ch07.1.htm
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2,844 行
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=47029">
</A>
.</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=47031">
</A>
(7.4)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=47033">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47035">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47037">
</A>
<SPAN CLASS="EquationVariables">
k</SPAN>
= 1</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47039">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=47041">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=47043">
</A>
</P>
</TD>
</TR>
</TABLE>
<P CLASS="Body">
<A NAME="pgfId=10352">
</A>
The error, <SPAN CLASS="EquationVariables">
E</SPAN>
, is a minimum when <SPAN CLASS="EquationVariables">
a</SPAN>
<SUB CLASS="Subscript">
k</SUB>
= 1 since initially <SPAN CLASS="EquationVariables">
V</SPAN>
<SUB CLASS="Subscript">
i</SUB>
(<SPAN CLASS="EquationVariables">
t</SPAN>
= 0) = <SPAN CLASS="EquationVariables">
V</SPAN>
<SUB CLASS="Subscript">
k</SUB>
(<SPAN CLASS="EquationVariables">
t</SPAN>
= 0) = 1 V (we normalized the voltages) and <SPAN CLASS="EquationVariables">
V</SPAN>
<SUB CLASS="Subscript">
i</SUB>
(<SPAN CLASS="EquationVariables">
t</SPAN>
= <SPAN CLASS="Symbol">
•</SPAN>
) = <SPAN CLASS="EquationVariables">
V</SPAN>
<SUB CLASS="Subscript">
k</SUB>
(<SPAN CLASS="EquationVariables">
t</SPAN>
= <SPAN CLASS="Symbol">
•</SPAN>
) = 0. </P>
<P CLASS="Body">
<A NAME="pgfId=27489">
</A>
Now we can rewrite Eq. <A HREF="CH07.1.htm#13241" CLASS="XRef">
7.2</A>
, setting <SPAN CLASS="EquationVariables">
a</SPAN>
<SUB CLASS="Subscript">
k</SUB>
= 1, as follows: </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=47047">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47049">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47051">
</A>
<SPAN CLASS="EquationVariables">
n</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47053">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47055">
</A>
d<SPAN CLASS="EquationVariables">
V</SPAN>
<SUB CLASS="SubscriptVariable">
i</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=47057">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=47059">
</A>
</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=47061">
</A>
<SPAN CLASS="EquationVariables">
V</SPAN>
<SUB CLASS="SubscriptVariable">
i</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47063">
</A>
=</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47065">
</A>
–<SPAN CLASS="BigMath">
S</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47067">
</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="TableEqnCenter">
<A NAME="pgfId=47069">
</A>
–––</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=47071">
</A>
,</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=47073">
</A>
(7.5)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=47075">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47077">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47079">
</A>
<SPAN CLASS="EquationVariables">
k</SPAN>
= 1</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47081">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47083">
</A>
d<SPAN CLASS="EquationVariables">
t</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=47085">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=47087">
</A>
</P>
</TD>
</TR>
</TABLE>
<P CLASS="BodyAfterHead">
<A NAME="pgfId=10360">
</A>
This is a linear first-order differential equation with the following solution: </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=47167">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47169">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=47171">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=47173">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=47175">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47177">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47179">
</A>
<SPAN CLASS="EquationVariables">
n</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47181">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=47183">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=47185">
</A>
</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=47187">
</A>
<SPAN CLASS="EquationVariables">
V</SPAN>
<SUB CLASS="SubscriptVariable">
i </SUB>
(<SPAN CLASS="EquationVariables">
t</SPAN>
)</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47189">
</A>
=</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=47191">
</A>
exp (–<SPAN CLASS="EquationVariables">
t</SPAN>
/<SPAN CLASS="Symbol">
t</SPAN>
<SUB CLASS="SubscriptVariable">
Di</SUB>
)</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=47193">
</A>
;</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=47195">
</A>
<SPAN CLASS="Symbol">
t</SPAN>
<SUB CLASS="SubscriptVariable">
Di</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47197">
</A>
=</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47199">
</A>
<SPAN CLASS="BigMath">
S</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47201">
</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=47203">
</A>
.</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=47205">
</A>
<A NAME="34895">
</A>
(7.6)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=47207">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47209">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=47211">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=47213">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnRight">
<A NAME="pgfId=47215">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47217">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47219">
</A>
<SPAN CLASS="EquationVariables">
k</SPAN>
= 1</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=47221">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=47223">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=47225">
</A>
</P>
</TD>
</TR>
</TABLE>
<P CLASS="Body">
<A NAME="pgfId=10367">
</A>
The time constant t<SPAN CLASS="EquationNumber">
D</SPAN>
<SPAN CLASS="EquationVariables">
i</SPAN>
is often called the <A NAME="marker=10366">
</A>
Elmore delay and is different for each node. We shall refer to t<SPAN CLASS="EquationNumber">
D</SPAN>
<SPAN CLASS="EquationVariables">
i</SPAN>
as the <A NAME="marker=27296">
</A>
Elmore time constant to remind us that, if we approximate <SPAN CLASS="EquationVariables">
V</SPAN>
<SUB CLASS="Subscript">
i</SUB>
by an exponential waveform, the delay of the RC tree using 0.35/0.65 trip points is approximately t<SPAN CLASS="EquationVariables">
Di</SPAN>
seconds.</P>
</DIV>
<DIV>
<H2 CLASS="Heading2">
<A NAME="pgfId=1388">
</A>
7.1.3 <A NAME="31273">
</A>
RC Delay in Antifuse Connections</H2>
<P CLASS="BodyAfterHead">
<A NAME="pgfId=1390">
</A>
Suppose a single antifuse, with resistance <SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="Subscript">
1</SUB>
, connects to a wire segment with parasitic capacitance <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="Subscript">
1</SUB>
. Then a connection employing a single antifuse will delay the signal passing along that connection by approximately one time constant, or <SPAN CLASS="EquationVariables">
R</SPAN>
<SUB CLASS="Subscript">
1</SUB>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="Subscript">
1</SUB>
seconds. If we have more than one antifuse, we need to use the Elmore time constant to estimate the interconnect delay.</P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFigure">
<A NAME="pgfId=24264">
</A>
</P>
<DIV>
<IMG SRC="CH07-4.gif">
</DIV>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableFigureTitle">
<A NAME="pgfId=27086">
</A>
FIGURE 7.4 <A NAME="11501">
</A>
Actel routing model. (a) A four-antifuse connection. L0 is an output stub, L1 and L3 are horizontal tracks, L2 is a long vertical track (LVT), and L4 is an input stub. (b) An RC-tree model. Each antifuse is modeled by a resistance and each interconnect segment is modeled by a capacitance.</P>
</TD>
</TR>
</TABLE>
<P CLASS="Body">
<A NAME="pgfId=27494">
</A>
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