Animate B-H loop tracing with the Jiles-Atherton model. Compute coercivity, remanence, BH-max, and hysteresis loss in real time for soft iron, hard magnets, transformer cores, and ferrite.
What exactly is a hysteresis loop? I see the simulator draws a loop, but why doesn't the magnetization just follow the same path up and down?
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Basically, it's the "memory" of a magnetic material. When you apply and then remove a magnetic field, the material's internal magnetization lags behind and doesn't retrace its steps. This lag creates the loop you see. In practice, it's caused by tiny magnetic domains inside the material resisting change. Try moving the "Coercivity H" slider in the simulator—you'll see the loop get wider, meaning it takes a stronger reverse field to "reset" the magnetization.
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Wait, really? So the area inside that loop has a physical meaning? What happens if I change the "Field Amplitude H" control?
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Great observation! Yes, the loop area represents energy loss per cycle—energy turned into heat as domains snap and realign. If you reduce the Field Amplitude so the material isn't fully saturated, you'll see a smaller, skinnier loop. A common case is in power transformers, where you want this area as small as possible to save energy. The simulator calculates this loss using the integral $W = \mu_0 \oint H\,dM$.
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Okay, so what are "Remanence" and "Coercivity" on the parameter panel? They sound like the key specs for a magnet.
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Exactly! Remanence (Br) is how strong the magnet remains after you turn off the external field—its "stickiness." Coercivity (Hc) is how hard it is to demagnetize it—its "toughness." For instance, a fridge magnet needs high Br. A motor magnet needs high Hc to resist heat. Play with the "Saturation M" and "Coercivity H" sliders together. You'll see Br is the loop's height at zero field, and Hc is its width on the axis.
Physical Model & Key Equations
The simulator uses the Jiles-Atherton model to calculate the magnetization M. The core idea is separating the ideal, reversible anhysteretic magnetization from the irreversible, lossy process of domain wall motion. The effective field accounts for internal interactions.
Here, $H$ is the applied external field, $\alpha$ is a mean field parameter representing inter-domain coupling, $M_s$ is the saturation magnetization, and $a$ is a shape parameter. $M_{an}$ is the ideal magnetization if there were no hysteresis.
The actual magnetization M is found by solving a differential equation that pins the change in M to the change in the anhysteretic curve, with a term for irreversible losses. From M, we find the total magnetic flux density B and the energy loss per cycle W.
$$B = \mu_0 (H + M)$$
$$W = \mu_0 \oint H \, dM$$
$B$ is the flux density (what you'd measure with a gaussmeter), $\mu_0$ is the permeability of free space. The loss $W$ is the area enclosed by the M-H loop. In the simulator, adjusting parameters like coercivity directly changes the shape of this loop and thus the calculated loss.
Frequently Asked Questions
If the amplitude or frequency of the magnetic field H is extremely large, the convergence calculation of the Jiles-Atherton model may become unstable. First, try reducing the maximum value of H (e.g., to 500 A/m or less). Also, be careful not to change the parameters a and k too much from their initial values (e.g., a=1000, k=1500).
Ms is the saturation magnetization (material-specific value), a is the shape coefficient of domain wall interaction, k is the pinning strength, c is the ratio of reversible magnetization, and α is the mean field coefficient. Start with standard soft magnetic material values (e.g., Ms=1.6e6, a=1000, k=1500, c=0.1, α=1e-5), and fine-tune while observing the squareness and slope of the B-H loop.
The displayed loss is in J/m³/cycle (volume loss per cycle). Multiplying this by the material volume (m³) and operating frequency (Hz) gives the power loss in watts. For example, in transformer core design, the total loss is estimated by adding lamination factors and temperature corrections to this loss value.
The current version does not have a direct function to change the animation speed, but you can effectively change the speed by adjusting the H sweep rate (frequency) with the slider. Also, if you want to check the loop at a specific magnetic field value, fix the H amplitude to that value, set the number of sweeps to 1, and play.
Real-World Applications
Transformer & Inductor Cores: These use "soft" magnetic materials with very narrow hysteresis loops (low coercivity and remanence). The tiny loop area minimizes energy lost as heat every time the AC current reverses, crucial for efficiency in power grids and electronics.
Permanent Magnets: Used in motors, speakers, and fridge magnets, these are "hard" magnetic materials. They require a wide loop with high remanence (to stay strong) and high coercivity (to resist demagnetization from external fields or high temperatures).
Magnetic Recording Media: Hard disk drives and magnetic tapes use materials with a specific, square-shaped hysteresis loop. This allows tiny regions to be reliably "written" (magnetized in one direction) and "read" without accidental flipping.
Non-Destructive Testing & Sensors: Measuring the hysteresis loop of a steel structure, like a bridge or pipeline, can reveal internal stresses or material degradation. Changes in the loop's shape signal potential fatigue or damage before it becomes visible.
Common Misconceptions and Points to Note
When you start using this simulator, there are a few common pitfalls. First is the oversimplified idea that "a material with a high coercivity Hc is a good magnet". While it's true that hard magnets have a large Hc, the definition of "good" varies completely depending on the application. For example, a magnetic circuit in a speaker requires a material with a reasonably high Hc that is resistant to temperature changes (hard to demagnetize). On the other hand, a motor core absolutely requires a soft iron material with a small Hc (a thin loop); if you used a hard magnet here, it would just overheat and not rotate. When selecting a material, work backwards from "what you want to achieve".
Next is the pitfall of parameter settings. While you can freely change parameters like "saturation magnetization" and "molecular field coefficient α" in the simulator, in real materials these parameters are not independent of each other. For instance, if you arbitrarily increase α too much for a ferrite, you might get non-physical loop shapes (e.g., overly squared). When identifying J-A model parameters in practical work, the golden rule is to fit them to measured B-H loop data and determine them as a set of parameters. For example, for a certain ferrite material, you get a balanced set of values like $M_s=3.2\times10^5$ [A/m], $a=50$ [A/m], $α=0.001$.
Finally, be careful not to assume that "the simulation loop area directly equals the heat generation". It's true that hysteresis loss W is proportional to the loop area, but you must not forget that in actual devices, "eddy current loss" is added to this. Especially in an alternating magnetic field, eddy current loss becomes more dominant as the frequency increases. In this tool, if you select "transformer core" and increase the frequency, the loop shape won't change (because it doesn't consider eddy currents), but in reality, the reason for laminating thin silicon steel sheets is precisely to suppress these eddy currents.
Enter saturation magnetization (Ms) in A/m for your material—typical values: soft iron 1.7e6, nickel 0.5e6, ferrite 0.4e6
Set coercivity (Hc) in A/m and remanence (Br) in Tesla based on material datasheet or measurement
Define maximum applied field (Hmax) in A/m to establish loop extent—use 2–3× Hc for complete characterization
Run simulation to generate hysteresis loop and extract Jiles-Atherton parameters automatically
Read output metrics: remanence (Br in T), coercivity (Hc in A/m), maximum energy product (BHmax in kJ/m³), and hysteresis loss per cycle
Worked Example
For electrical steel (grain-oriented, M115-30S): Ms=1.94e6 A/m, Hc=43 A/m, Br=1.95 T, Hmax=150 A/m. Simulator outputs BHmax=418 kJ/m³, hysteresis loss=2.1 kJ/m³/cycle at 50 Hz fundamental frequency (105 W/m³ local power dissipation in transformer core). Maximum permeability μr≈6500. Compare against IEC 61800-9-1 limits: core loss must stay below 1.8 W/kg at 1.0 T, 50 Hz for efficient motor drives.
Practical Notes
Soft magnetic materials (mu-metal, silicon steel) show narrow loops (Hc<50 A/m); hard magnets (NdFeB, SmCo) display wide loops (Hc>800 kA/m)—adjust Hmax scaling accordingly to avoid numerical saturation
Hysteresis loss scales nonlinearly with frequency and field amplitude; doubling frequency at constant Hmax increases loss by ~2.3× due to eddy currents and domain wall damping
Temperature shifts Br and Hc significantly: ferrite materials lose ~0.2%/K remanence above room temperature; recalculate loops if operating above 50°C
For laminated cores, multiply computed loss by stacking factor (~0.96 for typical grain-oriented steel) to account for insulation layer thickness