📄 ch15.9.htm
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44</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="Table">
<A NAME="pgfId=23197">
</A>
$35</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="Table">
<A NAME="pgfId=23199">
</A>
L</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="Table">
<A NAME="pgfId=30258">
</A>
0.46</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="Table">
<A NAME="pgfId=23201">
</A>
2800</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="Table">
<A NAME="pgfId=23203">
</A>
84</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="Table">
<A NAME="pgfId=23205">
</A>
$50</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="Table">
<A NAME="pgfId=23207">
</A>
XL</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="Table">
<A NAME="pgfId=30260">
</A>
0.64</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="Table">
<A NAME="pgfId=23209">
</A>
4700</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="Table">
<A NAME="pgfId=23211">
</A>
84</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="Table">
<A NAME="pgfId=23213">
</A>
$90</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableLast">
<A NAME="pgfId=23215">
</A>
XXL</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableLast">
<A NAME="pgfId=30262">
</A>
0.84</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableLast">
<A NAME="pgfId=23217">
</A>
6200</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableLast">
<A NAME="pgfId=23219">
</A>
84</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableLast">
<A NAME="pgfId=23221">
</A>
$130</P>
</TD>
</TR>
</TABLE>
<UL>
<LI CLASS="ExercisePartFirst">
<A NAME="pgfId=23148">
</A>
a. Notice that the FPGAs come in different package sizes. To eliminate the effect of package price, multiply the price for the S chip by 106 percent, and the M chip by 113 percent. Now all prices are normalized for an 84-pin plastic package. All the chips are the same speed grade; if they were not, we could normalize for this too (a little harder to justify though).</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=23153">
</A>
b. Plot the normalized chip prices vs. gate count. What is the cost per gate?</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=23230">
</A>
c. The part cost ought to be related to the yield, which is directly related to die area. If the cost of a 6-inch-diameter wafer is fixed (approximately $1000), calculate the cost per die, assuming a yield <SPAN CLASS="EquationVariables">
Y </SPAN>
(in percent), as a function of the die area, <SPAN CLASS="EquationVariables">
A</SPAN>
(in cm<SUP CLASS="Superscript">
2</SUP>
). Assume you completely fill the wafer and you can have fractional die (i.e., do not worry about packing square die into a circular wafer).</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=23246">
</A>
d. There are many models for the yield of a process, <SPAN CLASS="EquationVariables">
Y</SPAN>
. Two common models are </LI>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=205774">
</A>
<SPAN CLASS="EquationVariables">
Y</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=205776">
</A>
=</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=205778">
</A>
exp [– <SPAN CLASS="Symbol">
÷</SPAN>
(AD) ] .</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=205780">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=205782">
</A>
(15.29)</P>
</TD>
</TR>
</TABLE>
</UL>
<P CLASS="Exercise">
<A NAME="pgfId=23239">
</A>
and </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=206001">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=206003">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=206005">
</A>
<SPAN CLASS="Symbol">
Ê</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=206007">
</A>
1 – exp (–<SPAN CLASS="EquationVariables">
AD</SPAN>
)</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=206009">
</A>
<SPAN CLASS="Symbol">
ˆ</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=206011">
</A>
2</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=206013">
</A>
</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=206015">
</A>
<SPAN CLASS="EquationVariables">
Y</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=206017">
</A>
=</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=206019">
</A>
<SPAN CLASS="Symbol">
Á</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=206021">
</A>
–––––––––––</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=206023">
</A>
<SPAN CLASS="Symbol">
˜</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=206025">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=206027">
</A>
(15.30)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=206029">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=206031">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=206033">
</A>
<SPAN CLASS="Symbol">
Ë</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=206035">
</A>
<SPAN CLASS="EquationVariables">
AD</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=206037">
</A>
<SPAN CLASS="Symbol">
¯</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=206039">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=206041">
</A>
</P>
</TD>
</TR>
</TABLE>
<P CLASS="Exercise">
<A NAME="pgfId=23245">
</A>
Parameter <SPAN CLASS="EquationVariables">
A</SPAN>
is the die area in cm<SUP CLASS="Superscript">
2</SUP>
and <SPAN CLASS="EquationVariables">
D</SPAN>
is the spot defect density in defects/cm<SUP CLASS="Superscript">
2</SUP>
and is usually around 1.0 defects/cm<SUP CLASS="Superscript">
2</SUP>
for a good submicron CMOS process (above 5.0 defects/cm<SUP CLASS="Superscript">
2</SUP>
is unusual). The most important thing is the yield; anything below about 50 percent good die per wafer is usually bad news for an ASIC foundry. Does the FPGA cost data fit either model?</P>
<UL>
<LI CLASS="ExercisePart">
<A NAME="pgfId=23254">
</A>
e. Now disregard the current pricing strategy of company Z. If you had to bet that physics would determine the true price of the chip, how much worse or better off are you using two small FPGAs rather than one larger FPGA (assume the larger die is exactly twice the area of the smaller one) under these two yield models?</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=23260">
</A>
f. What assumptions are inherent in the calculation you made in part e? How much do you think they might affect your answer, and what else would affect your judgment?</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=23249">
</A>
g. Give some reasons why you might select two smaller FPGAs rather than a larger FPGA, even if the larger FPGA is a cheaper solution.</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=23250">
</A>
h. Give some reasons why you would select a larger FPGA rather than two smaller FPGAs, even if the smaller FPGAs were a cheaper solution.</LI>
</UL>
<P CLASS="ExerciseHead">
<A NAME="pgfId=180211">
</A>
15.23 <A NAME="18796">
</A>
(Constructive partitioning, 30 min.) We shall use the simple network with 12 blocks shown in <A HREF="CH15.7.htm#14578" CLASS="XRef">
Figure 15.7</A>
to experiment with constructive partitioning. This example is topologically equivalent to that used in [Goto and Matsud, 1986].</P>
<UL>
<LI CLASS="ExercisePartFirst">
<A NAME="pgfId=180216">
</A>
a. We shall use a gain function, <SPAN CLASS="EquationVariables">
g(m)</SPAN>
, calculated as follows: Sum the number of the <SPAN CLASS="Emphasis">
nets</SPAN>
(not <SPAN CLASS="Emphasis">
connections</SPAN>
) from the selected logic cell, <SPAN CLASS="EquationVariables">
m</SPAN>
, that connect to the current partition—call this <SPAN CLASS="EquationVariables">
P(m).</SPAN>
Now calculate the number of <SPAN CLASS="Emphasis">
nets</SPAN>
that connect logic cell <SPAN CLASS="EquationVariables">
m</SPAN>
to logic cells which are not yet in partitions—call this <SPAN CLASS="EquationVariables">
N(m).</SPAN>
Then <SPAN CLASS="EquationVariables">
g(m)</SPAN>
<SPAN CLASS="EquationVariables">
=</SPAN>
<SPAN CLASS="EquationVariables">
P(m)</SPAN>
<SPAN CLASS="EquationVariables">
–</SPAN>
<SPAN CLASS="EquationVariables">
N(m) </SPAN>
is the gain of adding the logic cell <SPAN CLASS="EquationVariables">
m</SPAN>
to the partition currently being filled.</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=180217">
</A>
b. <A NAME="41871">
</A>
Partition the network using the seed growth algorithm with logic cell C as the seed. Show how this choice of seed can lead to the partitioning shown in <A HREF="CH15.7.htm#14578" CLASS="XRef">
Figure 15.7</A>
(c). Use a table like <A HREF="CH15.9.htm#33501" CLASS="XRef">
Table 15.11</A>
as a bookkeeping aid (a spreadsheet will help too). Each row corresponds to a pass through the algorithm. Fill in the measures, P(m) – N(m), equal to the gain, g(m). Once a logic cell is assigned to a partition, fill in the name of the partition (X, Y, or Z) in that column. The first row shows you how logic cell L is selected; proceed from there. What problems do you encounter while completing the algorithm, and how do you resolve them?</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=180224">
</A>
c. <A NAME="37431">
</A>
Now partition using logic cell F as the seed instead—the logic cell with the highest number of nets. When you have a tie between logic cells with the same gain, or you are starting a new partition, pick the logic cell with the largest <SPAN CLASS="EquationVariables">
P(m)</SPAN>
. Use a copy of <A HREF="CH15.9.htm#42269" CLASS="XRef">
Table 15.12</A>
as a bookkeeping aid. How does your partition compare with those we have already made (summarized in <A HREF="CH15.9.htm#23475" CLASS="XRef">
Table 15.13</A>
)?</LI>
<LI CLASS="ExercisePart">
<A NAME="pgfId=180231">
</A>
d. Comment on your results.</LI>
</UL>
<P CLASS="Exercise">
<A NAME="pgfId=180940">
</A>
<A HREF="CH15.9.htm#29588" CLASS="XRef">
Table 15.14</A>
will help in constructing the gain function at each step of the algorithm. </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="14">
<P CLASS="TableTitle">
<A NAME="pgfId=180239">
</A>
TABLE 15.11 <A NAME="33501">
</A>
Bookkeeping table for Problem <A HREF="CH15.9.htm#18796" CLASS="XRef">
15.23</A>
(<A HREF="CH15.9.htm#41871" CLASS="XRef">
b</A>
).</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFirst">
<A NAME="pgfId=180270">
</A>
Pass</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFirst">
<A NAME="pgfId=180272">
</A>
Gain</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFirst">
<A NAME="pgfId=180274">
</A>
A</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFirst">
<A NAME="pgfId=180276">
</A>
B</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFirst">
<A NAME="pgfId=180278">
</A>
C</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFirst">
<A NAME="pgfId=180280">
</A>
D</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFirst">
<A NAME="pgfId=180282">
</A>
E</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFirst">
<A NAME="pgfId=180284">
</A>
F</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFirst">
<A NAME="pgfId=180286">
</A>
G</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFirst">
<A NAME="pgfId=180288">
</A>
H</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFirst">
<A NAME="pgfId=180290">
</A>
I</P>
</TD>
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