Lecture Digital logic design - Lecture 22: Sequential circuits analysis

The following will be discussed in this chapter: Combinational vs. sequential, synchronous vs. asynchronous, general models for sequential circuits, sequential logic (why)? analysis of clocked sequential circuits, the current “state”, sequential circuit analysis,...

Sequential Circuits Analysis

 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

 Uses Flip-Flops to implement memory.

 Synchronous Sequential Logic Circuit

 All Flip-Flops use the same clock and

change state on the same triggering edge.

 Asynchronous Sequential Logic Circuit

 Can change state at any instance in

time.

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.

° 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 can’t be

accommodated by combinational logic

° 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.

° 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

° 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 circuit’s 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).

° In a digital circuit, state variables have binary

values.

° A circuit with n binary state variables has 2 n

possible states.

° 2 n is always finite, so sequential circuits are

sometimes called finite-state machines.

° 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(t+1) 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

° Example (using D flip-flop)

State equation

Output Function

° 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 => 2 m+n line

State table for circuit in Example 1

° 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 => 2 m+n line

State equation Output function

Other method

° 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-flop=> until 2 m state.

° 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

° 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= (A+B)x’

° Circuit part that generate input to flip-flop is

represented by using Boolean equation and is known

as flip-flop input’s function

° Flip-flop input function determine next state

° From flop input function and criteria table for

flip-flop, next state of the flip-flop is obtained

° 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

namely A, B and one input x

° Flip-flop input function obtained from the circuit is

° 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 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

between current state and next state to determine flop input

flip-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

° Given state diagram as follows, get the sequential

circuit using JK flip-flop

° State/excitation table using JK flip-flop

For example, in the first row of Table (bottom right), we have a transition for flop A from 0 in the present state to 0 in the next state In Table (excitation table),

flip-we find that a transition of states from 0 to 0 requires that input J= 0 and input K

= X

° Block diagram

° From state table, get input flip-flop function

° Logic Diagram

table below.

° Determine input expression for flip-flop and y

output

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° How if using JK flip-flop (Homework)

° 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 2 n -1.

° Example: 3-bit binary counter (using T flip-flop)

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° 3-bit binary counter

° 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

° 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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• Compare this design with the synchronous counter

design

Thanks