Library / Circuits, electronics and noise / ELN-22
Circuits, electronics and noise
NMOS drain current in saturation
Drain current of a long-channel NMOS transistor in saturation, where Vds is at least Vgs − Vth.
- Ref
- ELN-22
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- ID [A]
In saturation (Vds ≥ Vgs − Vth) the drain current of a long-channel MOSFET depends on the gate overdrive only: Id = ½k(Vgs − Vth)². Doubling the overdrive quadruples the current. This is the basis of the current mirror and of the square-law amplifier model. The drain voltage has no effect, apart from the small slope covered by channel-length modulation.
Drain current of a long-channel NMOS transistor in saturation, where Vds is at least Vgs − Vth. The parameter k = μₙCox W/L is in A/V². Channel-length modulation and body effect are ignored.
Calculator
- Overdrive voltage, Vgs − Vth2 V
- Transconductance, k(Vgs − Vth)0.004 S
- Smallest Vds that keeps it in saturation2 V
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With an overdrive of 2 V the device carries 0.004 A, and its transconductance is 0.004 S.
- Raising Vgs by 10 % of the overdrive raises the current by about 21 %, because of the square.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (A) | Local step | Change in result |
|---|---|---|---|---|---|---|
| k Transconductance parameter μₙCox W/L | 1e-5 to 0.1 A/V² | 2e-5 | 0.2 | 0.2 | +10% | +10 % |
| Vgs Gate–source voltage | 1.2 to 5 V | 4e-5 | 0.016 | 0.01596 | +10% | +32.3 % |
| Vth Threshold voltage | 0.3 to 2.8 V | 4e-5 | 0.00729 | 0.00725 | +10% | −9.75 % |
Sweep of Gate–source voltage
| Gate–source voltage (V) | Drain current (A) |
|---|---|
| 1.2 | 4e-5 |
| 1.675 | 4.556e-4 |
| 2.15 | 0.001322 |
| 2.625 | 0.002641 |
| 3.1 | 0.00441 |
| 3.575 | 0.006631 |
| 4.05 | 0.009302 |
| 4.525 | 0.01243 |
| 5 | 0.016 |
Sweep one variable
Variables and how each one contributes
Drain current
What the equation gives you. Everything else feeds it.
Transconductance parameter μₙCox W/L
The transconductance parameter k = μₙCox W/L, in A/V². Scales the drain current in direct proportion. Raising it by 10 % (to 0.0022) moves the result by +10 %, from 0.004 to 0.0044 A.
Gate–source voltage
The gate–source voltage, in V. Raises the overdrive and so the current, in a square-law way. Raising it by 10 % (to 3.3) moves the result by +32.3 %, from 0.004 to 0.00529 A.
Threshold voltage
The threshold voltage, in V. A larger Vth lowers the overdrive and so the current, in a square-law way. Raising it by 10 % (to 1.1) moves the result by −9.75 %, from 0.004 to 0.00361 A.
Worked example with the values above
- Overdrive: Vgs − Vth = 3 − 1 = 2 V.
- Square: 4 V².
- Id = ½ × 0.002 × 4 = 0.004 A.
Assumptions and limits
- A long-channel NMOS device in strong inversion and the square-law model.
- Triode and saturation formulas used only in their stated regions, and body effect ignored.
Common mistakes
- Using the triode formula when Vds is above Vgs − Vth, or the saturation formula when it is below.
- Forgetting that k here has no factor of one half in the triode formula.
- Applying the saturation gm formula to a device in weak inversion or velocity saturation.
Statistics and uncertainty
Real transistors depart from the square-law model, especially at short channel lengths, where velocity saturation changes the current dependence. The threshold voltage varies from part to part, so results with small overdrive are very sensitive to it.
Where it is used
- Bias and gain estimates for MOSFET amplifiers, switches and current sources.
- Quick checks of simulation results by hand.
Maths you need
- MOSFET regions: cut-off, triode and saturation
- Overdrive voltage: Vov = Vgs − Vth
- Small-signal model: gm and ro at a bias point