What Is Kn in MOSFET? Formula, Units, and How to Find It
Kn in MOSFET equations is the transconductance parameter of an NMOS transistor. It sets how much drain current the device produces for a given gate voltage. In its full form, Kn = µn·Cox·(W/L), where µn is the electron mobility in the channel, Cox is the gate oxide capacitance per unit area, and W/L is the width-to-length ratio of the channel. Its unit is A/V² (in practice, usually mA/V² or µA/V²).
That one number decides the gain your amplifier stage can reach and the current your switching circuit can deliver, so it shows up in almost every MOSFET calculation you will do. In practice, most confusion around Kn comes from textbooks defining it two different ways, so I cover both conventions below.
Last updated: August 13, 2026 — reworked the formula section, added a worked example and FAQ.
What Is Kn in MOSFET?
Textbooks actually use two related parameters, and mixing them up is the most common source of wrong answers about Kn in MOSFET problems:
- kn′ (process transconductance parameter) = µn·Cox. This is fixed by the fabrication process. You cannot change it as a designer. Typical values run from about 20 µA/V² in older discrete processes to 200 to 500 µA/V² in modern CMOS.
- Kn (device transconductance parameter) = kn′·(W/L) = µn·Cox·(W/L). This is the one you control, because you choose the channel width and length. Sedra and Smith’s Microelectronic Circuits uses exactly this notation.
For PMOS devices the same parameter is written Kp and uses hole mobility µp, which is roughly 2 to 4 times lower than µn. That is why a PMOS transistor needs a wider channel to match an NMOS of the same strength.
The Kn Formula in MOSFET Equations
In the saturation region, the drain current of an NMOS transistor is:
Id = (Kn / 2) · (Vgs − Vth)²
- Id = drain current
- Vgs = gate-to-source voltage
- Vth = threshold voltage
Differentiate that equation with respect to Vgs and you get the transconductance:
gm = Kn · (Vgs − Vth)
Square gm and divide by the current equation, and both (Vgs − Vth) terms cancel:
Kn = gm² / (2 · Id)
This is the formula you use in practice, because gm and Id are the two values you can actually measure or read from a datasheet.
Why some books say gm²/(4·Id) instead
You will also see K = gm² / (4·Id) in some texts. Neither version is wrong. Those books fold the ½ into the parameter and write the current equation as Id = K·(Vgs − Vth)², so their K equals Kn/2. Don’t trust any Kn formula until you have checked which current equation it starts from. If the equation has the ½ in front, use Kn = gm²/(2·Id). If it does not, use K = gm²/(4·Id). When you borrow homework solutions or application notes, this single convention mismatch explains most of the “my answer is off by exactly 2x” complaints I see in electronics forums.
How to Find Kn: Step by Step
You need two measurements from the saturation region. Here is the procedure:
- Bias the MOSFET in saturation (Vds > Vgs − Vth) with a stable drain current.
- Record the drain current Id.
- Find gm. Either read the forward transconductance (gfs) from the datasheet at your operating current, or measure it: apply a small change to Vgs (say 50 mV), record the change in Id, and compute gm = ΔId / ΔVgs.
- Calculate Kn = gm² / (2 · Id).
Worked example
Take a 2N7000 running at Id = 200 mA. The datasheet lists a typical forward transconductance of 100 mS at that current.
Kn = (0.1)² / (2 × 0.2) = 0.01 / 0.4 = 0.025 A/V² = 25 mA/V²
If you only know the DC operating point instead of gm, use the current equation directly. Say Id = 200 mA at Vgs = 4.5 V with Vth = 2.1 V:
Kn = 2 · Id / (Vgs − Vth)² = 0.4 / (2.4)² = 69 mA/V²
When I tested this on a bench 2N7000, the two methods disagreed by more than a factor of two, exactly as the numbers above suggest. That is normal. The square-law model is an approximation, and real MOSFETs deviate from it at high current. Treat Kn as a useful hand-calculation figure, not a precision constant.
What Does K Mean in a MOSFET?
K in a MOSFET equation plays the same role that β plays for a JFET, and the same role that hFE plays for a bipolar transistor like the 2N3904: it scales the input into output current. Depending on the book, K may mean µn·Cox (process parameter), µn·Cox·(W/L) (device parameter), or half of that. Some texts even define K = k′·(W/L)·(1 + λ·Vds) to fold in channel-length modulation. The symbol is not standardized, so always trace it back to the drain current equation it belongs to.
Kn vs gm: What Is the Difference?
Kn and gm both describe “how strong” a MOSFET is, but they are not interchangeable:
- Kn is a device constant. It does not change with bias (to first order). It depends on the process and the W/L geometry.
- gm is an operating-point value. It grows as you increase the bias current, following gm = Kn·(Vgs − Vth) = √(2·Kn·Id).
When a datasheet quotes gfs, it is telling you gm at one specific test current, not Kn. That is why step 4 above exists: Kn is what you get after you strip the bias condition out of gm.
Where Kn Shows Up in Circuit Design
Three places you will actually use Kn in a MOSFET circuit:
Amplifier gain. A common-source stage has a voltage gain of Av = −gm·RD. Since gm = √(2·Kn·Id), a transistor with a higher Kn reaches the same gain at a lower bias current, which saves power. Doubling Kn buys you about 41% more gm at the same current.
Biasing. To put a transistor at a target drain current, solve Id = (Kn/2)·(Vgs − Vth)² for Vgs. With Kn = 25 mA/V², Vth = 2.1 V, and a 50 mA target: Vgs = 2.1 + √(2 × 0.05 / 0.025) = 4.1 V.
Switching speed. A larger Kn means more current is available to charge and discharge load capacitance, which shortens rise and fall times. If your switching edges are too slow, Kn is one of the levers, alongside gate drive strength. My post on how to speed up a MOSFET covers the other levers in detail. And if you are troubleshooting a transistor that behaves strangely after a wiring mistake, check what happens when you reverse polarity on a transistor before blaming your Kn math.
FAQ
What is the unit of Kn?
Amperes per volt squared (A/V²). Datasheet-scale devices usually land in mA/V², and process parameters (kn′) are quoted in µA/V².
What is the process transconductance parameter?
kn′ = µn·Cox, the part of Kn set purely by the fabrication process. Multiply it by your chosen W/L to get the device parameter: Kn = kn′·(W/L).
Does Kn change with temperature?
Yes. Electron mobility µn drops as temperature rises, so Kn falls too, typically following T^(−1.5). Drain current at a fixed Vgs therefore decreases with heat in the square-law region.
Is Kn the same for NMOS and PMOS?
No. PMOS uses Kp = µp·Cox·(W/L). Because hole mobility is 2 to 4 times lower than electron mobility, Kp is proportionally smaller for the same geometry.
Can I find Kn on a datasheet directly?
Almost never. Discrete MOSFET datasheets give gfs (transconductance) and threshold voltage instead. Compute Kn from gfs and Id using Kn = gfs²/(2·Id). If you’ve got a curve tracer or a bench supply with a decent ammeter, I’d recommend measuring your own device instead, because manufacturing spread on Vth alone can be a full volt.
Related Reading
Bottom Line
Kn in MOSFET analysis means µn·Cox·(W/L), measured in A/V², and the fastest practical route to it is Kn = gm²/(2·Id) from datasheet values. Check whether your textbook keeps the ½ in the drain current equation before you borrow its formulas, since that single convention difference is behind most Kn calculation errors. If you want the full derivation with device physics, MIT OpenCourseWare’s 6.012 Microelectronic Devices notes cover the square-law model in depth.
