## 1 Introduction  
In recent years, power-electronics-based power conditioning devices have been used in the area of induction heating (IH) technologies. IH technologies are energy efficient [1], and they are used extensively for heating applications (e.g., cooking stoves, etc.) [2, 3]. IH is also used in office automation appliances. IH systems are fundamentally composed of electrical power converter circuits, digital control schemes and magnetic components [4]. There are different types of circuit topology for IH applications. Most of them are categorized as full-bridge inverters [5, 6], half-bridge inverters [7, 8, 9, 10, 11], resonant single-ended inverters [12] and inverters employing active voltage clamping [13]. The half-bridge inverters are widely used for IH applications due to their cost effectiveness and robustness. Additionally, the techniques of zero-voltage and zero-current soft-switching (ZVS, ZCS) are employed in the circuit topologies, resulting in the reduction of losses during switching [8; 14].  
Generally, pulse modulation methodology is employed in the high-frequency inverter to achieve output power regulation for IH application. Mainly, there are three types of modulations, which are as follows: (i) pulse frequency modulation (PFM) [15; 16], (ii) pulse width modulation (PWM) [17] and (iii) pulse density modulation (PDM) [18; 19; 20; 21]. The simplest modulation technique is PFM. Hence, it is the most widely employed with a single-switched inverter, and a variable switching frequency is used for controlling the injected currents from the voltage power source. However, the variation of the switching frequency for control of the IH load parameters is a disadvantage for domestic applications. As a result, the inverters with PFM-controlled schemes emit some annoying acoustic sounds [22]. By contrast, the sound of inverters with PWM-controlled schemes is quieter. However, the power conversion efficiency of PWM inverters is lower when they are operated at low power output due to soft-switching commutations, which are likely to fail [23].  
In order to improve the functionality and efficiency of both conventional PFM and PWM high-frequency resonant inverters, PDM control schemes have been widely used in recent years. Most early works on implementing PDM control schemes in high-frequency resonant inverters are usually for industrial applications [24]. These inverters are proven to be effective because they are operated with fixed operating frequency, which guarantees the achievement of soft-switching operations at a wider range of output power. Generally, the power requirement of domestic IH applications is less than 2 kW, though this needs further investigation with different control topologies. There are some efforts towards implementing PDM into various modulation control schemes so that high-frequency resonant inverters can be operated in dual modes with the PDM operations at low-output-power regulations [18; 25; 26; 27; 28]. Such techniques significantly improve the efficiency of high-frequency resonant inverters. As a result, ZCS and ZVS operations are used at a wider range of output power, especially when power output is lower. An effort to improve high-frequency resonant inverter operation at low power output uses the discontinuous mode to establish the soft-switching commutations at a lower power output range [7]. It demonstrates improvement in the [analyzed. Lastly, a conventional halogen lamp heater and the proposed IH fixed roller application are presented from a practical viewpoint.