📄 ch13.6.htm
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</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=132144">
</A>
<A NAME="10560">
</A>
(13.14)</P>
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<P CLASS="TableEqnLeft">
<A NAME="pgfId=132146">
</A>
and</P>
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<P CLASS="TableEqn">
<A NAME="pgfId=132148">
</A>
</P>
</TD>
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<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=132150">
</A>
<SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="SubscriptVariable">
t</SUB>
<SUB CLASS="Subscript">
1</SUB>
= <SPAN CLASS="EquationVariables">
A</SPAN>
<SUB CLASS="Subscript">
1</SUB>
+<SPAN CLASS="EquationVariables">
B</SPAN>
<SPAN CLASS="EquationVariables">
I</SPAN>
<SUB CLASS="SubscriptVariable">
R</SUB>
+ <SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="Subscript">
1</SUB>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
L</SUB>
.</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=132152">
</A>
(13.15)</P>
</TD>
</TR>
</TABLE>
<P CLASS="Body">
<A NAME="pgfId=69560">
</A>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
R</SUB>
is the critical ramp that separates two regions of operation, we call these slow ramp and fast ramp. A sensible definition for <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
R </SUB>
is the point at which the end of the input ramp occurs at the same time the output reaches the 0.5 trip point. This leads to the following equation for <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
R</SUB>
: </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=125365">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=125353">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=125428">
</A>
<SPAN CLASS="EquationVariables">
A</SPAN>
<SUB CLASS="Subscript">
0</SUB>
+ <SPAN CLASS="EquationVariables">
A</SPAN>
<SUB CLASS="Subscript">
1</SUB>
+ (<SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="Subscript">
0</SUB>
+ <SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="Subscript">
1</SUB>
) <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
L</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=125329">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=125332">
</A>
</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=125367">
</A>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
R</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=125355">
</A>
=</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=125430">
</A>
––––––––––––––––––––––</P>
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<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=125345">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=125347">
</A>
(13.16)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=125369">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=125357">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=125432">
</A>
2 (1 – <SPAN CLASS="EquationVariables">
B</SPAN>
)</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnCenter">
<A NAME="pgfId=125349">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=125351">
</A>
</P>
</TD>
</TR>
</TABLE>
<P CLASS="Body">
<A NAME="pgfId=65232">
</A>
It is convenient to define two more parameters: </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=132191">
</A>
<SPAN CLASS="EquationVariables">
d</SPAN>
<SUB CLASS="SubscriptVariable">
A</SUB>
= <SPAN CLASS="EquationVariables">
A</SPAN>
<SUB CLASS="Subscript">
1</SUB>
– <SPAN CLASS="EquationVariables">
A</SPAN>
<SUB CLASS="Subscript">
0</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=132193">
</A>
and</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=132195">
</A>
<SPAN CLASS="EquationVariables">
d</SPAN>
<SUB CLASS="SubscriptVariable">
D</SUB>
= <SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="Subscript">
1 </SUB>
– <SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="Subscript">
0</SUB>
.</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=132168">
</A>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=132170">
</A>
<A NAME="12564">
</A>
(13.17)</P>
</TD>
</TR>
</TABLE>
<P CLASS="Body">
<A NAME="pgfId=69622">
</A>
<SPAN CLASS="EquationVariables">
</SPAN>
In the region that <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
R</SUB>
> <SPAN CLASS="EquationVariables">
I</SPAN>
<SUB CLASS="SubscriptVariable">
R</SUB>
, we can simplify Eqs. <A HREF="CH13.6.htm#10560" CLASS="XRef">
13.14</A>
and by using the definitions in Eq. <A HREF="CH13.6.htm#12564" CLASS="XRef">
13.17</A>
, as follows: </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=132271">
</A>
<SPAN CLASS="EquationVariables">
D</SPAN>
= (<SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="SubscriptVariable">
t</SUB>
<SUB CLASS="Subscript">
1</SUB>
+ <SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="SubscriptVariable">
t</SUB>
<SUB CLASS="Subscript">
0</SUB>
–<SPAN CLASS="EquationVariables">
I</SPAN>
<SUB CLASS="SubscriptVariable">
R</SUB>
)/2 = <SPAN CLASS="EquationVariables">
A</SPAN>
<SUB CLASS="Subscript">
0</SUB>
+ <SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="Subscript">
0</SUB>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
L</SUB>
+ <SPAN CLASS="EquationVariables">
d</SPAN>
<SUB CLASS="SubscriptVariable">
A</SUB>
/2 + <SPAN CLASS="EquationVariables">
d</SPAN>
<SUB CLASS="SubscriptVariable">
D</SUB>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
L</SUB>
/2</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=132273">
</A>
(13.18)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=132275">
</A>
and</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=132277">
</A>
</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=132279">
</A>
<SPAN CLASS="EquationVariables">
O</SPAN>
<SUB CLASS="SubscriptVariable">
R</SUB>
= <SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="SubscriptVariable">
t</SUB>
<SUB CLASS="Subscript">
1</SUB>
– <SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="SubscriptVariable">
t</SUB>
<SUB CLASS="Subscript">
0</SUB>
= <SPAN CLASS="EquationVariables">
d</SPAN>
<SUB CLASS="SubscriptVariable">
A</SUB>
+ <SPAN CLASS="EquationVariables">
d</SPAN>
<SUB CLASS="SubscriptVariable">
D</SUB>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
L</SUB>
.</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=132281">
</A>
(13.19)</P>
</TD>
</TR>
</TABLE>
<P CLASS="Body">
<A NAME="pgfId=65225">
</A>
Now we can understand the timing parameters in the primitive model in <A HREF="CH13.5.htm#40162" CLASS="XRef">
Section 13.5.1</A>
. For example, the following parameter, <SPAN CLASS="BodyComputer">
tA1D_fr</SPAN>
, models the falling input to rising output waveform delay for the logic cell (the units are a consistent set: all times are measured in nanoseconds and capacitances in picofarads):</P>
<P CLASS="ComputerOneLine">
<A NAME="pgfId=65228">
</A>
A0 = 0.0015;dA = 0.0789;D0 = -0.2828;dD = 4.6642;B = 0.6879;Z = 0.5630;</P>
<P CLASS="Body">
<A NAME="pgfId=69715">
</A>
The input-slope model predicts delay in the fast-ramp region, <SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="SubscriptVariable">
ISM</SUB>
(50 %, FR), as follows (0.5 trip points): </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=132293">
</A>
<SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="SubscriptVariable">
ISM</SUB>
(50 %, FR)</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=132295">
</A>
</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=132297">
</A>
= <SPAN CLASS="EquationVariables">
A</SPAN>
<SUB CLASS="Subscript">
0</SUB>
+ <SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="Subscript">
0</SUB>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
L</SUB>
+ 0.5<SPAN CLASS="EquationVariables">
O</SPAN>
<SUB CLASS="SubscriptVariable">
R</SUB>
= <SPAN CLASS="EquationVariables">
A</SPAN>
<SUB CLASS="Subscript">
0</SUB>
+ <SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="Subscript">
0</SUB>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
L</SUB>
+ <SPAN CLASS="EquationVariables">
d</SPAN>
<SUB CLASS="SubscriptVariable">
A</SUB>
/2 + <SPAN CLASS="EquationVariables">
d</SPAN>
<SUB CLASS="SubscriptVariable">
D</SUB>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
L</SUB>
/2</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=132299">
</A>
</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="EquationAlign">
<A NAME="pgfId=132316">
</A>
= 0.0015 + 0.5 <SPAN CLASS="Symbol">
¥ </SPAN>
0.0789 + (–0.2828 + 0.5 <SPAN CLASS="Symbol">
¥ </SPAN>
4.6642) <SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
L</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=132303">
</A>
</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="EquationAlign">
<A NAME="pgfId=132321">
</A>
= 0.041 + 2.05<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
L</SUB>
.</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=132323">
</A>
(13.20)</P>
</TD>
</TR>
</TABLE>
<P CLASS="Body">
<A NAME="pgfId=69684">
</A>
We can adjust this delay to 0.35/0.65 trip points as follows: </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=132499">
</A>
<SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="SubscriptVariable">
ISM</SUB>
(65 %, FR)</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=132501">
</A>
</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=132503">
</A>
= <SPAN CLASS="EquationVariables">
A</SPAN>
<SUB CLASS="Subscript">
0</SUB>
+ <SPAN CLASS="EquationVariables">
D</SPAN>
<SUB CLASS="Subscript">
0</SUB>
<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
L</SUB>
+ 0.65<SPAN CLASS="EquationVariables">
O</SPAN>
<SUB CLASS="SubscriptVariable">
R</SUB>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqn">
<A NAME="pgfId=132505">
</A>
</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="EquationAlign">
<A NAME="pgfId=132507">
</A>
= 0.0015 + 0.65 <SPAN CLASS="Symbol">
¥ </SPAN>
0.0789 + ( –0.2828<SPAN CLASS="EquationVariables">
C</SPAN>
<SUB CLASS="SubscriptVariable">
L</SUB>
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