Parameter Estimation for an N-Channel Enhancement MOSFET - Maple Help

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Parameter Estimation for an N-Channel Enhancement MOSFET

The following data of drain current versus gate-to-source voltage has been obtained for an n-channel enhancement mode MOSFET. This worksheet uses this data to determine the least squares curve fit estimate of the SPICE parameters KP (the transconductance parameter) and VTO (zero-bias threshold voltage). It assumes that the device is operating in its saturation mode and that the channel length modulation factor may be neglected.

If the device is operating in the saturation region and the channel length modulation factor can be neglected, the drain current, ID, is related to the gate-to-source voltage, VGS, according to the following equation:

Where K = K'/2, and K' is the SPICE parameter KP.

 > $\mathrm{restart}:\mathrm{with}\left(\mathrm{plots}\right):\mathrm{with}\left(\mathrm{Statistics}\right):$

Experimental Data

 >
 > $\mathrm{ID_exp}≔\left[0.306,1.393,3.141,6.618,11.242,15.687,20.93,25.389\right]:$

Let's plot the data.

 >
 > $\mathrm{display}\left(\mathrm{p1}\right)$

Parameter Estimation via Least-Squares Curve Fitting

 > $\mathrm{ID}≔\mathrm{K}\cdot {\left(\mathrm{VGS}-\mathrm{VTO}\right)}^{2}$
 ${\mathrm{ID}}{:=}{K}{}{\left({\mathrm{VGS}}{-}{\mathrm{VTO}}\right)}^{{2}}$ (2.1)

For a basic fit:

 > $\mathrm{eq}≔\mathrm{NonlinearFit}\left(\mathrm{ID},\mathrm{VGS_exp},\mathrm{ID_exp},\mathrm{VGS}\right)$
 ${\mathrm{eq}}{:=}{1.58405874756309}{}{\left({\mathrm{VGS}}{-}{1.42182812976269}\right)}^{{2}}$ (2.2)

You can also customize the output:

 > $\mathrm{res}≔\mathrm{NonlinearFit}\left(\mathrm{ID},\mathrm{VGS_exp},\mathrm{ID_exp},\mathrm{VGS},\mathrm{output}=\mathrm{parametervalues}\right)$
 ${\mathrm{res}}{:=}\left[{K}{=}{1.58405874756309}{,}{\mathrm{VTO}}{=}{1.42182812976269}\right]$ (2.3)
 > $\mathrm{NonlinearFit}\left(\mathrm{ID},\mathrm{VGS_exp},\mathrm{ID_exp},\mathrm{VGS},\mathrm{output}=\mathrm{residualsumofsquares}\right)$
 ${3.250638736}$ (2.4)

Visualizing the Solution

Let's plot the best fit equation against the experimental data.

 > $\mathrm{p2}≔\mathrm{plot}\left(\mathrm{eq},\mathrm{VGS}=\mathrm{min}\left(\mathrm{VGS_exp}\right)..\mathrm{max}\left(\mathrm{VGS_exp}\right)\right):$
 > $\mathrm{display}\left(\mathrm{p1},\mathrm{p2}\right)$

Fitting a Procedure

You can also fit a procedure (that includes, for example, conditional statements or piecewise equations). The first parameter(s) should be the independent variable(s), while the others should be the empirical parameters

 >
 > $\mathrm{NonlinearFit}\left(\mathrm{f},\mathrm{VGS_exp},\mathrm{ID_exp}\right)$
 $\left[\begin{array}{c}{1.58405874756309}\\ {1.42182812976269}\end{array}\right]$ (4.1)

Minimizing the SSE - the hard way

 >
 ${\mathrm{ID_f}}{:=}\left({\mathrm{VGS}}{,}{K}{,}{\mathrm{VTO}}\right){→}{K}{}{\left({\mathrm{VGS}}{-}{\mathrm{VTO}}\right)}^{{2}}$ (5.1)
 > 
 >
 > $\mathrm{Optimization}:-\mathrm{Minimize}\left(\mathrm{sse}\left(K,\mathrm{VTO}\right)\right)$
 $\left[{3.25063873633627010}{,}\left[{K}{=}{1.58405878529888}{,}{\mathrm{VTO}}{=}{1.42182800867805}\right]\right]$ (5.2)
 >