Week 3 — Energy Storage: Capacitors and Inductors from Field to Component

Overview

Resistors dissipate; capacitors and inductors store. A capacitor stores energy in the electric field between its plates; an inductor stores energy in the magnetic field around its winding. This is the point where the Griffiths field theory pays off most directly: the capacitance \(C\) and inductance \(L\) that appear as single numbers in circuit equations are derived from field configurations, and understanding that derivation tells you why a capacitor’s value depends on geometry and dielectric, why an inductor resists current change, and where the energy actually lives. The defining feature of these elements is that their \(i\)–\(v\) relations are differential — \(i = C\,dv/dt\) and \(v = L\,di/dt\) — which is what makes circuits have memory, transients (Week 4), and frequency-dependent behavior (Weeks 5–6).

The bench half introduces the FNIRSI LCR meter, and the central lesson is that real components are not ideal. A real capacitor has series resistance (ESR), series inductance (ESL), and leakage; a real inductor has winding resistance and self-capacitance. The LCR meter measures these parasitics directly, and — importantly — it measures them at a test frequency, because the effective value changes with frequency. Watching a capacitor’s apparent C and ESR change as you switch the LCR test frequency from 100 Hz to 100 kHz is the most concrete possible introduction to the frequency domain you will formalize in Weeks 5–6.

This week extends Week 1’s field-energy thread and Week 2’s network laws (KCL/KVL still hold; the element relations are now differential). It is the direct prerequisite for Week 4’s transient response, where these \(i\)–\(v\) relations generate the differential equations we solve and capture on the scope.

Readings

Griffiths Ch. 2.5, Ch. 4: Energy stored in an electrostatic field, \(W=\tfrac{\varepsilon_0}{2}\int E^2\,d\tau\); capacitance of a parallel-plate geometry; the role of a dielectric (permittivity \(\varepsilon=\varepsilon_r\varepsilon_0\), bound charge). Extract: \(C\) is geometry × permittivity, and the energy is in the field.
Griffiths Ch. 5, 7.2: Magnetic fields of currents, inductance as flux per current \(L=\Phi/I\), energy \(W=\tfrac{1}{2}LI^2\) stored in the magnetic field, and self-inductance from \(\oint\) relations. Extract: inductance opposes change in current via Faraday/Lenz.
CAD Ch. 5: The circuit-level \(i\)–\(v\) relations \(i_C=C\,dv/dt\) and \(v_L=L\,di/dt\), energy expressions, series/parallel combination rules (note they are opposite to resistors for capacitors), and continuity of capacitor voltage / inductor current.
PEI Ch. 3 (capacitor & inductor sections): Real-world parts — capacitor types (ceramic, electrolytic, film), polarity, ESR, ESL, dielectric absorption, voltage rating; inductor types and winding resistance. Extract: how to choose and not destroy real parts (electrolytic polarity!).

Key Concepts

Capacitance from the field

For a parallel-plate capacitor (plate area \(A\), separation \(d\), dielectric permittivity \(\varepsilon\)), Gauss’s law gives a uniform field \(E=\sigma/\varepsilon=Q/(\varepsilon A)\) and a voltage \(V=Ed\), so

\[ C=\frac{Q}{V}=\frac{\varepsilon A}{d}=\frac{\varepsilon_r\varepsilon_0 A}{d}. \]

\(C\) depends only on geometry and material — never on \(Q\) or \(V\) (for a linear dielectric). The stored energy, equal to the work to assemble the charge, is

\[ W=\tfrac{1}{2}CV^2=\tfrac{1}{2}\frac{Q^2}{C}=\frac{\varepsilon_0}{2}\int E^2\,d\tau, \]

the last form making explicit that the energy resides in the field, not “on the plates.” A dielectric increases \(C\) by \(\varepsilon_r\) because bound charges partially cancel the field, lowering \(V\) for the same \(Q\).

Inductance from the field

A current \(I\) produces magnetic flux \(\Phi\) linking the circuit; inductance is the proportionality \(L=\Phi/I\) (or \(N\Phi/I\) for \(N\) turns). By Faraday’s law a changing current induces a back-EMF \(v=L\,di/dt\) that opposes the change (Lenz). Energy stored in the magnetic field:

\[ W=\tfrac{1}{2}LI^2=\frac{1}{2\mu_0}\int B^2\,d\tau. \]

The duality with the capacitor is exact and worth internalizing: \(C\leftrightarrow L\), \(V\leftrightarrow I\), \(E\leftrightarrow B\), electric energy \(\leftrightarrow\) magnetic energy.

The differential i–v relations and continuity

\[ i_C(t)=C\frac{dv_C}{dt}, \qquad v_L(t)=L\frac{di_L}{dt}. \]

Two consequences that drive everything in Week 4: capacitor voltage cannot change instantaneously (a step in \(v_C\) would demand infinite current), and inductor current cannot change instantaneously (a step in \(i_L\) would demand infinite voltage). These continuity conditions are the initial conditions for transient analysis. Integrated forms: \(v_C(t)=v_C(0)+\frac1C\int_0^t i\,d\tau\) and \(i_L(t)=i_L(0)+\frac1L\int_0^t v\,d\tau\) — capacitors integrate current, inductors integrate voltage.

Series/parallel — the dual of resistors

Capacitors combine like resistor conductances; inductors combine like resistors:

\[ \frac{1}{C_\text{series}}=\sum\frac{1}{C_i},\quad C_\text{parallel}=\sum C_i; \qquad L_\text{series}=\sum L_i,\quad \frac{1}{L_\text{parallel}}=\sum\frac1{L_i}. \]

Real components: ESR, ESL, leakage

The ideal model is a starting point. A real capacitor is better modeled as \(C\) in series with ESR (equivalent series resistance, from leads/plates/dielectric loss) and ESL (series inductance), with a large parallel leakage resistance. These cause the dissipation factor \(D=\tan\delta=\omega C\cdot\text{ESR}\) and quality factor \(Q=1/D\). Crucially the measured C and ESR depend on the test frequency — this is exactly what the LCR meter’s selectable frequency exposes. Electrolytics have high ESR and capacitance that drifts; ceramics have low ESR but value that varies with voltage/temperature (Class II dielectrics). This is why Week 4’s measured time constants won’t perfectly match nominal values, and why decoupling and filtering choices in real designs care about ESR, not just C.

Theory Exercises

Derive \(C=\varepsilon A/d\) for a parallel-plate capacitor from Gauss’s law, and \(W=\tfrac12 CV^2\) from the work to charge it. Show \(W=\tfrac{\varepsilon_0}{2}\int E^2\,d\tau\) gives the same result for the parallel-plate field.
Derive \(v_L=L\,di/dt\) from Faraday’s law and \(L=\Phi/I\). Derive \(W=\tfrac12 LI^2\) from \(\int vi\,dt\).
Prove that capacitor voltage and inductor current must be continuous, and state precisely the conditions under which they could jump (idealized impulse sources).
Derive the series and parallel combination rules for capacitors and for inductors from the \(i\)–\(v\) relations.
For a capacitor with ESR, derive \(D=\tan\delta=\omega C\cdot\text{ESR}\) and \(Q=1/D\). Explain why \(D\) rises with frequency for fixed ESR.
A \(100\,\mu\text{F}\) capacitor is charged to 5 V. How much energy is stored? If discharged through \(1\,\text{k}\Omega\), what is the initial current?

Lab / Bench Work

LCR characterization: Pick several kit capacitors (a small ceramic, a large electrolytic) and inductors. Measure C/L, ESR, and D/Q at each available LCR test frequency (100 Hz / 1 kHz / 10 kHz / 100 kHz). Tabulate how each value drifts with frequency. Compare measured C against the printed value and the 4-band/printed tolerance.

Polarity and rating discipline: Identify electrolytic polarity (stripe = negative) and voltage rating; never exceed the rating or reverse-bias an electrolytic (it can vent). Confirm the ceramic caps are non-polar.

Build the integrator intuition: Drive a capacitor with a (roughly) constant current — e.g. a large resistor from 5 V — and watch \(v_C\) ramp on the scope, confirming \(v_C=\frac1C\int i\,dt\) produces a linear ramp for constant \(i\). (Full transient capture is Week 4; this is the qualitative preview.)

Qucs-S (ngspice): Model the electrolytic as ideal C + ESR using your measured ESR, and confirm the simulated impedance vs frequency matches the LCR readings at the four test points.

Measurement Methodology

Test frequency matters: always record the LCR test frequency alongside each value. A capacitor measured at 100 Hz and 100 kHz can read meaningfully different C and very different ESR; this is physics, not meter error.
Lead effects: for small capacitances and inductances, lead length adds stray L and C; keep leads short and use the meter’s relative/zeroing mode if available.
Self-heating / settling: electrolytics need a moment to settle; take the reading after the display stabilizes.
Reconcile: nominal value, LCR-measured value (with frequency noted), and Qucs-S impedance. Explain any gap via tolerance, dielectric class, or parasitics — do not hand-wave it.

Expected baselines: Ceramics close to nominal with low ESR (m\(\Omega\)–\(\Omega\)) and high Q. Electrolytics within their (wide, often −20%/+80%) tolerance, with ESR of ohms that falls with frequency, and noticeable value drift across the test frequencies.

Connections

The differential \(i\)–\(v\) relations established here are the engine of Week 4’s transient differential equations and Week 5’s impedances (\(Z_C=1/j\omega C\), \(Z_L=j\omega L\) are just these relations in phasor form). The frequency dependence you measured on the LCR meter is the experimental seed of the frequency-domain thinking formalized in Weeks 5–6 and 8. ESR will reappear as the real-world reason filter responses (Week 6) deviate from ideal and why decoupling capacitor choice matters in any embedded design (Courses 1, 4). The field-energy duality (\(\tfrac12CV^2 \leftrightarrow \tfrac12LI^2\)) underlies the energy oscillation in the RLC circuit you build next week.