# W/L Ratio of a MOSFET

In the schematic builder that calculates voltages/currents/transient analysis that we have been provided for the electrical engineering course, MOSFETs have the parameter "W/L Ratio".

I figured that it was probably Width:Length, but how does that affect the constant K, and the voltage threshold? I realise that there is not enough information for absolute values, but really I'm looking for a formula that relates the variables.

So my question is, what is the relationship between W/L, K, and the threshold voltage of the MOSFET?

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You're going to need to know a lot more about a transistor than its geometry to characterise it. Also, we don't know which of the many constants that can describe a transistor is the one you call "K". You should be able see for yourself this question is too vague with too much missing information. –  Olin Lathrop Apr 14 '12 at 12:18
There is nothing wrong with his question. His reference to K may be a little vague that is all. –  Konsalik Apr 14 '12 at 12:39
I updated the question to make my intentions clearer –  Kian Apr 14 '12 at 12:59
Ah, you're talking about designing the transistor itself, not building circuits out of them. The question still needs to be clearer though. –  JustJeff Apr 14 '12 at 13:04

The K constant you refer to (more specifically $K_n$) is called the conduction parameter of the n-channel device.

$K_n$ is given by:

$$K_n = \frac{k_n'}{2}\cdot\frac{W}{L}$$

Where

$$k_n' = \mu_nC_{ox}$$

$\mu_n$ is the mobility of the electrons in the inversion layer and $C_{ox}$ is the oxide capacitance per unit area. According to Neamen the $k_n'$ parameter is called the "process conduction parameter" and is considered to be a constant for a fabrication technology. Therefore the ratio $\frac{W}{L}$ is the transistor design variable.

Neamen goes on to say that the design variable is used to design MOSFETS to produce specific current-voltage characteristics in MOSFET circuits.

EDIT:

Yes w refers to width and L to length. It relates to the geometry of the semiconductor.

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What about VT, the threshold voltage? –  Kian Apr 14 '12 at 13:01
As I understand $V_T$ has little relation to the conductance parameter and is more reliant on the doping of the substrate however there should be some correlation between the two. In my course $V_T$ was always given as a separate variable. Here is some more information: en.wikipedia.org/wiki/Threshold_voltage –  Konsalik Apr 14 '12 at 13:12

Just a few additions to Konsalik's answer:

$V_T$ (threshold voltage) is not affected by the W/L ratio of the transistor, as it depends on other parameters, such as the gate insulator thickness and dielectric constant; it also depends on Source-Bulk voltage, in what is called Body effect:

$$V_{TB} = V_{T_0} + \gamma (\sqrt{V_{SB} + 2 \phi_{B} - \sqrt{2\phi_{B}}})$$

Just as note: usually in integrated circuits, L is limited by the technology (as small as possible) and the conductivity is increased with bigger W; in this way, though, also the gate capacitance is increased, so often it doesn't bring any advantage.

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This is college stuff, too long ago for me. I never needed this in my professional career. –  stevenvh Apr 14 '12 at 15:51
@stevenvh true, as it's also true that K is useless, since no datasheet gives that parameter :) and I'm not sure about W and L. But it makes sense in IC design –  clabacchio Apr 14 '12 at 16:41
k' isn't given in IC design either - you usually end up deriving it from the level 3 SPICE models. Also, VT does/can change with different W/L ratios, but this is often ignored for simplicity. –  W5VO Apr 14 '12 at 18:03