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Voltages below are magnitudes. For PMOS, swap the signs of Vgs, Vds and Vth and the equations stay the same. Vov = Vgs − Vth.

• Weak inversion (Vgs below Vth): the current is a diffusion current. Id grows exponentially with Vgs, one decade for every n·Vt·ln10 (about 80 mV with n = 1.3). Very efficient, very slow, and the device needs a large W/L.
• Triode (Vds below Vov): the channel reaches the drain and the device acts as a voltage-controlled resistor. Use it as a switch or an active resistor.
• Saturation (Vds at or above Vov): the channel pinches off at the drain. Id depends mainly on Vgs and weakly on Vds. This is where amplifiers and current sources live.
• Moderate inversion: the blend between the two. This calculator uses one smooth expression, Veff = 2nVt·ln(1 + e^(Vov/2nVt)), in place of Vov: it becomes Vov in strong inversion and an exponential in weak inversion, so gm/Id falls gradually through this region.

• gm = ∂Id/∂Vgs. How much current change one volt of gate drive buys. In saturation gm = k'(W/L)·Vov = 2Id/Vov = √(2k'(W/L)Id).
• gmb = η·gm. The body acts as a second, weaker gate. η = γ / (2√(2φF + Vsb)), typically 0.1 to 0.3.
• ro = 1/gds ≈ 1/(λId) = VA/Id. The finite slope of the saturation curve. Doubling L doubles VA, so ro doubles at the same current.
• Intrinsic gain gm·ro ≈ 2/(λVov). The most gain one transistor can give with an ideal current-source load. It rises with L and falls with Vov, which is why analog designers use longer channels and moderate overdrive.

Body effect. A positive source-to-body voltage widens the depletion region and raises the threshold: Vth = Vth0 + γ(√(2φF + Vsb) − √(2φF)). It lowers a source follower's gain further below 1, to about gm/(gm + gmb), and it weakens stacked transistors and pass gates. Tie the source to the body where the process allows it, for example with a separate well.

Channel-length modulation. As Vds rises, the pinch-off point moves toward the source and the effective channel shortens, so Id creeps up. λ scales roughly as 1/L, which is why the calculator takes VA per micron.

gm/Id tells you how much transconductance you get per microamp. It depends only on how far into inversion the device is, not on W directly, so it is a handy design variable.

• Weak inversion: the maximum, 1/(nVt), about 25 to 30 V⁻¹ at room temperature. Best efficiency, lowest fT, biggest area.
• Strong inversion: 2/Vov. With Vov = 0.2 V that is 10 V⁻¹.
• Moderate inversion (roughly 10 to 20 V⁻¹): a common compromise between power, speed and area for low-power analog.

The gm/Id–Vgs curve in this tool shows the square-law hyperbola dotted and the weak-inversion ceiling as a flat line.

fT. The frequency where the current gain drops to 1: fT = gm / (2π(Cgs + Cgd + Cgb)). In strong inversion this is about 3µVov / (4πL²), so speed grows with overdrive and falls fast with L. In saturation Cgs ≈ ⅔·W·L·Cox plus overlap.

Thermal noise. Drain-current noise is 4kTγ·gm with γ ≈ ⅔ in strong inversion and about ½ in weak inversion. Referred to the gate it is 4kTγ/gm, so a larger gm lowers input noise. Short channels often show higher γ.

Flicker (1/f) noise falls with gate area W·L·Cox and mismatch falls as 1/√(WL). Neither is modeled here, but both push analog devices toward large area.

• Common-source amplifier: Av = −gm·(ro ∥ RL). With an ideal current-source load it tops out at −gm·ro.
• Current mirror: use long L for a high ro and low λ mismatch, moderate Vov (150 to 250 mV) for matching, and equal Vds on both sides.
• Differential pair: each side carries half the tail current, so gm = 2·(Itail/2)/Vov = Itail/Vov.
• Switch: Ron ≈ 1 / (k'(W/L)·(Vgs − Vth)) for small Vds. Note that Vth rises with Vsb as the source voltage rises.
• Headroom: keep Vds at least 50 to 100 mV above Vov so process and temperature shifts do not push the device into triode.

This is a first-pass hand-calculation model: square law with channel-length modulation, body effect and a smooth weak-inversion join. It does not include velocity saturation (short channels behave closer to Id ∝ Vov than Vov²), DIBL, mobility degradation, gate leakage, parasitic source and drain resistance, temperature dependence of Vth and µ, or layout effects. Use it for sizing and sanity checks, then verify with SPICE and the foundry's BSIM models.