Inductors & Capacitors

⚖️ Inductor vs Capacitor — Side-by-Side

Complete Comparison

Property	Inductor (L)	Capacitor (C)
Unit	Henrys (H)	Farads (F)
Stores energy in	Magnetic field	Electric field
v–i relationship	v = L · di/dt	i = C · dv/dt
Energy stored	W = ½Li²	W = ½Cv²
DC steady state	Short circuit (v=0)	Open circuit (i=0)
Cannot change instantly	current i	voltage v
Series combination	L_eq = L₁+L₂+…	1/C_eq = 1/C₁+1/C₂
Parallel combination	1/L_eq = 1/L₁+1/L₂	C_eq = C₁+C₂+…
At high frequency	Open circuit (Z→∞)	Short circuit (Z→0)
At low/zero frequency	Short circuit (Z=0)	Open circuit (Z=∞)

🧲 Inductor

Solenoid Inductor — Structure & Equations

Inductor v–i Relationship

v = L · di/dt [V]

Voltage is proportional to the rate of change of current. A constant DC current produces zero voltage across an inductor (short circuit behavior).

Current from Voltage

i(t) = (1/L)∫v dt + i(t₀)

Current cannot change instantaneously — it requires a finite time to change, proportional to L.

💡 Key rule: i(t) right after a switch closes = i(t) just before it closes.

Energy & Power

W = ½L·i² [J]
P = L·i·(di/dt) [W]

Energy is stored in the magnetic field. Released when current decreases.

Inductance of Solenoid

L = μN²A / l [H]
μ = μ₀μᵣ = 4π×10⁻⁷·μᵣ

Increase L by: more turns N, higher μᵣ (ferromagnetic core), larger area A, shorter length l.

Series Inductors

L_eq = L₁ + L₂ + L₃ + …

Same current through each. Total voltage = sum of each inductor's voltage.

Parallel Inductors

1/L_eq = 1/L₁ + 1/L₂ + 1/L₃

Same voltage across each. Total current = sum of branch currents.

📝 Example — Inductor Waveform

i = 10t·e^(−5t) A for t≥0, L = 100 mH. Find v(t) and max current time.

di/dt = 10e^(−5t) − 50t·e^(−5t) = (10−50t)e^(−5t)
v = L·di/dt = 0.1 × (10−50t)e^(−5t) = (1−5t)e^(−5t) V
Max i: di/dt=0 → t = 1/5 = 0.2 s
Voltage changes polarity at t = 0.2 s (energy stored → released)

🔋 Capacitor (Dynamic Behavior)

Capacitor v–i Relationship

i = C · dv/dt [A]

Current is proportional to the rate of change of voltage. A constant DC voltage produces zero current across a capacitor (open circuit behavior).

Voltage from Current

v(t) = (1/C)∫i dt + v(t₀)

Voltage cannot change instantaneously. A strong current can charge a capacitor rapidly, but the voltage still rises continuously.

Energy in Capacitor

W = ½C·v² [J]
P = C·v·(dv/dt) [W]

Energy is stored in the electric field between the plates.

📝 Tutorial — 0.5 μF Capacitor

v(t) = 4t V for 0≤t≤1s, v(t) = 4e^(−(t−1)) V for t≥1s. Find i, p, w and state energy intervals.

For 0<t<1s: i = C·dv/dt = 0.5×10⁻⁶ × 4 = 2 μA
p = vi = 8t μW (positive → storing energy)
w = ½Cv² = 4t² μJ
For t≥1s: i = C·dv/dt = −2e^(−(t−1)) μA
p = vi = −8e^(−2(t−1)) μW (negative → releasing)
Energy stored: 0<t<1s. Delivered: t>1s.

📐 Impedance Z in AC Circuits

Resistor Impedance

Z_R = R [Ω] (purely real)

Resistance is the same in AC and DC. No imaginary part.

Inductor Impedance

X_L = ωL (reactance)
Z_L = jωL = jX_L [Ω]

Purely imaginary, positive. ω = 2πf. Higher frequency → higher impedance → blocks AC more.

Capacitor Impedance

X_C = 1/(ωC) (reactance)
Z_C = 1/(jωC) = −jX_C [Ω]

Purely imaginary, negative. Higher frequency → lower impedance → passes AC more easily.

Frequency Behavior Summary

Condition	Inductor	Capacitor
DC (ω=0)	Short (Z=0)	Open (Z=∞)
AC (ω=∞)	Open (Z=∞)	Short (Z=0)