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f>
<P><BIG><STRONG><BIG><BIG>State Tables and State=20
Diagrams</BIG></BIG></STRONG></BIG></P>
<P><FONT face=3DArial>We have examined a general model for sequential =
circuits. In=20
this model the effect of all previous inputs on the outputs is =
represented by a=20
state of the circuit. Thus, the output of the circuit at any time =
depends upon=20
its current state and the input. These also determine the next state of =
the=20
circuit. The relationship that exists among the inputs, outputs, present =
states=20
and next states can be specified by either the <STRONG>state =
table</STRONG> or=20
the <STRONG>state diagram</STRONG>.</FONT></P>
<P><FONT color=3D#0000ff><BIG><BIG><STRONG>State=20
Table</STRONG></BIG></BIG></FONT></P>
<P><FONT face=3DArial>The state table representation of a sequential =
circuit=20
consists of three sections labelled <EM>present state</EM>, <EM>next =
state</EM>=20
and <EM>output</EM>. The present state designates the state of =
flip-flops before=20
the occurrence of a clock pulse. The next state shows the states of =
flip-flops=20
after the clock pulse, and the output section lists the value of the =
output=20
variables during the present state.</FONT></P>
<P><FONT color=3D#0000ff><BIG><BIG><STRONG>State=20
Diagram</STRONG></BIG></BIG></FONT></P>
<P><FONT face=3DArial>In addition to graphical symbols, tables or =
equations,=20
flip-flops can also be represented graphically by a state diagram. In =
this=20
diagram, a state is represented by a circle, and the transition between =
states=20
is indicated by directed lines (or arcs) connecting the circles. An =
example of a=20
state diagram is shown in Figure 3 below.</FONT></P>
<TABLE border=3D0 width=3D"64%">
  <TBODY>
  <TR>
    <TD width=3D"40%"><IMG alt=3D"sd.gif (6231 bytes)"=20
      src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/sd.gif" =
width=3D380=20
      height=3D325></TD>
    <TD width=3D"60%">
      <P align=3Dcenter><BIG>Figure 3. State =
Diagram</BIG></P></TD></TR></TBODY></TABLE>
<P align=3Dleft><FONT face=3DArial>The binary number inside each circle =
identifies=20
the state the circle represents. The directed lines are labelled with =
two binary=20
numbers separated by a slash (/). The input value that causes the state=20
transition is labelled first. The number after the slash symbol / gives =
the=20
value of the output. For example, the directed line from state 00 to 01 =
is=20
labelled 1/0, meaning that, if the sequential circuit is in a present =
state and=20
the input is 1, then the next state is 01 and the output is 0. If it is =
in a=20
present state 00 and the input is 0, it will remain in that state. A =
directed=20
line connecting a circle with itself indicates that no change of state =
occurs.=20
The state diagram provides exactly the same information as the state =
table and=20
is obtained directly from the state table.</FONT></P>
<P align=3Dleft><FONT =
color=3D#0000ff><BIG><STRONG>Example</STRONG></BIG></FONT>:=20
This example is taken from P. K. Lala, <EM>Practical Digital Logic =
Design and=20
Testing</EM>, Prentice Hall, 1996, p.155.</P>
<P align=3Dleft><FONT face=3DArial>Consider a sequential circuit shown =
in Figure 4.=20
It has one input x, one output Z and two state variables Q1Q2 (thus =
having four=20
possible present states 00, 01, 10, 11).</FONT></P>
<TABLE border=3D0 width=3D"68%">
  <TBODY>
  <TR>
    <TD width=3D"50%"><IMG alt=3D"sq-expl.gif (4827 bytes)"=20
      =
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/sq-expl.gif"=20
      width=3D366 height=3D272></TD>
    <TD width=3D"50%">
      <P align=3Dcenter><BIG>Figure 4. A Sequential=20
Circuit</BIG></P></TD></TR></TBODY></TABLE>
<P align=3Dleft><FONT face=3DArial>The behaviour of the circuit is =
determined by the=20
following Boolean expressions:</FONT></P>
<TABLE border=3D0 width=3D"41%">
  <TBODY>
  <TR>
    <TD width=3D"100%">&nbsp;&nbsp;&nbsp;&nbsp; <BIG><STRONG><FONT=20
      face=3D"Times New Roman">&nbsp; Z =3D =
x*Q1</FONT></STRONG></BIG></TD></TR>
  <TR>
    <TD width=3D"100%">&nbsp;&nbsp;&nbsp; <STRONG><BIG><FONT=20
      face=3D"Times New Roman">D1 =3D x' + =
Q1</FONT></BIG></STRONG></TD></TR>
  <TR>
    <TD width=3D"100%">&nbsp;&nbsp;&nbsp; <STRONG><BIG><FONT=20
      face=3D"Times New Roman">D2 =3D x*Q2' +=20
  x'*Q1'</FONT></BIG></STRONG></TD></TR></TBODY></TABLE>
<P align=3Dleft><FONT face=3DArial>These equations can be used to form =
the state=20
table. Suppose the present state (i.e. Q1Q2) =3D 00 and input x =3D 0. =
Under these=20
conditions, we get Z =3D 0, D1 =3D 1, and D2 =3D 1. Thus the next state =
of the circuit=20
D1D2 =3D 11, and this will be the present state after the clock pulse =
has been=20
applied. The output of the circuit corresponding to the present state =
Q1Q2 =3D 00=20
and x =3D 1 is Z =3D 0. This data is entered into the state table as =
shown in Table=20
2.</FONT></P>
<TABLE border=3D0 width=3D"97%">
  <TBODY>
  <TR>
    <TD width=3D"100%">
      <TABLE border=3D1 width=3D"52%">
        <TBODY>
        <TR>
          <TD bgColor=3D#ff8040 width=3D"30%" align=3Dmiddle>Present =
State
            <TABLE border=3D0 width=3D"100%">
              <TBODY>
              <TR>
                <TD width=3D"100%">
                  <P =
align=3Dcenter>Q1Q2</P></TD></TR></TBODY></TABLE></TD>
          <TD bgColor=3D#ff8040 width=3D"35%" align=3Dmiddle>Next State
            <TABLE border=3D0 width=3D"100%">
              <TBODY>
              <TR>
                <TD width=3D"50%" align=3Dmiddle>x =3D 0</TD>
                <TD width=3D"50%" align=3Dmiddle>x =3D =
1</TD></TR></TBODY></TABLE></TD>
          <TD bgColor=3D#ff8040 width=3D"35%" align=3Dmiddle>Output
            <TABLE border=3D0 width=3D"100%">
              <TBODY>
              <TR>
                <TD width=3D"50%" align=3Dmiddle>x =3D 0</TD>
                <TD width=3D"50%" align=3Dmiddle>x =3D =
1</TD></TR></TBODY></TABLE></TD></TR>
        <TR>
          <TD width=3D"30%">
            <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"35%">
            <TABLE border=3D0 width=3D"100%">
              <TBODY>
              <TR>
                <TD width=3D"50%" align=3Dmiddle>1 1</TD>
                <TD width=3D"50%" align=3Dmiddle>0 1</TD></TR>
              <TR>
                <TD width=3D"50%" align=3Dmiddle>1 1</TD>
                <TD width=3D"50%" align=3Dmiddle>0 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 0</TD>
                <TD width=3D"50%" align=3Dmiddle>1 =
0</TD></TR></TBODY></TABLE></TD>
          <TD width=3D"35%">
            <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>1</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></TD>=
</TR></TBODY></TABLE>
<P><BIG>Table 2. State table for the sequential circuit in Figure =
4.</BIG></P>
<P><FONT face=3DArial>The state diagram for the sequential circuit in =
Figure 4 is=20
shown in Figure 5.</FONT></P>
<TABLE border=3D0 width=3D"63%">
  <TBODY>
  <TR>
    <TD width=3D"50%"><IMG alt=3D"state-ex.gif (6230 bytes)"=20
      =
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/state-ex.gif" =

      width=3D342 height=3D295></TD>
    <TD width=3D"50%">
      <P align=3Dleft><BIG>Figure 5. State Diagram of circuit in Figure=20
      4.</BIG></P></TD></TR></TBODY></TABLE>
<P align=3Dleft>&nbsp;</P>
<P align=3Dleft><FONT color=3D#0000ff><BIG><BIG><STRONG>State Diagrams =
of Various=20
Flip-flops</STRONG></BIG></BIG></FONT></P>
<P align=3Dleft><FONT face=3DArial>Table 3 shows the state diagrams of =
the four=20
types of flip-flops.</FONT></P>
<TABLE border=3D0 width=3D"100%">
  <TBODY>
  <TR>
    <TD width=3D"100%" align=3Dmiddle>
      <TABLE border=3D1 width=3D"47%" height=3D551>
        <TBODY>
        <TR>
          <TD bgColor=3D#ff8040 height=3D22 width=3D"22%">
            <P align=3Dcenter>NAME</P></TD>
          <TD bgColor=3D#ff8040 height=3D22 width=3D"78%">
            <P align=3Dcenter>STATE DIAGRAM</P></TD></TR>
        <TR>
          <TD bgColor=3D#69e2c1 height=3D125 width=3D"22%">
            <P align=3Dcenter><STRONG>SR</STRONG></P></TD>
          <TD bgColor=3D#ffffff height=3D125 width=3D"78%">
            <P align=3Dcenter><IMG alt=3D"SR-st.gif (3269 bytes)"=20
            =
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/SR-st.gif"=20
            width=3D334 height=3D108></P></TD></TR>
        <TR>
          <TD bgColor=3D#69e2c1 height=3D140 width=3D"22%">
            <P align=3Dcenter><STRONG>JK</STRONG></P></TD>
          <TD bgColor=3D#ffffff height=3D140 width=3D"78%" =
align=3Dmiddle><IMG=20
            alt=3D"JK-st.gif (3429 bytes)"=20
            =
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/JK-st.gif"=20
            width=3D333 height=3D124></TD></TR>
        <TR>
          <TD bgColor=3D#69e2c1 height=3D113 width=3D"22%">
            <P align=3Dcenter><STRONG>D</STRONG></P></TD>
          <TD bgColor=3D#ffffff height=3D113 width=3D"78%" =
align=3Dmiddle><IMG=20
            alt=3D"D-st.gif (3325 bytes)"=20
            =
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/D-st.gif"=20
            width=3D334 height=3D99></TD></TR>
        <TR>
          <TD bgColor=3D#69e2c1 height=3D121 width=3D"22%">
            <P align=3Dcenter><STRONG>T</STRONG></P></TD>
          <TD bgColor=3D#ffffff height=3D121 width=3D"78%" =
align=3Dmiddle><IMG=20
            alt=3D"T-st.gif (3292 bytes)"=20
            =
src=3D"http://www.eelab.usyd.edu.au/digital_tutorial/part3/T-st.gif"=20
            width=3D334 =
height=3D100></TD></TR></TBODY></TABLE></TD></TR></TBODY></TABLE>
<P align=3Dcenter><BIG>Table 3. State diagrams of the four types of=20
flip-flops.</BIG></P>
<P align=3Dleft><FONT face=3DArial>You can see from the table that all =
four=20
flip-flops have the same number of states and transitions. Each =
flip-flop is in=20
the set state when Q=3D1 and in the reset state when Q=3D0. Also, each =
flip-flop can=20
move from one state to another, or it can re-enter the same state. The =
only=20
difference between the four types lies in the values of input signals =
that cause=20
these transitions.</FONT></P>
<P align=3Dleft><FONT face=3DArial>A state diagram is a very convenient =
way to=20
visualise the operation of a flip-flop or even of large sequential=20
components.</FONT></P>
<P align=3Dleft><A =
href=3D"javascript:loadSession('quiz4.htm')"><BIG><STRONG>Short=20
Quiz</STRONG></BIG></A></P>
<P align=3Dleft>&nbsp;</P>
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