7.6 Self-inductance

Definition of Self-Inductance

Self-inductance describes how a changing current in a circuit induces a voltage across that same circuit. It quantifies a circuit's tendency to oppose changes in its own current flow by generating a counteracting emf. This property is what makes inductors work, and it shows up everywhere from power supplies to radio tuners.

Faraday's Law of Induction

Faraday's law states that the induced electromotive force (emf) in a closed loop equals the negative rate of change of magnetic flux through that loop:

$\varepsilon = -\frac{d\Phi_B}{dt}$

The negative sign matters. It tells you the induced emf always acts to oppose the flux change that created it. This single law is the foundation for both self-inductance (one coil inducing emf in itself) and mutual inductance (one coil inducing emf in another).

Lenz's Law

Lenz's law gives you the direction of the induced current. The induced current always flows in whatever direction creates a magnetic field that opposes the change in flux that caused it. If the flux through a loop is increasing, the induced current creates a field that fights the increase. If flux is decreasing, the induced current tries to maintain it.

This is really just a consequence of energy conservation. If the induced current helped the change instead of opposing it, you'd get runaway energy creation from nothing.

Self-Induced EMF

When current changes in a circuit, the magnetic field it produces also changes, which changes the flux through the circuit itself. That changing flux induces a voltage right back in the same circuit. This self-induced emf is given by:

$\varepsilon = -L\frac{di}{dt}$

where $L$ is the inductance and $\frac{di}{dt}$ is the rate of current change. The negative sign (from Lenz's law) means this voltage always opposes the current change. If current is increasing, the induced emf pushes back against it. If current is decreasing, the emf tries to keep it going.

Inductance in Circuits

Inductance measures a circuit's ability to store energy in a magnetic field when current flows through it. It plays a central role in AC circuits, filters, and oscillators, and it governs both the transient response and frequency behavior of electrical systems.

Inductors vs Capacitors

These two components are complementary in almost every way:

• Inductors store energy in a magnetic field; capacitors store energy in an electric field
• Inductors oppose changes in current; capacitors oppose changes in voltage
• Inductors have low impedance at low frequencies and high impedance at high frequencies; capacitors behave the opposite way

Because of these mirror-image properties, inductors and capacitors are often paired together in resonant circuits and filters.

Series vs Parallel Inductors

Inductors combine the same way resistors do (and opposite to how capacitors combine):

• Series: Total inductance adds up directly.

$L_{total} = L_1 + L_2 + L_3 + ...$ Use series connections when you need a larger inductance value.

• Parallel: Total inductance follows the reciprocal rule.

$\frac{1}{L_{total}} = \frac{1}{L_1} + \frac{1}{L_2} + \frac{1}{L_3} + ...$ Parallel connections reduce total inductance and can handle higher currents.

These formulas assume negligible mutual inductance between the inductors. If the inductors are close enough to share magnetic flux, coupling effects change the result.

RL Circuits

An RL circuit consists of a resistor and inductor connected in series (or parallel). These circuits exhibit a first-order transient response, meaning the current doesn't jump instantly to its final value but instead rises or falls exponentially.

The time constant is:

$\tau = \frac{L}{R}$

After one time constant, the current reaches about 63% of its final value. After about 5 time constants, the transient is essentially complete. RL circuits are used in filters, timing applications, and power supplies to smooth out current fluctuations.

Calculation of Self-Inductance

The self-inductance of a component depends on its geometry and the magnetic properties of the surrounding medium. Calculating it tells you how strongly the component will oppose current changes.

Inductance Formula

For a solenoid (the most common textbook geometry), the inductance is:

$L = \frac{\mu N^2 A}{l}$

where:

• $\mu$ = permeability of the core material
• $N$ = number of turns
• $A$ = cross-sectional area of the coil
• $l$ = length of the coil

Notice that $N$ is squared. Doubling the number of turns quadruples the inductance. This is because more turns means more flux and more loops for that flux to pass through. More complex geometries (toroids, planar coils) require more advanced methods to calculate.

Units of Inductance

Inductance is measured in henries (H). One henry means that a current changing at 1 ampere per second induces an emf of 1 volt. In practice:

• Power system inductors might be on the order of henries
• Common circuit inductors are typically in the millihenry (mH) range
• High-frequency electronic inductors are often in the microhenry ($\mu$H) or nanohenry (nH) range

Factors Affecting Inductance

Looking at the solenoid formula, you can see how each factor contributes:

• Number of turns: Inductance scales with $N^2$, so this has the strongest effect
• Core material permeability: Higher $\mu$ means higher inductance (iron cores vs. air cores)
• Cross-sectional area: Larger area captures more flux, increasing inductance
• Coil length: A longer coil spreads the turns out, decreasing inductance

Nearby magnetic materials can also alter the effective permeability around the inductor, changing its inductance from the expected value.

Energy Stored in Inductors

Inductors store energy in their magnetic field while current flows through them. When the current decreases, that energy gets released back into the circuit.

Magnetic Field Energy

The energy stored in an inductor is:

$E = \frac{1}{2}LI^2$

where $L$ is the inductance and $I$ is the current. The quadratic dependence on current is important: doubling the current stores four times the energy. This also explains why inductors can produce dangerously high voltage spikes when current is suddenly interrupted. All that stored energy has to go somewhere, and if the current drops to zero very quickly, $\frac{di}{dt}$ becomes enormous, producing a large voltage spike ($\varepsilon = -L\frac{di}{dt}$).

Energy Density

The energy stored per unit volume in a magnetic field is:

$u = \frac{1}{2}\mu H^2$

where $\mu$ is the permeability of the medium and $H$ is the magnetic field strength. Higher energy density comes from stronger fields or materials with higher permeability.

Inductor Charging and Discharging

In an RL circuit with a DC source:

• Charging (source connected): Current rises exponentially toward its maximum value.

$i(t) = I_{max}(1 - e^{-t/\tau})$

• Discharging (source removed, circuit closed through resistor): Current decays exponentially.

$i(t) = I_0 e^{-t/\tau}$

In both cases, the time constant $\tau = \frac{L}{R}$ controls how fast the transition happens. After one time constant, charging reaches ~63% of max; after about 2.3 time constants, it reaches ~90%.

Applications of Self-Inductance

Transformers and Power Transmission

Transformers rely on mutual inductance between two coils wound on a shared core. A changing current in the primary winding creates a changing magnetic flux that induces a voltage in the secondary winding. The voltage ratio follows the turns ratio:

$\frac{V_2}{V_1} = \frac{N_2}{N_1}$

By stepping voltage up for long-distance transmission (reducing current and therefore $I^2R$ losses in the wires), transformers make the modern power grid possible.

Electromagnetic Relays

A relay uses an inductor coil to create a magnetic field strong enough to physically move a mechanical switch (the armature). A small control current through the coil can switch a much larger current in a separate circuit, providing electrical isolation between the two. You'll find relays in automotive systems, industrial controls, and safety interlocks.

Induction Motors

Induction motors use a rotating magnetic field from the stator to induce currents in the rotor (via Faraday's law). The interaction between these induced currents and the stator field produces torque. The rotor always turns slightly slower than the rotating field; this difference is called slip, and it's necessary for the motor to work, since without changing flux there would be no induced current.

Mutual Inductance

Mutual inductance occurs when a changing current in one coil induces a voltage in a nearby coil. It's the operating principle behind transformers and wireless power transfer, and it depends on the geometry of the coils and how they're positioned relative to each other.

Coupling Coefficient

The coupling coefficient $k$ measures how much of the magnetic flux from one coil actually links with the other:

$k = \frac{M}{\sqrt{L_1 L_2}}$

where $M$ is the mutual inductance and $L_1$, $L_2$ are the self-inductances of each coil. It ranges from 0 (no shared flux at all) to 1 (every field line from one coil passes through the other). Well-designed transformers with shared iron cores can achieve $k > 0.99$.

Transformer Principle

The transformer works by mutual inductance between primary and secondary windings on a common core. A changing current in the primary creates changing flux in the core, which induces voltage in the secondary. The voltage ratio is set by the turns ratio:

$\frac{V_2}{V_1} = \frac{N_2}{N_1}$

This enables efficient voltage conversion and provides electrical isolation between circuits.

Mutual vs Self-Inductance

• Self-inductance involves a single coil opposing changes in its own current
• Mutual inductance involves two (or more) coils, where a current change in one induces voltage in the other

Both are governed by Faraday's law. Self-inductance stores energy within a single circuit; mutual inductance enables energy transfer between circuits.

Transient Behavior in Inductors

When circuit conditions change suddenly (a switch opens or closes), inductors don't respond instantly. The current changes gradually, following exponential curves determined by the circuit's time constant.

Time Constant in RL Circuits

The time constant for an RL circuit is:

$\tau = \frac{L}{R}$

A larger inductance or smaller resistance means a slower response. Here's a useful reference:

• After $1\tau$: current reaches ~63% of its final value
• After $3\tau$: ~95%
• After $5\tau$: ~99% (effectively at steady state)

Rise and Decay of Current

Current rise (when a voltage source is connected): $i(t) = I_{max}(1 - e^{-t/\tau})$

Current decay (when the source is removed): $i(t) = I_0 e^{-t/\tau}$

The time to reach 90% of the final value (or drop to 10%) is approximately $2.3\tau$. These curves are mirror images of each other, and they show up constantly in circuit analysis problems.

Steady-State Conditions

After the transient dies away (roughly $5\tau$), the circuit reaches steady state:

• In DC circuits, an ideal inductor acts like a short circuit (just a wire) because $\frac{di}{dt} = 0$, so the voltage across it is zero.
• In AC circuits, the inductor's behavior depends on frequency, which brings us to inductive reactance.

Inductors in AC Circuits

In AC circuits, inductors behave very differently than in DC. The continuously changing current means the inductor is always generating a self-induced emf, which introduces frequency-dependent opposition to current flow.

Inductive Reactance

The opposition an inductor presents to AC current is called inductive reactance:

$X_L = 2\pi f L$

where $f$ is the frequency and $L$ is the inductance. Reactance is measured in ohms, just like resistance, but there's a key difference: reactance doesn't dissipate energy as heat. At higher frequencies, the current changes faster, so the inductor opposes it more strongly. At DC ($f = 0$), the reactance is zero.

Phase Relationships

In an ideal inductor, the current lags the voltage by 90 degrees. You can remember this with the mnemonic ELI (voltage E leads current I in an inductor L). This phase difference means that energy sloshes back and forth between the source and the inductor's magnetic field rather than being consumed.

Power in Inductive Circuits

Because voltage and current are 90° out of phase in an ideal inductor, the average power dissipated over a full cycle is zero. Energy flows into the magnetic field during one quarter-cycle and flows back out during the next.

Real inductors have winding resistance, so they do dissipate some power. The reactive power (the power that oscillates without being consumed) is:

$Q = I^2 X_L$

Reactive power doesn't do useful work, but it does load the source and affect the current in the circuit.

Resonance in LC Circuits

When an inductor and capacitor are connected together, energy oscillates between the magnetic field of the inductor and the electric field of the capacitor. At a specific frequency, the inductive and capacitive reactances are equal and cancel each other out. This is resonance.

Natural Frequency

The resonant (natural) frequency of an LC circuit is:

$f_0 = \frac{1}{2\pi\sqrt{LC}}$

At this frequency, $X_L = X_C$, and the impedance of the LC combination is minimized (in a series circuit) or maximized (in a parallel circuit). You can tune the resonant frequency by changing either $L$ or $C$.

Quality Factor

The quality factor (Q) measures how "sharp" the resonance is, or equivalently, how little energy is lost per cycle:

$Q = \frac{\omega_0 L}{R}$

for a series RLC circuit, where $\omega_0 = 2\pi f_0$. A high Q means the circuit rings for many cycles before the oscillation dies out, producing a narrow, sharp resonance peak. A low Q means heavy damping and a broad peak.

Bandwidth and Resonance Curves

The bandwidth is the range of frequencies over which the circuit's response stays within 3 dB of its peak value:

$BW = \frac{f_0}{Q}$

A higher Q gives a narrower bandwidth, meaning the circuit is more selective. This is exactly what you want in a radio tuner (pick out one station) but not in an audio amplifier (pass a wide range of frequencies). The resonance curve (amplitude vs. frequency) gets taller and narrower as Q increases.

Practical Considerations

Real vs Ideal Inductors

No real inductor behaves perfectly. The main deviations from ideal behavior are:

• Series resistance: The wire in the coil has resistance, which dissipates power
• Parasitic capacitance: Small capacitances exist between adjacent turns of the coil
• Core losses: Magnetic core materials introduce energy losses, especially at high frequencies
• Self-resonant frequency: The parasitic capacitance and inductance form an LC circuit, creating a frequency above which the inductor actually behaves like a capacitor

Inductor Core Materials

The choice of core material involves trade-offs:

• Air core: Low inductance, but no core losses and usable at very high frequencies
• Ferrite cores: High permeability gives large inductance in a small package, good for moderate frequencies
• Powdered iron cores: A middle ground between air and solid ferromagnetic cores
• Laminated iron cores: Used in power-frequency (50/60 Hz) applications like transformers

Limitations and Non-Ideal Behavior

• Saturation: When the core material reaches its maximum magnetic flux density, increasing current no longer increases flux proportionally. The inductance drops.
• Hysteresis losses: Energy is lost each cycle as the core's magnetic domains are repeatedly realigned. These losses increase with frequency.
• Skin effect: At high frequencies, current crowds toward the surface of the conductor, increasing its effective resistance.
• Temperature dependence: Both the winding resistance and core permeability change with temperature, shifting the inductor's characteristics.