## 2 Schematic of Induction Heating Fixed Roller and Its Equivalent Circuit
### Transformer Circuit Model for IH Load  
The conventional \(R\)-\(L\) series or parallel model for IH applications uses equivalent impedance models. However, the practical model of IH applications requires distorted voltage and current to be applied to the IH load in case of operation with a high-frequency inverter. In order to analyze and simulate the characteristics of a high-frequency inverter, including IH load, an analytical model of the real IH load is required.  
Time-varying load parameters are used in IH load applications requiring high temperatures. Applications such as thermal processing, heat treatment and metal melting use these time-varying load parameters. However, in the case of applications for low temperatures, load parameters vary slowly around an operating point. Hence, in this work, load parameters are considered to be constant. To provide an accurate representation of the IH load, the transformer model is used for simulation analysis, as shown in Figure 2a. The model of the transformer has self-inductances \(L_{1}\) and \(L_{2}\), the winding resistance \(R_{1}\) and the effective resistance \(R_{2}\), considering the skin effect of the heating object. The equivalent circuit parameters of Figure 2a are assumed to be constant as the quasi-steady state. Resistance \(R_{1}\) is small enough and uses the litz wire, which reduces the skin effect, and therefore, it is ignored. Resistance \(R_{2}\) is based on the current penetration depth, and the skin effect is related to the high-frequency inverter switching frequency.  
The state equation for the equivalent circuit of the IH load in Figure 2a is expressed as follows:  
\[\left\{\begin{array}{l}L_{1}\frac{d{\dot{i}}_{11}}{dt}+M\frac{d{\dot{i}}_{2 }}{dt}=v_{L1}\\ M\frac{d{\dot{i}}_{11}}{dt}+L_{2}\frac{d{\dot{i}}_{21}}{dt}+R_{2}i_{L2}=0\end{array}\right. \tag{1}\]  
Magnetic fluxes \(\varphi_{1}\) and \(\varphi_{2}\) are expressed as follows:  
\[\left\{\begin{array}{l}\varphi_{1}=L_{1}i_{L1}\\ \varphi_{2}=M{\dot{i}}_{L2}\end{array}\right. \tag{2}\]  
By substituting \(\varphi_{1}\) and \(\varphi_{2}\) from Equation (2) into Equation (1), the following equation can be obtained:  
\[\left\{\begin{array}{l}\frac{d{\varphi_{1}}}{dt}+\frac{d{\varphi_{2}}}{dt}=v _{L1}\\ k^{2}\frac{d{\varphi_{1}}}{dt}+\frac{d{\varphi_{2}}}{dt}+\frac{{\varphi_{2}}}{ \tau}=0\end{array}\right. \tag{3}\]  
where  
\[\tau=\frac{L_{2}}{R_{2}} \tag{4}\]  
\[k=\frac{M}{\sqrt{L_{1}L_{2}}} \tag{5}\]  
By solving Equation (3), the IH load parameters can be represented by the electromagnetic coefficient of coupling \(k\) and time constant \(\tau\) of the high-frequency inverter. This inverter operation for an IH load is independent of the values of \(R_{2}\) and \(L_{2}\) if these values'  
Figure 2: Transformer circuit model for IH fixed roller. (**a**) Transformer model. (**b**) Transformer equivalent circuit model. (**c**) \(L\)-\(R\) series equivalent circuit model.  
ratio is constant for a certain value of \(k\). The electromagnetic coupling degree with the heating coil and the heating body is represented by the parameter \(k\). The IH load time constant is represented by parameter \(\tau\). The \(L_{1}\) value is the heating coil self-inductance as measured on the transformer model primary side, with the secondary side open-circuited. Further, this \(L_{1}\) value equals the heating coil self-inductance with a heating object with zero conductivity and a nonmagnetic heating body under the no-load condition using materials such as stainless steel or aluminium.