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Subject: Design of Counters - Example 1.5
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<P><BIG><STRONG><BIG>Design of Counters</BIG></STRONG></BIG></P>
<P>This example is taken from T. L. Floyd, <EM>Digital =
Fundamentals</EM>, Fourth=20
Edition, Macmillan Publishing, 1990, p.395.</P>
<P><BIG><FONT color=3D#0000ff><STRONG>Example=20
1.5</STRONG></FONT></BIG>&nbsp;&nbsp; <FONT face=3DArial>A counter is =
first=20
described by a state diagram, which is shows the sequence of states =
through=20
which the counter advances when it is clocked. Figure 18 shows a state =
diagram=20
of a 3-bit binary counter.</FONT></P>
<TABLE border=3D0 width=3D"62%">
  <TBODY>
  <TR>
    <TD width=3D"37%"><IMG alt=3D"fig-18.gif (5027 bytes)"=20
      =
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/fig-18.gif"=20
      width=3D270 height=3D260></TD>
    <TD width=3D"63%"><BIG>Figure 18. State diagram of a 3-bit binary=20
      counter.</BIG></TD></TR></TBODY></TABLE>
<P><FONT face=3DArial>The <A name=3Dcircuit>circuit</A> has no inputs =
other than the=20
clock pulse and no outputs other than its internal state (outputs are =
taken off=20
each flip-flop in the counter). The next state of the counter depends =
entirely=20
on its present state, and the state transition occurs every time the =
clock pulse=20
occurs. Figure 19 shows the sequences of count after each clock=20
pulse.</FONT></P>
<P><IMG alt=3D"fig-19.gif (36958 bytes)"=20
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/fig-19.gif" =
width=3D427=20
height=3D330></P>
<P><FONT face=3DArial>Once the sequential circuit is defined by the =
state diagram,=20
the next step is to obtain the next-state table, which is derived from =
the state=20
diagram in Figure 18 and is shown in Table 15.</FONT></P>
<P><BIG>Table 15. State table</BIG></P>
<TABLE border=3D1 width=3D"48%">
  <TBODY>
  <TR>
    <TD bgColor=3D#ff8040 width=3D"50%" align=3Dmiddle>Present State
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%">
            <P align=3Dcenter>Q2 Q1 =
Q0</P></TD></TR></TBODY></TABLE></TD>
    <TD bgColor=3D#ff8040 width=3D"50%" align=3Dmiddle>Next State
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%">
            <P align=3Dcenter>Q2 Q1 =
Q0</P></TD></TR></TBODY></TABLE></TD></TR>
  <TR>
    <TD width=3D"50%" align=3Dmiddle>
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp;&nbsp; 0&nbsp;&nbsp; =
0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp;&nbsp; 0&nbsp;&nbsp; =
1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp;&nbsp; 1&nbsp;&nbsp; =
0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp;&nbsp; 1&nbsp;&nbsp; =
1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp;&nbsp; 0&nbsp;&nbsp; =
0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp;&nbsp; 0&nbsp;&nbsp; =
1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp;&nbsp; 1&nbsp;&nbsp; =
0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp;&nbsp; 1&nbsp;&nbsp;=20
        1</TD></TR></TBODY></TABLE></TD>
    <TD width=3D"50%" align=3Dmiddle>
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp;&nbsp; 0&nbsp;&nbsp; =
1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp;&nbsp; 1&nbsp;&nbsp; =
0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp;&nbsp; 1&nbsp;&nbsp; =
1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp;&nbsp; 0&nbsp;&nbsp; =
0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp;&nbsp; 0&nbsp;&nbsp; =
1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp;&nbsp; 1&nbsp;&nbsp; =
0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp;&nbsp; 1&nbsp;&nbsp; =
1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp;&nbsp; 0&nbsp;&nbsp;=20
        0</TD></TR></TBODY></TABLE></TD></TR></TBODY></TABLE>
<P><FONT face=3DArial>Since there are eight states, the number of =
flip-flops=20
required would be three. Now we want to implement the counter design =
using JK=20
flip-flops.</FONT></P>
<P><FONT face=3DArial>Next step is to develop an excitation table from =
the state=20
table, which is shown in Table 16.</FONT></P>
<P>Table 16. Excitation table</P>
<TABLE border=3D1 width=3D"68%">
  <TBODY>
  <TR>
    <TD bgColor=3D#6feccd width=3D"58%" colSpan=3D2>
      <P align=3Dcenter>Output State Transitions</P></TD>
    <TD bgColor=3D#ff8040 rowSpan=3D2 width=3D"42%">
      <P align=3Dcenter>Flip-flop inputs</P>
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"33%" align=3Dmiddle>J2 K2</TD>
          <TD width=3D"33%" align=3Dmiddle>J1 K1</TD>
          <TD width=3D"34%" align=3Dmiddle>J0 =
K0</TD></TR></TBODY></TABLE></TD></TR>
  <TR>
    <TD bgColor=3D#ff8040 width=3D"28%" align=3Dmiddle>Present State
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%">
            <P align=3Dcenter>Q2 Q1 =
Q0</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>Q2 Q1 =
Q0</P></TD></TR></TBODY></TABLE></TD></TR>
  <TR>
    <TD width=3D"28%">
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp; 0&nbsp; 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp; 0&nbsp; 1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp; 1&nbsp; 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp; 1&nbsp; 1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp; 0&nbsp; 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp; 0&nbsp; 1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp; 1&nbsp; 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp; 1&nbsp;=20
    1</TD></TR></TBODY></TABLE></TD>
    <TD width=3D"30%">
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp; 0&nbsp; 1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp; 1&nbsp; 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp; 1&nbsp; 1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp; 0&nbsp; 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp; 0&nbsp; 1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp; 1&nbsp; 0</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>1&nbsp; 1&nbsp; 1</TD></TR>
        <TR>
          <TD width=3D"100%" align=3Dmiddle>0&nbsp; 0&nbsp;=20
    0</TD></TR></TBODY></TABLE></TD>
    <TD width=3D"42%">
      <TABLE border=3D0 width=3D"100%">
        <TBODY>
        <TR>
          <TD width=3D"33%" align=3Dmiddle>0 X</TD>
          <TD width=3D"33%" align=3Dmiddle>0 X</TD>
          <TD width=3D"34%" align=3Dmiddle>1 X</TD></TR>
        <TR>
          <TD width=3D"33%" align=3Dmiddle>0 X</TD>
          <TD width=3D"33%" align=3Dmiddle>1 X</TD>
          <TD width=3D"34%" align=3Dmiddle>X 1</TD></TR>
        <TR>
          <TD width=3D"33%" align=3Dmiddle>0 X</TD>
          <TD width=3D"33%" align=3Dmiddle>X 0</TD>
          <TD width=3D"34%" align=3Dmiddle>1 X</TD></TR>
        <TR>
          <TD width=3D"33%" align=3Dmiddle>1 X</TD>
          <TD width=3D"33%" align=3Dmiddle>X 1</TD>
          <TD width=3D"34%" align=3Dmiddle>X 1</TD></TR>
        <TR>
          <TD width=3D"33%" align=3Dmiddle>X 0</TD>
          <TD width=3D"33%" align=3Dmiddle>0 X</TD>
          <TD width=3D"34%" align=3Dmiddle>1 X</TD></TR>
        <TR>
          <TD width=3D"33%" align=3Dmiddle>X 0</TD>
          <TD width=3D"33%" align=3Dmiddle>1 X</TD>
          <TD width=3D"34%" align=3Dmiddle>X 1</TD></TR>
        <TR>
          <TD width=3D"33%" align=3Dmiddle>X 0</TD>
          <TD width=3D"33%" align=3Dmiddle>X 0</TD>
          <TD width=3D"34%" align=3Dmiddle>1 X</TD></TR>
        <TR>
          <TD width=3D"33%" align=3Dmiddle>X 1</TD>
          <TD width=3D"33%" align=3Dmiddle>X 1</TD>
          <TD width=3D"34%" align=3Dmiddle>X=20
1</TD></TR></TBODY></TABLE></TD></TR></TBODY></TABLE>
<P><FONT face=3DArial>Now transfer the JK states of the flip-flop inputs =
from the=20
excitation table to Karnaugh maps to derive a simplified Boolean =
expression for=20
each flip-flop input. This is shown in Figure 20.</FONT></P>
<TABLE border=3D0 width=3D"70%">
  <TBODY>
  <TR>
    <TD width=3D"62%"><IMG alt=3D"fig-20.gif (18403 bytes)"=20
      =
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/fig-20.gif"=20
      width=3D417 height=3D382></TD>
    <TD width=3D"38%"><BIG>Figure 20. Karnaugh =
maps</BIG></TD></TR></TBODY></TABLE>
<P><FONT face=3DArial>The 1s in the Karnaugh maps of Figure 20 are =
grouped with=20
"don't cares" and the following expressions for the J and K inputs of =
each=20
flip-flop are obtained:</FONT></P>
<DIV align=3Dleft>
<TABLE border=3D0 width=3D287 height=3D0>
  <TBODY>
  <TR>
    <TD height=3D0 width=3D287><STRONG><BIG>&nbsp;&nbsp;&nbsp;&nbsp; J0 =
=3D K0 =3D=20
      1</BIG></STRONG></TD></TR>
  <TR>
    <TD height=3D0 width=3D287><STRONG><BIG>&nbsp;&nbsp;&nbsp;&nbsp; J1 =
=3D K1 =3D=20
      Q0</BIG></STRONG></TD></TR>
  <TR>
    <TD height=3D0 width=3D287><STRONG><BIG>&nbsp;&nbsp;&nbsp;&nbsp; J2 =
=3D K2 =3D=20
      Q1*Q0</BIG></STRONG></TD></TR></TBODY></TABLE></DIV>
<P><FONT face=3DArial>The final step is to implement the combinational =
logic from=20
the equations and connect the flip-flops to form the sequential circuit. =
The=20
complete logic of a 3-bit binary counter is shown in Figure =
21.</FONT></P>
<TABLE border=3D0 width=3D"68%" height=3D117>
  <TBODY>
  <TR>
    <TD height=3D113 width=3D"49%"><IMG alt=3D"fig-21.gif (3527 bytes)"=20
      =
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/fig-21.gif"=20
      width=3D371 height=3D115></TD>
    <TD height=3D113 width=3D"51%"><BIG>Figure 21. Logic diagram of a =
3-bit binary=20
      counter.</BIG></TD></TR></TBODY></TABLE>
<P>&nbsp;</P>
<P><A=20
href=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/example1-6.ht=
m"><BIG><STRONG>Example=20
1.6</STRONG></BIG></A></P>
<P>&nbsp;</P>
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