It is known that a MOS transistor with the source-body voltage different from zero has the threshold voltage modified by the body effect, that is if , with the source-body voltage (let's remember that for a n-MOSFET and for a p-MOSFET ), then . The initial conditions of the chain nodes are set by the initial condition on the output. So if the output node is discharging, then one (and only one) n-MOSFET is switching from off to on. It means that all the other MOSFETs are already on, and while the starting voltage of the output node is , all the internal nodes have as a starting voltage .
With the notations of previous paragraphs, the N-th (topmost) n-MOS transistor has , with source potential and the threshold voltage modified by the body effect. All the internal transistors have , while the first one has and .
The threshold voltage variation as a function of is given by:
The source potential of the top transistor is
We can find an analogue equation for p-MOSFETs: knowing that, for the p-MOS chain depicted in figure 3.7(b), the drain potential of transistor is , while ; for the middle transistors ; and for the first (top-MOSt) transistor and .
The threshold voltage variation function of again is:
The threshold variation is approximated in the model by a linear approximation given by:
In figure 3.8(a) and 3.8(b) the actual threshold variation (of a n-MOS transistor and a p-MOS transistor) when a voltage is applied is compared with the linear approximation used in our model, for a technology.
The max error due to the linear approximation is limited to .