Page - 328 - in Emerging Technologies for Electric and Hybrid Vehicles

Energies 2017,10, 315 PL= I2VL= I22RL= ( jωM RL+ 1jωCL+ jωLL I1 )2 RL (3) where RS, CS, LS represent the resistor, capacitor, or inductor components of the source coil, respectively, andRL, CL, LL represent the resistor, capacitor, or inductor components of the load coil, respectively.ω represents the frequency,and I1, I2 represent thecurrentsflowinginthesource coil and loadcoil, respectively. M is themutual inductancebetween the source coil and load coil, which isexpressedas follows: M= k √ LSLL (4) wherek isacouplingcoefficient. Figure3.General circuitmodel forwirelesspower transfer (WPT). Themutual inductancebetweenthesourcecoilandtheloadcoilgreatlyaffects thereceivedpower at the loadcoilaswellas theoverall efficiencyof theWPTsystem,asnoted inpreviousworks [10,25]. Therefore, it is important foranyWPTsystemtomaintain thehighestmutual inductance tomaximize the received power and efficiency to the load coil. Based on theNeumann formula, themutual inductanceshownin(4) canalsobedescribedas thenumberofflux linkages in the loadcoil resulting fromthecurrentflowingfromthesourcecoil [10]: M= μ0 4π ∮ ∮ dlSdlL |rSL| (5) whereμ0, dlS anddlL represent thepermeabilityof free-space,andthe infinitesimalsegmentsof the sourcecoilandloadcoil, respectively,while |rSL| represents thedistancebetweenthesourcecoiland the loadcoil segments,whichcanalsobeseenasmisalignment.Assumingthataconstantcurrent is applied, theonlyvariable thatwillaffectM is |rSL|. Therefore, from(1) to (5), it canbedeterminedthat increasedmisalignmentreducesM, thusreducingthemagnitudeofreceivedpowerat the loadcoil andtheoverallWPTefficiency. Basedon (5), thereare several factors thatwill change themagnitudeof |rSL|: vertical, lateral, andangularmisalignment.However,notallof these factorsare likely tobeobservedinpractice, for the followingreasons: because the loadcoil isfixedto theunderbodyofavehicle, themisalignment betweenthesourcecoil and loadcoil ishighlydependentonvehiclemovement.Whenavehicle is operatingonhighwaysoronroads thatareflat, it canbeassumedthat theverticalmisalignmentand angularmisalignmentrelative to thexandyaxeswillhavesmallvariations. The two factors that remain are the lateral, and angularmisalignment relative to the z axis. Betweenthese twofactors, thedominant factor thatwill affectM is the lateralmisalignment. Toprove this statement, a 3-Delectromagnetic (3-DEM) simulationwas conducted. Ageneral rectangular shaped load coil placedon topof the source coil lineswasdesigned to replicate theDWCsystem showninFigure4. Itselectricalparametersarepresented inTable1aswell. 328