Power transferred between two circular coils

[EDITED]

I am interested in studying the physics behind wireless charging for autonomous vehicles. Therefore, I want to compute the mutual inductance between two coaxial circular coils.

I have read that the mutual inductance between two coaxial circular cables (i.e. individual cables, not coils) is

$$M = 4\pi \sqrt{Aa}((\frac{2}{k}-k)F-\frac{2}{k}E)$$

Where we have $$a$$ is the radius of the first circle, $$A$$ is the radius of the second circle, $$h$$ the distance between them, and $$k^2 = \frac{4Aa}{h^2+(a+A)^2}$$.

F and E are elliptic integrals of the first and second type (which must be evaluated numerically), or

$$F(k) = \int_0^{\pi/2} \frac{d\phi}{\sqrt{1-k^2\sin^2\phi}}$$

$$E(k) = \int_0^{\pi/2} d\phi\sqrt{1-k^2\sin^2\phi}$$

I can do these integrals numerically in Matlab no problem, which gives me an M value.

Now, I am interested in computing the voltage/power input from the first (driving) coil into the second (receiving) coil. Once I have M, how do i get the emf in the secondary coil? Where does the number of turns in the coil come into play? Or the frequency of the AC signal? The constant for the permeability of free space?

Finally, should I account for back-emf (self-inductance), capacity (resonant transfer), or anything else I haven't considered?

• If I have understood correctly, you are triying to compute the mutual inductance of two circular current loop. The analytic computation is not easy. For the case of two parallel coaxial circular loops, I think you could easily find the result in a textbook (for example, a very old reference is Becker, electromagnetism and interaction). The result use elliptic integrals. For an efficient wireless transfert, you also have to take into account the capacity of the coils (resonant transfer). Jan 31, 2019 at 6:33
• Thank you Vincent. I've been looking online for more information and have also stumbled across some elliptic integrals. ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20010038494.pdf For example, in the above paper, do you think I could just numerically integrate equation (14) over a circle's area to get the magnetic flux through it? Alternatively, I will also look into the textbook you mentioned, although my old reference is in storage right now. Maybe I can pay for it online. Jan 31, 2019 at 6:56
• Forget my most recent comment. After your suggestion, I changed my search queries to 'mutual inductance of coaxial circular loops', and have been getting much more useful results. I'm in the process of reviewing them, and will reply soon. Jan 31, 2019 at 7:07
• BTW, elliptic integrals of the 1st kind can be computed using the arithmetic-geometric mean, which converges very quickly. I assume Matlab uses this algorithm, since it is superior to a power series approach, or a generic numerical integrator. Jan 31, 2019 at 11:34

That depends on the units you are using. For SI units you will use the equation without the $$c^2$$ on the bottom, and if you are using Gaussian or electrostatic units then you will use the one with it.