ch17.6.htm

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 &#8211;<SPAN CLASS="EquationVariables">
E</SPAN>
/<SPAN CLASS="EquationNumber">
k</SPAN>
<SPAN CLASS="EquationVariables">
T</SPAN>
 ,</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85250">
 </A>
<SPAN CLASS="Symbol">
</SPAN>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=85252">
 </A>
(17.18)</P>
</TD>
</TR>
</TABLE>
<P CLASS="ExerciseNoIndent">
<A NAME="pgfId=9658">
 </A>
where MTTF is the mean time to failure, <SPAN CLASS="EquationNumber">
A</SPAN>
 is a constant, J is the current density, <SPAN CLASS="EquationVariables">
E</SPAN>
 is an activation energy, <SPAN CLASS="EquationNumber">
k</SPAN>
 is Boltzmann&#8217;s constant, and <SPAN CLASS="EquationVariables">
T</SPAN>
 is absolute temperature. You also remember the rule that you can have about 1 mA of current for every <SPAN CLASS="Symbol">
l</SPAN>
 of metal width for a reasonable time to failure of more than 10 years. Since this chip is in 0.5 <SPAN CLASS="Symbol">
m</SPAN>
m CMOS (<SPAN CLASS="Symbol">
l</SPAN>
 = 0.25 <SPAN CLASS="Symbol">
m</SPAN>
m), you guess that the metal is about 0.5 <SPAN CLASS="Symbol">
m</SPAN>
m thick, and the resistance is at least 50 m <SPAN CLASS="Symbol">
W </SPAN>
/square. </P>
<UL>
<LI CLASS="ExercisePartFirst">
<A NAME="pgfId=9670">
 </A>
a.&nbsp;How much current do you estimate you need to make your initials fail so that you can test the chip before your boss gets back in a week&#8217;s time?</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=9694">
 </A>
b.&nbsp;What else could you do to speed things up?</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=9676">
 </A>
c.&nbsp;How are you going to do this? (P.S. This sometimes actually works.)</LI>
</UL>
<P CLASS="ExerciseHead">
<A NAME="pgfId=9591">
 </A>
17.14&nbsp;(**Routing problems, 20 min.) We have finished the third iteration on the new game chip and are having yield problems in production. This is what we know:</P>
<P CLASS="Exercise">
<A NAME="pgfId=9592">
 </A>
1. We changed the routing on v3 by using an ECO mechanism in the detailed router from Shortem. We just ripped up a few nets and rerouted them without changing anything else.</P>
<P CLASS="Exercise">
<A NAME="pgfId=9593">
 </A>
2. The ASIC vendor, Meltem, is having yield problems due to long metal lines shorting&#8212;but only in one place. It looks as though they are the metal lines we changed in v3. Meltem blames the mask vendor&#8212;Smokem.</P>
<P CLASS="Exercise">
<A NAME="pgfId=9594">
 </A>
3. To save money we changed mask vendors after completing the prototype version v1, so that v2 and v3 uses the new mask vendor (Smokem). Smokem confirms there is a problem with the v3 mask&#8212;the lines we changed are shifted very slightly toward others and have a design rule violation. However, the v2 mask was virtually identical to v3 and there are no problems with that one, so Smokem blames the router from Shortem.</P>
<P CLASS="Exercise">
<A NAME="pgfId=9601">
 </A>
4. Shortem checks the CIF files for us, claims the mask data is correct, and they suggest we blame Meltem.</P>
<P CLASS="Exercise">
<A NAME="pgfId=9595">
 </A>
We do not care (yet) who is to blame, we just need the problem fixed. We need suggestions for the source of the problem (however crazy), some possible fixes, and some ideas to test them. Can you help?</P>
<P CLASS="ExerciseHead">
<A NAME="pgfId=6221">
 </A>
17.15&nbsp;<SPAN CLASS="Emphasis">
</SPAN>
(*Coupling capacitance, 30 min.) Suppose we have three interconnect lines running parallel to each other on a bus. Consider the following situations (VDD = 5 V, VSS = 0 V):</P>
<UL>
<LI CLASS="ExercisePartFirst">
<A NAME="pgfId=23241">
 </A>
a.&nbsp;The center line switches from VSS to VDD. The neighbor lines are at VSS.</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=23242">
 </A>
b.&nbsp;The center line switches from VSS to VDD. At the same time the neighbor lines switch from VDD to VSS.</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=23246">
 </A>
c.&nbsp;The center line switches from VSS to VDD. At the same time the neighbor lines also switch from VDD to VSS.</LI>
</UL>
<P CLASS="Exercise">
<A NAME="pgfId=23247">
 </A>
How do you define capacitance in these cases? In each case what is the effective capacitance from the center to the neighboring lines using your definition? </P>
<P CLASS="ExerciseHead">
<A NAME="pgfId=6225">
 </A>
17.16&nbsp;(**2LM and 3LM routing, 10 min.) How would you attempt to measure the difference in die area obtained by using the same standard-cell library with two-level and three-level routing?</P>
<P CLASS="ExerciseHead">
<A NAME="pgfId=35741">
 </A>
17.17&nbsp;(***SPF, 60 min) </P>
<UL>
<LI CLASS="ExercisePartFirst">
<A NAME="pgfId=76511">
 </A>
a.&nbsp;Write a regular SPF file for the circuit shown in <A HREF="CH17.1.htm#28148" CLASS="XRef">
Figure&nbsp;17.3</A>
(b), using the lumped-C model and the Elmore constant for the pin-to-pin timings. </LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=76512">
 </A>
b.&nbsp;Write the equivalent RSPF file. </LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=76513">
 </A>
c.&nbsp;Write a DSPF file for the same circuit. </LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=76514">
 </A>
d.&nbsp;Calculate the PI segment parameters for the circuit shown in <A HREF="CH17.1.htm#28148" CLASS="XRef">
Figure&nbsp;17.3</A>
(b). <SPAN CLASS="Emphasis">
Hint:</SPAN>
 You may need to consult [<A NAME="O誃rien89b">
 </A>
O&#8217;Brien and Savarino, 1989] if you need help.</LI>
</UL>
<P CLASS="ExerciseHead">
<A NAME="pgfId=6226">
 </A>
17.18&nbsp;(***Standard-cell aspect ratio, 30 min.) How would you decide the optimum value for the logic cell height of a standard-cell library?</P>
<P CLASS="ExerciseHead">
<A NAME="pgfId=35619">
 </A>
17.19&nbsp;(Electromigration, 20 min.) </P>
<UL>
<LI CLASS="ExercisePartFirst">
<A NAME="pgfId=88698">
 </A>
a.&nbsp;What is the current density in a 1 <SPAN CLASS="Symbol">
m</SPAN>
m wide wire that is 1 <SPAN CLASS="Symbol">
m</SPAN>
m thick and carries a current of 1 mA? </LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=88703">
 </A>
b.&nbsp;Using Eq. <A HREF="CH17.3.htm#14020" CLASS="XRef">
17.9</A>
, can you explain the temperature behavior of the parameters in <A HREF="CH17.3.htm#28224" CLASS="XRef">
Table&nbsp;17.1</A>
? </LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=76517">
 </A>
c.&nbsp;Using Eq. <A HREF="CH17.3.htm#41231" CLASS="XRef">
17.10</A>
, can you explain the dependence on current direction?</LI>
</UL>
<P CLASS="ExerciseHead">
<A NAME="pgfId=37415">
 </A>
17.20&nbsp;<SPAN CLASS="Emphasis">
</SPAN>
<A NAME="17227">
 </A>
(***SPF parameters, 120 min.). <SPAN CLASS="Emphasis">
Hint:</SPAN>
 You may need help from [<A NAME="O誃rien89c">
 </A>
O&#8217;Brien and Savarino, 1989] for this question.</P>
<UL>
<LI CLASS="ExercisePartFirst">
<A NAME="pgfId=37416">
 </A>
a.&nbsp;Find an expression for <SPAN CLASS="EquationVariables">
Y</SPAN>
(<SPAN CLASS="EquationVariables">
s</SPAN>
), where <SPAN CLASS="EquationVariables">
s</SPAN>
 = <SPAN CLASS="EquationVariables">
jw</SPAN>
, the <A NAME="marker=52212">
 </A>
driving-point admittance (the reciprocal of the driving-point impedance), for the interconnect network shown in <A HREF="CH17.4.htm#23689" CLASS="XRef">
Figure&nbsp;17.22</A>
(a), in terms of <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
A</SUB>
, <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
B</SUB>
, <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
C</SUB>
, <SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="SubscriptVariable">
AB</SUB>
, and <SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="SubscriptVariable">
BC</SUB>
.</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=37420">
 </A>
b.&nbsp;Find the first three terms of the Taylor-series expansion for <SPAN CLASS="EquationVariables">
Y</SPAN>
(<SPAN CLASS="EquationVariables">
s</SPAN>
).</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=37421">
 </A>
c.&nbsp;Derive expressions for <SPAN CLASS="EquationVariables">
Y</SPAN>
<SUB CLASS="Subscript">
1</SUB>
(<SPAN CLASS="EquationVariables">
s</SPAN>
), <SPAN CLASS="EquationVariables">
Y</SPAN>
<SUB CLASS="Subscript">
2</SUB>
(<SPAN CLASS="EquationVariables">
s</SPAN>
), and <SPAN CLASS="EquationVariables">
Y</SPAN>
<SUB CLASS="Subscript">
3</SUB>
(<SPAN CLASS="EquationVariables">
s</SPAN>
) for the lumped-C, the lumped-RC, and the PI segment network models (<A HREF="CH17.4.htm#23689" CLASS="XRef">
Figure&nbsp;17.22</A>
b&#8211;d).</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=37425">
 </A>
d.&nbsp;Comparing your answers to parts b and c, derive the values of the parameters of the lumped-C, the lumped-RC, and the PI segment network models in terms of <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
A</SUB>
, <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
B</SUB>
, <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
C</SUB>
, <SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="SubscriptVariable">
AB</SUB>
, and <SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="SubscriptVariable">
BC</SUB>
.</LI>
</UL>
<P CLASS="ExerciseHead">
<A NAME="pgfId=35683">
 </A>
17.21&nbsp;<A NAME="21193">
 </A>
(**Distributed-delay routing, 120 min. [Kahng and Robins, 1995])&nbsp;The Elmore constant is one measure of net delay,  </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=85271">
 </A>
<SPAN CLASS="Symbol">
t</SPAN>
<SUB CLASS="SubscriptVariable">
Di</SUB>
<SUB CLASS="Subscript">
</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85273">
 </A>
=</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85275">
 </A>
<SPAN CLASS="BigMath">
S</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85277">
 </A>
<SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="SubscriptVariable">
ki</SUB>
 <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
k</SUB>
 .</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85279">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=85281">
 </A>
<A NAME="14070">
 </A>
(17.19)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=85283">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85285">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85287">
 </A>
<SPAN CLASS="EquationVariables">
k</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85289">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85291">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=85293">
 </A>
&nbsp;</P>
</TD>
</TR>
</TABLE>
<P CLASS="Exercise">
<A NAME="pgfId=37276">
 </A>
The <SPAN CLASS="Definition">
distributed delay</SPAN>
<A NAME="marker=37487">
 </A>
, defined as follows, is another measure of delay in a network:  </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=85416">
 </A>
<SPAN CLASS="Symbol">
t</SPAN>
<SUB CLASS="SubscriptVariable">
p</SUB>
<SUB CLASS="Subscript">
</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85418">
 </A>
=</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85420">
 </A>
<SPAN CLASS="BigMath">
S</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85422">
 </A>
<SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="SubscriptVariable">
kk</SUB>
 <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
k</SUB>
 .</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85424">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=85426">
 </A>
<A NAME="21031">
 </A>
(17.20)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=85428">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85430">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85432">
 </A>
<SPAN CLASS="EquationVariables">
k</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85434">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85436">
 </A>
&nbsp;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=85438">
 </A>
&nbsp;</P>
</TD>
</TR>
</TABLE>
<P CLASS="Exercise">
<A NAME="pgfId=32891">
 </A>
We can write this equation in terms of network components as follows:  </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=85455">
 </A>
<SPAN CLASS="Symbol">
t</SPAN>
<SUB CLASS="SubscriptVariable">
p</SUB>
<SUB CLASS="Subscript">
</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85457">
 </A>
=</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=85459">
 </A>
<SPAN CLASS="BigMath">
S</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=85461">
 </A>
(<SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="Subscript">
0</SUB>
 <SPAN CLASS="EquationVariables">
L</SPAN>
<SUB CLASS="SubscriptVariable">
kn</SUB>
 + <SPAN CLASS="EquationVariables">