Subject :- VLSI Design

Topic :- Semiconductor Memories

Difficulty :- Medium

• R= driver resistance
• C= total interconnect capacitance + loading capacitance.
• Sink Delay: $t_d$=R*C
• 50% delay under step input=0.7 RC.
• Valid when driver resistance >> interconnect resistance.
• All sinks have equal delay.

Lumped RC Delay Model

• Minimize delay <=> minimize wire length
  $\hspace{0.5cm}t_D=R_d*C_{load}=R_d*(C_{int}+C_g)$
  $\hspace{1cm}=R_d*(C_0*L+C_g)$

Delay of Distributed RC Lines:

Distributed Interconnect Models

• Distributed RC circuit model – L,T or $\pi$ circuits
• Distributed RCL circuit model
• Tree of transmission lines

Why Elmore Delay ?

1) Elmore delay is easier to compute analytically in most cases:

• Elmore's insight [Elmore, J. App. Phy 1948]
• Verified later on by many other researchers, e.g.
  $\hspace{1.9cm}$Elmore delay for RC trees []Penfield- Rubinstein, DAC'18]
  $\hspace{1.9cm}$Elmore delay for RC networks with ramp input[Chan, T-CAS'86]

2) for RC trees: [krauter- Tatuianu-Willis-Pileggi, DAC'95]

3) Note: Elmore delah is not 50% value delay in general !

Elmore Delay for RC Trees

• Defination : $h(t)= impuse response$
  $\hspace{1.9cm}T_D=mean of h(t) $
  $\hspace{2.5cm}=\int_{0}^{\infty} h(t)*t\,dt$
• Interpretation
  $\hspace{1.9cm}H(t)$=output response (step process)
  $\hspace{1.9cm}h(t)$= rate of change of H(t)
  $\hspace{1.9cm}T_{50} $ =median of h(t)
  $\hspace{1.9cm}$ Elmore delay approximates the median of h(t) by the mean of h(t)

Elmore Delay Model

Advantages

• Simple closed-form expression
• Useful for interconnect optimization
• Upper bound of 50% delay [Gupta et al., DAC’95, TCAD’97]
• Actual delay asymptotically approaches Elmore delay as input signal rise time increases
• High fidelity [Boese et al., ICCD’93],[Cong-He, TODAES’96]
• Good solutions under Elmore delay are good solutions under actual (SPICE) delay

Disadvantages

• Low accuracy, especially poor for slope computation
• Inherently cannot handle inductance effect
• Elmore delay is first moment of impulse response
• Need higher order moments