Predicting thermal runaway – Part 2

In Part 1 of this two-part article, we explored the theory of thermal runaway and some settings in which is it of concern (and some in which it is not). In Part 2, we will discuss power-law devices, straight-line systems, and how to predict thermal runaway.

Thermal runaway occurs in semiconductors when the power dissipation of the device in question increases as a function of temperature. As we showed in part 1 of this article, under a certain set of conditions, a device may be unable to establish a nominally steady-state operating point. In such a case, it may either fail to operate all or it may slide into thermal runaway.

Before we can discuss how to predict thermal runaway, it is worthwhile to consider when we can consider a device power characteristic to be a power law at all. Simple devices such as diodes and rectifiers, and even many transistors under certain conditions, typically have a current vs. temperature characteristic at fixed voltages that follows a power law. Leakage current in diodes, for instance, is often described using the rule of thumb that leakage doubles for every increase in temperature of 10°C. This particular power law could be expressed as:
where I is current, I_o is leakage current at 0°C, and T is junction temperature. If the behavior is really a power law, obviously we can figure out what the leakage is at 0°C, given its value at any other temperature.

Alternatively, we can express Equation 4 in terms of the base of the natural logarithms, as in:
So we see that any power-law behavior, that is, behavior described by a geometric increase in the dependent variable (e.g., a factor of 2) for a linear increase in the independent variable (e.g., every 10°C), is just another exponential function, as in Equation 6:
From this example, it may be seen that we can define the “strength” of the power law in terms of that parameter λ. If device behavior is governed by a power law, then knowing the value of the independent variable (in this case, current) at any two corresponding values of the independent variable (in this case temperature), we can say:
So far in this example, we’ve only talked about current, and we need power. Given the premise that a diode under reverse bias (i.e., leakage mode) sees a constant voltage (reasonably assumed to be independent of the temperature of the particular diode), its power dissipation Q, therefore, is also a power law:
where V_R is runaway voltage and Q_o  is power dissipation at 0°C. Note that under constant current, which is the primary operating condition for many applications of diodes, the power law relationship does not hold with temperature. Indeed, it is typical for diodes that at a fixed current, voltage goes down linearly with temperature. As a result, diodes in constant-current operation display linearly decreasing power with increasing temperature, and, as we showed in figure 2 of part 1, are essentially immune to thermal runaway concerns.

For the power law situation, however, we may calculate the slope of the device curve by taking its derivative, that is:
Thus we see that the slope increases with increasing temperature, and all the previous discussions based on device curves with positive second derivatives apply. In particular, Figure 10 of part 1 provides the setting for the subsequent mathematical development.

Perfect runaway
We are now in a position to find the mathematical solution to “perfect runaway’’ in a power-law device cooled by a linear thermal system. To begin, let’s recap the two pertinent equations.

the linear system line:
where θ_Jx steady state thermal resistance of the system and T_x is thermal ground for runaway based on θ_Jx or °C and

the power-law device line:
In this system of two equations, we have two variables, junction temperature T and power dissipation Q. We also have four independent parameters: the thermal ground reference, T_x, the thermal resistance of the cooling system, θ_Jx, the reference power level Q_o, and the strength λ of the power-law function itself. By finding an exact solution to this set of equations, that is, a pair of values (T, Q) that satisfies both equations, we will also create some relationships between the various parameters.

For clarity in solving the equations, it will be helpful to substitute variables. Let us define:
where z is non-dimensional junction temperature

With this new variable, we can rewrite our two equations as follows. From Equation 2, we have:
that is:
the linear system line, and from Equation 8 we have:
or:
in the power-law device line (curve).

Equation 14 suggests that we might choose to define the non-dimensional power q as:
thus, we can make a final restatement of the two governing equations as follows.

the device line:
the system line:
where:
Eliminating q from between Equation 16 and Equation 17, we are left with a remarkably simple relationship defining points z that are the intersection points (i.e., the operating points) of our system
We can interpret our transformed problem, originally seen in Figure 10 of part 1, as the pure exponential function and its intersection with a straight line of slope k, passing through the origin of the (z, q) coordinate system (see figure 11).