Chapter5: Synchronous Sequential
Logic – Part 2
Originally by Reham S. Al-Majed
Imam Muhammad Bin Saud University
Outline

Analysis of clocked SC




Analysis with D FF.
Analysis with JK FF.
Analysis with T FF.
Mealy and Moore Models of FSMs.
2
SC Analysis

Analysis describes what a given circuit will do under certain
conditions.

How changes in input with the current state will affect next state and
output.

Given: logic diagram of a SC consists of FF and gates.

Goal:
Find State Equations ( transition equations).
1.


2.
3.
They describe the behavior of SC algebraically.
They specify the next state as a function of the present state and inputs.
Find State Table.  need state equations.
Find State Diagram.  need state table.
3
SC Analysis

To achieve any of the previous goals of SC Analysis we need to find:

Input equations( also called excitation equations): Boolean functions that
describe the part of the circuit that generate inputs to FF algebraically.

Output equations: Boolean functions that describe algebraically the part of the
CC that generate external output.

Characteristic equations of the used FF:

The characteristic equation for the D FF:
Q(t+1) = D
 The characteristic equation for the JK FF:
Q(t+1)= JQ’+K’Q
 The characteristic equation for the T FF:
Q(t+1)= T⨁Q

Input and output equations provide the necessary information to draw the
logic diagram of SC.
4
Example

Find the input and output equations of the following SC logic diagram.

FFs input equations:
DA= Ax + Bx
DB= A’x

Output equation:
y=(A+B)x’
5
SC Analysis

To achieve the first goal ( finding state equations):
1.
Determine input equations.
2.
Substitute the input equations into FF characteristic equations to obtain the
next state of all FFs.
3.
Write down the output equations.
6
Example
Analysis with D FF

Analyze the logic diagram by finding the state equations:

FFs input equations:
DA= Ax + Bx
DB= A’x

The characteristic equation for D FF : Q(t+1) = D
Substitute to find state equations:
A(t+1) = DA  A(t+1) = Ax + Bx
B(t+1)= DB  B(t+1) = A’x


Output equation:
y=(A+B)x’
7
SC Analysis

To achieve the second goal ( finding state table)
1.
After finding the state equations and output equations.
2.
Draw a table that:

Consists of four sections: present state, input, next state, and output.

Has all possible binary combination of present states (#m) and inputs(#n)  the
table has 2m+n rows.

Fill the next state and output sections of the table according to

The state equations you obtained in step 1.
OR

The binary values of inputs to FFs along with characteristic table of a FF.
8
Example
Analysis with D FF

Analyze the logic diagram by finding the state table:

From previous slide:

State equations:
A(t+1) = Ax + Bx
B(t+1) = A’x

Output equation:
y=(A+B)x’
9
Another form of state table
SC Analysis

To achieve the third goal ( finding state diagram):
Draw circles that represent states of the FFs
1.


The number of circles you need depend on the number of the FFs the circuit
has  if the SC has n FFs, then you need 2n circles/states.
Each circle is labeled with a binary number that identifies the state of FFs.
Draw directed lines the connect circles according to the state table
2.


The source is the present state.
The destination is the next state.
Label each line with two binary numbers separated by a slash.
3.


The first number is the input value during the present state.
The second number is the output.
input/output
AB
11
AB
Example
Analysis with D FF

Analyze the logic diagram by finding the state diagram:
1/0
0/0
0/1
00
1/0
10
0/1
0/1
01
1/0
11
1/0
12

Example
Analysis with JK FF
Find state equations of the following SC logic diagram.

FFs input equations:
JA= B
JB= x’
KA= Bx’
KB= A⨁ 𝑥
The characteristic equation for D FF :Q(t+1)= JQ’+K’Q
 Substitute to find state equations:
A(t+1) = JAA’ +K’A A
 A(t+1) = BA’ + (Bx’)’A = A’B+AB’+Ax

B(t+1) = JBB’ +K’B B
 B(t+1) = x’B’ + (A⨁ 𝑥)’ = B’x’ + ABx+ A’Bx’

Output equation:
There is no external output.
13

Example
Analysis with JK FF
Find state table of the previous SC logic diagram.

Fill Next state A, B from State equations:
A(t+1) = A’B+AB’+Ax
B(t+1) = B’x’ + ABx+ A’Bx’
OR

Use characteristics table of FF
FFs input equations:
JA= B
JB= x’
KA= Bx’
KB= A⨁ 𝑥
14

Example
Analysis with JK FF
Find state table of the previous SC logic diagram.

Fill Next state A, B from State equations:
A(t+1) = A’B+AB’+Ax
B(t+1) = B’x’ + ABx+ A’Bx’
OR

Use characteristics table of FF
FFs input equations:
JA= B
JB= x’
KA= Bx’
KB= A⨁ 𝑥
15

Example
Analysis with JK FF
Find state diagram of the previous SC logic diagram.
16
Exercise 1

Find the state equations, state table, and state diagram of the
following SC logic diagram.( page 219)
y
17
Exercise 2

Find the state equations, state table, and state diagram of the
following SC logic diagram. (page 214)
18
Mealy and Moore Models of FSM

The most general model of SC has inputs, outputs, and internal states.

However, there are two different models of SC:

The Mealy model:


The output is a function of both the present state and the input.
The Moore model:

The output is a function of only the present state.
19
Mealy and Moore Models of FSM
 In

Mealy model:
The outputs may change if the inputs change during the
clock cycle  outputs may have false values.

 In

Synchronize the inputs with a clock.
Moore model:
The outputs are synchronized with the clock since they
depend on FF outputs only.
20
Mealy Model Example
21
Moore Model Examples
y
22