Power in Electric Circuits — Question 7

Question 7

A heater with a positive temperature coefficient responds differently to fixed current and voltage

A heater has $R=R_0(1+\alpha\Delta T)$ and loses heat at rate $H\Delta T$, with H and alpha positive. Treat its temperature as uniform.

1. At fixed current I, find steady temperature rise and its stability condition.

2. At fixed voltage V, find the positive steady temperature rise and compare thermal feedback signs.

Question 7 - Solution

A heater with a positive temperature coefficient responds differently to fixed current and voltage

(a) Method and calculation

$$I^2R_0(1+\alpha\Delta T)=H\Delta T\Rightarrow\Delta T=\frac{I^2R_0}{H-\alpha I^2R_0}.$$$$H>\alpha I^2R_0\quad\text{for a positive stable equilibrium}.$$

(b) Interpretation and further calculation

$$\frac{V^2}{R_0(1+\alpha\Delta T)}=H\Delta T.$$$$\Delta T=\frac{\sqrt{1+4\alpha V^2/(HR_0)}-1}{2\alpha}.$$Current drive increases heating with temperature; voltage drive decreases it, giving negative feedback.

Physical check

Stability follows the slope of heating minus cooling; the fixed-current denominator must remain positive.