# The Design of a System for the Induction Hardening of Steels Using Simulation Parameters
## 2 Materials and Methods
### Inductor and Batch Modeling  
For a suitable calculation, the required information is the dimensions and materials of the inductor and the batch, the number of turns of the inductor, the frequency of the supply and the operating temperature. The data are as follows:  
* Batch: carbon steel;
* Inductor diameter: D\({}_{\text{ind}}\) = 100 mm;
* Inductor length: h\({}_{\text{ind}}\) = 800 mm;
* Number of inductor turns: 60;
* Batch diameter: D\({}_{\text{ch}}\) = 35 mm;
* Power supply frequency: 10-25 kHz;
* Maximum batch temperature: 800 \({}^{\circ}\)C.  
A transformer model was adopted, where the inductor represents the primary, with N turns, and the batch is the secondary, with one short-circuited turn. Therefore, the inductor and the load can be represented by a series branch, equivalent resistance and inductance [40; 41].  
The inductance of an inductor can be measured with an L-meter, with several items to consider. Since the charge is ferromagnetic, a higher inductance is expected when it is placed inside the inductor. Also, with an increase in temperature (up to Curie for steel), depending on the power supply frequency, the inductance can take values higher than those measured. Therefore, the measured value of the inductance will be taken as the initial one for the simulation, and corrections will be made exclusively in the direction of increasing that value, by using an iterative method, until a result of satisfactory accuracy is reached. The initial (measured) value of the equivalent series inductance was 32 uH.  
For the calculation of the equivalent resistance of the inductor and the batch, it is necessary to know the specific resistance of the materials at the appropriate temperatures and the depth of current penetration at the proper operating frequency for the given materials.  
In simple terms, the inductor will be considered to be at a temperature of 80 \({}^{\circ}\)C (because it is separated from the batch with ceramics and cooled with water), while the batch temperature is 800 \({}^{\circ}\)C. The specific electrical resistance of copper (\(\rho_{\text{find}}\)) at the working temperature of the coil is 2.07\(\cdot\)10\({}^{-8}\)\(\Omega\)m. The specific resistance of carbon steel (\(\rho_{\rm ch}\)), at 800 \({}^{\circ}\)C, at the indicated power supply frequency is 115.1\(\cdot\)10\({}^{-8}\)\(\Omega\)m.  
The equivalent series resistance of the inductor and the ferromagnetic charge is calculated according to the following equation [23]:  
\[\rm R_{ind,ch}\approx 1.3\cdot\pi\cdot\rho_{ind}\cdot N_{ind}^{2}\cdot\frac{F_{ind }}{z_{ind}\cdot h_{ind}}+\rho_{ch}\cdot N_{ind}^{2}\cdot\frac{F_{ch}\cdot\pi}{z_ {ch}\cdot h_{ind}} \tag{1}\]  
where \(\rm N_{ind}\) represents the number of turns of the inductor, while \(\rm z_{ind}\) and \(\rm z_{ch}\) represent the depths of Foucault's currents in the conductor of the inductor and in the batch, respectively. The first addend represents the equivalent series resistance of the inductor at 80 \({}^{\circ}\)C, and the second addend represents the equivalent resistance of the steel batch (reduced to the inductor side) at 800 \({}^{\circ}\)C.  
Based on the tables and calculations [41], \(\rm z_{ind}\approx 1\) mm and \(\rm z_{ch}\approx 3.3\) mm were obtained. This determined the serial resistance value of the LR branch \(\rm R_{ind,ch}=205.4\) m\(\Omega\). That resistance value was taken (as in the case of inductance) as the initial value for the simulation. Then, it was increased using an iterative method until satisfactory accuracy was achieved.