Semiconductor Transistor Current Fluctuation (Electrical Engineering)

In integrated circuit design, an advanced field-effect transistor (FET) regulates output current based on gate-to-source voltage ( $V_g$ ) and drain-to-source voltage ( $V_d$ ). In the saturation regime, the electrical drain current $I$ (in milliamperes, $\text{mA}$ ) is modeled by the non-linear constitutive equation:

I(V_g, V_d) = \sqrt{V_g^3 + V_d^2}

The transistor's nominal operating bias point (Q-point) is established at $V_g = 2.0\text{ V}$ and $V_d = 4.0\text{ V}$ , providing a baseline current of:

I(2, 4) = \sqrt{2^3 + 4^2} = \sqrt{8 + 16} = \sqrt{24} \approx 4.8990\text{ mA}

During high-frequency switching, power rail switching noise causes the terminal voltages to shift slightly to an off-nominal state of $V_g = 2.02\text{ V}$ and $V_d = 3.95\text{ V}$ .

State the multivariable total differential formula $dI$ for the transistor current in terms of voltage differentials $dV_g$ and $dV_d$ .
Compute the analytical partial derivatives $\frac{\partial I}{\partial V_g}$ and $\frac{\partial I}{\partial V_d}$ , and evaluate them at the nominal bias point $(2, 4)$ .
Use the linear approximation $L(V_g, V_d)$ centered at $(2, 4)$ to estimate the perturbed current $I(2.02, 3.95)\text{ mA}$ .
If manufacturer power rail tolerances guarantee $|dV_g| \le 0.05\text{ V}$ and $|dV_d| \le 0.10\text{ V}$ , compute the maximum worst-case current fluctuation $|dI|_{\max}$ .