Internal nodes approximation

Figure 3.2: Mos chain with proper numbering

Let be $N$ the number of n-MOSFETs in the n-chain and $P$ as the number of p-MOSFETs in the p-chain, and let's label the transistor in the chain from 1 to $N$ or from 1 to $P$ (figure 3.2). Let's assume that the label 1 comes with the driving transistor (i.e. the n-MOSFET with source connected to $V_{SS}$ as the p-MOSFET with source connected to $V_{DD}$), as in figure 3.2. This hypothesis is only for the develop of the discussion; in our model any (but only one) transistor can be a driving transistor, that is a transistor with a changing gate voltage.

Notation 4.1   In the following equations the superscript index refers to the node number (with the variable $i$ always for the n-MOSFETs and $j$ always for the p-MOSFETs), and the small-letter subscript indexes $n$ and $p$ refer, respectively, to n-MOSFETs and p-MOSFETs, both for the voltage variables or for the time variables; for the voltage variables the capital subscript indexes $G$ and $D$ refer to the drain node and the gate node, while the small-letter index $d$ refers to the initial conditions of the drain nodes.

So, for example, $V_{G_n}^i(t)$ is the gate voltage at the node $i$ for the n-MOSFETs (function of time), and $V_{d_p}^j$ is the initial condition of the drain voltage at node $j$ for the p-MOSFETs.

The wave forms of the voltage are shown in figure 3.4 and figure 3.5, with the hypothesis $t_{0_n}^1 = t_{0_n}^2 = \dots = t_{0_n}^N$ and $t_{0_p}^1 = t_{0_p}^2 =\dots = t_{0_p}^P$; that is because we suppose the start of conduction of all the MOSFETs in a chain contemporary.

We can write, referring to figures 3.4, 3.5:

Figure 3.3: The $i$-th and $i+1$-th MOSFETs with node voltages

Figure 3.4: Voltages wave forms in the n-mos chain

Figure 3.5: Voltages wave forms in the p-mos chain

Figure 3.6: Drain-source ($V_{DS}$) and gate-source ($V_{GS}$) voltages of th $i$-th n-MOS

It is also possible to define $\tau_{i_{n,p}}^i=\tau_{o_{n,p}}^{i-1}$ and the source voltage $V_s^i=V_d^{i+1}$, as shown in figure 3.3 for the $i$-th n-MOS. The same is valid for the p-MOSFETs.

The starting level $V_{d_{n,p}}$ are determined with a static analysis, described in §3.1.3.