01 · Foundational → intermediate

Capacitors, inductors & stored energy

Build the two ideal energy-storage elements from their time-domain constitutive laws before solving any transient circuit. Derive q=Cv and i=C dv/dt for a linear capacitor, v=L di/dt for a linear inductor, integrate power to obtain wC=½Cv² and wL=½Li², prove the ideal continuity of capacitor voltage and inductor current under finite excitation, distinguish state from instantaneous power flow, and mark the physical limits of the lumped ideal models.

150–190 minFoundational → intermediate

In this lesson

Define the ideal linear capacitor through q=Cv and derive i=C dv/dt with the AoN reference convention.
Define the ideal linear inductor through flux linkage λ=Li and derive v=L di/dt.
Use p=vi and integration to derive stored energies wC=½Cv² and wL=½Li² from an unenergized reference state.
Show from the constitutive laws why ideal capacitor voltage and ideal inductor current cannot jump under finite current/voltage impulses respectively.
Distinguish energy storage from energy dissipation and explain why ideal C and L can both absorb and return energy.
Identify the natural state variables of isolated ideal capacitive and inductive storage as capacitor voltage and inductor current.
Interpret DC steady-state open/short statements as limiting consequences of dv/dt=0 or di/dt=0 rather than as definitions of the components.
Check dimensions and signs of C, L, power and stored-energy expressions.
State important model limits: ESR/leakage/dielectric loss and breakdown for capacitors; winding resistance/core loss/saturation/parasitic capacitance for inductors; distributed effects at sufficiently high frequency.

Starting question

A resistor can dissipate energy, but after sources are removed an ideal resistor has no internal electrical state to remember. Capacitors and inductors are different: they can store recoverable energy, so the present circuit state can depend on the past.

Before solving any transient, we derive exactly what is stored, which variable carries the memory, and what continuity constraints follow from the ideal constitutive laws.

01.1The ideal linear capacitor

Capacitance relates charge separation to terminal voltage. For an ideal linear capacitor with the AoN passive reference convention—current entering the terminal marked positive for voltage—write

q(t)=C v_C(t)

where C is constant and measured in farads:

[C]=C/V=A·s/V=F

Current is the time rate of charge entering the positive terminal, so

i_C(t)=dq/dt=C dv_C/dt

This differential law is the source of capacitor memory: current determines how quickly voltage changes; it does not generally determine voltage from the instantaneous current alone.

For a nonlinear capacitor one may instead need a charge–voltage relation q(v) and differential capacitance dq/dv. DY-01 begins with the linear constant-C abstraction.

01.2The ideal linear inductor

For an ideal linear inductor, magnetic flux linkage λ is proportional to current:

λ(t)=L i_L(t)

Faraday’s law applied to the lumped element gives terminal voltage, with passive references, as

v_L(t)=dλ/dt=L di_L/dt

The henry has units

[L]=V·s/A=H

Here voltage determines how quickly current changes. The instantaneous current cannot in general be inferred from instantaneous voltage without knowing the previous current.

Real magnetic components can be nonlinear or hysteretic, so λ=Li with constant L is a model, not a universal material law.

01.3Derive capacitor energy from power

With passive sign convention, instantaneous absorbed power is

p_C=v_Ci_C=C v_C dv_C/dt

The change in stored energy from voltage v₁ to v₂ is therefore

Δw_C=∫p dt=∫_v₁^v₂C v dv=½C(v₂²−v₁²)

Choose the uncharged state v=0 as zero stored energy:

w_C=½C v_C² ≥ 0

The square makes stored energy independent of the chosen voltage polarity. Reversing voltage reverses charge labels but not the positive amount of ideal electrostatic energy.

01.4Derive inductor energy from power

p_L=v_Li_L=L i_L di_L/dt

Integrate from current i₁ to i₂:

Δw_L=∫_i₁^i₂L i di=½L(i₂²−i₁²)

Taking i=0 as the zero-energy state gives

w_L=½L i_L² ≥ 0

Again the energy depends on the square of the state variable, not its chosen positive direction.

01.5Stored energy is not the same as instantaneous power

Power tells us the instantaneous direction and rate of energy transfer. Stored energy tells us how much recoverable energy is currently associated with the ideal element.

p=viInterpretation under passive convention

p>0the element is absorbing energy at that instant

p<0the element is returning energy to the rest of the circuit

p=0stored energy may still be nonzero

An ideal charged capacitor with constant voltage has i=0 and p=0 but can have wC>0. An ideal inductor carrying constant current has v=0 and p=0 but can have wL>0.

This differs fundamentally from an ideal resistor, whose passive power p=Ri² is nonnegative and represents dissipation rather than recoverable storage.

Figure 1.1 — Which quantity carries the element’s memory?

The one question is: which variable must retain its previous value to predict what happens next?

In the ideal linear model, capacitor energy is completely determined by capacitor voltage and inductor energy by inductor current. Those quantities therefore become natural state variables for first-order circuits.

01.6Why capacitor voltage cannot jump under finite current

Integrate the capacitor law across a short interval from t₁ to t₂:

v_C(t₂)−v_C(t₁)=1/C ∫_t₁^t₂ i_C(t)dt

If current remains finite as the interval shrinks to zero, the integral tends to zero. Therefore

v_C(0⁺)=v_C(0⁻)

for an ordinary switching event with no current impulse.

A mathematical jump Δv is possible only if current contains an impulse whose charge area is

∫i dt=C Δv

The precise rule is therefore not “capacitor voltage can never jump,” but “an ideal capacitor voltage cannot jump under finite non-impulsive current.”

01.7Why inductor current cannot jump under finite voltage

Integrating the inductor law gives

i_L(t₂)−i_L(t₁)=1/L ∫_t₁^t₂ v_L(t)dt

For finite voltage over a vanishing time interval, the integral vanishes:

i_L(0⁺)=i_L(0⁻)

A current jump Δi requires an ideal voltage impulse with volt-second area

∫v dt=L Δi

This continuity rule is the key initial-condition constraint for RL dynamics.

01.8DC steady state: open and short are consequences, not definitions

If an ideal capacitor has reached a finite constant voltage in DC steady state,

dv_C/dt=0 ⇒ i_C=0

so its terminal behavior is an open circuit for that settled DC condition.

If an ideal inductor has reached a finite constant current,

di_L/dt=0 ⇒ v_L=0

so its terminal behavior is a zero-voltage short for that settled ideal DC condition.

These substitutions are not valid during a transient, and they are not definitions of C and L. Their defining laws remain the differential constitutive equations.

01.9Worked energy and continuity checks

Example A — a pedagogical 10 µF capacitor is at 5 V:

w_C=½(10 µF)(5 V)²=125 µJ

If 2 mA flows into its positive terminal for 1 ms and C remains constant,

Δv=1/C ∫i dt=(2 mA)(1 ms)/(10 µF)=0.2 V

so the voltage rises continuously from 5.0 V to 5.2 V.

Example B — a pedagogical 20 mH inductor carries 0.30 A:

w_L=½(20 mH)(0.30 A)²=0.90 mJ

Applying +4 V with passive polarity for 0.5 ms changes current by

Δi=1/L ∫v dt=(4 V)(0.5 ms)/(20 mH)=0.10 A

so current rises continuously to 0.40 A.

Both examples use declared teaching values. Real component voltage/current ratings and parasitics are separate design constraints.

01.10Model limits & mastery exercises

The ideal lumped models omit capacitor ESR, leakage, dielectric absorption/loss and breakdown; and inductor winding resistance, core loss, saturation, hysteresis and parasitic capacitance. At sufficiently high frequency, package/interconnect/distributed effects also matter. Use the ideal laws inside their validated range, then enrich the model when evidence demands it.

Starting from q=Cv, derive the capacitor current law for constant C.
Starting from λ=Li, derive the inductor voltage law for constant L.
Derive wC from p=vi and check its SI units.
Derive wL and check its SI units.
A 4.7 µF capacitor is at −3 V. What ideal energy is stored?
A 15 mH inductor carries −0.2 A. What ideal energy is stored?
What current impulse area is required to change a 2 µF capacitor voltage instantaneously by +1.5 V in the ideal model?
What voltage impulse area is required to change a 5 mH inductor current instantaneously by −0.4 A?
Why is “capacitor=open, inductor=short” dangerous as a universal statement?

Solutions

1. i=dq/dt=C dv/dt for constant C.

2. v=dλ/dt=L di/dt for constant L.

3. Integrating Cv dv gives ½Cv²; F·V²=J.

4. Integrating Li di gives ½Li²; H·A²=J.

5. ½(4.7 µF)(3²)=21.15 µJ; the sign of voltage does not make energy negative.

6. ½(15 mH)(0.2²)=0.300 mJ.

7. ∫i dt=CΔv=(2 µF)(1.5 V)=3 µC.

8. ∫v dt=LΔi=(5 mH)(−0.4 A)=−2 mV·s.

9. Those are settled ideal DC limits. During transients C and L obey differential laws; real components also include loss and parasitics.

Chapter 02 will place one storage element inside a resistive network and show how Kirchhoff plus its constitutive law creates a first-order differential equation. The goal is to derive the general solution structure before memorizing any RC or RL waveform.