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Subject: Design of Sequential Circuits - Example 1.4
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<HTML><HEAD><TITLE>Design of Sequential Circuits - Example 1.4</TITLE>
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<P><BIG><BIG><STRONG>Design of Sequential =
Circuits</STRONG></BIG></BIG></P>
<P><FONT color=3D#000000>This example is taken from P. K. Lala, =
<EM>Practical=20
Digital Logic Design and Testing</EM>, Prentice Hall, 1996, =
p.176.</FONT></P>
<P><BIG><FONT color=3D#0000ff><STRONG>Example 1.4</STRONG></FONT></BIG>=20
&nbsp;&nbsp;&nbsp; <FONT face=3DArial>Design a sequential circuit whose =
state=20
tables are specified in Table 12, using D flip-flops.</FONT></P>
<P><BIG>Table 12. State table of a sequential circuit.</BIG></P>
<TABLE border=3D1 width=3D"56%">
  <TBODY>
  <TR>
    <TD bgColor=3D#ff8040 width=3D"35%">&nbsp;&nbsp;&nbsp; Present State
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%">
            <P align=3Dcenter>Q0 Q1</P></TD></TR></TBODY></TABLE></TD>
    <TD bgColor=3D#ff8040 width=3D"37%">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; =
Next State
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"50%">
            <P align=3Dcenter>x =3D 0</P></TD>
          <TD width=3D"50%">
            <P align=3Dcenter>x =3D 1</P></TD></TR></TBODY></TABLE></TD>
    <TD bgColor=3D#ff8040 =
width=3D"28%">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=20
      Output
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"50%">
            <P align=3Dcenter>x =3D 0</P></TD>
          <TD width=3D"50%">
            <P align=3Dcenter>x =3D =
1</P></TD></TR></TBODY></TABLE></TD></TR>
  <TR>
    <TD width=3D"35%">
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0 1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1 =
1</TD></TR></TBODY></TABLE></TD>
    <TD width=3D"37%">
      <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 0</TD>
          <TD width=3D"50%" align=3Dmiddle>1 0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>1 1</TD>
          <TD width=3D"50%" align=3Dmiddle>1 0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0 0</TD>
          <TD width=3D"50%" align=3Dmiddle>0 =
1</TD></TR></TBODY></TABLE></TD>
    <TD width=3D"28%">
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0</TD>
          <TD width=3D"50%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0</TD>
          <TD width=3D"50%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0</TD>
          <TD width=3D"50%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0</TD>
          <TD width=3D"50%"=20
align=3Dmiddle>1</TD></TR></TBODY></TABLE></TD></TR></TBODY></TABLE>
<P><BIG>Table 13. Excitation table for a D flip-flop.</BIG></P>
<TABLE border=3D1 width=3D"50%" height=3D87>
  <TBODY>
  <TR>
    <TD bgColor=3D#ff8040 height=3D52 width=3D"50%" =
align=3Dmiddle>Output Transitions
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%">
            <P align=3Dcenter>Q<FONT=20
            =
face=3DWingdings>=E0</FONT>Q<SMALL>(next)</SMALL></P></TD></TR></TBODY></=
TABLE></TD>
    <TD bgColor=3D#ff8040 height=3D52 width=3D"50%" =
align=3Dmiddle>Flip-flop inputs
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%">
            <P align=3Dcenter>D </P></TD></TR></TBODY></TABLE></TD></TR>
  <TR>
    <TD height=3D27 width=3D"50%">
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp;&nbsp; <FONT=20
            face=3DWingdings>=E0</FONT> &nbsp; 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0 &nbsp; <FONT =
face=3DWingdings>=E0</FONT>=20
            &nbsp; 1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1 &nbsp; <FONT =
face=3DWingdings>=E0</FONT>=20
            &nbsp;&nbsp;0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp;&nbsp; <FONT=20
            face=3DWingdings>=E0</FONT>&nbsp;&nbsp; =
1</TD></TR></TBODY></TABLE></TD>
    <TD height=3D27 width=3D"50%">
      <TABLE border=3D0 width=3D"100%" height=3D116>
        <TBODY>
        <TR>
          <TD height=3D24 width=3D"100%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD height=3D24 width=3D"100%" align=3Dmiddle>1</TD></TR>
        <TR>
          <TD height=3D24 width=3D"100%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD height=3D28 width=3D"100%"=20
  align=3Dmiddle>1</TD></TR></TBODY></TABLE></TD></TR></TBODY></TABLE>
<P><FONT face=3DArial>Next step is to derive the excitation table for =
the design=20
circuit, which is shown in Table 14. The output of the circuit is =
labelled=20
Z.</FONT></P>
<TABLE border=3D1 width=3D"75%">
  <TBODY>
  <TR>
    <TD bgColor=3D#ff8040 width=3D"34%" align=3Dmiddle>Present State
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%">
            <P align=3Dcenter>Q0 Q1</P></TD></TR></TBODY></TABLE></TD>
    <TD bgColor=3D#ff8040 width=3D"30%" align=3Dmiddle>Next State
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%">
            <P align=3Dcenter>Q0 Q1</P></TD></TR></TBODY></TABLE></TD>
    <TD bgColor=3D#ff8040 width=3D"15%" align=3Dmiddle>Input
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%">
            <P align=3Dcenter>x</P></TD></TR></TBODY></TABLE></TD>
    <TD bgColor=3D#ff8040 width=3D"40%" align=3Dmiddle>Flip-flop Inputs
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>
            <P align=3Dcenter>D0</P></TD>
          <TD width=3D"50%" align=3Dmiddle>
            <P align=3Dcenter>D1</P></TD></TR></TBODY></TABLE></TD>
    <TD bgColor=3D#ff8040 width=3D"16%" 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 width=3D"34%" align=3Dmiddle>
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0 1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0 1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1 1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1 =
1</TD></TR></TBODY></TABLE></TD>
    <TD width=3D"30%" align=3Dmiddle>
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0 1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1 1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0 =
1</TD></TR></TBODY></TABLE></TD>
    <TD width=3D"15%" align=3Dmiddle>
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"100%" =
align=3Dmiddle>1</TD></TR></TBODY></TABLE></TD>
    <TD width=3D"40%" align=3Dmiddle>
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0</TD>
          <TD width=3D"50%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0</TD>
          <TD width=3D"50%" align=3Dmiddle>1</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0</TD>
          <TD width=3D"50%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>1</TD>
          <TD width=3D"50%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>1</TD>
          <TD width=3D"50%" align=3Dmiddle>1</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>1</TD>
          <TD width=3D"50%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0</TD>
          <TD width=3D"50%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"50%" align=3Dmiddle>0</TD>
          <TD width=3D"50%" =
align=3Dmiddle>1</TD></TR></TBODY></TABLE></TD>
    <TD width=3D"16%" align=3Dmiddle>
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0</TD></TR>
        <TR>
          <TD width=3D"100%"=20
align=3Dmiddle>1</TD></TR></TBODY></TABLE></TD></TR></TBODY></TABLE>
<P><BIG>Table 14. Excitation table</BIG></P>
<P><FONT face=3DArial>Now plot the flip-flop inputs and output functions =
on the=20
Karnaugh map to derive the Boolean expressions, which is shown in Figure =

16.</FONT></P>
<P><IMG alt=3D"fig-16.gif (7355 bytes)"=20
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/fig-16.gif" =
width=3D427=20
height=3D196></P>
<P><BIG>Figure 16. Karnaugh maps</BIG></P>
<P><FONT face=3DArial>The simplified Boolean expressions are:</FONT></P>
<DIV align=3Dleft>
<TABLE border=3D0 width=3D468 height=3D5>
  <TBODY>
  <TR>
    <TD height=3D5 width=3D468><BIG><STRONG>&nbsp;&nbsp;&nbsp;&nbsp; D0 =
=3D Q0*Q1' +=20
      Q0'*Q1*x</STRONG></BIG></TD></TR>
  <TR>
    <TD height=3D5 width=3D468><BIG><STRONG>&nbsp;&nbsp;&nbsp;&nbsp; D1 =
=3D=20
      Q0'*Q1'*x + Q0*Q1*x + Q0*Q1'*x'</STRONG></BIG></TD></TR>
  <TR>
    <TD height=3D5 =
width=3D468><BIG><STRONG>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Z=20
      =3D Q0*Q1*x</STRONG></BIG></TD></TR></TBODY></TABLE></DIV>
<P><FONT face=3DArial>Finally, draw the logic diagram.</FONT></P>
<P><IMG alt=3D"fig-17.gif (7367 bytes)"=20
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/fig-17.gif" =
width=3D499=20
height=3D238></P>
<P><BIG>Figure 17. Logic diagram of the sequential circuit.</BIG></P>
<P>&nbsp;</P>
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