Step 1: Sketching the 3-input NAND Gate with Transistor Widths

A 3-input NAND gate has three NMOS transistors in series and three PMOS transistors in parallel. To match the effective rise and fall resistances with that of a unit inverter, we need to adjust the transistor widths.

• NMOS Network: In a NAND gate, the NMOS transistors are in series. For three transistors in series, the resistance would add up. To achieve the same resistance $R$ as a unit inverter, each NMOS transistor's width $W_n$ should be increased by a factor of 3. Hence, $W_{n, \text{NAND}} = 3W_n$.

• PMOS Network: In a NAND gate, the PMOS transistors are in parallel. To achieve the same resistance $R$, each PMOS transistor's width $W_p$ can be kept the same as in the unit inverter because parallel transistors reduce the effective resistance. Hence, $W_{p, \text{NAND}} = W_p$.

Step 2: Estimating the Worst-Case Rising and Falling Delay

Worst-Case Falling Delay:

• Falling Delay is dominated by the NMOS transistors, which are in series.

• With three NMOS transistors in series, the effective resistance is approximately $R$.

• The load capacitance is $h \times C_L$ (where $C_L$ is the load capacitance of one gate, and $h$ is the number of identical gates driven).

• The worst-case falling delay $t_f$ is given by:

  $t_f = R \times hC_L$

Worst-Case Rising Delay:

• Rising Delay is dominated by the PMOS transistors, which are in parallel.

• Even though the PMOS transistors are in parallel, only one of them will be active during the worst-case scenario, making the effective resistance $R$.

• The load capacitance remains $h \times C_L$.

• The worst-case rising delay $t_r$ is given by:

  $t_r = R \times hC_L$

Summary:

• The transistor widths for the 3-input NAND gate:
  • $W_{n, \text{NAND}} = 3W_n$ (for NMOS)
  • $W_{p, \text{NAND}} = W_p$ (for PMOS)
• The worst-case rising and falling delays are both estimated to be: $t_f \approx t_r \approx R \times hC_L$

This indicates that the worst-case delay is linearly proportional to the number of identical gates $h$ being driven.

Would you like more details on any step, or do you have any further questions?

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Here are some additional questions for further exploration:

1. What are the effects of increasing the transistor width on power consumption?
2. How would you modify the design if the load is not identical gates but a capacitive load?
3. What is the impact of threshold voltage on the delay of the NAND gate?
4. How does the delay of a NAND gate compare to that of a NOR gate for the same transistor sizes?
5. What role does body effect play in the performance of a NAND gate?

Tip: In CMOS design, balancing power consumption and speed is crucial. Increasing transistor width reduces delay but increases dynamic power due to larger charging and discharging currents.