Question #43350

Consider a Wheatstone bridge circuit powered by an ideal DC voltage source $V_0$. The left branch consists of a temperature-independent resistor $R$ in series with a p-n junction diode $D$ (connected to ground). The right branch consists of two temperature-independent resistors, $R$ and $R_v$. The output voltage is measured between the midpoints: $V_{out} = V_{right} - V_{left}$, where $V_{left}$ is the voltage across the diode.

The diode operates in the forward-bias region with current $I = I_s(T) e^{qV_D/kT}$, where $V_D$ is the diode voltage, $q$ is the elementary charge, and $k$ is the Boltzmann constant. The reverse saturation current depends on absolute temperature $T$ as $I_s(T) = C T^3 e^{-E_g/kT}$, where $C$ is a material constant and $E_g$ is the bandgap energy.

At a specific temperature $T$, the bridge is balanced ($V_{out} = 0$) by adjusting $R_v$. At this operating point, the diode has a dynamic resistance $r_d = \frac{kT}{qI}$. Determine the temperature sensitivity of the bridge, defined as $S = \frac{dV_{out}}{dT}$, at this balanced state.