Inductors and Transformers

Inductor

Basic Concepts

An inductor is a passive component that stores energy in a magnetic field, consisting of wire wound into a coil.

      ┌──────┐
  ───┤  ⊗⊗⊗  ├───
      │  ⊗⊗⊗  │    ⊗ = Coil cross-section
      │  ⊗⊗⊗  │
      └──────┘

Parameters

Common Units

• 1 H = 10³ mH = 10⁶ μH = 10⁹ nH

Core Formulas

Inductive Reactance (AC Impedance)

Xl = 2πfL

f: Frequency (Hz)
L: Inductance (H)

Characteristics: Higher frequency results in higher reactance → "Blocks AC, passes DC"

Energy Storage

W = ½LI²

Induced Electromotive Force (EMF)

V = -L × (dI/dt)

The faster the current changes, the higher the induced voltage.
This is the core principle of a Boost converter!

Key Characteristics

1. Blocks AC, Passes DC

• DC: Acts as a short circuit once stable (only DCR remains)
• AC: Impedance increases with frequency

2. Current Cannot Change Instantly

• Current builds up or decays gradually
• Time Constant: τ = L/R

3. Frequency Characteristics

Common Types

Selection Guidelines

1. Inductance — Based on switching frequency and ripple requirements
2. Saturation Current (Isat) — Must be > peak current! Inductance drops sharply after saturation, leading to uncontrolled current
3. DCR — Affects efficiency (I²R losses)
4. Q Factor — High Q is required for high-frequency resonant circuits
5. SRF — Operating frequency must be < Self-Resonant Frequency
6. Magnetic Shielding — Use shielded inductors for sensitive circuits

Transformer

Basic Concepts

A transformer uses electromagnetic induction to transfer energy and transform voltage between two (or more) windings.

       Primary N1             Secondary N2
  ┌─────────────────┐  ┌─────────────┐
  │  ═══════════    │  │  ════════   │
  │  ═══ Core ═══  │  │  ═══  ═══  │
  │  ═══════════    │  │  ════════   │
  └─────────────────┘  └─────────────┘
       ↑                      ↑
      Vin                   Vout

Core Principles

Faraday's Law of Induction:
  V = N × dΦ/dt

Ideal Transformer:
  Vout/Vin = N2/N1 = n  (Turns Ratio)
  Iout/Iin = N1/N2 = 1/n
  Pin = Pout  (Ideal, no losses)

Impedance Transformation: Zin = ZL / n²  (Reflected Impedance)

Real-World Transformers

Non-ideal Factors:
- Leakage Inductance: Uncoupled magnetic flux → Voltage spikes
- Magnetizing Inductance: Required to excite the core → No-load current
- Copper Loss: I²R winding heating
- Core Loss: Eddy current + Hysteresis losses in the core
- Parasitic Capacitance: Inter-winding and turn-to-turn capacitance → Resonance

Transformer Types

Key Formulas

Flyback Converter Energy Storage:
  W = ½ × Lp × Ip²

Duty Cycle D (CCM):
  Vout/Vin = n × D/(1-D)  (Flyback)
  Vout/Vin = D             (Forward)

Saturation Issues

Core Saturation = Disaster!
  B = V×t / (N×Ae)
  B > Bsat → Permeability (μ) drops sharply → L drops → Current goes out of control

Prevention:
- Add an air gap (for energy storage types: Flyback)
- Limit Volt-Seconds (V×t)
- Select a sufficiently large Ae (Core Cross-Sectional Area)

Keywords: Inductor, Inductive Reactance, Blocks AC Passes DC, Transformer, Turns Ratio, Leakage Inductance, Magnetic Saturation, Flyback