## 1 Introduction
However, GaN HEMTs are usually discretely packaged due to unique fabrication processes, which are preferable in 3D-SPS [19; 20; 21]. For low-power and high-integrated applications, the Si lateral diffused MOSFET (LDMOSFET) is usually applied in PSiP and PwrSoC because of compati  
Figure 1: Roadmap for high-frequency PoL power supplies. The power inductor dominates core and coil losses and primarily determines integration levels. Although the magnetic core helps increase the inductance, the core loss increases rapidly with frequency that harms the efficiency. For magnetic cores, their magnetic permeability, \(\mu_{i}\), of soft magnetic material decreases dramatically when the operating frequency reaches a critical value, which causes the dramatic decrement of the inductance value. In TDK Mn-Zn ferrites, MAGNETICS nanomaterial and alloy powders, their initial relative magnetic permeability, \(\mu_{i}\), is shown in Figure 3[23; 24; 25]. Figure 3 shows that the permeability of magnetic materials decreases rapidly above 1 MHz. As the switching frequency increases, high-frequency inductors evolve from magnetic-core inductors to air-core inductors [26; 27; 28]. Without magnetic cores, air-core inductors do not suffer core loss and have good linearity to frequency. The inductance value decreases dramatically due to the removal of the magnetic core. In the literature and products, the typical inductance values of solenoid and planar spiral inductors are given in Table 2. All the investigated inductance values are less than 500 nH, which pushes the PoL power supply to operate at a frequency above 1 MHz. To address the above challenges, this paper analyzes integrated Si LDMOSFTs and air-core inductors in PSiPs. In terms of the integrated Si LDMOSFET, with consideration of parasitic resistors and capacitors, the size and parallel quantity optimization of integrated Si LDMOSFET are provided to reduce the switching and conduction losses. In terms of air-core inductors, with consideration of skin effect, quality factors of various air-core inductors are calculated for the winding structure optimization to reduce coil loss. Furthermore, combining the above optimizations, this paper predicts the efficiency boundaries of PSiP approaches based on power losses of switches and inductors. This paper is organized as follows. Section 2 analyzes power losses of an integrated Si LDMOSFET to provide a size and parallel quantity optimization. Section 3 calculates the quality factors of solenoid inductors and planar spiral inductors to optimize wind \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Inductor Type** & **Literature** & **Operating Frequency** & **Inductance** \\ \hline \multirow{4}{*}{Solenoid Inductor} & 0806 SQ, 0807 SQ & 1-1000 MHz & 5.5 nH-27.3 nH \\  & 0908 SQ [29] & 1111 SQ [29] & 1-500 MHz & 27 nH-47 nH \\  & 1515 SQ, 2222 SQ, 2929 SQ & 1-100 MHz & 47 nH-500 nH \\  & [29] & 100 MHz & 47 nH-500 nH \\ \hline \multirow{4}{*}{Planar Spiral Inductor} & [30] & 150 MHz & 1.5 nH \\  & [31] & 10 MHz & 2.7 nH \\ \cline{1-1}  & [32] & 100 MHz & 5.8 nH \\ \cline{1-1}  & [33] & 10 MHz & 220 nH \\ \cline{1-1}  & [34] & 550 MHz & 1.54 nH \\ \cline{1-1}  & [12] & 450 MHz & 0.85 nH \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Typical Inductance Values of Solenoid and Planar Spiral Inductors. Figure 2: FOMs of GaN HEMT and Si MOSFET. ing structures. Section 4 predicts the efficiency boundaries of PSiP. Section 5 concludes this paper.