Chapter 2 - Part 1 - PPT - Mano & Kime
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Transcript Chapter 2 - Part 1 - PPT - Mano & Kime
Logic and Computer Design Fundamentals
Chapter 6 – Sequential
Circuits
Part 1 – Storage Elements and Sequential
Circuit Analysis
Charles Kime & Thomas Kaminski
© 2004 Pearson Education, Inc.
Terms of Use
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Overview
Part 1 - Storage Elements and Analysis
• Introduction to sequential circuits
• Types of sequential circuits
• Storage elements
Latches
Flip-flops
• Sequential circuit analysis
State tables
State diagrams
• Circuit and System Timing
Part 2 - Sequential Circuit Design
• Specification
• Assignment of State Codes
• Implementation
Chapter 6 - Part 1
2
Introduction to Sequential Circuits
Inputs
A Sequential
circuit contains:
Combinational
Logic
Storage
Elements
• Storage elements:
Latches or Flip-Flops
• Combinatorial Logic:
State
Outputs
Next
State
Implements a multiple-output
switching function
Inputs are signals from the outside.
Outputs are signals to the outside.
Other inputs, State or Present State, are
signals from storage elements.
The remaining outputs, Next State are
inputs to storage elements.
Chapter 6 - Part 1
3
Introduction to Sequential Circuits
Inputs
Storage
Elements
Combinatorial Logic
• Next state function
Combinational
Logic
State
Outputs
Next
State
Next State = f(Inputs, State)
• Output function (Mealy)
Outputs = g(Inputs, State)
• Output function (Moore)
Outputs = h(State)
Output function type depends on specification and affects
the design significantly
Chapter 6 - Part 1
4
Types of Sequential Circuits
Depends on the times at which:
• storage elements observe their inputs, and
• storage elements change their state
Synchronous
• Behavior defined from knowledge of its signals at discrete
instances of time
• Storage elements observe inputs and can change state only in
relation to a timing signal (clock pulses from a clock)
Asynchronous
• Behavior defined from knowledge of inputs an any instant of
time and the order in continuous time in which inputs change
• If clock just regarded as another input, all circuits are
asynchronous!
• Nevertheless, the synchronous abstraction makes complex
designs tractable!
Chapter 6 - Part 1
5
Discrete Event Simulation
In order to understand the time behavior of a
sequential circuit we use discrete event
simulation.
Rules:
• Gates modeled by an ideal (instantaneous) function
and a fixed gate delay
• Any change in input values is evaluated to see if it
causes a change in output value
• Changes in output values are scheduled for the fixed
gate delay after the input change
• At the time for a scheduled output change, the
output value is changed along with any inputs it
drives
Chapter 6 - Part 1
6
Simulated NAND Gate
Example: A 2-Input NAND gate with a 0.5 ns. delay:
F(Instantaneous)
A
DELAY 0.5 ns.
F
B
Assume A and B have been 1 for a long time
At time t=0, A changes to a 0 at t= 0.8 ns, back to 1.
t (ns)
A
B
–
1
1
0
1 0 1
0 1
0.5
0.8 1 0 1
0.13
1
1
F(I)
0
1 0
F
Comment
0
A=B=1 for a long time
0
F(I) changes to 1
1 0 F changes to 1 after a 0.5 ns delay
1
F(Instantaneous) changes to 0
1 0 1
0 1 0 F changes to 0 after a 0.5 ns delay
Chapter 6 - Part 1
7
Gate Delay Models
Suppose gates with delay n ns are
represented for n = 0.2 ns, n = 0.4 ns,
n = 0.5 ns, respectively:
0.2
0.4
0.5
Chapter 6 - Part 1
8
Circuit Delay Model
Consider a simple A
2-input multiplexer:
With function:
• Y = A for S = 1
• Y = B for S = 0
0.4
0.2
S
B
0.5
Y
0.4
A
B
S
S
Y
“Glitch” is due to delay of inverter
Chapter 6 - Part 1
9
Storing State
What if A connected to Y?
Circuit becomes:
With function:
• Y = B for S = 1, and
Y(t) dependent on
Y(t – 0.9) for S = 0
0.4
0.2
S
B
0.5
Y
0.4
B
S
S
Y
The simple combinational circuit has now become a
sequential circuit because its output is a function of a time
sequence of input signals!
Y is stored value in shaded area
Chapter 6 - Part 1
10
Storing State (Continued)
Simulation example as input signals change with time.
Changes occur every 100 ns, so that the tenths of ns delays
are negligible.
Time
B
1
1
1
0
0
0
1
S
0
1
0
0
1
0
0
Y Comment
0 Y “remembers” 0
1 Y = B when S = 1
1 Now Y “remembers” B = 1 for S = 0
1 No change in Y when B changes
0 Y = B when S = 1
0 Y “remembers” B = 0 for S = 0
0 No change in Y when B changes
Y represent the state of the circuit, not just an output.
Chapter 6 - Part 1
11
Storing State (Continued)
Suppose we place
an inverter in the
“feedback path.”
0.2
0.4
0.2
S
B
The following behavior results:
The circuit is said
B
S
Y
0
1
0
to be unstable.
1
1
1
For S = 0, the
1
0
1
circuit has become 1
0
0
what is called an
1
0
1
oscillator. Can be
1
0
0
used as crude clock.
0.5
Y
0.4
Comment
Y = B when S = 1
Now Y “remembers” A
Y, 1.1 ns later
Y, 1.1 ns later
Y, 1.1 ns later
Chapter 6 - Part 1
12
Basic (NAND) S – R Latch
“Cross-Coupling”
two NAND gates gives
the S -R Latch:
Which has the time
sequence behavior:
S (set)
Q
Q
R (reset)
Time R S Q Q Comment
S = 0, R = 0 is
forbidden as
input pattern
1
1
1
0
1
0
1
1
0
1
1
1
0
1
?
1
1
0
0
1
?
?
0
0
1
1
1
?
Stored state unknown
“Set” Q to 1
Now Q “remembers” 1
“Reset” Q to 0
Now Q “remembers” 0
Both go high
Unstable!
Chapter 6 - Part 1
13
Basic (NOR) S – R Latch
Cross-coupling two
NOR gates gives the
S – R Latch:
Which has the time
sequence
Time R S
behavior:
0
0
0
0
1
0
1
0
1
0
0
0
1
0
R (reset)
S (set)
Q
?
1
1
0
0
0
?
Q
?
0
0
1
1
0
?
Q
Q
Comment
Stored state unknown
“Set” Q to 1
Now Q “remembers” 1
“Reset” Q to 0
Now Q “remembers” 0
Both go low
Unstable!
Chapter 6 - Part 1
14
Clocked S - R Latch
Adding two NAND
gates to the basic
S - R NAND latch
gives the clocked
S – R latch:
S
Q
C
Q
R
Has a time sequence behavior similar to the basic S-R
latch except that the S and R inputs are only observed
when the line C is high.
C means “control” or “clock”.
Chapter 6 - Part 1
15
Clocked S - R Latch (continued)
The Clocked S-R Latch can be described by a table:
Q(t) S R
S
Q
C
Q
R
The table describes
what happens after the
clock [at time (t+1)]
based on:
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
Q(t+1)
0
0
1
???
1
0
1
???
Comment
No change
Clear Q
Set Q
Indeterminate
No change
Clear Q
Set Q
Indeterminate
• current inputs (S,R) and
• current state Q(t).
Chapter 6 - Part 1
16
D Latch
Adding an inverter
to the S-R Latch,
gives the D Latch:
Note that there are
no “indeterminate”
states!
Q
0
0
1
1
D
0
1
0
1
Q(t+1)
0
1
0
1
D
Q
C
Comment
No change
Set Q
Clear Q
No Change
Q
The graphic symbol for a
D Latch is:
D
Q
C
Q
Chapter 6 - Part 1
17
Flip-Flops
The latch timing problem
Master-slave flip-flop
Edge-triggered flip-flop
Standard symbols for storage elements
Direct inputs to flip-flops
Flip-flop timing
Chapter 6 - Part 1
18
The Latch Timing Problem
In a sequential circuit, paths may exist through
combinational logic:
• From one storage element to another
• From a storage element back to the same storage
element
The combinational logic between a latch output
and a latch input may be as simple as an
interconnect
For a clocked D-latch, the output Q depends on
the input D whenever the clock input C has
value 1
Chapter 6 - Part 1
19
The Latch Timing Problem (continued)
Consider the following circuit:
Suppose that initially Y = 0.
Clock
D
Q
C
Q
Y
Clock
Y
As long as C = 1, the value of Y continues to change!
The changes are based on the delay present on the loop
through the connection from Y back to Y.
This behavior is clearly unacceptable.
Desired behavior: Y changes only once per clock pulse
Chapter 6 - Part 1
20
The Latch Timing Problem (continued)
A solution to the latch timing problem is
to break the closed path from Y to Y
within the storage element
The commonly-used, path-breaking
solutions replace the clocked D-latch
with:
• a master-slave flip-flop
• an edge-triggered flip-flop
Chapter 6 - Part 1
21
S-R Master-Slave Flip-Flop
Consists of two clocked S
S
S
Q
Q
S-R latches in series
C
C
C
with the clock on the
R
R
Q
R
Q
second latch inverted
The input is observed
by the first latch with C = 1
The output is changed by the second latch with C = 0
The path from input to output is broken by the
difference in clocking values (C = 1 and C = 0).
The behavior demonstrated by the example with D
driven by Y given previously is prevented since the
clock must change from 1 to 0 before a change in Y
based on D can occur.
Q
Q
Chapter 6 - Part 1
22
Flip-Flop Problem
The change in the flip-flop output is delayed by
the pulse width which makes the circuit slower or
S and/or R are permitted to change while C = 1
• Suppose Q = 0 and S goes to 1 and then back to 0 with
R remaining at 0
The master latch sets to 1
A 1 is transferred to the slave
• Suppose Q = 0 and S goes to 1 and back to 0 and R
goes to 1 and back to 0
The master latch sets and then resets
A 0 is transferred to the slave
• This behavior is called 1s catching
Chapter 6 - Part 1
23
Flip-Flop Solution
Use edge-triggering instead of master-slave
An edge-triggered flip-flop ignores the pulse
while it is at a constant level and triggers only
during a transition of the clock signal
Edge-triggered flip-flops can be built directly at
the electronic circuit level, or
A master-slave D flip-flop which also exhibits
edge-triggered behavior can be used.
Chapter 6 - Part 1
24
Edge-Triggered D Flip-Flop
The edge-triggered
D flip-flop is the
same as the masterslave D flip-flop
D
D
Q
S
Q
Q
Q
Q
C
C
C
Q
R
It can be formed by:
• Replacing the first clocked S-R latch with a clocked D latch or
• Adding a D input and inverter to a master-slave S-R flip-flop
The delay of the S-R master-slave flip-flop can be
avoided since the 1s-catching behavior is not present
with D replacing S and R inputs
The change of the D flip-flop output is associated with
the negative edge at the end of the pulse
It is called a negative-edge triggered flip-flop
Chapter 6 - Part 1
25
Positive-Edge Triggered D Flip-Flop
Formed by
adding inverter
to clock input
D
D
Q
S
Q
Q
Q
Q
C
C
C
Q
R
Q changes to the value on D applied at the
positive clock edge within timing constraints to
be specified
Our choice as the standard flip-flop for most
sequential circuits
Chapter 6 - Part 1
26
Standard Symbols for Storage
Elements
S
S
D
D
R
R
C
C
SR
Master-Slave:
Postponed output
indicators
Edge-Triggered:
Dynamic
indicator
D with 1 Control D with 0 Control
SR
(a) Latches
S
S
C
C
R
R
Triggered SR
Triggered D
Triggered SR
(b) Master-Slave Flip-Flops
D
D
C
D
D
C
C
Triggered D
C
Triggered D
Triggered D
(c) Edge-Triggered Flip-Flops
Chapter 6 - Part 1
27
Direct Inputs
At power up or at reset, all or part
of a sequential circuit usually is
S
D
Q
initialized to a known state before
it begins operation
This initialization is often done
C
Q
R
outside of the clocked behavior
of the circuit, i.e., asynchronously.
Direct R and/or S inputs that control the state of the
latches within the flip-flops are used for this
initialization.
For the example flip-flop shown
• 0 applied to R resets the flip-flop to the 0 state
• 0 applied to S sets the flip-flop to the 1 state
Chapter 6 - Part 1
28
Flip-Flop Timing Parameters
ts - setup time
th - hold time C
tw - clock
S/R
pulse width
tpx - propaQ
gation delay
• tPHL - High-toLow
C
• tPLH - Low-toHigh
• tpd - max (tPHLD
,
tPLH)
twH $ twH,min
twL$ twL,min
ts
th
tp-,min
t p-,max
(a) Pulse-triggered (positive pulse)
twH$ t wH,min
twL $ twL,min
ts
th
t p-,min
t p-,max
Q
(b) Edge-triggered (negative edge)
Chapter 6 - Part 1
29
Flip-Flop Timing Parameters (continued)
ts - setup time
• Master-slave - Equal to the width of the triggering
pulse
• Edge-triggered - Equal to a time interval that is
generally much less than the width of the the
triggering pulse
th - hold time - Often equal to zero
tpx - propagation delay
• Same parameters as for gates except
• Measured from clock edge that triggers the output
change to the output change
Chapter 6 - Part 1
30
Sequential Circuit Analysis
General Model
Inputs
Outputs
• Current State
Combinaat time (t) is
tional
Logic
stored in an Storage
Elements
array of
Next
flip-flops.
State
State
• Next State at time (t+1)
is a Boolean function of CLK
State and Inputs.
• Outputs at time (t) are a Boolean function of
State (t) and (sometimes) Inputs (t).
Chapter 6 - Part 1
31
Example 1 (from Fig. 6-17)
Input:
x(t)
Output: y(t)
State:
(A(t), B(t))
What is the Output
Function?
x
D
CP
What is the Next State
Function?
Q
A
C Q
A
Q
B
D
C Q
y
Chapter 6 - Part 1
32
Example 1 (from Fig. 6-17) (continued)
Boolean equations
for the functions:
• A(t+1) = A(t)x(t)
+ B(t)x(t)
x
Q
A
C Q
A
Q
B
D
Next State
• B(t+1) = A(t)x(t)
• y(t) = x(t)(B(t) + A(t))
D
CP
C Q'
y
Output
Chapter 6 - Part 1
33
Example 1(from Fig. 6-17) (continued)
Where in time are inputs, outputs and
states defined?
1
0
1
0
0
0
1
0
Chapter 6 - Part 1
34
State Table Characteristics
State table – a multiple variable table with the
following four sections:
• Present State – the values of the state variables for
each allowed state.
• Input – the input combinations allowed.
• Next-state – the value of the state at time (t+1) based
on the present state and the input.
• Output – the value of the output as a function of the
present state and (sometimes) the input.
From the viewpoint of a truth table:
• the inputs are Input, Present State
• and the outputs are Output, Next State
Chapter 6 - Part 1
35
Example 1: State Table (from Fig. 6-17)
The state table can be filled in using the next state and
output equations: A(t+1) = A(t)x(t) + B(t)x(t) B(t+1)
=A (t)x(t)
y(t)
=x (t)(B(t) + A(t))
Present State Input
A(t)
0
0
0
0
1
1
1
1
B(t)
0
0
1
1
0
0
1
1
x(t)
0
1
0
1
0
1
0
1
Next State Output
A(t+1) B(t+1)
0
0
0
1
0
0
1
1
0
0
1
0
0
0
1
0
y(t)
0
0
1
0
1
0
1
0
Chapter 6 - Part 1
36
Example 1: Alternate State Table
2-dimensional table that matches well to a K-map.
Present state rows and input columns in Gray code
order.
• A(t+1) = A(t)x(t) + B(t)x(t)
• B(t+1) =A (t)x(t)
• y(t) =x (t)(B(t) + A(t))
Present
Next State
Output
State
x(t)=0
x(t)=1
x(t)=0 x(t)=1
A(t) B(t) A(t+1)B(t+1) A(t+1)B(t+1) y(t) y(t)
0 0
0 0
0 1
0
0
0 1
0 0
1 1
1
0
1 0
0 0
1 0
1
0
1 1
0 0
1 0
1
0
Chapter 6 - Part 1
37
State Diagrams
The sequential circuit function can be
represented in graphical form as a state
diagram with the following components:
• A circle with the state name in it for each state
• A directed arc from the Present State to the Next
State for each state transition
• A label on each directed arc with the Input values
which causes the state transition, and
• A label:
On each circle with the output value produced,
or
On each directed arc with the output value
produced.
Chapter 6 - Part 1
38
State Diagrams
Label form:
• On circle with output included:
state/output
Moore type output depends only on state
• On directed arc with the output
included:
input/output
Mealy type output depends on state and
input
Chapter 6 - Part 1
39
Example 1: State Diagram
x=0/y=0
Which type?
Diagram gets
confusing for
x=1/y=0
large circuits
For small circuits,
usually easier to
understand than
the state table
x=0/y=1
AB
00
x=1/y=0
x=0/y=1 1 0
x=1/y=0
x=0/y=1
11
01
x=1/y=0
Chapter 6 - Part 1
40
Moore and Mealy Models
Sequential Circuits or Sequential Machines are
also called Finite State Machines (FSMs). Two
formal models exist:
Moore Model
• Named after E.F. Moore.
• Outputs are a function
ONLY of states
• Usually specified on the
states.
Mealy Model
• Named after G. Mealy
• Outputs are a function of
inputs AND states
• Usually specified on the
state transition arcs.
In contemporary design, models are sometimes
mixed Moore and Mealy
Chapter 6 - Part 1
41
Moore and Mealy Example Diagrams
Mealy Model State Diagram
maps inputs and state to outputs
x=0/y=0
x=1/y=0
1
0
Moore Model State Diagram x=0
maps states to outputs
x=0/y=0
0/0
x=1/y=1
x=0
x=1
x=1
x=0
1/0
2/1
x=1
Chapter 6 - Part 1
42
Moore and Mealy Example Tables
Mealy Model state table maps inputs and
state to outputs
Present
State
0
1
Next State
x=0 x=1
0
1
0
1
Output
x=0 x=1
0
0
0
1
Moore Model state table maps state to
Present Next State Output
outputs
State
0
1
2
x=0 x=1
0
1
0
2
0
2
0
0
1
Chapter 6 - Part 1
43
Example 2: Sequential Circuit Analysis
Logic Diagram:
D
Q
A
Z
C RQ
D
Q
B
C RQ
D
Clock
Reset
Q
C
CR Q
Chapter 6 - Part 1
44
Example 2: Flip-Flop Input Equations
Variables
• Inputs: None
• Outputs: Z
• State Variables: A, B, C
Initialization: Reset to (0,0,0)
Equations
• A(t+1) =
• B(t+1) =
• C(t+1) =
Z=
Chapter 6 - Part 1
45
Example 2: State Table
X’ = X(t+1)
ABC
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
A’B’C’
Z
Chapter 6 - Part 1
46
Example 2: State Diagram
Reset
111
ABC
000
100
001
011
010
Which states are used?
What is the function of
the circuit?
101
110
Chapter 6 - Part 1
47
Circuit and System Level Timing
Consider a system
comprised of ranks
of flip-flops
connected by logic:
If the clock period is
too short, some
data changes will not
propagate through the
circuit to flip-flop
inputs before the setup
time interval begins
D Q
D Q
C Q'
C Q'
D Q
D Q
C Q'
C Q'
D Q
D Q
C Q'
C Q'
D Q
D Q
C Q'
C Q'
D Q
D Q
C Q'
C Q'
CLOCK
CLOCK
Chapter 6 - Part 1
48
Circuit and System Level Timing
(continued)
Timing components along a path from flip-flop
to flip-flop
tp
C
tpd,FF
tpd,COMB
ts
tslack
(a) Edge-triggered (positive edge)
tp
C
tpd,FF
tpd,COMB
tslack
ts
(b) Pulse-triggered (negative pulse)
Chapter 6 - Part 1
49
Circuit and System Level Timing
(continued)
New Timing Components
• tp - clock period - The interval between occurrences
of a specific clock edge in a periodic clock
• tpd,COMB - total delay of combinational logic along the
path from flip-flop output to flip-flop input
• tslack - extra time in the clock period in addition to
the sum of the delays and setup time on a path
Can be either positive or negative
Must be greater than or equal to zero on all paths for
correct operation
Chapter 6 - Part 1
50
Circuit and System Level Timing
(continued)
Timing Equations
tp = tslack + (tpd,FF + tpd,COMB + ts)
• For tslack greater than or equal to zero,
tp ≥ max (tpd,FF + tpd,COMB + ts)
for all paths from flip-flop output to flip-flop input
Can be calculated more precisely by using tPHL
and tPLH values instead of tpd values, but
requires consideration of inversions on paths
Chapter 6 - Part 1
51
Calculation of Allowable tpd,COMB
Compare the allowable combinational delay for a
specific circuit:
a) Using edge-triggered flip-flops
b) Using master-slave flip-flops
Parameters
•
•
•
•
tpd,FF(max) = 1.0 ns
ts(max) = 0.3 ns for edge-triggered flip-flops
ts = twH = 1.0 ns for master-slave flip-flops
Clock frequency = 250 MHz
Chapter 6 - Part 1
52
Calculation of Allowable tpd,COMB
(continued)
Calculations: tp = 1/clock frequency = 4.0 ns
• Edge-triggered: 4.0 ≥ 1.0 + tpd,COMB + 0.3, tpd,COMB ≤ 2.7 ns
• Master-slave: 4.0 ≥ 1.0 + tpd,COMB + 1.0, tpd,COMB ≤ 2.0 ns
Comparison: Suppose that for a gate, average tpd =
0.3 ns
• Edge-triggered: Approximately 9 gates allowed on a path
• Master-slave: Approximately 6 to 7 gates allowed on a path
Chapter 6 - Part 1
53
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Chapter 6 - Part 1
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