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Subject: Example 1.2
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD><TITLE>Example 1.2</TITLE>
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f>
<P>This example is taken from D. D. Gajski, <EM>Principles of Digital=20
Design</EM>, Prentice Hall, 1997, p.234.</P>
<P><FONT color=3D#0000ff><BIG><STRONG>Example =
1.2</STRONG></BIG></FONT></P>
<P><FONT face=3DArial>Derive the next state, the output table and the =
state=20
diagram for the sequential circuit shown in Figure 10.</FONT></P>
<TABLE border=3D0 width=3D"68%">
  <TBODY>
  <TR>
    <TD width=3D"61%"><IMG alt=3D"example1-2.gif (5687 bytes)"=20
      =
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/example1-2.gif=
"=20
      width=3D399 height=3D259></TD>
    <TD width=3D"39%">&nbsp;
      <P align=3Dleft><BIG>Figure 10. Logic schematic of a sequential=20
      circuit.</BIG></P></TD></TR></TBODY></TABLE>
<P align=3Dleft><BIG><STRONG>SOLUTION</STRONG></BIG>:</P>
<P align=3Dleft><FONT face=3DArial>The input combinational logic in =
Figure 10 is the=20
same as in <A=20
href=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/example1-1.ht=
m">Example=20
1.1</A>, so the excitation and the next-state equations will be the same =
as in=20
Example 1.1.</FONT></P>
<P align=3Dleft><FONT face=3DArial><STRONG>Excitation=20
equations:</STRONG>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </FONT></P>
<DIV align=3Dleft>
<TABLE border=3D0 width=3D520 height=3D2>
  <TBODY>
  <TR>
    <TD height=3D2 width=3D520>
      <P align=3Dleft><STRONG><BIG>&nbsp;&nbsp;&nbsp;&nbsp; D0 =3D =
Cnt</BIG><IMG=20
      alt=3D"ex-or.gif (942 bytes)"=20
      =
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/ex-or.gif"=20
      width=3D18 height=3D17><FONT face=3D"Times New Roman"><BIG>Q0 =3D =
Cnt'*Q0 +=20
      Cnt*Q0'</BIG></FONT></STRONG></P></TD></TR>
  <TR>
    <TD height=3D2 width=3D520><BIG><FONT=20
      face=3D"Times New Roman"><STRONG>&nbsp;&nbsp;&nbsp;&nbsp; D0 =3D =
Cnt'*Q1 +=20
      Cnt*Q1'*Q0 + =
Cnt*Q1*Q0'</STRONG></FONT></BIG></TD></TR></TBODY></TABLE></DIV>
<P align=3Dleft><STRONG><FONT face=3DArial>Next-state=20
equations:</FONT></STRONG>&nbsp;&nbsp; <FONT face=3DArial>&nbsp; =
</FONT></P>
<DIV align=3Dleft>
<TABLE border=3D0 width=3D588 height=3D2>
  <TBODY>
  <TR>
    <TD height=3D2 width=3D588>
      <P align=3Dleft><STRONG><BIG>&nbsp;&nbsp;&nbsp;&nbsp;=20
      Q0</BIG><SMALL>(next)</SMALL><BIG> =3D D0 =3D <FONT=20
      face=3D"Times New Roman">Cnt'*Q0 + =
Cnt*Q0'</FONT></BIG></STRONG></P></TD></TR>
  <TR>
    <TD height=3D2 width=3D588><FONT=20
      face=3D"Times New Roman"><STRONG><BIG>&nbsp;&nbsp;&nbsp;&nbsp;=20
      Q1</BIG><SMALL>(next)</SMALL><BIG> =3D D0 =3D Cnt'*Q1 + Cnt*Q1'*Q0 =
+=20
      Cnt*Q1*Q0'</BIG></STRONG></FONT></TD></TR></TBODY></TABLE></DIV>
<P><FONT face=3DArial>In addition, however, we have computed the output=20
equation.</FONT></P>
<P><STRONG><FONT face=3DArial>Output=20
equation</FONT>:</STRONG>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<BIG><STRONG>Y=20
=3D Q1Q0</STRONG></BIG></P>
<P><FONT face=3DArial>As this equation shows, the output Y will equal to =
1 when=20
the counter is in state Q1Q0 =3D 11, and it will stay 1 as long as the =
counter=20
stays in that state.</FONT></P>
<P><STRONG><FONT face=3DArial>Next-state and output =
table:</FONT></STRONG></P>
<DIV align=3Dleft>
<TABLE border=3D1 width=3D559 height=3D1>
  <TBODY>
  <TR>
    <TD bgColor=3D#ff8040 height=3D1 width=3D112 align=3Dmiddle>Present =
State
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%">
            <P align=3Dcenter>Q1 Q0</P></TD></TR></TBODY></TABLE></TD>
    <TD bgColor=3D#ff8040 height=3D1 width=3D87 align=3Dmiddle>Next =
State
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>Cnt=3D0</TD>
          <TD width=3D"50%" =
align=3Dmiddle>Cnt=3D1</TD></TR></TBODY></TABLE></TD>
    <TD bgColor=3D#ff8040 height=3D1 width=3D73 align=3Dmiddle>Output
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%">
            <P align=3Dcenter>Z</P></TD></TR></TBODY></TABLE></TD></TR>
  <TR>
    <TD height=3D1 width=3D112>
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0 0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0 1</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>1 0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>1 =
1</TD></TR></TBODY></TABLE></TD>
    <TD height=3D1 width=3D87>
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0 0</TD>
          <TD width=3D"50%" align=3Dmiddle>0 1</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0 1</TD>
          <TD width=3D"50%" align=3Dmiddle>1 0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>1 0</TD>
          <TD width=3D"50%" align=3Dmiddle>1 1</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>1 1</TD>
          <TD width=3D"50%" align=3Dmiddle>0 =
0</TD></TR></TBODY></TABLE></TD>
    <TD height=3D1 width=3D73>
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"50%"=20
align=3Dmiddle>1</TD></TR></TBODY></TABLE></TD></TR></TBODY></TABLE></DIV=
>
<P><STRONG><FONT face=3DArial>State diagram:</FONT></STRONG></P>
<TABLE border=3D0 width=3D"70%">
  <TBODY>
  <TR>
    <TD width=3D"45%">
      <P align=3Dcenter><A =
href=3D"javascript:loadSession('state1-2.htm')"><IMG=20
      alt=3D"state1-2.gif (5991 bytes)"=20
      =
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      width=3D342 height=3D258></A></P></TD>
    <TD width=3D"55%">
      <P align=3Dcenter>&nbsp;</P>
      <P>Figure 11. State diagram of sequential circuit in Figure =
10.</P>
      <P>To see how the states move from one to another click on the =
image.</P>
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href=3D"javascript:loadSession('state1-2.htm')"><IMG=20
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<P><STRONG><FONT face=3DArial>Timing diagram:</FONT></STRONG></P>
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  <TBODY>
  <TR>
    <TD width=3D"61%">
      <P align=3Dcenter><A =
href=3D"javascript:loadSession('timing1-2.htm')"><IMG=20
      alt=3D"timing1-2.gif (8450 bytes)" align=3Dleft=20
      =
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/timing1-2.gif"=
=20
      width=3D423 height=3D334></A></P></TD>
    <TD width=3D"39%">
      <P align=3Dcenter>&nbsp;</P>
      <P>Figure 12. Timing diagram of sequential circuit in Figure =
10.</P>
      <P>Click on the image to see its timing behaviour.</P>
      <P align=3Dcenter><A =
href=3D"javascript:loadSession('timing1-2.htm')"><IMG=20
      alt=3D"signal.gif (1039 bytes)"=20
      =
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<P><FONT face=3DArial>Note that the counter will reach the state Q1Q0 =
=3D 11 only in=20
the third clock cycle, so the output Y will equal 1 after Q0 changes to =
1. Since=20
counting is disabled in the third clock cycle, the counter will stay in =
the=20
state Q1Q0 =3D 11 and Y will stay asserted in all succeeding clock =
cycles until=20
counting is enabled again.</FONT></P>
<P>&nbsp;</P>
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