Circuits with Capacitors and Inductors

SECTION 1

Historical Context & Motivation

The study of circuits containing capacitors and inductors grew from two centuries of experiments that progressively revealed how energy could be stored, released, and exchanged within electrical systems. Unlike purely resistive circuits—where energy is simply dissipated as heat—circuits with capacitors and inductors exhibit time-dependent, transient behavior that underpins technologies from radio broadcasting to modern power electronics. The intellectual lineage connects Leyden jars, Faraday's induction experiments, and Maxwell's unifying electromagnetic theory into a coherent framework for understanding how circuits store and shuttle energy between electric and magnetic fields.

1745

The Leyden Jar

Pieter van Musschenbroek and Ewald Georg von Kleist independently invent the Leyden jar, the first practical capacitor. It demonstrated that electrical charge could be accumulated and later discharged in a sudden burst—an early hint at energy storage in circuits.

1831

Faraday's Law of Induction

Michael Faraday discovers that a changing magnetic flux through a conducting loop produces an electromotive force. This principle forms the physical basis for inductance, the tendency of a coil to oppose changes in current.

1853

Thomson's LC Oscillation Theory

William Thomson (Lord Kelvin) mathematically predicts that a circuit containing only a capacitor and an inductor will oscillate at a natural frequency f = 1/(2π√(LC)). This analysis laid the groundwork for all resonant-circuit applications.

1887

Hertz Generates Electromagnetic Waves

Heinrich Hertz uses an LC oscillator to generate and detect radio-frequency electromagnetic waves, experimentally confirming Maxwell's predictions and demonstrating the practical importance of LC circuits.

1893

Steinmetz and AC Phasor Analysis

Charles Proteus Steinmetz introduces complex-number (phasor) methods for analyzing AC circuits with capacitors and inductors, providing the mathematical toolkit still used in electrical engineering today.

The central question this lesson addresses is: how do circuits behave when they include elements that store energy in electric fields (capacitors) or magnetic fields (inductors)? The answer involves differential equations whose solutions describe exponential charging and discharging, sinusoidal oscillations, and the interplay between resistive damping and reactive energy storage. Mastering these ideas is essential for understanding AC power systems, filters, oscillators, and the electromagnetic waves that carry wireless signals.

SECTION 2

Core Principles & Definitions

Circuits with capacitors and inductors are governed by a small set of foundational ideas that distinguish them from purely resistive networks. Each passive element—resistor, capacitor, inductor—has a unique voltage–current relationship, and it is the time-derivative nature of the capacitor and inductor relationships that gives rise to the rich transient and oscillatory phenomena we observe. Understanding these principles provides the vocabulary and conceptual framework needed for the quantitative analysis that follows.

Capacitance (C)

A capacitor stores energy in an electric field between its plates. The charge Q on a capacitor is related to the voltage V across it by Q = CV, and the current flowing into it is i = C(dV/dt). Energy stored: U_C = ½CV².

Inductance (L)

An inductor stores energy in the magnetic field generated by current flowing through its coil. The voltage across an inductor is V = L(di/dt), meaning it opposes rapid changes in current. Energy stored: U_L = ½Li².

Time Constants (τ)

In RC circuits, the time constant τ = RC sets the exponential rate of charging or discharging. In RL circuits, τ = L/R governs how quickly current rises or decays. After one time constant, the variable reaches approximately 63.2% of its final value.

Natural Frequency (ω₀)

An LC circuit oscillates at the natural angular frequency ω₀ = 1/√(LC). This is the electrical analog of a mass-spring system, with energy shuttling between the capacitor's electric field and the inductor's magnetic field.

Damping in RLC Circuits

When resistance is present in an LC circuit, energy is dissipated and oscillations decay over time. The circuit may be underdamped (oscillatory decay), critically damped, or overdamped, depending on the relative magnitudes of R, L, and C.

✦ KEY TAKEAWAY

Think of a capacitor as a spring and an inductor as a flywheel. A spring stores potential energy when compressed (analogous to charge on a capacitor), while a flywheel stores kinetic energy through its rotational inertia (analogous to current through an inductor). Connect them, and energy bounces back and forth—potential to kinetic and back—just as in a mechanical oscillator. A resistor acts like friction, gradually draining the oscillation until equilibrium is reached.

SECTION 3

Visual Explanation: RC & RL Transient Behavior

The transient response of RC and RL circuits follows exponential curves whose shapes are governed by the circuit's time constant τ. The diagram below illustrates the canonical charging and discharging behaviors side by side, providing a visual reference for the mathematical expressions developed in Section 4.

Top panels: Voltage across a charging/discharging capacitor (left, cyan/pink curves) and current through an energizing/de-energizing inductor (right, violet/amber curves), plotted against time normalized to the time constant τ. Bottom panels: corresponding circuit schematics with the time constant expression for each configuration.

Several features of the diagram merit attention. First, the charging (growth) curves and discharging (decay) curves are mirror images of one another: the sum of the charging voltage and the discharging voltage equals V₀ at every instant. Second, both RC and RL circuits exhibit the same mathematical structure—the exponential function e^−t/τ—despite the fact that one stores energy in an electric field and the other in a magnetic field. Third, after approximately 5τ, the transient has decayed to less than 1% of its initial amplitude, and the circuit is considered to have reached steady state. This 5τ rule of thumb is widely used in engineering practice.

SECTION 4

Mathematical Framework

The behavior of circuits with capacitors and inductors is described by first-order or second-order ordinary differential equations obtained by applying Kirchhoff's voltage law (KVL) around each loop. Because the voltage–current relationships for C and L involve derivatives, the resulting equations are fundamentally different from the algebraic equations of resistive networks. We develop the key results for RC, RL, and LC/RLC circuits below.

RC Circuit Equations

RC CHARGING — CAPACITOR VOLTAGE

V_C(t) = V₀ (1 − e^(−t/RC))

V₀ = applied EMF, R = resistance (Ω), C = capacitance (F), τ = RC = time constant (s). Derived from KVL: V₀ = iR + q/C, where i = dq/dt.

RC DISCHARGING — CAPACITOR VOLTAGE

V_C(t) = V₀ e^(−t/RC)

When the source is removed and the capacitor discharges through R, the voltage decays exponentially. The current during discharge is i(t) = −(V₀/R) e^(−t/RC), with the negative sign indicating direction reversal.

RL Circuit Equations

RL CURRENT GROWTH

i(t) = (V₀/R)(1 − e^(−Rt/L))

V₀ = applied EMF, L = inductance (H), R = resistance (Ω), τ = L/R = time constant (s). Derived from KVL: V₀ = iR + L(di/dt). The inductor initially opposes the full current, which then exponentially approaches V₀/R.

LC Oscillation

LC NATURAL FREQUENCY

ω₀ = 1/√(LC) → f₀ = 1/(2π√(LC))

In an ideal LC circuit (R = 0), charge oscillates sinusoidally: q(t) = Q₀ cos(ω₀t + φ). Energy is continuously exchanged between U_C = q²/(2C) and U_L = ½Li², with total energy conserved.

RLC Damping

RLC CHARACTERISTIC EQUATION

L(d²q/dt²) + R(dq/dt) + q/C = 0

The nature of the solution depends on the discriminant Δ = R² − 4L/C. If R² < 4L/C → underdamped (oscillatory decay). If R² = 4L/C → critically damped. If R² > 4L/C → overdamped (non-oscillatory decay).

📐 Derivation Sketch

For the series RLC circuit, apply KVL around the loop: V_L + V_R + V_C = 0, yielding L(d²q/dt²) + R(dq/dt) + q/C = 0. This is mathematically identical to the damped harmonic oscillator equation m(d²x/dt²) + b(dx/dt) + kx = 0, with the correspondences L ↔ m (inertia), R ↔ b (damping), 1/C ↔ k (restoring force). The trial solution q = Q₀ e^αt leads to the characteristic equation Lα² + Rα + 1/C = 0, whose roots determine the three damping regimes.

SECTION 5

Energy Exchange in LC Oscillations

The oscillatory behavior of an ideal LC circuit is best understood by tracking energy flow between the capacitor and the inductor over one complete cycle. At t = 0, suppose the capacitor is fully charged to Q₀ and the current is zero. All energy resides in the electric field: U_total = Q₀²/(2C). As the capacitor begins to discharge, current builds in the inductor, transferring energy to the magnetic field. At t = T/4 (one-quarter period), the capacitor is fully discharged and all energy has migrated to U_L = ½LI_max². The current then recharges the capacitor with reversed polarity, and the cycle repeats.

Upper panel: Electric-field energy U_C (cyan) and magnetic-field energy U_L (violet) as functions of time over one full period T. The two curves are complementary: their sum equals U_total at every instant. Lower panel: Four snapshots showing the energy distribution, charge, and current at quarter-period intervals.

The diagram makes clear that U_C and U_L oscillate as cos²(ω₀t) and sin²(ω₀t), respectively, and their sum is constant—energy conservation in its most elegant electrical form. The maximum current is related to the maximum charge by I_max = ω₀ Q₀ = Q₀/√(LC), which follows directly from differentiating q(t) = Q₀ cos(ω₀t). This relationship is the electrical counterpart of v_max = ωA in simple harmonic motion, reinforcing the deep analogy between mechanical and electrical oscillators.

Analogy between mechanical simple harmonic motion and LC electrical oscillation

Mechanical SHM	LC Oscillation	Physical Role
Mass m	Inductance L	Inertia — resists changes in motion/current
Spring constant k	1/C (inverse capacitance)	Restoring force / voltage
Displacement x	Charge q	Quantity that oscillates
Velocity v = dx/dt	Current i = dq/dt	Rate of change of displacement/charge
Damping b	Resistance R	Dissipative element
½kx²	q²/(2C)	Potential / electric-field energy
½mv²	½Li²	Kinetic / magnetic-field energy

SECTION 6

Worked Example: RLC Transient Analysis

Consider a series RLC circuit with R = 10 Ω, L = 50 mH, and C = 200 μF. The capacitor is initially charged to V₀ = 12 V and then allowed to discharge through the resistor and inductor at t = 0. Determine the damping regime, the oscillation frequency (if applicable), and the charge on the capacitor as a function of time.

Series RLC Discharge

Step 1 — Identify Given Values and Compute the DiscriminantWe have R = 10 Ω, L = 50 × 10⁻³ H = 0.050 H, and C = 200 × 10⁻⁶ F = 2.00 × 10⁻⁴ F. The initial charge is Q₀ = CV₀ = (2.00 × 10⁻⁴ F)(12 V) = 2.40 × 10⁻³ C. The characteristic equation Lα² + Rα + 1/C = 0 has discriminant Δ = R² − 4L/C.

Δ = (10)² − 4(0.050)/(2.00 × 10⁻⁴) = 100 − 1000 = −900

Step 2 — Determine the Damping RegimeSince Δ < 0 (i.e., R² < 4L/C), the circuit is in the underdamped regime. The charge will oscillate with exponentially decaying amplitude.

Underdamped (oscillatory decay)

Step 3 — Compute the Damping Coefficient and Angular FrequencyThe damping coefficient is γ = R/(2L) = 10/(2 × 0.050) = 100 s⁻¹. The natural angular frequency is ω₀ = 1/√(LC) = 1/√(0.050 × 2.00 × 10⁻⁴) = 1/√(10⁻⁵) = 1/(3.162 × 10⁻³) ≈ 316.2 rad/s. The damped angular frequency is ωd = √(ω₀² − γ²) = √(316.2² − 100²) = √(100,000 − 10,000) = √90,000 ≈ 300 rad/s.

γ = 100 s⁻¹, ω₀ ≈ 316 rad/s, ω_d ≈ 300 rad/s

Step 4 — Write the General SolutionFor the underdamped case, q(t) = Q₀ e^(−γt) cos(ω_d t + φ). With initial conditions q(0) = Q₀ and i(0) = dq/dt|₀ = 0, one can show that φ = tan⁻¹(γ/ω_d), which in practice requires a slight modification of the amplitude. Using the standard form with these initial conditions:

q(t) = (2.40 × 10⁻³ C) e^(−100t) [cos(300t) + (100/300)sin(300t)]

Step 5 — Interpret the ResultThe charge oscillates at a frequency f_d = ω_d/(2π) ≈ 300/(6.283) ≈ 47.7 Hz with an exponentially decaying envelope governed by e^(−100t). After t = 1/γ = 10 ms, the amplitude has fallen to about 37% of its initial value. After about 50 ms (5/γ), oscillations are essentially negligible.

f_d ≈ 47.7 Hz, decay time ≈ 10 ms per e-folding

SECTION 7

Comparing RC, RL, LC, and RLC Circuits

Each circuit topology exhibits distinct behavior determined by which energy-storage elements are present and whether resistance provides damping. The table below summarizes the key contrasts, helping to clarify when each model is appropriate and what physical insights each provides.

Comparison of first- and second-order passive circuit topologies

Property	RC	RL	LC	RLC
ODE Order	1st order	1st order	2nd order	2nd order
Time Constant / Frequency	τ = RC	τ = L/R	ω₀ = 1/√(LC)	ω_d = √(ω₀² − γ²)
Transient Shape	Exponential	Exponential	Sinusoidal (undamped)	Decaying sinusoid or pure decay
Energy Stored In	Electric field (C)	Magnetic field (L)	Both (alternating)	Both, with dissipation in R
Steady-State (DC)	C is open circuit	L is short circuit	No steady state	All energy dissipated
Typical Applications	Filters, timers, coupling	Relay circuits, snubbers	Oscillators, tuned circuits	Bandpass filters, radio receivers

✦ KEY TAKEAWAY

The distinction between first-order (RC, RL) and second-order (LC, RLC) circuits mirrors the difference between overdamped and oscillatory systems in all of physics. First-order circuits relax monotonically toward equilibrium, much like a piston settling in a viscous fluid. Second-order circuits can oscillate, analogous to a pendulum swinging back and forth. The resistance in an RLC circuit plays exactly the role of air resistance for the pendulum—it determines whether the system rings many times before stopping (underdamped), settles without overshoot (overdamped), or returns to rest in the fastest possible time without oscillating (critically damped).

SECTION 8

Connections to Advanced Theory

The transient analysis of RC, RL, and RLC circuits provides the foundation for several advanced topics in physics and engineering. Understanding the behavior of these simple circuits prepares you for more complex analyses involving driven (forced) oscillations, resonance phenomena, and the frequency-domain methods that dominate electrical engineering practice.

How this lesson connects to more advanced physics and engineering topics

This Lesson	Advanced Topic	Key Extension
Free RC/RL transients	AC steady-state analysis (phasors)	Replace transients with sinusoidal steady-state; introduce impedance Z = R + j(ωL − 1/(ωC))
Free LC oscillation (ω₀)	Driven RLC resonance	When driven at ω = ω₀, the circuit exhibits resonance: maximum current, minimum impedance, and quality factor Q = ω₀L/R
RLC damping regimes	Laplace-transform circuit analysis	Transfer functions H(s) in the s-domain unify transient and frequency response; poles of H(s) correspond to the roots α from the characteristic equation
Energy oscillation in LC	Electromagnetic wave propagation	Maxwell's equations predict E- and B-fields oscillating out of phase, directly analogous to V_C and i_L in the LC circuit
Exponential decay with τ	Quantum tunneling and radioactive decay	Exponential decay laws appear across physics; the mathematical structure e^(−t/τ) is universal for first-order relaxation processes

Perhaps the most profound forward connection is to electromagnetic waves. Maxwell showed that electromagnetic radiation is essentially an LC oscillation propagating through space: the electric field plays the role of charge on the capacitor, and the magnetic field plays the role of current in the inductor. The speed of light c = 1/√(μ₀ε₀) has the same structural form as ω₀ = 1/√(LC). Hertz's experimental verification of this prediction, using a spark-gap LC oscillator, demonstrated that the humble LC circuit contains within it the seed of all wireless communication.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

In a series RC circuit connected to a DC source, explain why the current is maximum at the instant the switch is closed and decreases thereafter. How does the physical behavior of the capacitor account for this?

PROBLEM 2 — BASIC CALCULATION

A 4.7 kΩ resistor is connected in series with a 10 μF capacitor and a 9 V battery. Calculate the time constant τ and determine the voltage across the capacitor at t = 47 ms.

PROBLEM 3 — INTERMEDIATE

An RL circuit consists of a 2.0 H inductor and a 40 Ω resistor connected to a 24 V source. (a) Find the time constant. (b) Calculate the current at t = 25 ms. (c) Determine the energy stored in the inductor's magnetic field at t = 25 ms.

PROBLEM 4 — APPLIED

An AM radio receiver uses an LC tuning circuit to select a station broadcasting at 1020 kHz. If the inductor in the circuit has L = 250 μH, what capacitance C is required? If the circuit also has a parasitic resistance of 5.0 Ω, determine whether the circuit is underdamped at this frequency and compute the quality factor Q.

PROBLEM 5 — CRITICAL THINKING

A capacitor C is charged to voltage V₀ and then connected at t = 0 to an inductor L and resistor R in series. Show that the total energy dissipated in the resistor as t → ∞ equals ½CV₀², regardless of the values of R and L and regardless of whether the circuit is underdamped, critically damped, or overdamped. What physical principle guarantees this result?

SUMMARY

Lesson Summary

Circuits with capacitors and inductors extend circuit analysis beyond Ohm's law into the domain of time-dependent behavior. A capacitor stores energy in an electric field (U = ½CV²) and relates charge to voltage through Q = CV, while an inductor stores energy in a magnetic field (U = ½Li²) and produces a voltage V = L(di/dt) that opposes changes in current. First-order RC circuits (τ = RC) and RL circuits (τ = L/R) exhibit exponential transient responses, reaching steady state after approximately 5τ.

Second-order LC circuits oscillate at the natural frequency ω₀ = 1/√(LC), continuously exchanging energy between electric and magnetic fields in exact analogy with a mass-spring system. Adding resistance creates an RLC circuit governed by L(d²q/dt²) + R(dq/dt) + q/C = 0, whose solutions fall into three damping regimes—underdamped, critically damped, and overdamped—depending on whether R² < 4L/C, R² = 4L/C, or R² > 4L/C. These circuits form the foundation for AC analysis, resonance, filters, oscillators, and ultimately the electromagnetic wave theory that underpins modern communications.