Wirelessly powered motor operation in dynamic scenarios using non-Hermitian parity-time symmetry

Motors arise as a heart of the mobility society, and wirelessly operated motors may improve our standard of living. Wireless power transfer in the kilohertz and megahertz range has been extensively explored, finding various potential applications in consumer electronics, electric vehicles, and medical implants. However, stable operation of wirelessly powered motors remains challenging due to voltage fluctuations for motors occurring in dynamic scenarios, e.g., the rotating speed of the motors is varied. Here, we theoretically and experimentally demonstrate the operation of a motor, where the power is wirelessly transferred via coils, is robust against the rotating speed by employing the analogy with non-Hermitian parity-time (PT) symmetry. In addition, our system is robust for misalignment of the coils. Our results open up opportunities for the robust operation of motors via wireless power transfer in dynamic scenarios towards autonomous vehicles.


S1. Circuit layouts of the differential amplifier and the phase compensator
The differential amplifier consists of an operational amplifier (LT1223) and four resistors with resistances being  1 =  2 = 1 kΩ and  3 =  4 = 2.1 kΩ [Fig.S1(a)] so that it has a gain of  = 2. Voltages of ±12 V are applied to the operational amplifier.
The phase compensator consists of two variable resistors with resistances of  V1 and  V2 , and a capacitor with a capacitance of 470 pF [Fig.S1

S2. Parameters of fabricated coils
Table S1 shows the parameters of the fabricated coils for the transmitting and receiving resonators.

S3. Circuit diagram of the conventional system
In the main text, a conventional system is presented, where the control scheme in the system of Fig. 2(a) is taken out.The circuit diagram of the conventional system is shown in Figure S2.The inverter and the DC/DC converter are incorporated into the conventional system.

Fig. S2.
Circuit layout of the conventional wireless power transfer system.
We derive the power, voltage, and current of the system of Fig. S2 from the circuit diagram above and the coupled mode theory formalism.Voltage  2 of the receiving coil is expressed with voltage  1 of the transmitting coil and load impedance   1 , where the resonant frequency  0 of the transmitting resonator is given as 1 and  2 are the intrinsic losses of the transmitting and receiving resonators, respectively, and  is the coupling coefficient between the resonators.Voltage  1 of the transmitting coil can be calculated using input voltage  dc of the half-bridge module.
Then current  2 flowing through the receiving coil is given as 1

S4. Relationship between the gain rate and half-bridge input voltage
The relationship between the gain rate  10 and the current flowing through the transmitting resonator  1 is discussed below.The relationship between the gain rate  10 and negative resistance  ℎ is shown as 2 , Since the half-bridge circuit generates a square wave voltage, its fundamental wave component is √2  1 .The negative resistance  ℎ is given from Ohm's law  h = √2 dc  ⁄  1 . (S9) Substituting Eq. (S9) into Eq.(S8), we have Thus, the gain value decreases as the current flowing through the transmitting resonator increases.The current depends on the load impedance changes.
Fig. S1. Circuit layouts of (a) the differential amplifier and (b) the phase compensator.

S5.
Currents of the motor for the misalignments of the coils in Fig.7FigureS3(a)-(c)shows the corresponding currents of the motor for the misalignments of the coils; (a) distance, (b) angle, and (c) offset position in Fig.7.The currents increase as the rotating speed of the motor decreases in the PT symmetric phase for all the three cases.The experiment results (symbols) agree well with the analytical results (lines).

Fig. S3 .
Fig. S3.Correspondin currents of the motor for the misali nments of the coils; (a) distance, (b) an le, and (c) offset position in Fi .7. he horizontal axis is the load impedance so that the measured results (black symbols) of the motor are compared with the analytical results obtained from Eqs. (3) (5) (black curves) as well as those of the conventional system (pink dashed curves).

Table S1 .
Parameters of fabricated coils.
Using Eq. (S5), voltage   and current   of load impedance   are expressed S4)Using Eqs.(S3) and (S4), the power flowing load impedance   is written as