SUT 25571
HW #3 - VLSI Interconnect Design
Due 10-03-89
Consider a transmission line with length x excited with arbitrary input vs through a source impedance Zs
and terminated with a load impedance Zl, as shown in Figure 1. A general transmission line can be
considered a distributed zy (in contrast with distributed rlc), where z(s) is the impedance per unit length
and y(s) is the admittance per unit length of the line as shown in Figure 2.
Zs
+
v(0,t)
-
Z0
+
vs(t)
-
+
v( ll,t)
-
+
v(x,t)
-
Zl
Figure 1.
x
x+dx
z(s)
ix
ix+dx
+
vx
-
+
vx+dx
-
y(s)
Figure 2.
1) Show that the transfer function of the T-line can be written as:
( )
( )=
( )
=
(1 +
)
;
=
−
+
;
;
=
/
(1)
where
=
+
=
=
2) As we discussed in the course, normalizing a line ( ( ) = +
−
+
, ( )=
) to a line with unity
time-of-flight (TF) and characteristic impedance makes the equations shorter and simpler, and
makes the comparisons easier. Suppose an rlc line with length x is driven by a source that has an
impedance of
+
is terminated by a load capacitance
. The concepts that are used for
SUT 25571
HW #3 - VLSI Interconnect Design
Due 10-03-89
the normalization are as follows:
i) An rlc line with length x, can be replaced by a (xr)(xl)(xc) line with unity length.
ii) Multiplication of all impedances of any linear circuit by a certain value does not change the
voltage-voltage transfer function of the system; therefore, all impedances can be multiplied by the
reciprocal characteristic impedance of the line, Z0.
iii) Finally, time-frequency duality implies that shrinking time by TF is the same as multiplying all
frequencies by the same factor (T F).
Show that, by using these three concepts, an rlc line with length x can be replaced by an r11 line
with unity length. The following relations between timing in the original and normalized cases,
however, should be considered (all normalized parameters are marked with a tilda sign):
1
t  t
TF
;
f  f TF
(2)
Other parameters should also be normalized as
1
Rs  Rs
;
Z0
Ls  Ls
1
;
TF Z 0
3) Considering the skin effect as ( ) = +
√ +
r  r 
x
TF Z 0
Z
Cl  Cl 0 ;
TF
r  r
x
Z0
(3)
, show that normalized r can be written as:
(4)
4) For n-tier design, two adjacent levels with orthogonal metal levels are called a pair, and a tier is a
collection of pairs with the same pitch (p). It is assumed that all the cross-sectional dimensions of
the wires are equal: w=t=s=h=p/2, where w, t, s, and h are the metal width, wire thickness, spacing
between interconnects, and height of the inter-level dielectrics, respectively, as shown in Figure 3.
Considering result for the n-tier design for year 2018 as shown in Table 1. And relations given in
(3) and (4) find the minimum and maximum values for r and r and fill up the Table. For Cl
consider that Cl is 10 times C0 of the minimum size gate extracted from ITRS for year 2018.
5) Consider an rlc line (with no skin effect) is driving with a source with
(
= 0 and is open ended
= ∞ ). Rewrite the transfer function (1) for the normalized parameters and plot the root locus
of this transfer function for different values of r . What is the minimum value for r such that the
SUT 25571
HW #3 - VLSI Interconnect Design
Due 10-03-89
T-line acts like an rc line (real roots). What is the minimum length in the de-normalized line
6) (Extra credit) Redo 5, now considering skin effect ( r ≠ 0). You may plot the root locus for
changes in r for different r values. Use data in 4, for choosing reasonable range of r and r .
pair
#4


w
s
t
tier
#3
h
pair
#3

pair
#2
pair
#1




tier
#2
tier
#1
Figure 3.
Table 1. Results of n-tier design methodology for year 2018 (18 nm node). Lmax is
normalized to the gate pitch (350 nm), (Lmin of each pair is equal to Lmax of the next pair)
pair #1
pair #2
pair #3
pair #4
pair #5
pair #6
pair #7
Pitch
Lmax
42
42
42
152
406
1199
1593
6
44
1077
5125
1368
4038
5366
rmin
rmax
min
r
max
r
C l  min
C l  max