The Bread knife was making an awful mess of cutting expanded polystyrene packing material so I came straight to "Instructables" to find out how to make one:
I am not doing any great profiles or precision carving so intended to use only a short piece of wire, about 6 inches so I needed to find out how much current I needed to make a clean cut.

Step 1: The "cutter"

This is what I was going to use, an old hacksaw handle.

Step 2: What to Do?

After looking through the many outstanding "Instructables" it was still not clear what I needed: It seemed that a current greater than 2 Amperes was requred but I could not find an explanation of what that temperature would be especially when I became very excited at the thought of cutting acrylic sheet!
It seems that most examples simply "found the right power supply" and ran it on the current limit, not giving a clue to the melting (cutting) temperature involved for any particular application.

So I decided that I would use a Rheostat and a Car Battery Charger or standard computer Power supply.

A Rheostat is a variable resistor not a potentiometer, just 2 active wires.

I already had the Nichrome wire and found that it was about 5 ohms per metre. I thought that would safely limit the current to about 1 Ampere on a 5 volt P.S.U and 2.4 Amperes on the Car Battery charger, which, with suitable tapping points would enable me to adjust the current to find out the best value for the work.

Step 3: The Parts for the Resistor = Rheostat

Here is how I went about making a home-made Rheostat:
This method means that you do not need to know the gauge or length of the wire any one that has salvaged their resistance wire from a toaster or hair-dryer will have enough spare to make a similar Rheostat.

I am going to show how I took about 1 metre of Nichrome wire (28 s.w.g not that it matters) and wrapped it around 4 times to give me 4 tapping points using long machine screws. Here are the things required:

Two (2) non-conducting sides e.g scrap composite panels
Five (5) 50mm long machine screws e.g M4x50
25 Nuts e.g M4
Optional M4 Washers
1+ Metre of Nichrome wire

Step 4: The Test Rig

This shows how I made the resistor mount which is only suitable for this demonstration.

What I mean by that statement is that the wire will be at the same temperature as the cutting (melting) element so if you intend to make one for regular use then it should be contained on a proper enclosure.

Safety measures to be observed are that it is only operated in a well ventilated area, a non-explosive atmosphere and away from inflammable objects.

The mounting board is about 30cm by 15cm meaning that I can zig-zig 1 metre of wire four times.

I measured the width and divided it by 4 to give me about 3.5cm which I used as the base border for measurements; at one end I measured 3.5cm spacing for 3 mounting holes and at the other end I measured about 5cm spacing for 2 mounting holes and drilled the pilot holes. The other panel was aligned and drilled to give a "mirror" matched pair.

The mounting holes were then drilled clear for my machine screws.

The five (5) machine screws were fitted and secured with nuts, ready for the fitting of the wire.

Step 5: Threading the Wire (resistor)

Extra nuts are fitted to hold the wire, in the middle space, away from the sides and leaving enough thread to go through the other panel with thread to spare for the final nut.

If you have enamel covered wire then it must be removed (scraped with a craft knife and sand-paper) so that there is good electrical contact at each post.

I started the wrap from the middle of the wire and central screw post to keep each length equal.

It would be best to use washers to clamp the wire but I did not have enough. Tighten the two nuts together to lock the wire in place.

Five (5) more nuts are fitted to the machine screws to support and secure the other panel with nuts on the outer side.

Step 6: Finishing

The other panel is fitted onto the end of the machine screws and nuts fitted to secure.

The picture here shows the panel fitted before scraping and attaching the wire end connections.

Good electrical practice is not to rely on a mechanical joint for an electrical connection: I have been casual with the tapping points but making-off the two ends electrically correct means that there will be proper continuity.

You can see that I have drilled two more holes through the board for the wire(s) to be attached under the securing nuts and also fitted an extra machine screw for attaching the other power terminal. You have seen the finished item in the Header picture but here it is again to save you scrolling. ;).

Step 7: Success!

My first result!

It turned out that 2.4 Amperes was sufficient to make this cut (melt).

To get this result I used the Car Battery charger on 12 volts and half of my resistance wire so that shows that 3 Amperes would be about right for general use and now I can experiment with different materials.

With the total resistance I measured 0.8 Amperes at 5 volts and 1.8 Amperes on the Car Battery Charger 12 volts so the computer power supply is a good option but this shows that any high current supply can be used with only minor changes.

Do be careful if you try your own experiments as that wire gets HOT!