The characteristics of a MOSFET including the subthreshold region can be described using the following equations:

At which of the following point(s) the equations are discontinous? (Select all that apply)

Question

The characteristics of a MOSFET including the subthreshold region can be described using the following equations:

At which of the following point(s) the equations are discontinous? (Select all that apply)

BlackTom AI · Accepted Answer

To tackle this question, I will examine where the given piecewise expressions could fail to be continuous as the operating point moves across boundaries.

Option A: VG = 0V
- In the subthreshold model, the first equation ID = ID0 e^{q VG/(nkT)} applies for VG ≤ VT. At VG = 0 V this is just a finite, well-defined current ID = ID0, and there is no inherent discontinuity introduced by VG = 0 in the exponential form. Since the piecewise boundary is defined at VG = VT, not at VG = 0, the current does not inherently jump discontinuously at VG = 0. Therefore this point is not a guaranteed discontinuity by the given equations.

Option B: VG = VT
- This is a primary boundary between the subthreshold (exponential) region and the above-threshold region. If we evaluate the two expressions at VG just below VT and just above VT, the currents need to match for continuity. The subthreshold expression yields ID = ID0 e^{q VT/(nkT)} at VG → VT−. The above-threshold region provides ID expressions (IDlin or IDsat) that depend on VD as well as VG. Unless the ID expressions are explicitly defined to match at VG = VT with VD chosen accordingly, there can be a jump discontinuity at VG = VT. In many MOSFET models, a mismatch occurs at this boundary unless a smoothing/transition is added, so VG = VT is a potential discontinuity point.

Option C: VD = 0V
- When VD = 0, the linear region expression IDlin = (μ W C_ox / L) (VG − VT − VD/2) VD becomes IDlin = 0, since VD multiplies the bracket. Meanwhile, the subthreshold expression for VG ≤ VT gives ID = ID0 e^{q VG/(nkT)}, which is generally nonzero for VG > 0 and even at VG = 0 yields ID = ID0. Thus there is a potential jump between the VD = 0 line (where IDlin predicts 0 in the above-threshold regime) and the subthreshold expression as VG crosses VT, indicating a discontinuity along VD = 0 for currents computed from the different branches. Hence VD = 0V is a discontinuity point.

Option D: VD = (VG − VT)
- This line is explicitly the boundary between the linear (IDlin) and saturation (IDsat) regions, defined by VD ≤ VG − VT for the linear regime and VD > VG − VT for saturation. At the boundary VD = VG − VT, the two expressions would have to yield the same ID for continuity. However, IDlin = (μ W C_ox / L) (VG − VT − VD/2) VD evaluated at VD = VG − VT gives IDlin = (μ W C_ox / L) (VG − VT)^2 / 2. The saturation expression IDsat = (μ W C_ox / (2L)) (VG − VT)^2 (1 + λ VD) evaluated at VD = VG − VT gives IDsat = (μ W C_ox / (2L)) (VG − VT)^2 (1 + λ (VG − VT)). These two values are generally not equal unless the lambda term is zero or VG − VT takes a specific value. Therefore, there is typically a discontinuity at VD = VG − VT unless a deliberate smoothing is applied. Hence VD = (VG − VT) is a discontinuity point.

Summary: Each option represents a potential boundary where the piecewise current expressions may fail to be continuous without care taken to enforce matching conditions. Among the given choices, VG = VT, VD = 0V, and VD = (VG − VT) are all situations where the modeled current expressions can exhibit discontinuities, whereas VG = 0V does not inherently introduce a discontinuity by the definitions of the regions unless additional constraints are imposed.

The characteristics of a MOSFET including the subthreshold region can be described using the following equations: At which of the following point(s) the equations are discontinous? (Select all that apply)多项选择题

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