Delay formula obtained by the Elmore model

The delay function obtainable by the Elmore's model (§2.1, page ) is a continuous function. Referring to figure 2.1 (page ), the delay of a single MOS is:

$$t_{d_i}=R_0C_{S_i}+(R_0+R_{d_i})C_{D_i}+(R_0+R_{d_i}+R_L)C_L$$

The drain and source capacitance, and the dynamic resistance of a MOS are function of the MOS width $W$:

$$C_{D_i} =C_jW_i$$

$$C_{S_i} =C_jW_i$$

$$R_{d_i} =\frac{R_j}{W_i}$$

where $C_j$ and $R_j$, are, respectively, the capacitance for unit length and the resistance for unit length. The delay function of the MOS width become:

$$t_{d_i}=R_0C_jW_i+\biggl(R_0+\frac{R_j}{W_i}\biggr)C_jW_i+\biggl(R_0+ \frac{R_j}{W_i}+R_L\biggr)C_L.$$

Separating the terms containing the width $W_j$ from the terms that are independent from $W_j$ we obtain:

$$t_{d_i}=2R_0C_jW_i+\frac{R_j}{W_i}C_L +R_jC_j+(R_0+R_L)C_L.$$

Summing the delay of all the MOS in a conducting path we obtain the total delay of this path:

$$t_d=\sum_{i}t_{d_i}=\sum_{i}\left(AW_i+\frac{B}{W_i} +C\right)$$

where $A,B,C$ are all independent from $W_i$.

The delay of a critical path is the sum¹⁶ of the delays of all the conducting path.

Figure 5.5: Elmore delay: convex function

As long as $A,B$ are not zero, the delay $t_d$ is a convex function (definition 4.12, page ) as in figure 5.5. If the term $A$ is zero, instead, then the delay is a monotonic decreasing function (figure 5.6).

Figure 5.6: Elmore delay: monotonic function

Note that the term $A$ is zero, practically, only if the the resistance $R_0$ is zero, that is the MOSFET chain is driven by an ideal voltage source.