PCB coil electromagnet driver
A coil energized with current can create a force of attraction to an iron object. It can also repel a magnet if the polarity is correct. That force is: -

Force = \$\dfrac{(amps\cdot turns)^2\cdot \mu_0\cdot A}{2\cdot g^2}\$

Where 


A is cross sectional area of the electromagnet
g is gap to your magnet
\$\mu_0\$ is 4\$\pi\$ x \$10^{-7}\$


Let's say the force needed is 0.2 newtons. Your required coil is 20mm in diameter and lets assume that all the turns are closely packed around a diameter of 20mm. The gap to the magnet is 10mm. Lets say you can get 100 turns wound at this diameter...

Re-writing: -

Amps = \$\sqrt{\dfrac{20 \cdot 2 \cdot 0.01^2}{100^2\cdot 4\pi \times 10^{-7}\cdot 3.14\times 10^{-4}}}
\$

That's about 32 amps. 

But, it's going to be a lot worse for a spiral design because as the coil diameter gets tighter (for the inner turns), the effective area of the electromagnet gets smaller.

I think you might be looking at something like 100 amps for a 100 turn coil and basically it looks unfeasible. Of course, if your force is significantly less than 0.2 newtons then the amperage would drop so plug your own numbers in to the formula.

Once you have got an idea how much current is needed then that is the time to work out how to drive your electromagnet.