Inductor Properties

Why This Matters

Inductors are one of the three fundamental passive components in electrical engineering, alongside resistors and capacitors. You're being tested on how inductors oppose changes in current, store energy magnetically, and behave differently across frequencies—concepts that directly connect to circuit analysis, filter design, and power systems. Mastering inductor properties means understanding the physics of electromagnetic induction, energy storage mechanisms, and frequency-dependent behavior.

Don't just memorize formulas like $V = L \frac{dI}{dt}$—know what each property tells you about circuit behavior. When you see an inductor in a circuit problem, you should immediately think: How does this affect current changes? What happens at different frequencies? How does it store and release energy? These conceptual connections are what separate students who ace exams from those who struggle with application questions.

---

Fundamental Definitions

These core properties define what an inductor is and how we quantify its behavior. Inductance measures the opposition to current change, expressed through the relationship between magnetic flux and current.

Inductance (L)

• Measured in henries (H)—one henry means one volt is induced when current changes at one ampere per second
• Quantifies energy storage capacity in the magnetic field surrounding the conductor
• Higher inductance results from more turns, larger coil area, or higher-permeability core materials

Self-Inductance

• Voltage induced in the same coil that carries the changing current—the coil opposes its own current changes
• Depends on coil geometry including number of turns ($N$), cross-sectional area, and core material
• Foundation for single-inductor circuits and the basis for understanding more complex magnetic coupling

Mutual Inductance

• Voltage induced in a neighboring coil due to changing current in the first coil—the principle behind all transformers
• Coupling coefficient (k) ranges from 0 to 1, indicating how much flux links both coils
• Depends on physical arrangement—distance, orientation, and shared core material all affect coupling strength

---

Energy and Voltage Relationships

Understanding how inductors store energy and relate voltage to current is essential for circuit analysis. The inductor's fundamental behavior stems from Faraday's law of electromagnetic induction.

Energy Storage in Magnetic Field

• Energy stored follows $W = \frac{1}{2}LI^2$—note the squared current term means doubling current quadruples stored energy
• Energy resides in the magnetic field, not in the inductor material itself
• Released back to the circuit when current decreases, enabling applications like boost converters and flyback supplies

Voltage-Current Relationship

• Governing equation $V = L\frac{dI}{dt}$ shows voltage is proportional to the rate of current change, not current itself
• Rapid current changes produce large voltage spikes—this is why inductors resist sudden current changes
• DC steady-state behavior—with constant current ($\frac{dI}{dt} = 0$), the inductor acts like a short circuit (ideal case)

---

AC Behavior and Frequency Response

Inductors behave very differently in AC circuits compared to DC. Inductive reactance creates frequency-dependent opposition to current flow.

Impedance (Inductive Reactance)

• Reactance formula $X_L = 2\pi fL = \omega L$ shows opposition increases linearly with frequency
• Measured in ohms but represents reactive (not resistive) opposition—current and voltage are 90° out of phase
• Higher frequencies see more opposition, making inductors useful for blocking high-frequency signals

Frequency Response

• Low frequencies—small $X_L$ means inductor approximates a short circuit (passes low-frequency signals)
• High frequencies—large $X_L$ means inductor approximates an open circuit (blocks high-frequency signals)
• Critical for filter design—inductors naturally create low-pass behavior, essential for power supply filtering and signal processing

Quality Factor (Q)

• Defined as $Q = \frac{X_L}{R} = \frac{\omega L}{R}$—ratio of energy stored to energy dissipated per cycle
• Higher Q means lower losses and sharper resonance peaks in tuned circuits
• Frequency-dependent—Q changes with operating frequency since $X_L$ varies while $R$ stays relatively constant

---

Circuit Combinations and Physical Construction

How inductors combine and what they're made of directly impacts circuit design choices. Series and parallel rules follow from magnetic flux relationships.

Series and Parallel Combinations

• Series connection $L_{total} = L_1 + L_2 + ...$ (assuming no mutual coupling)—total inductance increases
• Parallel connection $\frac{1}{L_{total}} = \frac{1}{L_1} + \frac{1}{L_2} + ...$—total inductance decreases below smallest individual value
• Mutual inductance complicates calculations—coupled inductors require adding or subtracting $2M$ terms depending on flux orientation

Core Materials

• Air-core inductors have lower inductance but zero core losses and no saturation—ideal for high-frequency RF applications
• Ferromagnetic cores (iron, ferrite) dramatically increase inductance by concentrating magnetic flux with high permeability
• Core saturation limits maximum current—exceeding this causes inductance to drop suddenly, a critical design constraint

---

Quick Reference Table

---

Self-Check Questions

1. An inductor and capacitor both store energy—what's the fundamental difference in how they store it, and how does this affect their voltage-current relationships?

2. If you double the frequency in an AC circuit containing an inductor, what happens to the inductive reactance? What happens to the quality factor (assuming resistance stays constant)?

3. Compare self-inductance and mutual inductance: which one is essential for transformer operation, and why can't a transformer work with just self-inductance?

4. Two inductors with values $L_1 = 10 \text{ mH}$ and $L_2 = 40 \text{ mH}$ are connected in parallel. Is the total inductance closer to 8 mH, 25 mH, or 50 mH? What's the reasoning?

5. An FRQ asks you to explain why inductors are used in power supply filters. Using the frequency response concept, explain why an inductor blocks switching noise while passing DC current to the load.