elct201: digital logic design - german university …...dr. eng. ahmed h. madian 3 flip-flop...

ELCT201: Digital Logic

Design Lecture 7

1

Dr. Eng. Rania.Swief e-mail: [email protected]

Dr. Eng. Haitham Omran e-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

Dr. Eng. Ahmed H. Madian 2

Outlines

• Sequential Circuit

– Flip-Flop (FF) characteristic table & equations

– FF Direct input

– Finite State Machine (FSM)

– Design of FSMs

– Analysis of FSMs


Flip-Flop Excitation Tables

Q(t+1) 0 1 0 1

Q(t) 0 0 1 1

J

0 1 X X

K

X X 1 0

S

0 1 0 X

R

X 0 1 0

T

0 1 1 0

D 0 1 0 1

You can use any FF type for your implementation

If you have the present and next outputs what’s the input

of the Flip-flops that leads to this output?


Direct input ( Asynchronous input )

– Direct preset :

Setting the flip-flop asynchronously.

– Direct clear :

Clearing the flip-flop asynchronously.

J K

Q’ Q

CLK

Clear

Inputs Outputs

Clear CLK J K Q Q’

0 x x x 0 1

1 0 0 Q0 Q0’

1 0 1 0 1

1 1 0 1 0

1 1 1 Q0’ Q0


NOW

• we have covered the memory elements

issue and we are ready to implement the

sequential circuits.

• We need to know how to Deal (analyze)

with a sequential circuit?


How could you describe

combinational circuit?

• Truth table

• Logic function between input and output • F = A+B => if A= 1 and B= 0 F= 1

if A =0 and B=0 F=0 (not depends on the pervious value of F)

How could you describe a sequential circuit?

• The sequential circuit output is now function in the inputs and past outputs

• So, we need a tool to help us describe the behavior of the circuit.


Finite State Machine (FSM)

• Finite State Machine is a tool to model the desired behavior of a

sequential system.

• The designer has to develop a finite state model of the system behavior and then designs a circuit that implements this model

• A FSM consists of several states. Inputs into the machine are combined with the current state of the machine to determine the new state or next state of the machine.

• Depending on the state of the machine, outputs are generated based on either the state or the state and inputs of the machine How to describe FSM?

– State equation ( transition equation ) input variables, present states, next states equation

– State table input variables, present states ，next states, truth table

– State diagram

FSM Example • Events:

– Wake up at fixed time every day

– Weekends: you don’t need alarm, so you wake up, turn off the alarm and resume sleep

• FSM modeling this chain of events, with: – Three states:

• Asleep

• Awake but still in bed

• Awake and up

– Inputs:

• Alarm

• Weekday (determines you how to react to alarm)

– Outputs:

• Turn off the alarm

State tables

• Similar to the truth table

• Doesn’t contain the system clock when specifying

its transitions (it is implicit that transitions occur

only when allowed by clock)

• Unless different stated, all the transitions are

occurring on the positive edge of the clock

Present

State

Inputs Next

State

Outputs

Alarm clock state table

• When you are asleep and alarm goes on, you go from being asleep to being awaked in bed; you also turn off the alarm

• The next two rows encode your actions: – You get up

– You go back to sleep

• This table doesn’t cover what you wouldn’t do…(i.e. if you are asleep and the alarm doesn't go off, you remain asleep, etc..)

Present

State

Alarm Weekday Next State Turn off alarm

Asleep On X Awake in

bed

Yes

Awake in

bed

Off Yes Awake and

up

No

Awake in

bed

Off No Asleep No

Alarm clock state table

• Covers all the cases – First row covers the situation you are asleep, the alarm

doesn’t go off and you remain asleep

– Last row covers the situation you are awake and up and you remain awake and up

– The third row covers the case you are already up and the alarm goes off. You turn it off and remain Awake in bed

Present State Alarm Weekday Next State Turn off alarm

Asleep Off X Asleep No

Asleep On X Awake in bed Yes

Awake in bed On X Awake in bed Yes

Awake in bed Off Yes Awake and up No

Awake in bed Off No Asleep No

Awake and up X X Awake and up No

State diagram

• Graphical representation of the state table

• Each state is represented by a circle vertex

• Each row of the state table is represented as a directed arc from present state vertex to the next state vertex

• In this diagram, the outputs are associated with the states

Initial State

State

0/0

Awake and up

Awake in bed Asleep

Alarm’ or Weekday/0

Alarm’/0 Input/Output

Alarm/1

Always/0

Alarm/0


Transitions ( how it works?)

• Triggered by input events the FSM moves from one state to

other based on the Transition Function.

• Transition Function produces the Output and Next State

depending on Current State and Input Event

• While in particular state FSM is not active, it is waiting for an

input to perform next activity


Design of Sequential Circuits The procedure of designing synchronous sequential circuits

can be summarized as follows:

1. From the word description and specs of the desired operation, derive a state diagram for the circuit

2. Assign binary values to the states.

3. Obtain the binary-coded state table

4. Choose the type of flip-flops to be used.

5. Derive the simplified flip-flop input and output equations.

6. Draw the logic diagram.


Example

• Using T-type FFs, design a 3-bits binary

counter that can count in binary from 0 to 7

with step 1


1-Derive the state diagram

As we need a 3 binary bits to represent the numbers from 0 to

7 so we need 1 T-FF to generate 1 binary bit

We need 3 T-FF to implement this design


2-Derive state table

T-FF excitation table


3-K-map for the FF output equations


Draw Circuit logic diagram


Types of FSM systems

There are two types of sequential systems:

•Moore FSM

•Mealy FSM


Moore FSM • Output Is a Function of a Present State Only

Present State

Register

Next State

function

Output

function

Inputs

Present State Next State

Outputs

clock reset

State

diagram state 1 /

output 1

state 2 /

output 2

transition

condition 1

transition

condition 2


Mealy FSM • Output Is a Function of a Present State and Inputs

Next State

function

Output

function

Inputs

Present State Next State

Outputs

Present State

Register clock reset

state 1 state 2

transition condition 1

/output 1

transition condition 2

/output 2

State

diagram

In this course we will consider

only Mealy FSM



• Design a FSM that detects 3 or more

consecutive ones.

Design Example 2

Our system

X Y 3 ones detector


1. Derive state diagram

S0

00

S1

01

1/0

0/0

• We will begin with assumption that no input entered to

system, we will give this state name S0

• Our system will stop at S0 if the input to the system is

always ‘0’

• If the system has ‘1’ input a new states created which

represent the state that you have only 1 we will call it S1


• If the system was in state S1 and a ‘0’ entered the system, it must be returned to S0 as we are searching for 3 consecutive ones.

• Else if another ‘1’ entered the system it will go to new state that represent existing of two ones consecutive we will call it S2.

• If the system was in state S2 and ‘0’ entered the system it will return to S0.

• Else if ‘1’ entered to the system it will move to a new state that represent the existing of three consecutive ones S3 and the output would be equal to y=‘1’.

• If the system has more ones as an input it will continue at S3 and output y=‘1’

• Finally, if we have input X=‘0’ to the system, it will goes to the starting state S0

0/0

S0

00

S1

01

S2

10

0/0

1/0 1/0

1/1 S3

11

0/0 0/0

1/1


Present State

Input Next State

Out FF Inputs

A B X A B y DA DB

0 0 0 0 0 0 0 0

0 0 1 0 1 0 0 1

0 1 0 0 0 0 0 0

0 1 1 1 0 0 1 0

1 0 0 0 0 0 0 0

1 0 1 1 1 1 1 1

1 1 0 0 0 0 0 0

1 1 1 1 1 1 1 1

2. Construct the state table

Q(t) Q(t+1) D

0 0 0

0 1 1

1 0 0

1 1 1

D–FF Excitation Table


7. Drive simplified State Equations

Note: Output here depends on the present

state (A(t)) and input (x)


8: Implement the FSM


FSM Circuit Analysis • Start with schematic diagram

• Need to determine how circuit works

– Trace schematic, determine equations of operation

• FF input equations

• sequential circuit output equations

– Create State transition table

• Sequential circuit inputs, FFs are comb. logic inputs

• Organize truth table as current state (FFs) and inputs

• Create FF input, seq. Circuit output columns

• From FF char. Tables, determine FF next state values


Sequential Circuit Analysis (cont.)

– Generate State Diagram

• Circles (nodes) represent current or present state values

• Lines (arcs) represent how state and output values change

– Given the current state and current inputs, the next state and output values are indicated by the associated arc

• State diagram can have different forms depending on the type of sequential circuit output.

Present State Value

Next State Value

Inputs/outputs


Example 3

• Analyze the following sequential circuit

• How to analyze any sequential circuit

• You need to know : – State equation

– State diagram

– State table


1- Determine State equation

• From the circuit get the logic equations of the input of flip-flops A(t+1) & B(t+1) and output y(t)


A(t+1) = A(t)X(t)+B(t)X(t)

B(t+1) = A’(t)X(t)

• Output equation

Y(t) = X’(t)(B(t)+A(t))

A(t+1)

B(t+1)

(t)

(t)

(t)


2-Create State table

• From logic equation & characteristics table of flip-flop create State table


A(t+1) = AX+BX

B(t+1) = A’X

• Output equation

Y = X’(B+A)

• D flip-flop

characteristic table


3-Construct the state diagram

00 10

01 11

0/0

Input/Output

1/0 0/1

1/0

0/1 1/0

0/1 1/0


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elct201: digital logic design - german university …...dr. eng. ahmed h. madian 3 flip-flop...

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