Higher-order Multipoles in the Electromagnetic Field Produced by a Wireless Power Transfer System Employing DD Polarized Couplers

The DD coupler for magnetic field wireless power transfer (WPT) systems has been analyzed previously., and it has been shown that its electromagnetic field is dominated by a horizontal magnetic dipole moment. However., a significant contribution from a higher-order multipole is also present and is apparent in the magnetic field even at distances of 10m. Here., the deviation of the electromagnetic field from that of a horizontal magnetic dipole is quantified by subtracting the properly weighted dipolar $\mathrm{TE}_{11^{-}}^{e}R$ field from numerically computed data leaving a residual field. The residual electromagnetic field clearly exhibits higher-order multipole behavior. It is anticipated that this higher-order multipole would have the form of two antiparallel vertical magnetic dipoles (one for each spiral winding) combined with two canceling images due to the effect of the conducting shield. This multipole is then an ensemble of four elementary magnetic dipole sources or two side-by-side linear., vertical., anti-phase magnetic quadrupoles and hence in terms of TE-z electric vector potential has a multipole order of $m=1$ and $n=2$. We show that such a multipole source would excite $\mathrm{TE}_{13^{-}}^{e} R, \mathrm{TE}_{11^{-}}^{e}R$:., and $\mathrm{TM}_{12^{-}}^{o}R$ spherical modes. In order to dissect the residual field it is useful to note that radial magnetic field must be due exclusively to TE-$R$ modes while radial electric field is due solely to TM-R modes. The radial component of the residual magnetic field clearly contains tesseral harmonics $\mathrm{T}_{11}^{e}$ and $\mathrm{T}_{13}^{{e}}$ that is of order of $m=1,\ \ n=1$ and $m=1$., $n=3$ and thus corresponds to the $\mathrm{TE}_{13^{-}}^{e}R$: and $\mathrm{TE}_{11^{-}}^{e}R$ spherical modes. However., investigation of the residual radial electric field indicates that it is composed of more than just the $\mathrm{TM}_{12}^{o}-R$ spherical mode. It appears that this additional component of the field is due to the spiral nature of the windings and the feed.