Generalized one-port elements

A-type energy storage elements

An element that stores energy as a function of its across-variable is called an A-type energy storage element. Sometimes we call it a generalized capacitor because a capacitor is an A-type energy storage element.

For generalized through-variable \(\mathcal{F}\), across-variable \(\mathcal{V}\), integrated through-variable \(\mathcal{H}\), and integrated across-variable \(X\) the ideal, linear constitutive equation is \[\begin{aligned} \label{eq:generalized_con_A} \mathcal{H} = C \mathcal{V} \end{aligned}\] for \(C\in\mathbb{R}\) called the generalized capacitance. Differentiating eq. ¿eq:generalized_con_A? with respect to time, the elemental equation is \[\begin{aligned} \frac{d \mathcal{V}}{d t} = \frac{1} {C} \mathcal{F}.\end{aligned}\]

A-type energy storage elements considered thus far are capacitors, translational masses, and rotational moments of inertia. As with generalized variables, the analogs among elements are more important than are generalized A-type energy storage elements.

T-type energy storage elements

An element that stores energy as a function of its through-variable is called a T-type energy storage element. Sometimes we call it a generalized inductor because an inductor is a T-type energy storage element.

The ideal, linear constitutive equation is \[\begin{aligned} \label{eq:generalized_con_T} \mathcal{X} = L \mathcal{F} \end{aligned}\] for \(L\in\mathbb{R}\) called the generalized inductance. Differentiating eq. ¿eq:generalized_con_T? with respect to time, the elemental equation is \[\begin{aligned} \frac{d \mathcal{F}}{d t} = \frac{1} {L} \mathcal{V}.\end{aligned}\]

T-type energy storage elements considered thus far are inductors, translational springs, and rotational springs. As with generalized variables, the analogs among elements are more important than are generalized T-type energy storage elements.

D-type energy dissipative elements

An element that dissipates energy from the system and has an algebraic relationship between its through-variable and its across-variable is called a D-type energy dissipative element. Sometimes we call it a generalized resistor because a resistor is a D-type energy dissipative element.

The ideal, linear constitutive and elemental equation is \[\begin{aligned} \label{eq:generalized_con_D} \mathcal{V} = R \mathcal{F} \end{aligned}\] for \(R\in\mathbb{R}\) called the generalized resistance.

D-type energy dissipative elements considered thus far are resistors, translational dampers, and rotational dampers. As with generalized variables, the analogs among elements are more important than are generalized D-type energy dissipative elements.

Sources

An ideal through-variable source is an element that provides arbitrary energy to a system via an independent (of the system) through-variable. The corresponding across-variable depends on the system. Current, force, and torque sources are the through-variable sources considered thus far.

An ideal across-variable source is an element that provides arbitrary energy to a system via an independent (of the system) across-variable. The corresponding through-variable depends on the system. Voltage, translational velocity, and angular velocity are the across-variable sources considered thus far.