Relationship between current and temperature

I am trying to find the specific current needed to heat up a wire to a specific temperature. The wire material is kanthal. I've calculated the resistance, from the resistivty and the dimensions of the wire. I also have the supply voltage.

But what is the relationship between Wattage and Temperature? How do I find out how much current needed to heat the wire to a specific temp?

• It depends upon the environment. Is it in air? If so can the air move? Is there a fan? What is the ambient temperature? Is it a short length of wire with huge metal connections on the ends or so long that it can be considered infinite from a thermal conductivity view point? – Andrew Sep 19 '16 at 16:27
• The actual answer to your question - the relationship between current and temperature of a cylindrical object in a fluid such as air is described (approximately) by King's law (developed about 100 years ago). It's a bit complex. – Spehro Pefhany Sep 19 '16 at 20:51

The part you are missing is the thermal resistance from the wire to ambient air (or whatever medium you have it in). This is commonly specified in °C/W for semiconductor devices. For wires, of course the total power depends on the length, so you are looking for figure like °C/(W m).

For example, let's say you want the wire temperature to be 100 °C and that you've found its thermal resistance to ambient air is 10 °C/(W m). Ambient air in this example is at 20 °C, so the temperature rise required is 80 °C. That means that you need to dump 8 W into every meter of wire. If your wire segment is 200 mm long, then you need to dump 1.6 W into it.

Once you know the power required, you can use the electrical resistance to compute the voltage and current that will attain that power.

Keep in mind the resistivity of many substances changes significantly over temperature. Old LEBs (light emitting bulbs) were a good example. The filament gets so hot it gives off significant light. It is several times more resistive at that temperature than at room temperature. LEBs therefore had large currents for a short time after being switched on until they got to glowing temperature.

One way to deal with varying resistivity is to measure both voltage and current in a microcontroller, do the multiply to find the power, and adjust the power supply accordingly to maintain the desired power. The power supply is likely a switcher controlled by a micro anyway, so this doesn't add much complexity.

A even better way is to regulate the temperature directly. This scheme makes use of the fact that the wire resistance changes with temperature. You measure voltage and current, but this time compute the resistance. The power supply is then regulated to keep this resistance constant. This method is tricky if the material doesn't have a lot of resistivity change per temperature at the operating point you care about.

• What is the base function for the relationship between resistance and temperature at a constant current? I would assume It would be $T = k * ln(a*R+b)$. where T is temp, R is resistance, and k,a,and b are arbitrary constants – user86234 Sep 19 '16 at 16:46
• @Tusk: There are lots of approximations out there. The Steinhart-Hart (sp?) equation is popular, for example. Various simpler versions can be useful too, especially the more limited the temperature range is. – Olin Lathrop Sep 19 '16 at 16:49
• Thank you, this is actually very useful. you have my vote! – user86234 Sep 19 '16 at 16:51
• Thank you! So the wire is just going to be 5-7 cm, with a cross section between 0.5-3mm. The thermal conductivity of the material is 22 W/mK. If I want the wire temperature to be 1000 degrees. How do I find how much current I need? How’s the relationship between the effect and the thermal? – WireHeating Sep 20 '16 at 12:03