmatrix} v_1(t) \\ \vdots \\ v_n(t) \end{bmatrix} \]

Most commonly, this reference point is taken to be one of the terminals.

Notation

\[ \boldsymbol{v}(t) \qquad \mathbf{v}(t) \]

I-V Characteristic#

Definition: I-V Characteristic

The I-V characteristic of an \(n\)-terminal network at time \(t\) is the set \(\mathcal{F}(t) \subseteq \mathbb{R}^{2n}\) of all pairs \((\boldsymbol{v}, \boldsymbol{i})\) of a voltage vector and a current vector which the network is allowed to have at time \(t\).