# Variable Turn Density Solenoid

I am working on a medical device application and would like to use a solenoid with a variable turn density such that the force $$\F\$$ on a metal piston within the solenoid cylinder is not constant but is proportional to the the overlap length $$\l\$$ between the piston and cylinder - effectively functioning as a spring with spring constant as a function of the current.

$$F(l)=k(I)l$$

My naive assumption is that I can make this work by using a wire turn density which is a function of the overlap $$\n(l)\$$, instead of the usual uniform turn density $$\n=N/L\$$, where $$\N\$$ is the total number turns and $$\L\$$ is the total length of the solenoid. I made some progress using the Ampere derivation of the magnetic field on infinitesimal slices of the cylinder but couldn't figure out how to do the line integral. Any insights would be much appreciated.

• If it's just the integral that's the problem, then you might be better on the maths site. You might also get to know FEA (finite element analysis) software, to model the situation. Unless your forces are tiny, you might be disappointed with air-core solenoids. Any introduction of ferrous material to improve things will take you out of the realm of simple integral geometries. – Neil_UK Dec 25 '20 at 5:31
• this may help with your assumptions mathworks.com/help/physmod/sps/ref/solenoid.html – Tony Stewart EE75 Dec 25 '20 at 8:05
• I have a suspicion you'll get a roughly linear variation with a long piston and a constant turn spacing. As the length of Fe entering the coil is proportional to displacement, so is the reluctance. But as Neil says, the forces will be pretty low. – user_1818839 Dec 25 '20 at 12:22