Sequential Circuit Analysis

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Sequential Circuit Analysis

• State tables show the inputs, outputs, and
  flip-flop state changes for sequential circuits.
• State diagrams are an alternative but equivalent
  way of showing the same information.

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How do you analyze a sequential circuit?

• For a combinational circuit we could find a truth
  table, which shows how the outputs are related to
  the inputs.
• A state table is the sequential analog of a truth
  table. It shows inputs and current states on the
  left, and outputs and next states on the right.
• For a sequential circuit, the outputs are
  dependent upon not only the inputs, but also the
  current state of the flip-flops.
• In addition to finding outputs, we also need to
  find the state of the flip-flops on the next
  clock cycle.

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Flip-flop input equations

• Finding the next states is harder. To do this, we
  have to figure out how the flip-flops are
  changing.
• Step 1
• Find Boolean expressions for the flip-flop
  inputs.
• I.e. How do the inputs (say, J K) to the
  flip-flops
• depend on the current state and input
• Step 2
• Use these expressions to find the actual
  flip-flop input values for each possible
  combination of present states and inputs.
• I.e. Fill in the state table (with new
  intermediate columns)
• Step 3
• Use flip-flop characteristic tables or
  equations to find the next states, based on the
  flip-flop input values and the present states.

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Derive the state table and state diagram.
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Derive the state diagram
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Step 3 Find the next states

• Finally, use the JK flip-flop characteristic
  tables or equations to find the next state of
  each flip-flop, based on its present state and
  inputs.
• The general JK flip-flop characteristic equation
  is
• Q(t1) KQ(t) JQ(t)
• In our example circuit, we have two JK
  flip-flops, so we have to apply this equation to
  each of them
• Q1(t1) K1Q1(t) J1Q1(t)
• Q0(t1) K0Q0(t) J0Q0(t)
• We can also determine the next state for
• each input/current state combination
• directly from the characteristic table.

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Sequential circuit design procedure

• Step 1
• Make a state table based on the problem
  statement. The table should show the present
  states, inputs, next states and outputs. (It may
  be easier to find a state diagram first, and then
  convert that to a table.)
• Step 2
• Assign binary codes to the states in the state
  table, if you havent already. If you have n
  states, your binary codes will have at least
• ?log2 n? digits, and your circuit will have at
  least ?log2 n? flip-flops.
• Step 3
• For each flip-flop and each row of your state
  table, find the flip-flop input values that are
  needed to generate the next state from the
  present state. You can use flip-flop excitation
  tables here.
• Step 4
• Find simplified equations for the flip-flop
  inputs and the outputs.
• Step 5
• Build the circuit!

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Class example revisited

• 1) Find non overlapping sequence 1001. Design
  using D-flipflops.

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Blank Sheet
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Design a twos complementer using J-K flip-flops

• InputX,Y. Input Sequence a binary number of
  arbitrary length. Sequence represented by
  sequential values of X. Output Z, the
  corresponding bit in the twos complement. When
  y1 gt the sequence is complete, Y0 gt
  Sequence ongoing.
• Solution The trick is in identifying that
  whenever we compute a 2's complement, we
  complement each bit and add a 1. So as long as we
  keep receiving 0's, we complement them to convert
  to 1's and then we add a 1. So until we hit our
  first 1, we stay on the same state outputting
  0's. As soon as we hit a 1, the complement for
  that is a 0 to which we add a 1 getting a 1. And
  then we have no carry so we move onto a state
  where we just complement each bit we receive.

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Blank Sheet
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Excitation tables for all flip-flops
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Summary

• The basic sequential circuit design procedure
• Make a state table and, if desired, a state
  diagram. This step is usually the hardest.
• Assign binary codes to the states if you didnt
  already.
• Use the present states, next states, and
  flip-flop excitation tables to find the flip-flop
  input values.
• Write simplified equations for the flip-flop
  inputs and outputs and build the circuit.
• Next, well look at common examples of sequential
  circuits, including different types of counters.