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Feedback control systems have always been my fascination and topic of interest. When I find that something as simple as a device failing by overheating can be described as a feedback loop I’m in awe of the way something in the physical world can be expressed by a few simple equations.

In the following I describe a few simple equations that govern the temperature rise of an actuator coil that is driven with a current source and show that thermal runaway can occur. I will show that a loop gain can be derived for the system and suggest design limits to prevent thermal run-away.

These are the simple mechanisms of self heating:

• Current dissipates power $Latex formula$
• Temperature rise is proportional to power, $Latex formula$
• Resistance increases with temperature $Latex formula$

These three elements form a positive feedback of power to temperature and can be analyzed using small signal partial derivatives. First though we will use the three equations to directly derive $Latex formula$. A control system person will recognize this as being the form of a positive feedback system. The numerator is the forward gain and the 2nd term in the denominator is the loop gain.

The stability is a function of the RMS current, thermal resistance and resistance thermal coefficient. The plot below is an example of a actuator coil temperature rise versus current given a Cu coil with initial resistance 10 ohms and a thermal resistance of 30 C/W.

At the highest current shown in the graph the loop gain is only about 5%.

Always keep in mind your operating range. Some parameters, such as resistance, may be non-linear at extreme temperatures.