Self-heating Feedback

The basic feedback equation encountered is Gcl = Gf/(1 + Gol) where Gcl is the closed loop gain of a system that has open loop gain Gol and forward loop gain Gf. Consider the case of a resistive heater. The source of power could be either a constant current source or a constant voltage. In this problem we are interested in the steady state temperature of the load.

Constant Voltage Source

For the first case lets look at constant voltage. The power dissipated in the load is P = V²/R. Here R is the load resistance and V is the applied voltage. However, R is a function of temperature, T, given by R = Ro*(1 + a*(T-To) ) where the resistance is linear with temperature and has a value Ro at temperature To. The steady-state temperature is also a function of the temperature given by the dissipation characteristics of the thermal system T = Tamb + theta*P where theta is the thermal resistance (C/W) and Tamb is the ambient temperature. Intuitively you can see that when the voltage is applied the power increases the temperature which in turn increases the resistance (assuming a positive temperature coefficient, a). The increase in resistance then reduces the power. This is inherent negative feedback at play that prevents a runaway thermal condition.
Combining these three equations we can solve for the temperature as

(dT)² + (1/a)*(dT) - V²*Theta/(a*R0) = 0

Where dT = (T-Tamb). This is a non-linear system. The block diagram is shown below as a feedback network. I also added a thermal time constant to the thermal resistance block. The loop gain is found using small signal gain theory to be -(a*Ro*Theta*I²(dT)) which is negative for a>0.









Constant Current Source

The second case we will look at is a constant current supply. In this case the power dissipation is given by P = I² * R. We can see intuitively that power dissipation raises the temperature which raises the resistance and therefore increases the power dissipation. This is a case of positive feedback and hence has the potential for thermal runaway. Solving the three equations for power, resistance and temperature we have the result for temperature rise

(T-Tamb) = (I²*R0*Theta) / (1 - (a*Theta*I²*R0))

This is a linear system and has the form of a closed loop equation with open loop gain a*Ro*Theta*I². In the block diagram below I added a thermal time constant to the thermal resistance block.


Note that for positive temperature coeffficents (a>0) this system has positive feedback. If the loop gain gets too close to unity there will be a thermal runaway condition. Some boundary conditions are: 1. If the temperature coefficient is negative (a<0) dt=I²*R0*Theta, 2. Perfect Heat sink (Theta=0) then dT= 0 and 3. Perfect Short R=0 then dT= 0. The loop gain can be used as a measure of the safety factor in a system design.

This basic analysis of the feedback mechanism in a self heating system has many applications. Who would of thought that a handful of variables would say so much about a system?