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In wireless mobile charging circuit, is there any formula of spacing which varifies that by increasing diameter and turns, spacing (range) can be increased? enter image description here

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The efficiency of the power transfer depends on the coupling between the inductors ”k” and their quality factor “Q”. (“Q” = the ratio of the inductance ”L” to the resistance “R” of a coil) The quality factor Q is mainly dependent on the shape and size of the coil as well as the materials used, usually you may expect values around 100.The coupling “k” is determined by the distance between the inductors (we mark the distance by “z”) and the ratio of two coils diameters D2 / D1. The coupling is further determined by the shape of the coils and the angle between them. The coupling factor is a value between 0 and 1. The value of 1 expresses perfect coupling, so all flux generated in the transmitter penetrates the receiver coil. By the other hand, a 0 value expresses a system where transmitter and receiver coils are independent of each other. Poor coupling can be linearly compensated by a better quality factor and vice versa. So the efficiency drops dramatically at larger distance (z/D > 1) or at a large size difference of the coil (D2/D1 < 0.3). A very high efficiency (>90%) can be achieved at close distance (the ratio z/D < 0.1) and for coils of similar size (D2/D1 = 0.5 ~ 1). For more details about this refer this link wireless charger

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The transferred power between two coils depends on the coupling inductance between them, which depends on several things, such as axial spacing, number of turns and radius of the coils, according to this formula:

$$ L_{12} = N_1.N_2.\frac{\mu_0}{2}.r_1.r_2.\int_0^{2\pi}\dfrac{cos\alpha}{\sqrt{b^2+r_1^2+r_2^2-2.r_1.r_2.cos\alpha}}d\alpha $$

Being \$b\$ the axial spacing, \$r_i\$ the radius of the coils and \$N_i\$ the number of turns. You can see how increasing the number of turns and radius will allow you to increase the axial distance to obtain the same coupling inductance, therefore the same transfered power.

See reference paper (page 5) Limitation of inductive power transfer for consumer applications

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