Question #43898

A biasing circuit for a transistor uses a voltage divider consisting of a fixed ideal resistor $R_1 = 2.0 \text{ k}\Omega$ and a temperature-sensitive resistor $R_2$ connected in series across a constant DC supply $V_{CC} = 10.0 \text{ V}$. The output voltage $V_{out}$ is taken across $R_2$. To provide thermal compensation, $R_2$ (initial resistance $R_0 = 2.0 \text{ k}\Omega$ at $T_0$) is a thin-film resistor perfectly bonded to the top surface of a bimetallic strip. The strip consists of two metals of equal thickness with linear expansion coefficients $\alpha_1 = 3.0 \times 10^{-5} \text{ K}^{-1}$ (top metal) and $\alpha_2 = 1.0 \times 10^{-5} \text{ K}^{-1}$ (bottom metal). When the temperature increases by $\Delta T$, the strip bends, inducing a longitudinal tensile strain $\epsilon = \frac{3}{4}(\alpha_1 - \alpha_2)\Delta T$ on its top surface. The resistor $R_2$ has an intrinsic temperature coefficient of resistance $\alpha_\rho = 1.0 \times 10^{-4} \text{ K}^{-1}$ and a strain gauge factor $G = \frac{\Delta R_2 / R_0}{\epsilon} = 4.0$. Assuming the total fractional change in resistance is the sum of the intrinsic thermal and mechanical strain contributions, what is the rate of change of the output voltage with respect to temperature, $\frac{dV_{out}}{dT}$, at $T = T_0$?