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There are quite a few different types of resistors, but they all do pretty much the same thing, they convert voltage into heat when a current flows through them.  Voltage, or electromotive force, is the potential difference in charge between two points.  It represents the electrical potential energy between those points.  A resistor is a device that resists the flow of charge or current. Ohms law is the equation that ties these ideas together.  It states that the current through a conductor is directly proportional to the voltage across that conductor.  The constant of proportionality is called resistance.  Capacitors store energy in an electric field, inductors store energy in a magnetic field, resistors don’t store energy they simply convert it to heat.  The power that is expended in heat is simply the voltage across the conductor times the current through the conductor.  For example, a 1 volt drop across a 1 ohm resistor will produce a current of 1 amp. This also means that 1 watt of power will be dissipated in the resistor.
How does resistance change over temperature?
It would be nice if the resistance of a resistor were precise and constant.  As the saying goes, it is good to have wants.  In the real world a resistor comes with a tolerance 1%, 5%, etc.  A 1% resistor will have a value within +/- 1% of the stated value, e.g. a 100 ohm, 1% resistor will have a value between 99 and 101 ohms.
The value specified holds for a specific temperature usually close to ambient, 20 degrees C is typical.  The sensitivity of the value of the resistor to changes in temperature is described by a parameter called the temperature coefficient of resistance or TCR.  It is usually expressed in ppm/degC (parts per million per degree Celsius).  If a resistor has a 50 ppm/degC TCR, that means a change of one-degree C will not cause the resistance to vary by more than 50 ohms for every 1,000,000 ohms of the resistors value (0.05 ohms for every 1Kohm of its value).

How much power is too much power?
A resistor that has a value of 100 ohms can carry a current of 50 mA if it is rated at 0.25 watts (ignoring the need for a safety factor for the moment). P = i2R = 0.05 x 0.05 x 100 = 0.25. But this is true only at ambient temperatures.  

As the temperature increases the power rating on the component decreases.  Below is a typical derating curve.

Figure 1 Typical Resistor de-rating curve.
A 0.25 watt resistor what is run at 110 degC will need to be de-rated by 50% or 0.125 watts.
There are other factors that will cause a de-rating of the maximum power of a resistor.  Her are just a few:

  1. How many other heat sources are located close to the resistor?
  2. What heat sinks are available. Heat can be removed from the area of a resistor both by the air and by the board it sits on.
  3. Air flow across the components.
  4. Air temperature.
  5. Pulse operation or continuous current flow?
  6. High frequency?  A resistor also has parasitic capacitance and inductance that come into play at higher frequencies.
  7. At what altitude will the device operate?  Higher altitudes have less air mass and therefore less heat carrying capacity.

The big question everyone wants to know is how close to the maximum power can you run the resistor?  This is called the safety margin.  Running a resistor hot will affect its value.  It will also effect the life of the device.  When deciding how hot to run a device you must consider what will happen of the device fails.

  1. Will failure cause the device to catch fire?
  2. Will it cause a safety risk?
  3. Is the device mission critical? 
  4. What are the safety and economic impacts if this device fails? 
  5. Is it just an inconvenient or will lives be at risk because of the failure?
  6. Will it create a large economic impact?

Engineering Design
I am sure you wanted a straight forward answer to the question of, “How hot can the resistor get?”  Unfortunately, the answer to that question is part of engineering design.  Specifying too broad of a safety margin will cost money, while too narrow a margin will cause failure.  Finding the right balance for a specific application is why we go to work each morning.
Final thoughts
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