Consider the circuit shown in Figure (a). A gate pulse v_g is applied between time instants t₀ and t₁. After t₁, during the MOSFET turn OFF process, it experi ences a voltage overshoot. Based on the v_ds waveforms shown in Figure (b), which one of the following options is correct?

The correct option is R3 > R2 > R1 (represented in the options as R3 ¿ R2 ¿ R1).

Step-by-Step Explanation and Derivation:

1. Circuit Analysis during Switch Turn-OFF:
Consider the circuit shown in Figure (a). When the gate pulse _vg is applied between time instants _t0 and _t1, the MOSFET is in the ON state, and current flows through the inductor _L, storing magnetic energy. At _{t = t1}, the gate voltage drops to 0 V, initiating the turn-off process of the MOSFET.

Because the current through the inductor _L cannot change instantaneously, the stored energy forces the inductor current to freewheel. The freewheeling path consists of the diode _D and the resistor _Rf connected in series, and this combined branch is in parallel with the inductor _L.

2. Expression for the Drain-to-Source Voltage:
The voltage across the MOSFET _vds after turn-off is determined by the voltage at the drain node with respect to ground. Since the top of the inductor is connected to the DC source _VDC, we can write:

$$v_{\mathrm{ds}}=V_{\mathrm{DC}}+v_L$$

Here, _vL is the voltage across the inductor. Because the freewheeling branch is in parallel with the inductor, when the diode _D is conducting, we have:

$$v_L=v_D+i_f{R}_f$$

Assuming an ideal diode with negligible forward voltage drop (_{vD ≈ 0}):

$$v_{\mathrm{ds}}=V_{\mathrm{DC}}+i_f{R}_f$$

3. Relating Damping and Overshoot to Resistor Values:
During the transition, the parasitic drain-to-source capacitance of the MOSFET _Cds interacts with the inductor _L and the resistor _Rf, forming a parallel resonant RLC network during the turn-off transient. The damping factor _α of this parallel RLC network is given by:

$$\alpha =\frac{1}{2R_f{C}_{\mathrm{ds}}}$$

This damping factor formula shows that the damping factor _α is inversely proportional to the resistance _Rf. Therefore:
- A larger resistance _Rf corresponds to a smaller damping factor (underdamped response), resulting in a larger peak voltage overshoot and ringing.
- A smaller resistance _Rf corresponds to a larger damping factor (closer to critically damped/overdamped response), resulting in a lower peak voltage overshoot.

From the waveforms in Figure (b), we observe the peak voltage overshoots for different values of _Rf:
- The waveform with _{Rf = R1} has the highest overshoot (least damping).
- The waveform with _{Rf = R3} has an intermediate overshoot.
- The waveform with _{Rf = R2} has the lowest overshoot (most damping).

Thus, by comparing the peak values and the damping levels, we establish the following relation for the resistances:
- _R1 (least damping/highest spike) is the smallest damping resistance.
- _R2 (most damping/lowest spike) is the largest damping resistance.

Therefore, we conclude that:

$$R_3>R_2>R_1$$