# How to estimate inductive coupling between distant air coils

I have two air coils. They both have diameter d. There is a distance D between their centres.

D is much greater than d (more than 10x greater)

Both coils are at different angles, a and b, relative to the line between their centers. There is an alternating current in one coil.

Is there some simple function which estimates the current induced in the second coil? I would be happy if the form of the function was:

k * f(D, a, b) (where k had to be measured)

For example: k * cos(a) * cos(b) * D^-2

• This question is more appropriate for a physics forum than it is for here. Although it's still a fun problem. – Adam P Apr 21 '11 at 0:32
• My guess is that it would be more something like k * cos(a-b) * D^-2 – stevenvh Apr 21 '11 at 7:46
• ... assuming a spherical cow on a frictionless surface ... – vicatcu May 28 '11 at 2:43
• This is not homework. I do not go to school, as I am 33 years old. – Rocketmagnet Jun 20 '11 at 11:29
• @Rocketmagnet - That's no reason not to go to school! – Kevin Vermeer Jun 21 '11 at 1:31

## 1 Answer

I think you are looking for Ampere's law and Faraday's law, parts of Maxwell's equations.

For a distance D between coils much greater than the coil diameter, The voltage induced on the second coil from a given current on the input coil is something like

• Vout = 2 π f N S Bo cos b,

where

• f = carrier frequency of transmitted signal in Hz
• N = number of turns in the receiving output coil
• S = area of output loop in m^2
• Bo = strength of the arrival magnetic field
• b = angle of the output coil away from "axis pointing directly at input coil"

and in turn,

• Bo = μ0 I N R^2 cos a / ( 2( R^2 + D^2 )^3/2 ),

where

• μ0 = the universal permeability of free space
• I = current in input wire
• N = number of turns in the transmitting coil
• R = radius of transmitting coil
• D = distance between the coils
• a = angle of the transmitting coil axis away from "axis pointing directly at output coil"

At long distances, the magnetic field (and hence the produced voltage) drops off with D^(-3).