In induction heated workpieces, the alternating current density is not homogeneously distributed within the cross-section of the billet, as shown in Fig. 1, where the currents are localized in the closest area to the outer surface. This occurs as a result of the so-called skin effect and the exponential decay of the current density can be described analytically [1]

$$\begin{aligned} \vec {J}(\vec {n})=J_S \ e^{-\vec {n}/\delta } \end{aligned}$$

(1)

This expression is valid for plane or curved billets where the radius of the billet is considerably greater than the skin depth [7] and does not considerate edge effects. The skin depth theoretically depends on the frequency of the current and the material properties of the workpiece and can be calculated as [6]

$$\begin{aligned} \delta = \sqrt{\dfrac{1}{\sigma \pi \mu _0 \mu _r f}} \end{aligned}$$

(2)

From the current density distribution inside the workpiece, it is possible to estimate the volumetric heat generation. Ohmic and hysteresis losses are the mechanisms that generate heat inside the workpiece. The later is represented by the enclosed area in the magnetic hysteresis curve. However, for high frequency systems, hysteresis losses are usually neglected in the calculations because of its computational complexity and low impact in the overall losses [1, 15]. The primary cause of heat generation for these cases are the Ohmic losses. The transfer of electric energy into heat is known as Joule effect and produces the so-called Ohmic losses. The ohmic losses are related to the electric conductivity and the current density as [15]

$$\begin{aligned} {\dot{Q}}=\dfrac{1}{2\sigma }\left| \vec {J} \right| ^2 \end{aligned}$$

(3)

In this expression \(\vec {J}\) is the peak current.

The aforementioned material properties are mainly temperature-dependent, but might also depend on chemical composition, microstructure or stress state of the material. The change of these properties during induction heating has been described by several empirically-obtained relations.

Ferritic microstructure in low alloy steels such as 42CrMo4 is generally ferromagnetic, thus, the magnet dipoles are oriented in the same direction, reacting strongly with the applied magnetic field. When the microstructure becomes austenitic, its ability to magnetize decreases abruptly and the material becomes paramagnetic, which is weakly attracted by the magnetic field. The transition between ferromagnetism and paramagnetism occurs at the so-called Curie temperature, which is usually between \(700\,^{\circ }\hbox {C}\) and \(800\,^{\circ }\hbox {C}\).

The magnetic permeability describes the response of a material to an applied magnetic field. It is defined as the derivative of the magnetic field with respect to the magnetic strength (\(\mu =\mu _0\mu _r=d|\vec {B}|/d|\vec {H}|\)). In the case of steel, the relative magnetic permeability varies enormously with the temperature and the applied field. See Fig. 2 for experimentally obtained magnetic hysteresis curves for 42CrMo4, shown as the averaged curves over a quarter of a period.

The Analytical Saturation Curve model describes the magnetization curve for non-linear materials, and is often presented as an alternative for the classical Fröhlich-Kennelly model, which does not properly describe the saturation of the materials at high magnetic fields [15,16,17]

$$\begin{aligned} B\left( \left| H \right| ,T \right) = \mu _0\left| H \right| + \dfrac{2B_S}{\pi }\arctan \left[ \dfrac{\pi \mu _0 \left( \mu _{r0}-1 \right) }{2B_S} \left| H \right| \right] \left[ 1- e^{\left( \dfrac{T-T_C}{C}\right) } \right] \end{aligned}$$

(4)

The fitted coefficients for 42CrMo4 are \(B_S=1.32\) T, \(\mu _{r0}=1860\), \(33.6^{\circ }\hbox {C}\) and \(T_C=783^{\circ }\hbox {C}\) using the nonlinear least square technique for the measured data set. Details of the measurements are given in [16].

The electrical resistivity, which is reciprocal to electric conductivity, is also an important material property that must be well described in order to obtain accurate results. The electrical resistivity of steel depends on the temperature and can be calculated as [1]

$$\begin{aligned} \rho _{\mathrm {W}}\left( T \right) =\rho _{\mathrm {T}_{\mathrm {ref}}}\left[ 1+\alpha \left( T-T_{ref} \right) \right] \end{aligned}$$

(5)

Common values for the resistivity coefficient \(\alpha \) can be found at room temperature and is usually assumed to be a constant for the sake of simplicity. In this study, a value of \(\alpha =0.005\) is used.

A model for determining the electrical resistivity at room temperature is used in reference [18], which depends on the chemical composition of the steel, provided in mass percentage.

$$\begin{aligned} \begin{aligned} \rho _{\mathrm {T}_{\mathrm {ref}}}=10^{-6} (0.001+0.283\,\mathrm {C} + 0.17\,\mathrm {Si} + 0.0387\,\mathrm {Mn} - 0.1295\,\mathrm {S} + 0.0702\,\mathrm {Al}\\ + 0.00272\,\mathrm {Cr} + 0.0335\,\mathrm {Cu} + 0.0333\,\mathrm {Mo} + 0.0193\,\mathrm {Ni}) \end{aligned} \end{aligned}$$

(6)

### Numerical simulation using the finite element method

The electromagnetic phenomena is described by a set of very well known differential equations called Maxwell’s equations [19]. These governing equations describe electric and magnetic fields from the generation of the magnetic field due to the currents in the coil to the induced ones inside the workpiece. By assuming that the displacement current is negligible, the combination of Maxwell’s equations gives the diffusion equation that describes the electromagnetic phenomena for the frequency range used in this study [15, 20]

$$\begin{aligned} \sigma \dfrac{\partial \vec {A}}{\partial t}-\nabla \left( \dfrac{1}{\mu }\nabla \vec {A}\right) =\vec {J}_s \end{aligned}$$

(7)

The common approach to solve Eq. (7) is to utilize the harmonic approximation, which assumes that the source current in the coil is time-harmonic, usually sinusoidal, therefore forcing the response to be time-harmonic as well. Therefore, we can simplify the diffusion equation equation (7) to

$$\begin{aligned} i \omega \sigma \vec {A}_0 -\nabla \left( \dfrac{1}{\mu }\nabla \vec {A}_0\right) =\vec {J} \end{aligned}$$

(8)

The harmonic approximation is used in many commercial eletromagnetic softwares. This steady-state solution is only valid if the magnetic permeability of the material is linear, which is not true for ferritic steels.

Since the relation between the electromagnetic field and flux density is nonlinear for ferromagnetic materials, several adaptations need to be introduced in the calculation of the required linear magnetic permeability. One of the methods to perform this is the calculation of an effective permeability by a fictitious linear material, as developed by [4] and used in several works such as [15, 16, 21]. The basis of the effective permeability method is that the linear fictitious material has the same eddy current average loss density as the true nonlinear material. The effective permeability at point *i* can be calculated as

$$\begin{aligned} \mu ^{e}_i=\dfrac{\mathrm {w}_{1i}+\mathrm {w}_{2i}}{(H^{e}_{mi})^2} \end{aligned}$$

(9)

where Eqs. (10) and (11) correspond to the upper and lower bounds of magnetic co-energy density of the real material, respectively

$$\begin{aligned} \mathrm {w}_{1i}= & {} \int ^{H^{e}_{mi}}_{0}{B\mathrm {dH}} \end{aligned}$$

(10)

$$\begin{aligned} \mathrm {w}_{2i}= & {} \dfrac{1}{2}B_{mi}H^{e}_{mi} \end{aligned}$$

(11)

The fictitious material is locally and instantly linear, thus its magnetic permeability changes from point to point according to local temperature and applied field. For a better understanding, Fig. 3 shows a real magnetization curve (blue dashed line), its linearization (red dotted line) and the linear effective permeability of the fictitious material (black solid line), fitted so that its magnetic co-energy (in orange) is equal to the one of the real magnetization curve (in blue).

Once the electromagnetic heat loss is computed, the system must be analyzed thermally. The workpiece faces three thermal mechanisms: conduction of heat inside the workpiece from the hot surface through its colder core, convection from the surface to the surrounding media and radiation. The heat conduction equation and details about its computation are not shown in this work for the sake of brevity.

Because of the nonlinear and coupled nature of the induction heating problem, several algorithms have been developed for the computation of the electromagnetic–thermal coupling. There are three main approaches to perform a multi-physical analysis; the simplest method is the unidirectional two-step approach, where the distribution of the heat source is calculated once and introduced to a thermal model, where the temperature profile is obtained. In this approach, the temperature and field dependence of electromagnetic properties are not considered. Thus, this approach is limited to low-temperature heating of linear or nearly-linear materials such as aluminum or copper. The most used approach to couple electromagnetic and thermal models is the indirect or staggered coupling method [22], where the electromagnetic diffusion equation is usually solved using the harmonic approximation, where the permeability has been linearized. Similarly to the two-step approach, both analyses are performed separately. However, there is an iterative process so that both calculations are performed at each step, enabling the nonlinearities and thermal-dependent electromagnetic properties to be taken into account [1]. The third method for coupling electromagnetic and thermal calculations is the direct or fully coupled approach, where the set of differential equations (including the electromagnetic diffusion equation (7)) is solved simultaneously and there is no need to use the harmonic approximation. This approach requires an intensive computational time and memory allocation and is not commonly used in finite element programs [22].

The main advantage when using FEM is that different geometries can be easily evaluated, opposite to the geometrically limited analytical equations. However, FEM calculations require much longer computational times. Authors such as Kolanska-Pluska [23] state that, for finite coils, numerical evaluation is required and only infinite coils can be analytically evaluated. It is important to mention that, when performing an electromagnetic–thermal coupled calculation for ferromagnetic materials, the simulation process should be two directional or iterative because of the material non-linearity. This means that, for each time step, the electromagnetic calculation needs to be re-evaluated due to the temperature-dependent material properties, which increases the computation time.