Chapter 5A – Sequential Logic Circuits (Part 1)

Report
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
Reading Assignment:
- Chapter 5 in Logic and Computer Design Fundamentals, 4th Edition by Mano
- Online supplement to text “Design and Analysis using JK and T Flip-Flops”
(www.prenhall.com/mano)
Ch. 5 - Sequential Circuits
There are two primary classifications of logic circuits:
1. Combinational logic circuits
• Chapters 1 – 4 dealt with combinational logic circuits
• Circuits of this type have outputs that are functions of the inputs
(illustrated below)
• The order in which the inputs are applied is not important.
Inputs
Combinational
Logic
Combinational Logic Circuit
Outputs = f(Inputs)
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Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
2. Sequential logic circuits
• Chapter 5 introduces sequential logic circuits
• Circuits of this type have outputs that are functions of both inputs and
previous outputs (illustrated below)
• Sequential circuits contain some type of memory elements. As an
example, a counter must “remember” that its previous output was 6 in
order to produce its new output 7.
• Sequential logic circuits typically also include some combinational logic
components.
Inputs
Combinational
Logic
Memory
Sequential Logic Circuit
Outputs = f(inputs + past outputs)
2
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EGR 270 – Fundamentals of Computer Engineering
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There are two main types of sequential circuits:
1) Synchronous Sequential Circuits (also called Clock Sequential Circuits)
– All signals are synchronized to some “master clock”
– The memory devices respond only when activated by the master clock
– The most common memory device: the flip-flop
– This course will primarily focus on this type of sequential circuit
– Circuits can be designed using systematic methods such as:
• Excitation table method
• State equation method
• One-hot method
2) Asynchronous Sequential Circuits
– Outputs depend solely on the order in which the inputs change, so timing
is critical.
– The design methods used for synchronous sequential circuits do not apply
to asynchronous circuits. As a result, design is difficult and is perhaps
covered in more advanced courses.
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Flip-flops and latches
• Binary memory cell capable of storing 1 bit of information (0 or 1)
• Have two outputs, Q and Q’
• Q = binary state = present state = value stored
• Maintain the present state Q indefinitely until inputs instruct it to change
• Flip-flops and latches behave as described above but have one key
difference:
– A latch can change states whenever the input signals change
– A flip-flop has a clocked input and can only change at certain times
specified by the type of clocking. So a flip-flop could be called a
clocked latch. See below.
S
Q
S
Q
R
Q’
Clock input
R
Q’
SR Latch
SR Flip-Flop
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
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Basic RS (or SR) Latch Operation
A simple SR latch can be constructed using either two NAND gates or two
NOR gates with feedback connections. The NOR circuit is shown below.
R (reset)
S (set)
Q
Q
Case 1: S = 1, R = 0 (Set)
R (reset)
S (set)
Determine the truth table for the RS
latch shown by analyzing the circuit
for various cases:
Case 2: S = 0, R = 1 (Reset)
Q
Q
R (reset)
S (set)
Q
Q
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
Case 3: S = 0, R = 0 (No Change)
A) Latch originally Reset
R (reset)
S (set)
6
B) Latch originally Set
R (reset)
Q
Q
Q
Q
S (set)
Case 4: S = 1, R = 1 (Illegal)
R (reset)
Q
Summary:
Truth Table for SR Latch
S R Q(t+1) Comment
S (set)
Q
0
0
Q
NC (no change)
0
1
0
R (reset)
1
0
1
S (set)
1
1
---
Illegal
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
7
SR Flip-flop
A flip-flop is essentially a clocked latch. In other words, a flip-flop is a latch
that is only allowed to change during certain portions of the clock cycle.
Adding AND gates to the SR latch below results in a latch that can only change
when Clock = 1.
Discuss the operation of the simple clocked RS latch shown below. Hint:
Find the values of R1 and S1 when Clock = 0 and when Clock = 1
R
R1
Q
Clock
S
S1
Q
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
Flip – flops - There are 4 primary types of flip-flops:
• SR flip-flop
• D flip-flop
For each type of flip-flop, show
the truth table and symbol.
• JK flip-flop
• T flip-flop
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EGR 270 – Fundamentals of Computer Engineering
Relationship between different types of flip – flops
• Show how to create a D flip-flop from an SR flip-flop or a JK flip-flop
• Show how to create a T flip-flop from a JK flip-flop
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When are each type of flip-flop used?
JK and D flip-flops will be most commonly used in this course.
• JK flip-flops are the most powerful and yield the simplest sequential circuits
• D flip-flops are easiest to design with and are commonly used with
programmable devices (PLDs and FPGAs)
• SR flip-flops are the simple to introduce using NOR gate circuits
• T flip-flops are sometimes used design simple counters by noting when
certain bits toggle.
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Triggering of flip-flops
A trigger is a momentary change in the input clock signal which allows for a
possible change in the state of the flip-flop. Flip-flops are typically triggered
by pulse transitions, using either the rising (positive) edge or the falling
(negative) edge of the clock waveform (see below).
positive
edge
negative
edge
positive
edge
negative
edge
positive
edge
negative
edge
EGR 270 – Fundamentals of Computer Engineering
Chapter 5A
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There are 3 common types of triggering:
1. Positive-edge triggering – the flip-flop output can only change on the
positive edges of the clock
2. Negative-edge triggering – the flip-flop output can only change on the
negative edges of the clock
3. Master-slave – Internally this flip-flop is constructed using two flip-flops
called the master and the slave. The master flip-flop “reads” the input
values on the positive edge of the clock, and the output Q of the master is
transferred to the slave on the negative clock edge.
The following symbols are sometimes used to indicate the type of triggering.
J
Q
J
Q
Clock
Clock
K
Q
Positive-edge
triggered
JK flip-flop
Clock
K
Q
Negative-edge
triggered
JK flip-flop
J
C
Q
K
Q
Master-slave
JK flip-flop
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Master-slave - Discuss how the master-slave flip-flop below works.
Master
Slave
J
Clock
J1
Q1
J2
Q2
Q
K
K1
Q1
K2
Q2
Q
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
Example: Given J, K, and Clock input waveforms, sketch the output Q for a
JK flip-flop if each flip-flop is initially LOW and the type of triggering is:
a) positive-edge triggering (labeled as Q1)
b) negative-edge triggering (labeled as Q2)
c) master-slave triggering (labeled as Q3)
Clock
J
K
Q1
Q2
Q3
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EGR 270 – Fundamentals of Computer Engineering
Asynchronous (Direct) Inputs
Two inputs are commonly available that can be used to initialize a flip-flop
independently of the clock:
• PRESET – used to initialize the flip-flop to Q = 1
• CLEAR – used to initialize the flip-flop to Q = 0
Sketch a JK flip-flop with active-LOW inputs for PRESET and CLEAR.
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EGR 270 – Fundamentals of Computer Engineering
Example: Given J, K, PRESET, CLEAR and Clock input waveforms for a
negative-edge triggered JK flip-flop, sketch Q. Assume that PRESET and
CLEAR are active-LOW inputs.
Clock
J
K
PR
CLR
Q
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EGR 270 – Fundamentals of Computer Engineering
Chapter 5A
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CLK Q
6
16
7474
K
747614
Q
3
Q
CLK
Q
K
2
5
4
CLK
J
4
6
5
7
74109
1
J
Q
CLK
K
3
1
U4A
PRE
Q
2
PRE
PRE
2
1
J
15
U5A
CLR
13
Q
7473
Q
4
3
K
3
D
5
CLR
CLK
2
CLR
12
1
3
Q
CLR
1
J
U1A
U2A
2
14
CLR
U3A
PRE
4
Commercially available flip-flop IC’s. A few are listed below and PSPICE
symbols are shown below also.
• 7473 Dual JK Master-Slave (pulse-triggered) Flip-flop with CLEAR
• 7473A Dual JK Negative-Edge Triggered Flip-flop with CLEAR
• 7474 Dual D Positive-Edge Triggered Flip-flop with PRESET and CLEAR
• 7476B Dual JK Master-Slave Flip-flop with PRESET and CLEAR
• 74109 Dual D Positive-Edge Triggered Flip-flop with PRESET and CLEAR
• 74111 Dual JK Master-Slave Flip-flop with PRESET and CLEAR
• 74279 Quadruple S’-R’ Latch
1
2
7 3
6 5
Q
74111 6
U7A
1R
1S1
1S2 1Q
4
2R
2S 2Q
7
74279
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
Synchronous Logic Circuits versus Combinational Logic Circuits
Let’s begin by comparing how we describe each type of circuit.
Descriptions of Combinational Logic Circuits
Recall that there are numerous ways to describe a combinational logic circuit,
including:
• Truth tables
• Karnaugh maps
• (minterms)
• (maxterms)
• Boolean expressions
• SOP expressions
• POS expressions
• Logic diagrams
• VHDL descriptions
Note: Given any one of the descriptions above, we could determine all of the
others.
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Descriptions of Sequential Logic Circuits
Similarly, there are numerous ways to describe a sequential logic circuit,
including:
• State diagrams
• State tables
• State equations and output equations
• Input equations (flip-flop input functions) and output equations
• Logic diagrams
• VHDL descriptions
Note: Given any one of the descriptions above, we could determine all of the
others.
Some of these ways to describe sequential circuits will now be introduced.
Finite State Machines (FSM) – Sequential circuits are also referred to as finite
state machines. The circuit operates by moving between a finite number of
pre-determined states.
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State Diagrams
This is the most common way to describe a sequential circuit.
A state diagram is somewhat like a flowchart that describes the sequence to
states through which the circuit might progress.
State – a distinct event that is to occur (one event in a sequence)
Example: A state might be a single count in a counter. If a counter counts
0, 1, 2, ….. , 9 and then repeats, then it has 10 unique states. If four flipflops were used to store the count, then the flip-flops would store the values
1001 corresponding to state 9.
Example: A state might be one step in a machining operation (there might
be 5 states corresponding to the operations drill, ream, counterbore,
countersink, and polish).
Example: A traffic light controller might have three states: Green, Yellow,
and Red. Under certain input conditions or at certain times, the controller
will change state.
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Encoding states – Show how a binary code can be stored in a set of flip-flops.
In most cases,
Number of flip-flops needed = log2(Number of states)
Example: Determine the number of states needed in each case below:
Description of circuit
Circuit with 20 states
Traffic light controller
3-bit UP/DOWN counter
Decade counter
Number of flip-flops
required
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There are two primary types of state diagrams (state machines):
Mealy state machine (Mealy model) – the output depends on the present state
and the inputs applied.
Moore state machine (Moore model) – the output only depends on the present
state.
We will
primarily
use Mealy
models
X
X/Y
A
B
A/Y
B/Y
X
X/Y
C
Mealy Model
• Transition from one state to another
depends on the Input, X
• Output, Y, is specified with the transition
• Output, Y, depends on both the Present
State and the Input, X
C/Y
Moore Model
• Transition from one state to another
depends on the Input, X
• Output, Y, is specified with the
Present State
• Output, Y, depends only on the
Present State
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
Examples of state diagrams:
A) Modulo-5 (mod-5) counter:
B) 3-bit Up/Down counter:
C) Traffic Light Controller
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Chapter 5A
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State Table - A state table provides the same information as the state diagram,
but in tabular form.
Example: Determine the state table for the state diagram shown below.
0/0
1/0
0/1
1/0
A
D
1/0
Is this a Mealy machine or
a Moore machine?
B
C
0/0
0/1
1/1
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EGR 270 – Fundamentals of Computer Engineering
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State Assignment
Some state diagrams have states indicated by letters or names because there is
no numeric value assigned to the states. As an example, there are no natural
numeric values for the states in a circuit that controls a traffic light (states,
RED, YELLOW, and GREEN). In such cases, numeric values must be
assigned to each state. In the case of the traffic light, 2 bits are needed to
encode the three states, but various possible codes could be used. For example,
RED = 00, YELLOW = 01, and GREEN = 10. There are many other possible
state assignments. Which is the best assignment to use? We don’t know. This
is a current research topic.
Example: List possible state assignments for the last problem and repeat
the state table using one of the state assignments.
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Determining the State Diagram from a Logic Diagram
The state table (or state diagram) can be found from the logic diagram using JK
(or other types) flip-flops as follows:
• Form a table similar to the one on the following slide
• List all possible present states
• Read expressions for each J and K input from the logic diagram and use these
expressions to complete the columns for each J and K (based on present
states)
• Use the truth table for the JK flip-flop to determine the next state for each
present state and the corresponding values of J and K
• Read expressions for any outputs from the logic diagram and add them to the
table (based on present states)
J K Q(t+1) Comment
Truth Table for 0 0
Q
NC (no change)
JK Flip-Flop
0 1
0
R (reset)
1
0
1
S (set)
1
1
Q’
Toggle
EGR 270 – Fundamentals of Computer Engineering
Chapter 5A
27
1
16
J
U1A
Q
15
A (MSB)
U5A
CLK
K
CLR
4
PRE
2
Example: Determine the state diagram for the logic diagram shown below.
1
Q
14
3
U7A
2
1
7476
3
U6A
x
7408
3
2
1
U3A
1
3
2
4
2
1
7404
U4A
7432
16
1
3
J
7432
U2A
7408
Q
15
B
CLK
K
CLR
U8A
1
PRE
2
3
2
Q
14
7476
3
2
7486
Present State/Inputs
x
A
B
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Flip-flop inputs
JA
KA
JB
Next State
KB
A
B
Output
y
y
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28
Example (continued): Draw the state diagram for the logic diagram on the
previous slide.
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Sequential Circuit Design Methods
Three specific methods will be covered for designing synchronous sequential
circuits:
• The Excitation Table method
• The State Equation method
• “One-Hot” method
Excitation Table Method (also read the Online supplement to text “Design
and Analysis using JK and T Flip-Flops” available at www.prenhall.com/mano
Before covering the excitation table method, it is useful to develop
the excitation tables for each type of flip-flop.
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Flip-flop Excitation Tables
Truth table – defines the output state based on the inputs
Excitation table – defines required inputs to cause a transition from one state to
another.
Example – Complete the excitation table below for the JK flip-flop
JK flip-flop truth table
JK flip-flop excitation table
J
K
Q(t + 1)
Q(t)
Q(t + 1)
0
0
Q
0
0
0
1
0
0
1
1
0
1
1
0
1
1
Q’
1
1
J
K
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Flip-flop Excitation Tables
Excitation tables could similarly be found for SR, D, and T flip-flops. The
excitation tables for all 4 types of flip flops have been summarized below.
Q(t)
Q(t+1)
J
K
S
R
D
T
0
0
0
X
0
X
0
0
0
1
1
X
1
0
1
1
1
0
X
1
0
1
0
1
1
1
X
0
X
0
1
0
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Excitation table method – Design Procedure
1. Problem description.
2. Obtain the state table.
3. Use state assignment to assign binary values to each state if they are
symbolic.
4. Determine the number of flip-flops required and assign a letter or number to
each.
5. Determine the type of flip-flop to use (SR, JK, D, or T).
6. Derive the circuit excitation table and output table from the state table.
7. Simplify
a) the circuit output functions
b) the flip-flop input functions
8. Draw the logic diagram.
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
Example: Design a modulo-7
counter (counts 0 to 6 and repeats)
using the excitation table method
and JK flip-flops. Treat unused
counts as “don’t cares.”
33
Flip-flop Excitation Tables
Q(t)
Q(t+1)
J
K
S
R
D
T
0
0
0
X
0
X
0
0
0
1
1
X
1
0
1
1
1
0
X
1
0
1
0
1
1
1
X
0
X
0
1
0
Circuit Excitation Table
Present States/Inputs
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
Next State
Flip-flop Inputs and Circuit Outputs
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
34
Example: (continued)
Fill out the K-maps from the excitation table to determine expressions for
each J and K input and for each output
Flip-flop Input Functions and Circuit Output Functions
00 01 11 10
00 01 11 10
00 01 11 10
00 01 11 10
0
0
0
0
1
1
1
1
00 01 11 10
00 01 11 10
00 01 11 10
00 01 11 10
0
0
0
0
1
1
1
1
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
Example: (continued)
Draw the logic diagram
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Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
Example: Use the excitation table method to design a counter as follows:
• Treat unused states as “don’t cares”
• Use JK flip-flops
• Include an input switch, x, such that the counter operates as follows:
• If x = 0: Counts 1, 2, 3, 4, 5 and repeats
• If x = 1: Counts 5, 4, 3, 2, 1 and repeats
A) Draw the state diagram
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EGR 270 – Fundamentals of Computer Engineering
Chapter 5A
B) Fill out the excitation table
Circuit Excitation Table
Present State/Circuit Inputs
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Next State
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Flip-flop Excitation Tables
Q(t)
Q(t+1)
J
K
S
R
D
T
0
0
0
X
0
X
0
0
0
1
1
X
1
0
1
1
1
0
X
1
0
1
0
1
1
1
X
0
X
0
1
0
Flip-flop Inputs and Circuit Outputs
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C) Fill out the K-maps to determine the J and K inputs
Flip-flop Input Functions and Circuit Output Functions
00 01 11 10
00 01 11 10
00 01 11 10
00 01 11 10
00
00
00
00
01
01
01
01
11
11
11
11
10
10
10
10
00 01 11 10
00 01 11 10
00 01 11 10
00 01 11 10
00
00
00
00
01
01
01
01
11
11
11
11
10
10
10
10
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
D) Draw the logic diagram
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Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
Counters
Counters are an important class of sequential circuits.
A counter changes state upon application of an input pulse. The input pulse is
not necessarily a periodic clock. For example, a traffic counter is “clocked”
each time a car passes.
Key uses of counters
1. Count occurrences of an event
• Examples:
 Traffic counter
 Line counter on ASEE autonomous vehicle (illustrate in class)
 Wheel revolution counter
2. Generate timing sequences to control operations
• Example: Use a mod-16 counter to control a traffic light (Green for 8
counts, Yellow for 1 count, and Red for 7 counts). More details later.
3. Frequency division – Use a “master “clock generator to create the highest
frequency and then create lower frequencies by dividing the master clock.
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Example: Show that a JK flip-flop in the toggle mode acts as a modulo-2
counter or a divide-by-2 circuit.
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Example: Show that a 3-bit counter can serve as a modulo-8 counter or a
divide-by-8 circuit.
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43
Example: The circuit below has a 1MHz master clock and uses counters (as
frequency dividers) to provide synchronized lower frequencies. Determine the
frequencies f1, f2, and f3.
1 MHz
Master
Clock
(MSB)A
B
4-bit
C
Counter
D
1
J
Q
1
J
1
K Q’
1
K Q’
Q
f1
f2
(MSB)A
f3
Answers:
f1 = _______
Divide-by-100
Circuit
f2 = _______
f3 = _______
Note: This same technique can be used in software. In Lab 7 we will use a
VHDL program that implements a “divide-by-50 million” circuit to produce a 1
Hz clock from the 50 MHz internal clock on an FPGA board. We will need the
1 Hz clock to display a designed counting sequence on a 7-segment display.
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Counters with multiple counting sequences
So far we have seen counters with single counting sequences and with 2
counting sequences (controlled by an input switch x). Additional input
switches allows for more counting sequences.
Example: Design a counter with 2 input switches, x and y, that can count in 4
possible sequences based on the switch positions. Use JK flip-flops.
x
0
0
1
1
y
0
1
0
1
Counting sequence
0, 1, 2, 3 and repeat
0, 3, 2, 1 and repeat
0, 2, 1, 3 and repeat
0, 2, 3, 1 and repeat
EGR 270 – Fundamentals of Computer Engineering
Chapter 5A
Example: (continued)
A) Draw the state diagram
B) Complete the state table below
x
0
0
1
1
State table
AB
AB
AB
AB
AB
xy=00
xy=01
xy=10
xy=11
01
10
11
Counting sequence
0, 1, 2, 3 and repeat
0, 3, 2, 1 and repeat
0, 2, 1, 3 and repeat
0, 2, 3, 1 and repeat
State diagram
Present
State
00
y
0
1
0
1
Next State
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Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
Example: (continued)
C) Fill out the excitation table
46
Flip-flop Excitation Tables
Q(t)
Q(t+1)
J
K
S
R
D
T
0
0
0
X
0
X
0
0
0
1
1
X
1
0
1
1
1
0
X
1
0
1
0
1
1
1
X
0
X
0
1
0
Circuit Excitation Table
Circuit Inputs/Present States
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Next State
Flip-flop Inputs and Circuit Outputs
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
Example: (continued)
D) Fill out the K-maps to find the flip-flop input functions
E) Draw the logic diagram
Logic diagram
Flip-flop Input Functions
00 01 11 10
00 01 11 10
00
00
01
01
11
11
10
10
00 01 11 10
00 01 11 10
00
00
01
01
11
11
10
10
47
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
48
Self-starting counters
Counters are considered to be self-starting if all unused counts eventually lead
to the correct counting sequence. Since the initial state for a flip-flop is
unpredictable upon powering up the IC, a counter that is not self-starting could
possibly power up into an unused state that would not eventually go into the
correct counting sequence (so the counter might “lock up” in an incorrect count
or counting pattern.
Recall that the next states for unused counts were sometimes treated as “don’t
cares.” With this method it is difficult to predict what will happen if the
counter powers up into an unused count (although it can be later determined by
analyzing the circuit). A safer technique it to let all unused counts have a valid
count for their next state.
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
49
Example: Consider the state diagrams for two modulo-5 counters below.
Are they self-starting?
Case 1: Counter is NOT self-starting.
Next states for unused counts 5, 6, and
7 were perhaps treated as don’t cares.
6
Case 2: Counter is self-starting.
Next states for unused counts 5, 6,
and 7 were all set to count 0.
5
7
6
7
5
0
0
4
1
3
2
4
1
3
2
Chapter 5A
EGR 270 – Fundamentals of Computer Engineering
Example: Determine the counting sequence
for the counter shown (begin with count 0).
Use a timing diagram to display the values
of Clock, JA, KA, JB, KB, JC, KC, A, B,
and C. Is the counter self-starting?
Also draw the state diagram.
1
JA
QA
KA
QA
JB
QB
KB
QB
JC
QC
KC
QC
Clock
Clock
JA
KA
JB
KB
JC
C
KC
A
B
C
Count
0
50
A (MSB)
B
C
Count

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