**Appendix: Equations for design of a transmitter and a receiver**

A transmitting antenna is surrounded by an electromagnetic field. This electromagnetic field is divided into two separate regions-the reactive near field and the radiating field. The energy is stored in the transmitting coil before it propagates as electromagnetic waves to the receiving coil [41].

The magnetic field experience between transmitter or receiver is called mutual inductance, which can be predicted through:

$$\mathbf{M} = \mathbf{L\_1} \,\, \mathbf{L\_2} \tag{A1}$$

Where, L1 is inductance of transmitter coil and L2 is inductance of receiver coil.

For circular loop coil the inductance can be calculated by using the following formula [42]:

$$L = N^2 \mu\_o \mu\_r \frac{D}{2} \left( \left( \ln \frac{8D}{d} \right) - 2 \right) \tag{A2}$$

Where

*N* – Number of turns of the coil

µ= 4π × 10−7permeability of vacuum, (H/m)

*D* – Diameter of loop coil (m).

*d* – Diameter of conductor cross-section (m).

The coil inductance (*L*) and optimal resonance frequency is determined based on operating frequency that has been used in the system, the capacitance C can be calculated by:

*Wireless Power Transfer – Recent Development, Applications and New Perspectives*

$$\mathbf{f\_r = 1/2\pi \left(1/\sqrt{\mathrm{LC}}\right)}\tag{A3}$$

Quality Factor, *Q* characterize the energy decay in an antenna coil which is inversely proportional with the energy loss in antenna coil before transfer to the receiving coil. The factor Q of coil can be determined using Quality factor:

$$\mathbf{Q} = \alpha \mathbf{L} \left( \mathbf{R}\_{\rm ac} + \mathbf{R}\_{\rm rad} \right) \tag{A4}$$

Where, Rac is AC resistivity; Rrad is radiation resistivity. The quality factor *Q* can have values ranging from 0 to infinity. It is difficult to obtain the values of Q far above 1000 for antenna coils in actual practice [43]. A high-*Q* antenna coil can be defined with *Q* greater than 100. These two coupling antenna coils should have Q greater than100 for each of the coils for transmission of the energy wirelessly [44]. The efficiency of the transfer system is very low for the antenna coils which have Q between 100 and 200 [45]. For obtaining a high efficiency of the wireless power transfer system, a high factor *Q* which approximates to 1000 is preferred for design purposes [46].
