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Circuits, electronics and noise
NMOS drain current in the triode region
Drain current of a long-channel NMOS transistor in the triode (linear) region, with k = μₙCox W/L in A/V² and no factor of one half.
- Ref
- ELN-21
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- ID [A]
In the triode (linear) region, with Vds ≤ Vgs − Vth, the drain current is Id = k[(Vgs − Vth)Vds − Vds²/2], where k = μₙCox W/L is in A/V². There is no factor of ½ in front of k here, unlike the saturation formula. For small Vds the device acts as a resistor of about 1/[k(Vgs − Vth)]. At Vds = Vgs − Vth it joins the saturation formula.
Drain current of a long-channel NMOS transistor in the triode (linear) region, with k = μₙCox W/L in A/V² and no factor of one half. It needs Vgs above Vth and 0 ≤ Vds ≤ Vgs − Vth.
Calculator
- Overdrive voltage, Vgs − Vth2 V
- Small-signal on-resistance, 1/[k(Vgs − Vth)]250 Ω
- Current at the edge of saturation, ½k(Vgs − Vth)²0.004 A
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.
- At Vds = 0.5 V the drain current is 0.00175 A, and the channel acts like 285.7 Ω; for small Vds it would be 250 Ω.
- The formula stops at Vds = 2 V; beyond that the current stays at 0.004 A.
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 | 8.75e-6 | 0.0875 | 0.08749 | +10% | +10 % |
| Vds Drain–source voltage | 0 to 2 V | 0 | 0.004 | 0.004 | +10% | +8.43 % |
| Vgs Gate–source voltage | 1.6 to 5 V | 3.5e-4 | 0.00375 | 0.0034 | +10% | +17.1 % |
| Vth Threshold voltage | 0.3 to 2.4 V | 3.5e-4 | 0.00245 | 0.0021 | +10% | −5.71 % |
Sweep of Drain–source voltage
| Drain–source voltage (V) | Drain current (A) |
|---|---|
| 0 | 0 |
| 0.25 | 9.375e-4 |
| 0.5 | 0.00175 |
| 0.75 | 0.002437 |
| 1 | 0.003 |
| 1.25 | 0.003438 |
| 1.5 | 0.00375 |
| 1.75 | 0.003938 |
| 2 | 0.004 |
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 0.00175 to 0.001925 A.
Gate–source voltage
The gate–source voltage, in V. A larger Vgs raises the overdrive and so the drain current. Raising it by 10 % (to 3.3) moves the result by +17.1 %, from 0.00175 to 0.00205 A.
Threshold voltage
The threshold voltage, in V. A larger Vth reduces the overdrive and so the drain current. Raising it by 10 % (to 1.1) moves the result by −5.71 %, from 0.00175 to 0.00165 A.
Drain–source voltage
The drain–source voltage, in V; it must not exceed Vgs − Vth. A larger Vds raises the current, but with falling returns as it nears the saturation edge. Raising it by 10 % (to 0.55) moves the result by +8.43 %, from 0.00175 to 0.001898 A.
Worked example with the values above
- Overdrive: Vgs − Vth = 3 − 1 = 2 V.
- Bracket: (Vgs − Vth)Vds − Vds²/2 = 1 − 0.125 = 0.875 V².
- Id = 0.002 × 0.875 = 0.00175 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