Simplified Theory of a Risk of Thermal Catastrophe in Air-Cooled Electronic Circuits

B2 - HVAC and Condenser Fans

The goal is to evidently illustrate a risk of thermal catastrophe in electronic circuits cooled by air. As known, the heat produced in an electric current, if not dissipated, may notably elevate the temperature of the circuit and damage its components. In circuits with a negative thermal coefficient of resistance (which is typical for the elements made of carbon and semiconductors), such a transient process can follow a pattern of thermal catastrophe. The theoretical model includes a simple resistor under constant voltage; the dominant mechanism of heat dissipation is a forced thermal convection in air; the temperature coefficient of resistance can be negative or positive. Despite a relative simplicity of the model, the author has never come across such a study in the literature for case of negative temperature coefficient of resistance.
The main results are the following:
(1) if the temperature coefficient of resistance is positive, a stable equilibrium can be achieved at some temperature;
(2) if the temperature coefficient of resistance is negative but the air cooling is efficient, the thermal equilibrium may formally exist at two temperatures, only one of such equilibrium states being stable;
(3) if the temperature coefficient of resistance is negative and the air cooling is relatively weak, a thermal equilibrium is not feasible at all and the temperature goes up unlimitedly for this mathematical model. It is noteworthy that such an effect might have happened in Edison's experiments on his first electric bulbs because the filaments were made of carbonized materials. Simple equations and plots have been derived to describe the key physical phenomena qualitatively. Certainly, the real-life conditions are more complex but the theory clearly describes the main trends and can help practical specialists and engineering students better understand the importance of adequate air cooling systems in computers and other electronic devices, in particular those utilized for real-time control.