## 2 Quasi-Resonance Circuit
### Constraints  
In Figure 3a, before switching the transistor, \(V_{BUS}\) is placed across the collector-emitter junction of the transistor (\(V_{CE}\)), and when it gets turned on (moment 1 in Figure 3b), the current of the inductor increases linearly with time (according to Equation (3)), and when the switching stops, the oscillation between the inductor and the capacitor causes the energy of the inductor to be stored in the form of capacitor voltage. Depending on its polarity, this voltage can inflate \(V_{CE}\) (moment 2) or reduce it to the point of the reverse conduction threshold at which the freewheeling diode gets activated (moment 3). After that, the circuit continues to oscillate with the energy that has been left in the capacitor (in Figure 3b, the difference between the peak voltages of the first period and the subsequent ones indicates an energy discharge during the diode conduction phase). Figure 3: Transistor voltage during circuit excitation (**a**) circuit (**b**) IGBT PWM output. Figure 2: Working basis for quasi-resonant circuit. In Figure 3, moment 2 is the maximum voltage that the transistor will experience, and we can express it as the sum of the source voltage (\(V_{BUS}\)) and the capacitor voltage (\(V_{C,max}\)), since the total energy of the system, including the capacitor and inductor, is cumulated solely in the capacitor. \[E_{L}=\frac{1}{2}LI_{L}^{2} \tag{4}\] \[E_{C}=\frac{1}{2}CV_{C}^{2}\] (5) \[E_{C}[\max]+0=E_{C}[t]+E_{L}[t] \tag{6}\]  
Equation (6) shows that, at this particular moment, the energy of the system is completely stored in the capacitor and is equal to the sum of the energy from the inductor and the capacitor, at all times (only the first cycle is considered here). Since their energy at moment 1 is known according to the following equations, the maximum voltage of the capacitor can be calculated:  
\[I_{L}=\frac{V_{bus}}{L}\Delta t,V_{C}=V_{bus} \tag{7}\]  
In the above equations, \(\Delta t\) denotes the time period in which the transistor is conducting and charging the inductor [Sec]. The obtained formula expresses a direct relation between the transistor's maximum voltage (during switching) with \(\Delta t\). And if it exceeds more than the permissible limit, it will lead to permanent damage to the transistor. The transferred induction power can be assessed by considering one switching cycle of the transistor. During this time, the energy is conveyed from the source to the inductor, which is then absorbed by the metallic load during oscillation. This statement heavily relies on the condition that the coil winding and the metallic recipient form a decent enough imperfect transformer [26; 27]. This means that all the magnetism produced by the primary coil gets lost in the core, which is of course our desired purpose in induction heating since it will maximize the magnetic engagement between the coil and the load. In induction heating systems, the transistor functions as a crucial switching element that controls the flow of current to the inductor. The time period \(\Delta t\) during which the transistor is in a conductive state is essential because it directly affects how much energy is transferred from the power source to the inductor. During this conduction phase, the inductor accumulates energy in the form of a magnetic field. The relationship between the maximum voltage across the transistor, VCE, max, during switching and \(\Delta t\) is critical; if the maximum voltage exceeds the transistor's breakdown voltage, which is the highest voltage the transistor can withstand without failure, it can result in irreversible damage. Therefore, effectively managing \(\Delta t\) is vital for ensuring the reliability and longevity of the device. Therefore, according to this ideal assumption, we can conclude the following:  
\[W=\frac{\frac{1}{2}LI_{L}^{2}+\frac{1}{2}CV_{C}^{2}}{T_{on}+T_{off}} \tag{8}\]  
To fully understand the energy transfer process in an induction heating system, one must consider a complete switching cycle of the transistor.