Electronic – How to Attach a Nichrome Wire to a Regular Copper Wire

edit: changed calculations to the diode's package

I would use SMT diodes (I'll explain in a minute why). 45 °C is not very hot, they can easily handle that without derating. You can use the BAS16 in SC79 package.

Thermal resistance for the SC79 package is 120 K/W. That means that without thermal losses supplying 200 mW will already heat the package up from room temperature (20 °C) to 45 °C.

You'll have to experiment a bit with power, because at equilibrium you supply as much power as is lost to the environment, and that last thing is not easy to calculate theoretically. Keep in mind that the diodes will lose more heat through conduction (to the PCB) than through convection (to the air).

Arduino can supply 100mW per pin, so you can't multiplex directly from the I/O. Three problems.

• Not only is the 100mW insufficient, if you want to multiplex 25 columns and have 200mW for each, that means you have to supply 5 A at a 4 % duty cycle. Arduino can't source nor sink that, so you'll need transistors for controlling both rows and columns.
• And that's where the diodes come in, you can't just multiplex an array of resistors, nor Nichrome wire. The resistors in adjacent colums/rows are parallel to them and they will also take a share in the load. You can't control them individually. With diodes you can do that, LED display multiplexing is done all the time.
• 25 diodes \$\times\$ 200 mA average = 5 A. Arduino can't supply that. Using an external 5 V supply is a Bad Idea™: the diodes drop only about 1 V, so the rest of the 5 V would cause 20 W dissipation in transistors and current limiting resistors. You want a power supply at around 2 V.

Further reading
Thermal Management in Surface-Mounted Resistor Applications

First, as Spehro points out in the comment below, you should make sure your coil matches the inductance of the original cooktop coil. This will minimize losses in the coil itself, some of which radiates as heat.

Then you should be using copper tubing for your coil, which you run coolant (often water) through. The currents going through the coil are high frequency anyway, so most of the current is in the outside of the conductor, which is fine for a tube. You'll have to make sure to properly isolate the coolant from the user, ground, and other metal and circuit parts as it will carry a small current from the coil.

There are a number of diy induction heater examples that use this technique. If you need an example, try a search for them. The image below shows more than you need. The transformer and huge capacitor in the upper left are already taken care of by your induction cooktop circuitry. The interesting part will be the copper tubing, which is connected to the plastic tubing on the lower left which then leads to a pump and a bucket or radiator.