Question #43468

A photo-diode is modeled as an ideal AC current source $i_{ph}(t) = I_0 \sin(\omega t)$ in parallel with its voltage-dependent junction capacitance $C_j$. The internal shunt resistance of the diode is assumed to be infinite. The junction capacitance varies with the applied DC reverse bias voltage $V_R$ (in Volts) according to the relation $C_j = \frac{C_0}{\sqrt{1 + V_R/V_{bi}}}$, where $C_0$ (in Farads) is the zero-bias capacitance and $V_{bi}$ (in Volts) is the built-in potential. The photo-diode is connected to an external series combination of an inductor $L$ (in Henrys) and a resistor $R$ (in Ohms). An ideal biasing network maintains the DC reverse bias $V_R$ across the diode without affecting the AC impedance of the circuit. The intensity of the incident light is modulated at an angular frequency $\omega$ (in rad/s). At a specific reverse bias voltage $V_R = V_{res}$, the power factor of the equivalent AC circuit, as seen by the internal photo-current source, becomes unity. Which of the following expressions correctly gives $V_{res}$?