Time-Varying, Nonlinear & Signed Equivalent Elements

Equivalent parameters are coordinates of a model, not automatically physical component values. When a device is nonlinear or changes operating mode, one number such as $R=V/I$ cannot describe both its present operating point and its response to perturbations. When a fitted $L(t)$ or $C(t)$ becomes negative, the distinction becomes impossible to ignore.

Static, differential, and dynamic quantities #

For a nonlinear resistor $v=f(i)$:

$$R_{static}=\frac{v}{i},\qquad r_{diff}=\frac{dv}{di}.$$

They answer different questions. $R_{static}$ relates the operating point to the origin; $r_{diff}$ governs small perturbations around that point. A device can have positive $v/i$ and negative slope over part of an active characteristic.

For nonlinear storage, use charge and flux linkage:

$$i=\frac{dq(v)}{dt},\qquad v=\frac{d\lambda(i)}{dt}.$$

The differential capacitance $dq/dv$ or inductance $d\lambda/di$ belongs to a local model. Energy must be computed from the constitutive curve, not by blindly inserting a varying parameter into $Cv^2/2$ or $Li^2/2$.

Explicit time variation adds an energy port #

If a parameter changes because of motion, control, temperature, or switching, the variation exchanges energy with another domain. Differentiating an expression such as $\tfrac12L(t)i^2$ produces a term containing $dL/dt$ in addition to $L i\,di/dt$. A circuit model that omits the mechanism driving $L(t)$ can balance terminal equations while hiding where that energy comes from.

Interpreting signed fits #

The correct response to nonphysical parameters is to revisit topology, state variables, excitation, and constraints. Treat signed values as diagnostics until a causal active or multi-domain realization is demonstrated.

For time-varying storage elements, energy derivatives contain parameter-rate terms: $d(Cv^2/2)/dt=Cv\,dv/dt+(v^2/2)dC/dt$. Omitting the second term assigns externally driven parameter modulation to an ordinary capacitor current and can create false power or passivity conclusions.