# A Three-Coil Constant Output Wireless Power Transfer System Based on Parity-Time Symmetry Theory
## 1 Introduction  
With technological development, electricity has become an indispensable part of life. Unlike wired power transmission, in which electrical energy is transmitted through wires or cables, wireless power transfer (WPT) has been widely used in consumer electronics, smart homes, physical robots, and implantable medical devices because of its high flexibility, convenience, and safety, without requiring any physical connections [1; 2; 3; 4]. One of the types of WPT, magnetic resonance wireless power transfer, was proposed by MIT's Kurs team in 2007 [5], and a large number of researchers subsequently invested in it [6; 7]. They used four coils coupled with a medium distance to wirelessly light a 60 W light bulb through the strong magnetic resonance theory in physics combined with coupled-mode modeling theory. However, the transmission efficiency and output power of the system decreased significantly with increasing distance. To ensure the robustness of the transmission efficiency of a WPT system toward coil coupling distance, [PERSON] et al. introduced a wireless power transfer system using parity-time (PT) symmetry theory in 2017 and constructed a negative resistor, using an operational amplifier to replace the traditional excitation source. When the system operated in the PT-exact symmetry region, the transmission efficiency of the system did not vary with the coupling coefficient between the coils [8]. Although the power of this system was only on the order of milliwatts, the output characteristics of this system and its circuit characteristics, namely, the lack of a requirement for additional complex control loops, have attracted many researchers to the study of WPT systems using PT symmetry theory [9]. [PERSON] et al. used a self-oscillating inverter to achieve a negative-resistance composition and, thus, increase the output power of a system to a level that can be appliedto electronic devices, and it could maintain a transmission efficiency of more than 90% within a distance of 100 mm from the coupling coil [10]. In Ref. [11], a PT-symmetric system was applied to multiple decoupled receiver coils to achieve high-power wireless energy transmission with low current stress in each receiver coil. In Ref. [12], a steady-state analysis of the PT-symmetric system was performed using the mutual inductance model of the circuit, and a primary-side output power control scheme was proposed according to the characteristics of the PT-symmetric interval. Unlike the coupled-mode theory modeling reported in [10], modeling based on circuit mutual inductance theory can more precisely describe the correlation between system transmission performance and circuit parameters. Although the PT-symmetric system is robust regarding transmission efficiency over the transmission distance, it exists only in the PT-exact symmetry region [8]. When the coupling coefficient of the system is less than the critical value, the system will lose its robustness toward the coupling distance because it operates in the PT-broken region. However, this property is only satisfied in the PT-exact symmetry region. As the coil transmission distance increases, the system begins operating in the PT-broken region, and the constant transmission characteristic of the system is not satisfied. In Ref. [9], the critical coupling coefficient of the studied system was reduced by connecting an inductor in series to the resonant tank on the secondary side that did not participate in the coupling. In Ref. [13], a DC-DC link was added to the secondary side of the system, reducing the critical coupling coefficient of the system by changing the equivalent AC resistance of the system and, thus, its necessary coupling coefficient. However, additional components increase the size of the receiving end. In addition to the methods above, a WPT system with multiple relay coils can effectively extend the transmission distance. In Refs.