How to find mutual inductance between planar rectangular coil and metal plate

Hello

I am designing variable inductor in MEMS, by moving metal shield above the coil, the inductance is varied. The lumped model for that design is shown in fig.

My question is how to find the mutual inductance between coil and ground metal plate. As shown in lumped model, coupling co-efft is needed to model.

Is this mutual inductance is frequency dependent then how to find it.

I need geometry dependent analytical formula to find mutual inductance.

Thanks in advance...

Frequency dependence can be expected due to variable penetration of current. If you are familar with the "analytical formulas" for simple electromagnetic problems, I guess you won't want to see the solution for your problem. A page of paper may be too small to hold it (if it exists at all).

People use numerical methods for similar problems. Fasthenry from Fast Field Solvers is free tool, that can basically handle it.

With a numerical field solver, it is indeed simple to get the inductance as a function of shield position/distance.
I had worked on a similar problem some years ago for a customer using the Sonnet EM solver.

If such a numerical approach is useful to you, you can try the free Sonnet student version:
Sonnet Lite

Thanks for support... can i get soft copy of this book

"Inductance Calculations: Working Formulas and Tables"
F W Grover, 1946 and 1973.
Dover Phoenix Edition 2004.
Dover Publications.