Ideal Sources, Resistors, Capacitors & Inductors
The basic one-port elements each impose one voltage-current constraint. Resistors dissipate energy, while capacitors and inductors store it and therefore carry state across time.
Every ideal circuit element contributes a constitutive equation: a constraint between its terminal voltage and current. Network topology supplies the remaining equations. The distinction matters because topology alone cannot tell a resistor from a capacitor, and an element law alone says nothing about what the element is connected to.
The element laws #
Under the passive sign convention:
| Element | Terminal law | Power or stored energy |
|---|---|---|
| Resistor | ||
| Capacitor | ||
| Inductor | ||
| Ideal voltage source | current is set by the network | |
| Ideal current source | voltage is set by the network |
An ideal resistor has no memory. A capacitor's voltage and an inductor's current are state variables because changing either instantaneously would require an impulse of current or voltage. Those continuity rules are the bridge from algebraic DC analysis to transient analysis.
Ideal sources are constraints, not batteries #
An ideal voltage source maintains its specified voltage for any current, including a current that makes it absorb power. An ideal current source does the dual operation. These are mathematical primitives; real sources have limits, internal impedance, saturation, thermal constraints, and protection behavior.
Some source connections are inconsistent:
- unequal ideal voltage sources cannot be placed directly in parallel;
- unequal ideal current sources cannot be placed directly in series;
- a zero-resistance loop of voltage sources must have an algebraic voltage sum of zero;
- a cut set of current sources must satisfy current conservation.
The simulator error produced by such a connection is often a model error, not a numerical one.
DC steady state is a special limit #
For constant signals after transients have died out, and . An ideal capacitor then carries zero current and looks open; an ideal inductor has zero voltage and looks short. This shortcut is valid for DC steady state only. It cannot be applied at the switching instant or to a sinusoidal circuit.
The ideal laws are limiting models. A real capacitor adds leakage, series resistance, and inductance; a real inductor adds winding resistance, core loss, capacitance, and saturation. Add those terms only when the operating frequency, amplitude, or accuracy target makes them observable.