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Circular Dependency Error: Modeling T-dependent BH Saturation in Frequency-Transient Study
Posted 2026年1月26日 GMT+1 16:08 3 Replies
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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!
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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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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.
Hello Magnus,
Thank you for the suggestions. I have tried both approaches, but unfortunately, I am still facing issues:
Defining permeability as μ(T, B): I inverted the function to depend on normB, but the solver failed immediately with: "Failed to find consistent initial values. Last time step is not converged." It seems the nonlinearity is too strong for the initial step.
Using Magnetic Field Formulation (MFH): This would solve the dependency, but my model relies on a multi-turn coil. The MFH interface does not seem to have the "Coil" domain feature (unlike the standard MF interface).
Is there a recommended way to represent a multi-turn coil in MFH, or is there another workaround for the circular dependency in the standard MF interface?
Thank you again for your help.
Martin
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Hello Martin,
There may be more/other reasons that you run into problems than the fundamental circular dependency that I addressed. In version 6.4, there is a working example on frequency-transient induction heating that should be spot on what you are after, see:
ACDC_Module/Electromagnetic_Heating/induction_heating_curie_movement https://www.comsol.com/model/induction-heating-including-curie-temperature-and-movement-51821
Have a look at that and compare to what you are doing and you may be able to spot the cause. I suspect that what you are missing is the frequency domain initialization step. That is having the EM variables (A) starting from zero in the frequency-transient step is not working well.
-------------------Magnus
