## Gm/ID versus IC

According to the EKV model, the inversion coefficient, IC, is defined by the ratio of drain current to a specified drain current, IDSspec, where VGS-VT = 2n*kT/q [1]. In order to know the IC, I have to set up a separate testbench to simulate the IDSspec. This is not efficient especially at the initial phase of design which might encounter many changes.

On the other hand, the annotation of the DC operating point provided by Cadence is really helpful. Now we can even have gm/ID annotated beside the transistor (it is called ‘gmoverid’ in the simulator). Hence, a curve showing the gm/ID-IC relationship will be informative, and this Mr.Sansen has [1]! It is plotted in Fig.1.

Fig.1 Gm/ID*nUT versus IC

In order to derive the relationship, we first need to recall the following equations:

$e^{\sqrt{IC}} = e^v+1$       $IC = \frac{I_{DS}}{I_{DSspec}}$         $v = \frac{V_{GS}-V_T}{2nU_T}$

Based on the above equations, the gm/ID can be derived:

$\frac{g_m}{I_{D}} = \frac{\partial{I_{DS}}}{\partial{V_{GS}}} \frac{1}{I_{DS}} = \frac{\partial{IC} \times I_{DSspec}}{\partial v \times 2nU_T} \frac{1}{I_{DS}} = \frac{\partial{IC}}{\partial v} \frac{1}{2nU_T \times IC} = \frac{1-e^{-\sqrt{IC}}}{nU_T \sqrt{IC}}$

Now we may have a rough idea of IC based on the annotated gm/ID (assuming nUT is about 35 mV).

gm/ID       25          18          9

IC            0.1          1           10

Reference

[1] W. Sansen, “Minimum power in analog amplifying blocks – presenting a design procedure ”, IEEE Solid-State Circuits Magazine, fall 2015.

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