Circular Dependency Error: Modeling T-dependent BH Saturation in Frequency-Transient Study

Posted 26.01.2026, 10:08 GMT-5 1 Reply

Martin Kadlec

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Hello everyone,

I am working on an Induction Heating simulation using the Frequency-Transient study step. My goal is to model a ferromagnetic workpiece heated above its Curie temperature (Tc), capturing two nonlinear effects simultaneously:

1. Magnetic Saturation: Dependent on ∣H∣ (B-H curve).
2. Phase Transition: The drop in permeability to unity as T>Tc.

I attempted to combine these effects by creating an analytic function for relative permeability:

mu_r_combined(T, H) = (mu_r_T(T) - 1) * mu_r_H(H) + 1

mu_r_T(T): Interpolation of relative permeability vs. Temperature.
mu_r_H(H): Interpolation of relative permeability derived from B-H curve data.
The logic is to scale the saturation curve by the temperature factor, ensuring μr→1 when T>Tc.

The Problem: When I define the material property using this function (passing the magnetic field norm, e.g., mf.normH or similar, as the argument), the solver fails with a Circular Dependency Error.

Is there a standard workaround for this error in the Frequency Domain? Or is there a better way to implement "Temperature-dependent Effective BH curve" to avoid manually defining μr?

Any advice on the correct setup to handle this nonlinearity would be appreciated.

Thank you!

Reply

1 Reply Last Post 28.01.2026, 11:46 GMT-5

Magnus Olsson COMSOL Employee

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Posted: 2 hours ago

Hi Martin,

The problem is that the Magnetic Fields interface is based on the magnetic vector potential, A, so B= curl(A) is the primary quantity. So to compute H from the solution, you need to use your permeability function and then you get a circular dependency. One way to get around this is to define the permeability as a function of B and T. Another way is to use the Magnetic Field Formulation (MFH) where H is the dependent variable. There is no built-in Frequency-Transient study for that but you can hard-wire the equation form to Frequency Domain and also set the frequency in the main node for the Magnetic Field Formulation node and use a regular Time Dependent study. Keep in mind that the MFH interface will also require that you treat air and other insulators as having a small conductivity e.g. 1 S/m.

-------------------
Magnus

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