Lecture 22 Sequential Circuits Analysis

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Lecture 22Sequential Circuits Analysis
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Combinational vs. Sequential

• Combinational Logic Circuit
• Output is a function only of the present inputs.
• Does not have state information.
• Does not require memory.
• Sequential Logic Circuit (Finite State Machine)
• Output is a function of the present state and at
  times present state and input.
• Has state information
• Requires memory.
• Uses Flip-Flops to implement memory.

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Synchronous vs. Asynchronous

• Synchronous Sequential Logic Circuit
• Clocked
• All Flip-Flops use the same clock and change
  state on the same triggering edge.
• Asynchronous Sequential Logic Circuit
• No clock
• Can change state at any instance in time.
• Faster but more complex than synchronous
  sequential circuits.

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General Models for Sequential Circuits
A sequential circuit can be divided conveniently
into two parts -- the flip-flops which serve as
memory for the circuit and the combinational
logic which realizes the input functions for the
flip-flops and the output functions. The
combinational logic may be implemented with
gates, with a ROM, or with a PLA.
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Sequential Logic (Why) ?

• Sequential circuit has additional dimension which
  is time
• Combinational logic only depends on current input
• Sequential circuit output depends on previous
  input other than current input
• More powerful than combinational logic
• Able to model condition that cant be
  accommodated by combinational logic

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Analysis of Clocked Sequential Circuits

• Analysis of a sequential circuit consists of
  obtaining a table or a diagram for the time
  sequence of inputs, outputs, and internal states.
• Sequential circuit behavior is determined from
  the inputs, the outputs, and the state of its
  flip-flops
• Boolean expressions that describe the behavior of
  the sequential circuit
• Outputs and the next state are both a function of
  the inputs and the present state
• A logic diagram is recognized as a clocked
  sequential circuit if it includes flip-flops.
• Logic diagram may or may not include
  combinational circuit gates.

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The Current State

• It is inconvenient, and often impossible, to
  describe the behaviour of a sequential circuit by
  means of a table that lists outputs as a function
  of the input sequence that has been received up
  until the current time.
• To know where you are going next, you need to
  know where you are now.
• With the TV channel selector, it is impossible to
  determine what channel is currently selected by
  looking only at the preceding sequence of
  presses, whether we look at the preceding 10
  presses or the preceding 1000.
• More information, the current state of the
  channel selector, is needed.

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State

• The state of a sequential circuit is a
  collection of state variables whose values at any
  particular time contain all the information about
  the past necessary to account for the circuits
  future behaviour.
• In the channel-selector example, the current
  channel number is the current state.
• Inside the TV, this state might be stored as
  seven binary state variables representing a
  decimal number between 1 and 9.
• Given the current state (channel number), we can
  always predict the next state as a function of
  the inputs (up/down pushes).

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Finite-State Machines

• In a digital circuit, state variables have binary
  values.
• A circuit with n binary state variables has 2n
  possible states.
• 2n is always finite, so sequential circuits are
  sometimes called finite-state machines.

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D Flip-Flop with Clock input

• Q(t1) Q?.D Q.D D.(Q? Q) D.1 D

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Boolean equation for D Flip-Flop

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Sequential Circuit Analysis

• Given sequential circuit diagram, behavioral
  analysis from state table and also state diagram
• Need state equations to get flip-flop input and
  output functions for circuit output other than
  flip-flop (if any)
• A(t) and A(t1) are used to represent current
  state and the next state for flip-flop.
• A and A can also be used in order to represent
  current state and the following state

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Sequential Circuit Analysis

• Example (using D flip-flop)
• State equation
• Output Function

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Sequential Circuit Analysis

• From the state equations and output function,
  state table can be derived that contains all
  combined binary combination for the current
  condition (present state) and input
• State table
• The same as Truth Table
• Input and condition pad on the left
• Output and next condition on the right
• Combined binary combination available for current
  state and input
• M flip-flop and n input gt 2mn line

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Sequential Circuit Analysis

• From the state equations and output function,
  state table can be derived that contains all
  combined binary combination for the current
  condition (present state) and input
• State table
• The same as Truth Table
• Input and condition pad on the left
• Output and next condition on the right
• combined binary combination available for current
  state and input
• M flip-flop and n input gt 2mn line

State table for circuit in Example 1
State equation Output function
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Sequential Circuit Analysis

• Other method

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Sequential Circuit Analysis

• From the truth table, we can draw state diagram
• State diagram
• Each state is represented by circle
• Each arrow (between two circle) represent
  transfer for sequential logic (i.e. line
  transition in truth table)
• a/b label for each arrow where a represent inputs
  and b represent output for circuit in transition
• Each flip-flop value combination represent state.
  Therefore, m flip-flopgt until 2m state.

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Sequential Circuit Analysis

• State diagram for circuit in previous example

• Each state is represented by circle
• Each arrow (between two circle) represent
  transfer for sequential logic (i.e. line
  transition in truth table)
• a/b label for each arrow where a represent inputs
  and b represent output for circuit in transition

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Flip-flop Input Function

• Output of sequential circuit is a function of the
  current state of the flip-flop and the input.
  This is explained using algebra by circuit output
  function
• In previous example y (AB)x
• Circuit part that generate input to flip-flop is
  represented by using Boolean equation and is
  known as flip-flop inputs function
• Flip-flop input function determine next state
• From flip-flop input function and criteria table
  for flip-flop, next state of the flip-flop is
  obtained

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Flip-flop Input Function

• Example circuit with JK flip flop
• 2 characters are used in order to represent
  flip-flop input first character represents the
  flip-flop input (J or K for JK flip-flop, S or R
  for SR flip-flop, D for D flip-flop, T for T
  flip-flop respectively) and the second character
  represents the name of the flip-flop

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Analysis Example

• Given a sequential circuit with two JK flip-flop,
  namely A, B and one input x
• Flip-flop input function obtained from the
  circuit is

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Analysis Example

• Input flip-flop function
• Fill the state table with the above function
  using criteria table for the used flip-flop

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Analysis Example

• Draw state diagram from the state table

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Flip-flop Excitation Tables

• Analysis Vs Design
• Analysis Start from circuit diagram, build state
  table or state diagram
• Design Start from specification set (i.e. in
  state equation form, state table or state
  diagram) build logic circuit.
• Criteria table is used in analysis
• Excitation tables is used in design

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Flip-flop Excitation Tables

• Excitation tables it give transition
  characteristic between current state and next
  state to determine flip-flop input

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Designing Sequential Circuit

• Design steps
• Start with circuit specification characteristic
  of circuit
• Build state table
• Perform state reduction if required
• State assignment
• Determine number of flip-flop ( that has to be
  used)
• Build circuit excitation and output table from
  state table
• Build circuit output function and flip-flop input
  function
• Draw logic diagram

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Design Example

• Given state diagram as follows, get the
  sequential circuit using JK flip-flop

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Design Example

• State/excitation table using JK flip-flop

For example, in the first row of Table (bottom
right), we have a transition for flip-flop A from
0 in the present state to 0 in the next state. In
Table (excitation table), we find that a
transition of states from 0 to 0 requires that
input J 0 and input K X
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Design Example

• Block diagram

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Design Example

• From state table, get input flip-flop function

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Design Example

• Input flip-flop function
• Logic Diagram

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Design Example

• Design, using D flip-flop, circuit is based on
  state table below.

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Design Example

• Determine input expression for flip-flop and y
  output

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Design Example

• From Boolean expressions built, draw logic diagram

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Design Example

• How if using JK flip-flop (Homework)

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Design a Synchronous Counter

• Counter sequential circuit cycle through state
  sequence
• Binary counter follow binary sequence. n-bit
  binary counter (with n flip-flop) able to count
  from 0 to 2n-1.
• Example 3-bit binary counter (using T flip-flop)

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Design a Synchronous Counter

• 3-bit binary counter (cont)

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Design a Synchronous Counter

• 3-bit binary counter

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Sequential Circuit Design

• Sequential circuit consists of
• A combinational circuit that produces output
• A feedback circuit
• We use JK flip-flops for the feedback circuit
• Simple counter examples using JK flip-flops
• Provides alternative counter designs
• We know the output
• Need to know the input combination that produces
  this output
• Use an excitation table
• Built from the truth table

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Sequential Circuit Design (contd)
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Sequential Circuit Design (contd)

• Build a design table that consists of
• Current state output
• Next state output
• JK inputs for each flip-flop
• Binary counter example
• 3-bit binary counter
• 3 JK flip-flops are needed
• Current state and next state outputs are 3 bits
  each
• 3 pairs of JK inputs

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Sequential Circuit Design (contd)
Design table for the binary counter example
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Sequential Circuit Design (contd)
Use K-maps to simplify expressions for JK inputs
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Sequential Circuit Design (contd)

• Final circuit for the binary counter example
• Compare this design with the synchronous counter
  design

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• Thanks