Other Single-Loop Circuits — Question 8

Question 8

Opposite temperature coefficients can stabilize a series current

Two resistors obey $R_1(T)=R_{10}(1+a\Delta T)$ and $R_2(T)=R_{20}(1-b\Delta T)$, with $a=.0040$ K$^{-1}$ and $b=.0020$ K$^{-1}$. They share a uniform temperature and an ideal 12.0 V source.

1. Choose positive R10 and R20 so the current is .100 A and its first-order temperature coefficient vanishes.

2. Within the supplied linear laws, find the exact total resistance and state the temperature range required for both individual resistances to remain positive.

Question 8 - Solution

Opposite temperature coefficients can stabilize a series current

(a) Method and calculation

$$R_{10}+R_{20}=12/.100=120\,\Omega,\quad aR_{10}-bR_{20}=0.$$$$R_{20}=2R_{10}\Rightarrow R_{10}=40.0\,\Omega,\quad R_{20}=80.0\,\Omega.$$

(b) Interpretation and further calculation

$$R_1+R_2=120+(aR_{10}-bR_{20})\Delta T=120\,\Omega.$$$$1+a\Delta T>0,\ 1-b\Delta T>0\Rightarrow-250<\Delta T<500\,\mathrm K.$$The empirical linear laws may be valid only over a much smaller interval.

Physical check

Cancellation is exact within the specified linear model, but is not guaranteed for real nonlinear temperature dependence.