Effects of Inductance on the Propagation Delayand Repeater Insertion in VLSI Circuits
Yehea I. Ismail and Eby G. Friedman, Fellow, IEEE
Agenda
Introduction
Propagation Delay Formula
Comparison to an RC Model
Dependence of Delay on wire Length
Repeater Insertion for an RLC Interconnect
Conclusions
Introduction
Propagation Delay Formula
npd wet /48.135.19.2
Close Form
2212......8202 1
0
nnn
x
x nnx
2/)1(......2100
nnnxn
x
Interconnect Model-RCL
A gate driving an RLC transmission line
The transfer function-a lossy transmission line
l
L
sl
L
sin
out
eZ
Z
Z
Ze
Z
Z
Z
ZsV
sV
1111
2
)(
)(
0
0
0
0
t
t
t
t
sL
R
C
LZ 10
t
ttt sL
RCLsl 1
trs RZ L
L sCZ
1
Zs is source impedance, ZL is load impedance
γ is propagation constant, Z0 is characteristic impedance
Laplace transform-1
Ltt
nCCL
W
1
'SWs n
Tline
t
CsC
sl
1
'
21
1
''
'
Tline
T
t CsC
LZZ 1
'
21'00
T
T
T
tLL C
C
sC
LZZ
1
'
1'
t
ttline
t
LT
L
CR
C
CC
2
Laplace transform-2
t
trT R
RR
t
LT C
CC
t
ttline L
CR
2
The parameter ζ
TTTTT
t
tt
C
CRCR
L
CR
1
5.0
2
TTTTT
lineC
CRCR
1
5.0
Three parameter:ζ,RT,CT
Laplace transform-3
The scaled 50% propagation delay t`pd can be calculated by solving
Thus, the propagation delay of an RCL line with a source Resistance Rtr and a load capacitance CL has the form
AS/X [20] simulations
the maximum error is 4.6% and the average error is 1.65%
AS/X [20] simulations
npd wet /48.135.19.2 Eq.(18):
Delay function is more accurate
npd wet /48.135.19.2
When CT And RT are high and ζ is lowEq(18) suffers high errors
This case can only occur for unreasonably high values of the inductance per unit length of the line as compared to the resistance and capacitance per unit lengthSuch a case does not exist in a practical VLSI circuit.
ζ characterize inductance effects
ζ characterize inductance effects
ζ characterize inductance effects
ζ characterize inductance effects
The Factor of Inductance
npd wet /48.135.19.2
TTTTT
t
tt
C
CRCR
L
CR
1
5.0
2
Comparison to an RC model
npd wet /48.135.19.2
Comparison to an RC model
copper interconnect line
copper interconnect line
copper interconnect line
Dependence of Delay on Interconnect Length
αasym is the asymptotic value at high frequencies of the attenuation per unit length of the signals
L 0 tpd =0.37RCl2
R 0 tpd = l(LC)0.5
Repeater Insertion for an RLC Interconnect
The buffer output impedance Rtr is R0/hThe input capacitance of the buffer CL is hC0
Curve Fitting Method
highly accurate
The error is less than 0.05%If Lt 0 hopt(RLC) = hopt(RC) kopt(RLC) = kopt(RC)
Comparison with RC model
Total delay in RC & RLC
When TL/R=3 ,tpdtotal increase by 10%When TL/R=5 ,tpdtotal increase by 20%When TL/R=10 ,tpdtotal increase by 30%
Repeater Insertion in RC & RLC
Area in RC & RLC
Amin is the area of a minimum size buffer
=
When TL/R=3 ,Area increase by 154%When TL/R=5 ,Area increase by 435%
conclusion
Propagation delay --- 35%
propagation delay has linear dependence on the length of the line.
Conclusion for repeater insertion
Insert less buffer
Reduce delay
Reduce buffer area