Library / Circuits, electronics and noise / ELN-24
Circuits, electronics and noise
MOSFET small-signal output resistance with channel-length modulation
Small-signal drain resistance of a saturated MOSFET with Id = Id0(1 + λVds), where Id is the actual biased current.
- Ref
- ELN-24
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- ro [Ω]
Channel-length modulation makes the saturated current rise slowly with Vds: Id = Id0(1 + λVds). The small-signal output resistance is the inverse slope, ro = (1 + λVds)/(λId) using the actual biased current, and roughly 1/(λId). It limits the best gain of one transistor to gm ro. A longer channel gives a smaller λ and so a larger ro.
Small-signal drain resistance of a saturated MOSFET with Id = Id0(1 + λVds), where Id is the actual biased current. It limits the intrinsic gain of the device. Vgs is held fixed.
Calculator
- Approximate ro, 1/(λId)11,360 Ω
- Early-type voltage, 1/λ50 V
- Current at Vds = 0, Id/(1 + λVds)0.004 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.
- With λ = 0.02 /V and 0.0044 A the output resistance is 12,500 Ω (12.5 kΩ).
- With the same current, a device with half the λ has twice the ro and, at a given gm, twice the intrinsic gain.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Ω) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ID Biased drain current | 1e-5 to 0.1 A | 550 | 5,500,000 | 5,499,000 | +10% | −9.09 % |
| λ Channel-length modulation parameter | 0.002 to 0.2 1/V | 2,273 | 114,800 | 112,500 | +10% | −8.26 % |
| Vds Drain–source voltage | 0.1 to 20 V | 11,390 | 15,910 | 4,523 | +10% | +0.909 % |
Sweep of Channel-length modulation parameter
| Channel-length modulation parameter (1/V) | Output resistance (Ω) |
|---|---|
| 0.002 | 114,800 |
| 0.02675 | 9,633 |
| 0.0515 | 5,549 |
| 0.07625 | 4,117 |
| 0.101 | 3,387 |
| 0.1258 | 2,944 |
| 0.1505 | 2,646 |
| 0.1753 | 2,433 |
| 0.2 | 2,273 |
Sweep one variable
Variables and how each one contributes
Output resistance
What the equation gives you. Everything else feeds it.
Channel-length modulation parameter
The channel-length modulation parameter λ, in 1/V. A larger λ means a steeper current slope, so ro falls. Raising it by 10 % (to 0.022) moves the result by −8.26 %, from 12,500 to 11,470 Ω.
Drain–source voltage
The drain–source voltage, in V. A larger Vds raises ro a little, through the factor 1 + λVds. Raising it by 10 % (to 5.5) moves the result by +0.909 %, from 12,500 to 12,610 Ω.
Biased drain current
The biased drain current, in A. More current means a steeper slope for the same λ, so ro falls in inverse proportion. Raising it by 10 % (to 0.00484) moves the result by −9.09 %, from 12,500 to 11,360 Ω.
Worked example with the values above
- Numerator: 1 + λVds = 1 + 0.02 × 5 = 1.1.
- Denominator: λId = 0.02 × 0.0044 = 8.8×10−5 S.
- ro = 1.1 / 8.8×10−5 = 12,500 Ω.
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