What do you mean by "input characteristics"?

Textbooks and datasheets describe the behavior of MOSFETs using two graphs:

**Output characteristics**: \$I_D\$ versus \$V_{DS}\$ with \$V_{GS}\$ as parameter.

**Transfer characteristic**: \$I_{D}\$ versus \$V_{GS}\$ at a given fixed \$V_{DS}\$ value (this latter is chosen so that the MOSFET is in saturation region).

There is no "input characteristic" (such as the \$I_B\$ versus \$V_{BE}\$ curve of a BJT) because the other input quantity besides \$V_{GS}\$, namely \$I_G\$, is virtually zero at DC (and all these curves assume DC operations). Therefore it wouldn't make much sense to plot \$I_G\$ versus \$V_{GS}\$, unless you wanted to analyze leakage gate current, but I assume you are not interested in that.

So it is clear (also by a comment of yours) that by *input characteristic* you mean the *transfer characteristic* (TC). Note that the TC is plotted with a fixed drain-source voltage that guarantees that the MOSFET is in saturation for each \$V_{GS}\$ value on the horizontal axis. This is done because the TC is useful when the MOSFET is in saturation, i.e. when the output current depends solely on the input voltage (not considering "Early effect"), for example when you want to use the MOSFET as an amplifier and you need to draw a load line to design its bias circuit.

If you plot the TC for different values of \$V_{DS}\$ you get a family of TC curves. For example consider this circuit simulation with LTspice:

Plotting the TC for different \$V_{DS}\$ values you get:

As you can see, the more you increase \$V_{DS}\$ the more the curve resembles a parabola, as you would expect for the TC in saturation. Notice that this part shows a threshold voltage \$V_{th} \approx 4V\$.

Let's consider what happens if \$V_{DS}\$ is not big enough to drive the MOSFET in saturation for every \$V_{GS}\$ value, like in the lowest blue curve (Note: to present a more revealing plot I selected the curve corresponding to \$V_{DS} = 2V\$, whereas the lowest blue curve above corresponds to \$V_{DS} = 1V\$):

As you can see, in saturation region you get a quadratic curve, whereas in triode region you get a linear curve. Everything as expected, except that real devices don't have an abrupt change between the two regions and that the linearity of the triode region is not perfect because of the device not being ideal (SPICE models usually take into account these effects).

If you see in your simulation an abrupt departure from this behavior it could be that you tried plotting the curves outside the range of the voltages/currents admissible for your device. Notice that I limited the first plot to max 14A/20V which are the absolute maximum ratings for the device I chose. If you don't keep this in mind you will destroy the device (in real life) or get odd results (in simulations).

**EDIT** (in response to a comment and a question edit)

You ask why the "perfectly" linear curve for \$I_D\$ versus \$V_{GS}\$ in ohmic region is not exploited. Here is some insight:

Why do you need a linear characteristic between input (\$V_{GS}\$) and output (\$I_D\$)? Usually to use the device as a (linear) amplifier. But what are the conditions that allows to have that linearity? \$V_{DS}\$ must be held constant. Therefore to make an amplifier this way you have to insert a load in the output circuit *and* still keep \$V_{DS}\$ constant. You can understand that such a load cannot be a simple resistor (which is the simplest kind of load). Therefore you need a much more complex circuit (with other active devices).

On the other side, you can use the same MOSFET biased in saturation and get a decent linear amplifier: even if the behavior *of the device* is not intrinsically linear, but quadratic, there are linearization techniques (e.g. employ simple feedback schemes, like a resistor in series with the source terminal) that allow *the overall amplifier* to become more linear.

## Best Answer

If you want to apply that formula to any real, discrete, device, then it is not going to work, even if the datasheet mentioned K.

The fact is, that real discrete device sport a wild variation of parameters, in particular Vth. Therefore, even if you knew a fairly precise value of K, it wouldn't matter much, because in the inverse formula you would have Vth as a parameter.

Consequently, any value of Vgs you would get would be highly inaccurate, unless you

measuredthe Vth of the specific device.Of course if you can measure quantities on a real specimen, you would be able also to measure Vgs, so the whole point in deriving that formula might be moot. In fact, with some patience and a relatively simple circuit, you could characterize the device yourself, by measuring Id vs Vds at different Vgs levels, and plotting the output curves of the device.

To be more specific, here is an excerpt from a datasheet of a jellybean enhancement MOSFET (2N7000G; emphasis mine):

As you can see, Vth is specified with very loose tolerance, therefore you could expect the same tolerance on the values of Vgs obtained by inverting that formula.