Question #43323

An engineer designs a circuit to deliver power to an ideal p-n junction diode from an AC voltage source $V(t) = V_0 \sin(\omega t)$ with amplitude $V_0 = 50 \text{ mV}$. Due to thermal drift, the diode stabilizes at $T = 300 \text{ K}$, where the thermal voltage is $V_T = \frac{kT}{q} = 25 \text{ mV}$. At this temperature, governed by the Law of Mass Action, the reverse saturation current is $I_s = 2.0 \mu\text{A}$.

Applying the Maximum Power Transfer principle linearly, the engineer assumes the diode acts as an Ohmic load with a constant resistance equal to its zero-bias dynamic resistance $r_d = \frac{V_T}{I_s}$, and calculates the expected average power as $P_{linear} = \frac{V_{rms}^2}{r_d}$.

However, the true average power $P_{true}$ is governed by the non-linear ideal diode equation $I(t) = I_s \left( e^{\frac{V(t)}{V_T}} - 1 \right)$.

Calculate the absolute error $\Delta P = |P_{true} - P_{linear}|$ in the engineer's calculation.

(To find $P_{true}$, approximate the exponential up to the cubic term: $e^x - 1 \approx x + \frac{x^2}{2} + \frac{x^3}{6}$. Use standard time averages $\langle \sin^2(\omega t) \rangle = \frac{1}{2}$ and $\langle \sin^4(\omega t) \rangle = \frac{3}{8}$.)