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Capacitors store energy in the electric field: E = ½CV², with i = C(dv/dt); voltage cannot change instantaneously because that would require infinite current. Inductors store energy in the magnetic field: E = ½LI², with v = L(di/dt); current cannot change instantaneously because that would require infinite voltage. Series and parallel combinations follow rules dual to resistors (capacitors in series combine like resistors in parallel, and vice versa). Initial conditions on capacitor voltage and inductor current at the moment of a switching event determine the starting state for all transient analysis.
Derive the i-v relationships from the definitions of capacitance (Q = CV) and inductance (λ = LI) rather than memorizing them. Practice computing energy stored and identifying initial and final conditions before writing any differential equations.
The i-v relationships for capacitors and inductors are not arbitrary formulas — they follow directly from the definitions of capacitance and inductance. Capacitance is defined by Q = CV: the charge stored equals capacitance times voltage. Differentiate both sides with respect to time and you get i = C(dv/dt). Similarly, inductance is defined through magnetic flux linkage λ = LI; Faraday's law gives v = dλ/dt = L(di/dt). These two relationships are the starting point for all capacitor and inductor analysis.
The most important consequence of these relationships is the instantaneous-change prohibition. If a capacitor's voltage jumped from 5 V to 10 V in zero time, then dv/dt would be infinite, requiring infinite current — an impossibility. Likewise, if an inductor's current jumped instantaneously, v = L(di/dt) would be infinite. This means capacitor voltage and inductor current are state variables: they carry memory of the past and cannot be reset by a switching event. The moment before and the moment after a switch closes or opens, these quantities must be equal.
The energy stored in each element is E = ½CV² for a capacitor and E = ½LI² for an inductor. Notice the symmetry: voltage plays the role for the capacitor that current plays for the inductor. This duality extends to how they combine in circuits. Two capacitors in series add reciprocally (just as resistors in parallel do), while capacitors in parallel add directly — the reverse of the resistor rule. Inductors follow the standard resistor pattern: series inductances add, parallel inductances add reciprocally.
When a switch opens or closes in a circuit, the initial conditions on capacitor voltage and inductor current at that instant determine the starting state of the transient solution. Before writing any differential equation, you should identify what V_C and I_L were just before the switching event. These values persist into t = 0⁺ and pin down the arbitrary constants in the homogeneous solution. Missing or incorrectly applying initial conditions is the most common source of error in transient circuit analysis.
Understanding these elements as energy-storage devices also builds physical intuition. A capacitor resists voltage change because changing voltage means moving charge, which takes current over time. An inductor resists current change because changing current means changing magnetic flux, which requires voltage over time. Together, these resistances to change give rise to oscillatory behavior when capacitors and inductors appear in the same circuit — the energy shuttles back and forth between electric and magnetic fields.