The functions used to model B-H curve of magnetic materials are mathematical equations—such as the Jiles-Atherton, Preisach, and Frohlich models—that map the non-linear relationship between applied magnetic field intensity (H) and resulting flux density (B), including hysteresis and saturation effects. In a real circuit, applying the correct model dictates whether your switch-mode power supply (SMPS) transformer operates efficiently or saturates, overheats, and destroys your switching MOSFETs. A common point of confusion among hobbyists and junior engineers is treating the initial magnetization curve as identical to the full hysteresis loop, or assuming permeability ($\mu$) is a fixed constant rather than a dynamic derivative ($dB/dH$) that collapses near saturation.
The Core Mathematical Models Compared
When you open a magnetics design tool or configure a SPICE simulation, you are implicitly choosing one of several mathematical frameworks. No single function perfectly captures every magnetic phenomenon across all frequencies and temperatures, so engineers select the model based on the specific failure mode they are trying to predict.
| Model / Function | Best Used For | Handles Hysteresis? | Computational Cost | Common Tool Implementation |
|---|---|---|---|---|
| Frohlich Equation | Initial magnetization, quick saturation checks | No (Single-valued) | Very Low | Custom SPICE behavioral sources |
| Jiles-Atherton | Hysteresis loop shape, core loss estimation | Yes | Medium | LTspice (non-linear inductor models) |
| Preisach Model | Complex minor loops, memory effects | Yes (Advanced) | High | Finite Element Analysis (FEA) like COMSOL |
| Langevin Function | Superparamagnetic materials, nanomaterials | No | Low | Material science research tools |
The Jiles-Atherton model is the undisputed workhorse for power electronics simulation. It breaks magnetization into reversible and irreversible components, mimicking the physical "pinning" of magnetic domain walls against impurities in the crystal lattice. If you are simulating a flyback transformer in LTspice, the non-linear inductor parameters (Lm, Vp, Ip) are essentially mapping to a simplified Jiles-Atherton derivative.
The Preisach model, on the other hand, treats the material as a collection of independent magnetic dipoles with distinct up-and-down switching thresholds. It is mathematically heavy but necessary when modeling asymmetric minor loops—such as when a transformer experiences a sudden DC offset fault and doesn't return to the origin point of the B-H plane.
Worked Numeric Example: Predicting Saturation with the Frohlich Equation
Let’s look at the Frohlich equation, which is highly practical for quick bench calculations when you need to verify if a core will saturate under peak current. The Frohlich function models the initial (anhysteretic) magnetization curve:
$B = \frac{H}{a + bH}$
Where B is flux density (Tesla), H is magnetic field intensity (A/m), and a and b are material constants derived from initial permeability and saturation flux density.
Step 1: Derive constant 'a'
At very low H, the curve is linear, so $B \approx H/a$. We know $B = \mu_i \mu_0 H$.
Therefore, $a = \frac{1}{\mu_i \mu_0} = \frac{1}{2300 \times (4\pi \times 10^{-7})} \approx 346$.
Step 2: Derive constant 'b'
At very high H, the core saturates, and $B \approx 1/b$.
Therefore, $b = \frac{1}{B_{sat}} = \frac{1}{0.40} = 2.5$.
Step 3: Calculate B at operational H levels
Our specific Frohlich function for this core is: $B = \frac{H}{346 + 2.5H}$
- At H = 10 A/m (light load): $B = \frac{10}{346 + 25} = 0.027$ T. (Core is highly linear, effective $\mu$ is near 2300).
- At H = 100 A/m (nominal load): $B = \frac{100}{346 + 250} = 0.168$ T. (Permeability has started to roll off).
- At H = 1000 A/m (fault/short circuit): $B = \frac{1000}{346 + 2500} = 0.351$ T. (Approaching the 0.40 T hard limit).
The Engineering Takeaway: If your SMPS controller allows the peak current to push H beyond 1500 A/m during a load transient, $B$ hits 0.375 T. At 100°C, the $B_{sat}$ of this ferrite drops to roughly 0.30 T. Your math just predicted a thermal fault condition that a simple linear $V = L(di/dt)$ calculation would have completely missed.
Where You Meet This in Practice
You rarely sit down with a calculator to solve Jiles-Atherton differential equations by hand. Instead, these functions dictate the parameters you enter into design software and the physical tests you perform on the bench.
1. Switch-Mode Power Supply (SMPS) Transformer Design
When using tools like Micrometals or Ferroxcube design software, the underlying algorithms use curve-fitted functions to calculate core losses ($P_v$). If you input the wrong material grade, the software applies the wrong B-H coefficients, underestimating hysteresis loss. This results in a transformer that passes electrical tests but melts the bobbin after 20 minutes of operation due to unmodeled thermal runaway.
2. Audio Output Transformers
In high-fidelity tube amplifiers, the B-H curve dictates harmonic distortion. The non-linear "knee" of the curve generates 3rd and 5th harmonics. Designers intentionally gap the core or use specialized grain-oriented silicon steel (like M6) to force the operating point into the most linear region of the Frohlich curve, trading inductance for lower total harmonic distortion (THD).
3. Current Transformers (CTs) and Protection Relays
For AC current sensing, the CT must not saturate during fault currents. Protection engineers use the hysteresis loop data to calculate the "Accuracy Limit Factor" (ALF). If the B-H model shows saturation at 20x nominal current, but the breaker requires 30x to trip reliably, the relay will see a clipped waveform and fail to clear the fault.
Frequently Asked Questions
Which functions used to model B-H curve of magnetic materials include hysteresis?
Single-valued functions like the Frohlich equation or simple polynomial fits only model the initial magnetization curve (the "virgin" curve). To model hysteresis—the energy lost as heat when the magnetic field reverses—you must use multi-valued or history-dependent functions. The Jiles-Atherton model is the most common for circuit simulation because it calculates the area inside the hysteresis loop to predict core loss. The Preisach model is used in advanced finite element analysis (FEA) when the material experiences complex, asymmetrical minor loops, such as in variable frequency drive (VFD) motor stators.
How do SPICE simulators apply functions used to model B-H curve of magnetic materials?
Standard SPICE inductors are linear. To model non-linear magnetics, simulators like LTspice or PSpice use behavioral voltage/current sources or specialized non-linear core models. In LTspice, you can define a non-linear inductor using the Flux=... parameter, which often implements a hyperbolic tangent (tanh) function or a simplified Jiles-Atherton derivative. You input parameters like saturation flux (Bs), remanence (Br), and coercive force (Hc). The simulator then calculates the instantaneous inductance ($L = N^2 \frac{d\Phi}{dI}$) at every time step, dynamically reducing the inductance as the simulated current pushes the core toward the B-H knee.
Why do the functions used to model B-H curve of magnetic materials fail at high frequencies?
Standard B-H functions assume quasi-static conditions (DC or low-frequency AC, typically < 1 kHz). At high frequencies (e.g., 100 kHz+ in modern LLC resonant converters), two phenomena break the standard models: eddy currents and dynamic hysteresis. Eddy currents create internal opposing magnetic fields that shield the center of the core lamination or ferrite grain, effectively reducing the usable cross-sectional area (skin effect in magnetics). Furthermore, domain wall movement lags behind the applied H field due to viscous damping. To model this accurately, engineers must augment standard B-H functions with frequency-dependent loss separation models (like the Improved Generalized Steinmetz Equation, or iGSE) rather than relying on the static B-H curve alone.
