Inductor Inductance, Magnetic Flux & Stored Energy Solver
Calculate inductor inductance, magnetic flux density (B), magnetic field intensity (H), flux linkage, stored energy, and core saturation for solenoids and toroids.
Inductor & Magnetic Core Specs
Switched-mode power supply (SMPS) power transformers and filters (25 kHz – 500 kHz)
Electromagnetic Field Outputs
361.28 mJ3.6128e-1 Joules (Exact)
W = ½ · L · I²
1.1561e-1 Henries (Exact)
L = (μ·N²·A) / l
Equivalent magnetic circuit resistance
Stored energy per unit core volume
Critical saturation! B field exceeds 1.4 T. Core will saturate, causing inductance drop and thermal runaway.
Electromagnetic Foundations: Inductance, Energy & Ampere's Law
An inductor is a passive two-terminal electrical component that stores energy in an electrodynamic magnetic field when electric current flows through its conductive windings. Named after the American physicist Joseph Henry and modeled rigorously under Maxwell's equations, self-inductance (L) quantifies the ratio of total magnetic flux linkage (λ) established within the winding structure per ampere of excitation current:
Under Faraday's Law of Induction and Lenz's Law, any temporal variation in current establishes a counter-electromotive force (back-EMF) that directly opposes the rate of current change: V(t) = -L · (dI/dt). To establish a steady current I through an inductor against this counter-EMF, the external power source must perform mechanical work on the charges. Integrating instantaneous power P(t) = V(t) · I(t) across time yields the foundational stored magnetic energy equation:
This quadratic dependence on current reveals that doubling the operating current quadruples the stored potential energy. Unlike an electrostatic component that stores potential energy within an electric field between charged plates—which you can calculate using our Capacitor Capacitance, Charge & Stored Energy Calculator—an inductor stores potential energy electrodynamically within the concentrated magnetic field created by moving charge carriers. To evaluate the driving voltage and steady-state currents that establish these fields, verify circuit parameters with our Ohm's Law Voltage, Current & Resistance Calculator.
| Physical Variable | Symbol | SI Standard Unit | CGS / Engineering Unit | Physical Significance |
|---|---|---|---|---|
| Self-Inductance | L | Henry (H = Wb/A = J/A²) | mH, µH, nH | Oppositional inertia to changes in electric current; flux linkage per unit current |
| Stored Magnetic Energy | WL | Joule (J = N·m = W·s) | mJ, µJ, erg | Net potential energy stored inside the inductor's magnetic field volume |
| Magnetic Flux | Φ | Weber (Wb = V·s = T·m²) | Maxwell (1 Wb = 10⁸ Mx) | Total surface integral of magnetic B-field vectors crossing a coil cross-section |
| Magnetic Flux Density | B | Tesla (T = Wb/m²) | Gauss (1 T = 10,000 G) | Concentration of magnetic flux per unit perpendicular area; dictates core saturation |
| Magnetic Field Intensity | H | Amperes per meter (A/m) | Oersted (Oe) | External magnetizing excitation force generated purely by current and turns: (N·I)/l |
| Magnetic Reluctance | Rm | Ampere-turns/Wb (H⁻¹) | Rel (A·t/line) | Opposition of the core medium to magnetic flux establishment (magnetic resistance) |
Geometric Inductance Synthesis & Hopkinson's Magnetic Ohm's Law
To calculate self-inductance from physical dimensions, electromagnetic theory invokes Hopkinson's Law—the direct magnetic analog of Ohm's Law for electrical circuits. In this analogy, the Magnetomotive Force (MMF = N · I) acts as the driving voltage, magnetic flux (Φ) acts as electric current, and magnetic reluctance (Rm) acts as electrical resistance:
Here, l represents the mean magnetic path length (in meters), A is the core cross-sectional area (in square meters), and μ is the total absolute magnetic permeability of the medium (μ = μr · μ0). Substituting the expression for magnetic flux Φ = (N · I) / Rm back into the self-inductance definition L = (N · Φ) / I yields the general geometric inductance formula:
For an elongated cylindrical bobbin of length l and cross-sectional area A, magnetic flux lines exit the core ends and return through external ambient space. The path length l corresponds directly to the physical bobbin coil length. When l is much greater than bobbin diameter (l » d), fringing effects at the extremities become minimal, making the standard solenoid formula highly accurate.
Toroids represent the ultimate closed magnetic loop topology. Because windings wrap continuously around an endless doughnut-shaped ring of mean radius rmean, the mean magnetic path length is exactly the circular circumference: l = 2 · π · rmean. Virtually all magnetic flux remains confined entirely inside the core material, preventing electromagnetic interference (EMI) radiation and parasitic coupling to neighboring traces.
Magnetic Core Saturation: Physics of the B-H Hysteresis Curve
In theoretical physics, relative permeability μr is frequently treated as a constant scalar. In physical electrical engineering, however, ferromagnetic and ferrimagnetic cores are non-linear materials governed by B-H hysteresis loops. Within an unmagnetized core, atomic magnetic dipoles group into microscopic Weiss domains oriented in random directions, canceling each other out.
As excitation current I increases, the external magnetizing force H = (N · I) / l rotates these microscopic domains into parallel alignment with the applied field, rapidly boosting magnetic flux density B = μ · H. However, once all available magnetic domains are completely aligned, the material reaches its saturation flux density (Bsat).
Beyond the saturation knee, differential permeability drops abruptly from thousands of units down to 1.0 (the permeability of vacuum). As a result, the inductor's incremental inductance collapses instantly to its bare air-core value. In switched-mode power supplies (such as buck, boost, and flyback converters), this sudden collapse causes current ramp rates (dI/dt = V/L) to spike catastrophically, blowing switching transistors, melting bond wires, and causing severe magnetic core overheating.
1. Power Ferrites (MnZn / NiZn)
Characterized by high electrical resistivity and near-zero eddy current losses up to MHz frequencies. However, their saturation flux density is relatively low (typically 0.38 T at 25°C, dropping to 0.30 T at 100°C). Design margins must prevent peak flux from exceeding 0.30 T under maximum load.
2. Powdered Iron & Sendust
Formed by compressing insulated magnetic metal grains with synthetic resin. The microscopic resin spaces act as distributed microscopic air gaps throughout the core volume, providing a gentle “soft saturation” characteristic that gracefully degrades inductance rather than collapsing sharply.
3. Silicon Steel & Metglas
Outstanding energy storage capacity and the highest saturation threshold in industrial use. Widely deployed in 50/60 Hz utility grid transformers, heavy vehicle propulsion chokes, and energy-dense pulsed power inductors. To learn how stored energy translates into AC resistive power losses in circuits, check our companion Stefan-Boltzmann Radiation Solver.
Magnetic Core Materials: Comprehensive Technical Comparison
Selecting the correct magnetic core involves balancing permeability, saturation limits, eddy current dissipation, and operating frequency:
| Core Material | Relative Permeability (μr) | Saturation (Bsat) | Optimal Frequency | Primary Engineering Use |
|---|---|---|---|---|
| Air / Vacuum / PTFE Former | 1.0 | Infinite (No Saturation) | 1 MHz – 10 GHz | RF transmitters, antenna matching tuners, crossover filters, MRI gradient coils |
| Nickel-Zinc Ferrite (NiZn) | 100 – 1,000 | 0.25 – 0.35 Tesla | 2 MHz – 250 MHz | High-frequency EMI beads, wideband RF transformers, cable ferrite clamps |
| Manganese-Zinc Ferrite (MnZn) | 2,000 – 15,000 | 0.35 – 0.48 Tesla | 20 kHz – 2 MHz | SMPS power transformers, common-mode AC line chokes, flyback inductors |
| Sendust / Kool Mµ (Al-Si-Fe) | 60 – 125 | 1.05 Tesla | 10 kHz – 500 kHz | PFC boost inductors, telecom power supplies, audio outputs (zero acoustic hum) |
| Molypermalloy (MPP) | 60 – 200 | 0.80 Tesla | 10 kHz – 1 MHz | Ultra-stable aerospace telemetry, precision loading coils, analog filters |
| Grain-Oriented Silicon Steel | 1,500 – 8,000 | 1.50 – 1.80 Tesla | 50 Hz – 1 kHz | AC utility line transformers, heavy industrial welding reactors, audio output iron |
| Amorphous Metglas / Nanocrystalline | 30,000 – 100,000 | 1.25 – 1.55 Tesla | 10 kHz – 100 kHz | Ultra-high-efficiency renewable inverters, EV chargers, precision current sensors |
Step-by-Step Practical Engineering Case Studies
Follow these two worked examples contrasting a high-frequency SMPS toroidal inductor with a heavy DC energy storage choke:
- Given Parameters:
- Turns N = 50 turns
- Toroid Mean Radius r = 15 mm = 0.015 m
- Core Area A = 40 mm² = 4.0 × 10⁻⁵ m²
- Core Material: MnZn Ferrite (μr = 2,300)
- Peak Ripple Current I = 2.0 A
- 1. Mean Magnetic Path Length (l):
- l = 2 · π · 0.015 m = 0.09425 m
- 2. Total Inductance (L):
- μ = 2,300 × 4π × 10⁻⁷ = 2.890 × 10⁻³ H/m
- L = (2.890e-3 × 50² × 4.0e-5) / 0.09425
- L = 0.2890 / 0.09425 = 3.066 mH (3,066 µH)
- 3. Stored Energy (WL):
- W = 0.5 × 3.066e-3 × (2.0)² = 6.132 mJ
- 4. Flux Density & Saturation Margin:
- B = (μ · N · I) / l = (2.890e-3 × 50 × 2.0) / 0.09425
- B = 0.2890 / 0.09425 = 3.066 Tesla (CRITICAL OVER-SATURATION!)
- Action: Insert discrete air gap (g) or switch to Kool Mµ powder.
- Given Parameters:
- Turns N = 250 turns
- Solenoid Core Length l = 150 mm = 0.15 m
- Core Area A = 600 mm² = 6.0 × 10⁻⁴ m²
- Core Material: Powdered Sendust (μr = 125)
- Continuous DC Current I = 8.0 A
- 1. Total Inductance (L):
- μ = 125 × 4π × 10⁻⁷ = 1.5708 × 10⁻⁴ H/m
- L = (1.5708e-4 × 250² × 6.0e-4) / 0.15
- L = 5.8905 / 0.15 = 39.27 mH (0.03927 H)
- 2. Stored Energy (WL):
- W = 0.5 × 0.03927 × (8.0)² = 1.256 Joules
- 3. Flux Density & Saturation Margin:
- B = (1.5708e-4 × 250 × 8.0) / 0.15 = 2.094 Tesla
- Flux per turn Φ = B · A = 2.094 × 6.0e-4 = 1.256 mWb
- Total Flux Linkage λ = N · Φ = 0.314 Weber-turns
Frequently Asked Questions (FAQ)
How is stored magnetic energy in an inductor calculated?
The potential energy stored in an inductor's magnetic field is calculated with the equation W = 0.5 · L · I², where W is energy in Joules, L is the inductance in Henries, and I is the instantaneous current in Amperes. Because current is squared, doubling the operating current quadruples the stored magnetic energy.
What is the difference between magnetic flux (Φ), flux density (B), and field intensity (H)?
Magnetic flux (Φ, measured in Webers) represents the total quantity of magnetic field lines passing through a cross-sectional area. Magnetic flux density (B, measured in Tesla or Gauss) is the concentration of flux per unit area (B = Φ / A). Magnetic field intensity (H, measured in Amperes per meter or Ampere-turns per meter) represents the magnetizing force produced by electric current flowing through wire turns (H = N · I / l). The two are coupled by the material permeability: B = μ · H.
How do core geometry and turns count influence inductance?
For standard cylindrical solenoids and toroidal cores, inductance scales with the square of the turns count: L = (μ · N² · A) / l. Doubling the number of wire turns quadruples the inductance, while doubling the magnetic core cross-sectional area doubles it. Conversely, doubling the magnetic path length halves the inductance.
What is magnetic core saturation and why is it dangerous in power electronics?
Magnetic core saturation occurs when the external magnetizing force H aligns virtually all magnetic domains in the ferromagnetic or ferrimagnetic core material. Once saturated, the differential permeability drops to that of free space (μr≈ 1). This causes the component's inductance to collapse abruptly, triggering massive current spikes that often destroy switching transistors (MOSFETs or IGBTs) in buck, boost, and flyback converters.
What is magnetic reluctance and how does it relate to Ohm's Law?
Reluctance (Rm) is the opposition that a magnetic circuit offers to the production of magnetic flux, analogous to electrical resistance in an electrical circuit. Under Hopkinson's Law (the magnetic counterpart of Ohm's Law), Magnetomotive Force (MMF = N · I) equals Magnetic Flux multiplied by Reluctance: MMF = Φ · Rm. Reluctance is calculated as Rm = l / (μ · A).
Why do ferrite cores have lower saturation flux density than silicon steel?
Ferrites are ceramic compounds composed of iron oxides blended with nickel, zinc, or manganese. Because they are ceramic insulators, they exhibit very high electrical resistivity which virtually eliminates high-frequency eddy current losses, but their magnetic domain density is lower, resulting in saturation flux densities (Bsat) typically between 0.35 T and 0.45 T. Silicon steel alloys have higher metallic atom densities and can reach 1.5 T to 2.0 T before saturation, but suffer massive eddy current losses at frequencies above several kilohertz.
What is the physical significance of magnetic flux linkage (λ)?
Flux linkage (λ = N · Φ = L · I) represents the total magnetic flux coupled across all turns of an inductive winding. Under Faraday's Law of Electromagnetic Induction, the instantaneous counter-electromotive force (back-EMF) induced across an inductor is directly proportional to the time rate of change of flux linkage: V = dλ / dt = L · (dI / dt).
Why is an air-core inductor preferred for high-frequency radio applications?
Air-core inductors have a relative permeability of exactly 1.0 and contain no physical magnetic domains. Consequently, they suffer zero magnetic hysteresis loss, zero eddy current losses in the core, and cannot be magnetically saturated regardless of how much current passes through them, making them ideal for high-Q RF resonant circuits and high-power transmitters.
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