If you have ever watched a power MOSFET explode into shrapnel because an inductor core saturated, you have met the physical limits of the magnetisation formula. In electrical engineering, magnetisation (often denoted as M) defines how a material responds to an applied magnetic field. The fundamental working formula linking measurable bench quantities is M = (B / μ₀) - H. This equation is the bridge between the physical material inside your transformer or inductor and the electrical waveforms you see on your oscilloscope.
The Core Magnetisation Formula and Symbol Definitions
While material scientists often use M = χH (where χ is magnetic susceptibility), power electronics engineers and bench builders rely on the macroscopic field relationship. This form allows you to calculate magnetisation using the magnetic flux density (B) and the magnetic field intensity (H) derived from your coil geometry and current.
| Symbol | Parameter | SI Unit | Typical Bench Measurement Method |
|---|---|---|---|
| M | Magnetisation (Magnetic dipole moment per unit volume) | Amperes per meter (A/m) | Calculated (not directly measured with a standard multimeter) |
| B | Magnetic Flux Density | Tesla (T) | Hall-effect gaussmeter or calculated via Faraday's law (V = N·A·dB/dt) |
| μ₀ | Vacuum Permeability | T·m/A (or H/m) | Constant: 4π × 10⁻⁷ (approx. 1.2566 × 10⁻⁶) |
| H | Magnetic Field Intensity | Amperes per meter (A/m) | Calculated via Ampere's Law: H = (N × I) / lₑ |
Rearranged Forms
Depending on your design constraints, you will need to isolate different variables. Here are the algebraic rearrangements:
- Solve for Flux Density (B): B = μ₀(H + M)
- Solve for Field Intensity (H): H = (B / μ₀) - M
- Solve for Permeability (Theoretical): μ₀ = B / (H + M)
Assumptions, Unit Traps, and Realistic Magnitudes
When the Formula Applies (and Its Assumptions)
The formula M = (B / μ₀) - H is a general macroscopic identity and applies to all materials. However, if you attempt to substitute M = χH into it to predict behavior, you are assuming the material is linear, homogeneous, and isotropic. This assumption holds true for air, vacuum, and weakly paramagnetic materials. It violently breaks down in ferromagnetic materials (like iron, silicon steel, and ferrites) once you approach the saturation knee of the B-H curve. At high fields, M caps out at a physical limit called saturation magnetisation (Mₛ).
The Unit Mistake That Breaks the Math
The most common way hobbyists and junior engineers break this formula is by mixing SI units with the legacy CGS (Centimeter-Gram-Second) system. If you measure B in Gauss and H in Oersteds, the constant μ₀ is effectively 1 in CGS, and the formula becomes M = B - H. If you plug Gauss and Oersteds into the SI formula (using μ₀ = 4π × 10⁻⁷), your calculated magnetisation will be off by orders of magnitude. Always convert to SI first: 1 Tesla = 10,000 Gauss, and 1 A/m = 0.01256 Oersteds.
What a Realistic Answer Magnitude Looks Like
If your calculator spits out M = 0.05 A/m for a ferrite core, you have made a math error. Here is what realistic magnitudes look like across different materials:
- Vacuum / Air: 0 A/m (or negligibly close to zero)
- Paramagnetic salts: 1 to 10 A/m
- Soft Ferrites (e.g., 3C90, N87): 200,000 to 400,000 A/m
- Silicon Electrical Steel: 1,000,000 to 1,500,000 A/m
Worked Problem 1: Calculating Magnetization in a Ferrite Core
Scenario: You are testing an ungapped toroidal ferrite core (Material: TDK N87). You apply a current that generates a magnetic field intensity (H) of 50 A/m. Your Hall-effect sensor measures a flux density (B) of 0.25 Tesla inside the core. What is the magnetisation (M) of the core material at this operating point?
- Identify the knowns and convert to SI:
B = 0.25 T
H = 50 A/m
μ₀ = 4π × 10⁻⁷ T·m/A ≈ 1.2566 × 10⁻⁶ T·m/A - Select the formula:
M = (B / μ₀) - H - Calculate the vacuum contribution term (B / μ₀):
B / μ₀ = 0.25 T / (1.2566 × 10⁻⁶ T·m/A)
B / μ₀ = 198,949 A/m - Subtract the applied field intensity (H):
M = 198,949 A/m - 50 A/m
M = 198,899 A/m - Sanity check the magnitude:
198.9 kA/m is perfectly in line with the expected 10⁵ magnitude range for manganese-zinc soft ferrites.
Worked Problem 2: Finding the Required Field Intensity
Scenario: You are designing a magnetic shield using Mu-metal, which has a known magnetisation of 800,000 A/m when exposed to a specific external field. You need the total flux density (B) inside the shield to be exactly 1.1 Tesla to protect a sensitive analog sensor. What magnetic field intensity (H) must be present?
- Identify the knowns:
B = 1.1 T
M = 800,000 A/m
μ₀ = 1.2566 × 10⁻⁶ T·m/A - Select the rearranged formula:
H = (B / μ₀) - M - Calculate the B / μ₀ term:
B / μ₀ = 1.1 / (1.2566 × 10⁻⁶) = 875,378 A/m - Subtract M to find H:
H = 875,378 A/m - 800,000 A/m
H = 75,378 A/m - Interpret the result:
Because Mu-metal has extremely high permeability, the material's internal magnetisation (M) does almost all the work. The applied external field intensity (H) required to reach 1.1 T is relatively small compared to the total flux generated.
Real-World Scenario: Why the 10A Buck Converter Inductor Melted
The Setup
A hobbyist was building a 12V-to-5V synchronous buck converter rated for 10A continuous output. To save space, they chose a compact EE16 ferrite core (3C90 material) without an air gap, winding 20 turns of 18 AWG wire. They calculated the required inductance based on the linear formula L = (N² × μ × Aₑ) / lₑ and assumed the core would handle the peak current of 12A.
The Numbers
Let us look at the magnetic field intensity (H) at the 12A peak current. The effective magnetic path length (lₑ) of an EE16 core is roughly 37.4 mm (0.0374 m).
- H = (N × I) / lₑ
- H = (20 turns × 12 A) / 0.0374 m
- H = 6,417 A/m
The designer assumed the material would remain linear, expecting B to scale proportionally with H via the initial permeability (μᵢ ≈ 2300). If linear, B would be roughly 1.8 Tesla.
The Outcome
Upon applying full load, the inductor emitted a high-pitched whine, the MOSFET temperature spiked to 150°C in seconds, and the high-side FET shorted, sending 12V directly into the 5V logic rail and destroying the microcontroller.
What Went Wrong (The Magnetisation Limit)
The designer forgot that the linear magnetisation formula (M = χH) is a lie at high field intensities. For 3C90 ferrite, the saturation magnetisation (Mₛ) physically caps out at roughly 3.1 × 10⁵ A/m. Using the fundamental identity B = μ₀(H + M), the absolute maximum flux density the core can support (Bₛₐₜ) is roughly μ₀ × Mₛ, which equals about 0.39 Tesla at room temperature (and drops to ~0.32 T at 100°C). When the current hit 12A, the core demanded a B-field far beyond 0.39 T. Because M could not increase any further, the relative permeability of the core collapsed from 2300 down to near 1 (the permeability of air). The inductance vanished, the current ramp rate (di/dt = V/L) went vertical, and the MOSFET died. The fix? The designer needed to introduce a physical air gap in the EE16 core to lower the effective permeability, increasing the H-field required to reach saturation, or switch to a larger core geometry like an EE25.
For deeper reading on core material limits and B-H curve non-linearities, consult the TDK Ferrite Cores Databook or the Texas Instruments Magnetics Design Guide (SNVA038). For fundamental physics definitions of magnetic moments, the Georgia State University HyperPhysics database remains an excellent, stable reference.
