UNIT - IV€¦ · UNIT - IV Sequential Circuits P.VIDYA SAGAR ( ASSOCIATE PROFESSOR) 1 VIDYA SAGAR...
Transcript of UNIT - IV€¦ · UNIT - IV Sequential Circuits P.VIDYA SAGAR ( ASSOCIATE PROFESSOR) 1 VIDYA SAGAR...
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Department of Electronics and Communication Engineering, VBIT
UNIT - IVSequential Circuits
P.VIDYA SAGAR ( ASSOCIATE PROFESSOR)
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Department of Electronics and Communication Engineering, VBITDepartment of Electronics and Communication Engineering, VBIT
Sequential logic circuits
The main characteristic of combinational logic circuits is that their output values depend on their
present input values.
Sequential logic circuits differ from combinational logic circuits because they contain memory
elements so that their output values depend on both present and past input values
Sequential circuits can be Asynchronous or synchronous.
Asynchronous sequential circuits change their states and output values whenever a change in
input values occurs.
Synchronous sequential circuits change their states and output values at fixed points of time,
i.e. clock signals.
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Sequential Circuit Models
Universal model
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Sequential Circuits
Combinational
CircuitMemory
Elements
Inputs Outputs
Asynchronous
Combinational
Circuit
Flip-flops
Inputs Outputs
Clock
Synchronous
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S-R latch operation
S
R
QN
Q
0
0
0=
=1
S
R
QN
Q
1
00=
=0
=1
0=S
R
QN
Q
0
0 =1
=0
=1
S
R
QN
Q
0
1 =1
=1
=0
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Latches
SR Latch
R
S
Q
Q
• If S = 1 (Set), Q+ = 1
• If R = 1 (Reset), Q+ = 0
• If S = R = 0, Q+ = Q (no change)
• S = R = 1 is not allowed.
Present
value
Next
value
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Latches
SR Latch
R
S
Q
Q
S R Q
0 0 Q0
0 1 0
1 0 1
1 1 Q=Q’=0
No change
Reset
Set
Invalid
S’ R’ Q
0 0 Q=Q’=1
0 1 1
1 0 0
1 1 Q0
Invalid
Set
Reset
No change
S
R
Q
Q
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Controlled Latches Or Gated Latch
SR Latch with Control Input
C S R Q
0 x x Q0
1 0 0 Q0
1 0 1 0
1 1 0 1
1 1 1 Q=Q’
No change
No change
Reset
Set
Invalid
S
R
Q
Q
S
R
C
S
RQ
QS
R
C
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Controlled Latches
D Latch (D = Data)
C D Q
0 x Q0
1 0 0
1 1 1
No change
Reset
Set
C
Timing Diagram
D
Q
t
Output may
change
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Controlled Latches
D Latch (D = Data)
C D Q
0 x Q0
1 0 0
1 1 1
No change
Reset
Set
C
Timing Diagram
D
Q
Output may
change
S
R
Q
Q
D
C
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Flip-Flops
Controlled latches are level-triggered
Flip-Flops are edge-triggered
C
CLK Positive Edge
CLK Negative Edge
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CLK
D
Qlatch
CLK
D
Q
Latch Vs Flip-Flops
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Flip-Flops
Master-Slave D Flip-Flop
D Latch(Master)
D
C
QD Latch(Slave)
D
C
Q QD
CLK
CLK
D
QMaster
QSlave
Looks like it is negative
edge-triggered
Master Slave
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Flip-Flops
Edge-Triggered D Flip-Flop
D
CLK
Q
Q
D Q
Q
D Q
Q
Positive Edge
Negative Edge
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Flip-Flops
JK Flip-Flop
D Q
Q
Q
QCLK
J
K
J Q
QK
D = JQ’ + K’Q
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Flip-Flops
T Flip-Flop
D = TQ’ + T’Q = T Q
J Q
QK
T D Q
Q
T
D = JQ’ + K’QT Q
Q
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Flip-Flop Characteristic Tables
D Q
Q
D Q(t+1)
0 0
1 1
Reset
Set
J K Q(t+1)
0 0 Q(t)
0 1 0
1 0 1
1 1 Q’(t)
No change
Reset
Set
Toggle
J Q
QK
T Q
Q
T Q(t+1)
0 Q(t)
1 Q’(t)
No change
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Flip-Flop Characteristic Equations
D Q
Q
D Q(t+1)
0 0
1 1Q(t+1) = D
J K Q(t+1)
0 0 Q(t)
0 1 0
1 0 1
1 1 Q’(t)
Q(t+1) = JQ’ + K’Q
J Q
QK
T Q
Q
T Q(t+1)
0 Q(t)
1 Q’(t)Q(t+1) = T Q
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Flip-Flop Characteristic Equations
Analysis / Derivation
J Q
QK
J K Q(t) Q(t+1)
0 0 0 0
0 0 1 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
No change
Reset
Set
Toggle
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Flip-Flop Characteristic Equations
Analysis / Derivation
J Q
QK
J K Q(t) Q(t+1)
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0
1 0 1
1 1 0
1 1 1
No change
Reset
Set
Toggle
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Flip-Flop Characteristic Equations
Analysis / Derivation
J Q
QK
J K Q(t) Q(t+1)
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0
1 1 1
No change
Reset
Set
Toggle
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Flip-Flop Characteristic Equations
Analysis / Derivation
J Q
QK
J K Q(t) Q(t+1)
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 0
No change
Reset
Set
Toggle
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Flip-Flop Characteristic Equations
Analysis / Derivation
J Q
QK
J K Q(t) Q(t+1)
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 0
K
0 1 0 0
J 1 1 0 1
Q
Q(t+1) = JQ’ + K’Q
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Flip-Flop Applications
Parallel Data Storage
Frequency Division
Counting
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Registers.
Shift Registers:
Serial in, serial out shift register
Serial in, parallel out shift register
Parallel in, serial out shift register
Parallel in, parallel out shift register
Shift Register Applications
Counters:
Ripple Counters
Synchronous Counters
Counter Applications
Registers & Counters
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Shift Registers
Multi-bit register that moves stored data bits left/right ( 1 bit position per clock cycle)
Shift Left is towards MSB
0 1 1 1 LSI
Q3 Q2 Q1 Q0
1 1 1 LSI
Q3 Q2 Q1 Q0
RSI 0 1 1 1
Q3 Q2 Q1 Q0
RSI 0 1 1
Q3 Q2 Q1 Q0
– Shift Right (or Shift Up) is towards MSB
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Serial In, Serial Out Shift Register
D Q
CLK
D Q
CLK
D Q
CLK
SERIN
CLOCK
SEROUT
For a n-bit SRG:Serial Out = Serial In delayed by n clock period
4-bit shift register example:serin: 1 0 1 1 0 0 1 1 1 0serout: - - - - 1 0 1 1 0 0clock:
SRG n>SI SO
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Serial In, Parallel Out Shift register
D Q
CLK
D Q
CLK
D Q
CLK
SERIN
CLOCK
nQ
2Q
1Q
Serial to Parallel Converter
4-bit shift register example:serin: 1 0 1 1 0 0 1 1 1 01Q: - 1 0 1 1 0 0 1 1 12Q: - - 1 0 1 1 0 0 1 13Q: - - - 1 0 1 1 0 0 14Q: - - - - 1 0 1 1 0 0clock:
SRG n>SI 1Q
2Q
nQ (SO)
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Parallel In, Serial Out Shift Register
SERIN
CLOCK
D Q
CLK
D Q
CLK
D Q
CLK
SEROUT
LOAD/SHIFT
1D
2D
ND
S
L
S
L
S
L
1Q
2Q
NQ
Parallel to Serial Converter
Load/Shift=1Di Qi
Load/Shift=0 Qi Qi+1
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Parallel In, Parallel Out Shift Register
SERIN
CLOCK
D Q
CLK
D Q
CLK
D Q
CLK
LOAD/SHIFT
1D
2D
ND
1Q
2Q
NQ
S
L
S
L
S
L
General Purpose:Makes any kind of (left) shift register
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Types of counters
Counters are classified according to the way they are clocked:
Synchronous Counter :
Common clock is connected to all of the f/f’s and thus they are clocked simultaneously.
Asynchronous Counter (Ripple Counter):
The first flip-flop is clocked by the external clock pulse and then each successive flip-flop is
clocked by the output of the preceding f/f.
a counter is a device which stores (and sometimes displays) the number of times a
particular event or process has occurred, often in relationship to a clock signal.
VIDYA SAGAR P33
COUNTERS
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Synchronous Counters
Binary Up Counters
A synchronous binary counter counts from 0 to 2N-1, where N is the number of bits/flip-flops in the
counter. Each flip-flop is used to represent one bit. The flip-flop in the lowest-order position is
complemented/toggled with every clock pulse and a flip-flop in any other position is complemented
on the next clock pulse provided all the bits in the lower-order positions are equal to 1.
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In a binary up counter, a particular bit, except for the first bit, toggles if all the lower-order bits
are 1's. The opposite is true for binary down counters. That is, a particular bit toggles if all the
lower-order bits are 0's and the first bit toggles on every pulse.
Binary Down Counters
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Binary Up/Down Counters :
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Binary Counter with Parallel Load:
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Ring Counters,Johnson/Twisted-Ring Counters :
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Binary Ripple Counter
A binary ripple counter is an asynchronous counter where only the first flip-flop is clocked by
an external clock. Each subsequent flip-flop is triggered by the transition occurring in the
preceding flip-flop.
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BCD Ripple Counter
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Analysis of Sequential Circuits
• Analysis is describing what a given circuit will do
• The behavior of a clocked (synchronous) sequential circuit is determined from the inputs, the
output, and the states of FF
Steps:
• Obtain state equations
• FF input equations
• Output equations
• Fill the state table
• Put all combinations of inputs and current states
• Fill the next state and output
• Draw the state diagram
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Analysis of Combinational vs Sequential Circuits
–Combinational :
•Boolean Equations
•Truth Table
•Output as a function of
inputs
–Sequential :
•State Equations
•State Table
•State Diagram
•Output as a function of input and current state
•Next state as a function of inputs and current state.
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State Equations
–A state equation is a Boolean expression which specifies the next state and output as a function of the present state and inputs.
–Example:
•The shown circuit has two D-FFs (A,B), an input x and output y.
•The D input of a FF determines the next state
•A(t+1) = A(t)x+B(t)x = Ax+Bx
•B(t+1) = A’(t)x = A’x
•Output:
•y = (A+B)x’
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State Table
–A state table is a table enumerating all present
states, inputs, next states and outputs.
•Present state, inputs: list all combinations
•Next states, outputs: derived from state equations
4 sections
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State Table
2-D Form
–A state table is a table enumerating all present
states, inputs, next states and outputs.
•Present state, inputs: list all combinations
•Next states, outputs: derived from state equations
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State Diagram
• The state diagram is a graphical representation of a state table (provides same information)
• Circles are states (FFs), Arrows are transitions between states
• Labels of arrows represent inputs and outputs
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Analysis of Sequential Circuits
• Analysis is describing what a given circuit will do
• The behavior of a clocked (synchronous) sequential circuit is determined from the inputs, the
output, and the states of FF
Steps:
• Obtain state equations
• FF input equations
• Output equations
• Fill the state table
• Put all combinations of inputs and current states
• Fill the next state and output
• Draw the state diagram
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Example 1
Analyze this circuit?
• Is this a sequential circuit? Why?
• How many inputs?
• How many outputs?
• How many states?
• What type of memory?
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Example 1 (cont.)
Q(t) D Q(t+1)
0 0 0
0 1 1
1 0 0
1 1 1
D Q(t+1)
0 0
1 1
Q(t+1) = D
Characteristic Tables and Equations
D Flip Flop (review)
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Example 1 (cont.)
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Example 1 (cont.)
State equations:
DA = AX + BX
DB = A’ X
Y = (A + B) X’
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Example 1 (cont.)
State equations:
DA = AX + BX
DB = A’ X
Y = (A + B) X’
State table:
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Example 1 (cont.)
State equations:
DA = AX + BX
DB = A’ X
Y = (A + B) X’
State table (2D):
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Example 1 (cont.)
State equations:
DA = AX + BX
DB = A’ X
Y = (A + B) X’
State table:
State diagram:
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Example 2
• Analyze this circuit.
• What about the output?
• This circuit is an example of a Moore machine (output depends only on current state)
• Mealy machines is the other type (output depends on inputs and current states)
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Example 2 (cont.)
Equation:
DA = A X Y
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Example 2 (cont.)
Equation:
DA = A X Y
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Example 3
Analyze this circuit?
• Is this a sequential circuit?
Why?
• How many inputs?
• How many outputs?
• How many states?
• What type of memory?
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Example 3 (cont.)
State equations:
JA = B, KA = B X’
JB = X’, KB = A X
by substitution:
A = JAA’ + KA’A
= A’ B + A B’ + A X
B = B’ X’ + A B X + A’ B X’
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Example 3 (cont.)
State equations:
JA = B, KA = B X’
JB = X’, KB = A X
by substitution:
A = JAA’ + KA’A
= A’ B + A B’ + A X
B = B’ X’ + A B X + A’ B X’
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Example 3 (cont.)
State equations:
JA = B, KA = B X’
JB = X’, KB = A X
by substitution:
A = JAA’ + KA’A
= A’ B + A B’ + A X
B = B’ X’ + A B X + A’ B X’
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Example 4
Analyze this circuit?
• Is this a sequential
circuit? Why?
• How many inputs?
• How many outputs?
• How many states?
• What type of memory?
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Example 4 (cont.)
State equations:
JA = BX’
KA = BX’ + B’X
DB = X
Y = X’AB
by substitution:
A(t+1) = JAA’ + KA’A
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Example 4 (cont.)
Current State Input Next State Output
A(t) B(t) X A(t+1) B(t+1) Y
0 0 0 0 0 0
0 0 1 0 1 0
0 1 0 1 0 0
0 1 1 0 1 0
1 0 0 0 0 0
1 0 1 1 1 0
1 1 0 1 0 1
1 1 1 0 1 0
State equations:
JA = BX’
KA = BX’ + B’X
DB = X
Y = X’AB
by substitution:
A(t+1) = JAA’ + KA’A
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Example 5
Analyze this circuit?
• Is this a sequential circuit?
Why?
• How many inputs?
• How many outputs?
• How many states?
• What type of memory?
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Example 5 (cont.)
State equations:
TA = BX
TB = X
Y = AB
by substitution:
A(t+1) = TAA’ + TA’A
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Example 5 (cont.)
State equations:
TA = BX
TB = X
Y = AB
by substitution:
A(t+1) = TAA’ + TA’A
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Example 5 (cont.)
State equations:
TA = BX
TB = X
Y = AB
by substitution:
A(t+1) = TAA’ + TA’A
The output depends only on current state.
This is a Moore machine
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Mealy vs Moore Finite State Machine (FSM)
–Mealy FSM:
• Output depends on current state and input
• Output is not synchronized with the clock
–Moore FSM:
• Output depends on current state only
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Asynchronous Sequential Circuits
Asynchronous sequential circuits basics
No clock signal is required
Internal states can change at any instant of time when there is a change in the input
variables
Have better performance but hard to design due to timing problems
Why Asynchronous Circuits?
Accelerate the speed of the machine (no need to wait for the next clock pulse).
Simplify the circuit in the small independent gates.
Necessary when having multi circuits each having its own clock.
Analysis Procedure
The analysis consists of obtaining a table or a diagram that describes the sequence of
internal states and outputs as a function of changes in the input variables.
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Transition Table
Transition table is useful to analyze an asynchronous circuit from the circuit
diagram. Procedure to obtain transition table:
1. Determine all feedback loops in the circuits
2. Mark the input (yi) and output (Yi) of each feedback loop
3. Derive the Boolean functions of all Y’s
4. Plot each Y function in a map and combine all maps into one table (flow table)
5. Circle those values of Y in each square that are equal to the value of y in the same row
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Asynchronous Sequential Circuit
The excitation variables: Y1 and Y2
Y1 = xy1+ xy2
Y2 = xy1 + xy2
CUT
CUT
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Transition Table
Combine the internal state with
input variables
Stable total states:
y1y2x = 000, 011, 110 and 101
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Transition Table
In an asynchronous sequential circuit, the internal
state can change immediately after a change in the
input.
It is sometimes convenient to combine the internal
state with input value together and call it the Total
State of the circuit. (Total state = Internal state +
Inputs)
In the example , the circuit has
4 stable total states: (y1y2x= 000, 011, 110, and 101)
4 unstable total states: (y1y2x= 001, 010, 111, and 100)
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Transition Table
If y=00 and x=0 Y=00 (Stable state)
If x changes from 0 to 1 while y=00, the circuit
changes Y to 01 which is temporary unstable
condition (Yy)
As soon as the signal propagates to make Y=01,
the feedback path causes a change in y to 01.
(transition form the first row to the second row)
If the input alternates between 0 and 1, the
circuit will repeat the sequence of states
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Flow Table
A flow table is similar to a transition table except that the internal state are
symbolized with letters rather than binary numbers.
It also includes the output values of the circuit for each stable state.
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Flow Table
In order to obtain the circuit
described by a flow table, it is
necessary to convert the flow table
into a transition table from which we
can derive the logic diagram.
This can be done through the
assignment of a distinct binary value
to each state.
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Race condition Two or more binary state variables will change value when one input variable changes.
Cannot predict state sequence if unequal delay is encountered.
Non-critical race: The final stable state does not depend on the change order of state
variables
Critical race: The change order of state variables will result in different stable states. Must
be avoided !!
VIDYA SAGAR P77
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Race Solution
It can be solved by making a proper binary assignment to the state variables.
The state variables must be assigned binary numbers in such a way that only one state
variable can change at any one time when a state transition occurs in the flow table.
VIDYA SAGAR P78
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Latches in Asynchronous Circuits
The traditional configuration of asynchronous circuits is using one or more feedback loops
No real delay elements.
It is more convenient to employ the SR latch as a memory element in asynchronous circuits
Produce an orderly pattern in the logic diagram with the memory elements clearly visible.
SR latch is an asynchronous circuit
So will be analyzed first using the method for asynchronous circuits.
VIDYA SAGAR P79
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SR Latch with NOR Gates
VIDYA SAGAR P80
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SR Latch with NOR Gates
S=1, R=1 (SR = 1)
should not be
used
⇒ SR = 0 is
normal mode
should be
carefully
checked first
VIDYA SAGAR P81
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SR Latch with NAND Gates
S=0, R=0 (S+R=0) should not be
used ⇒ S+R=1 is normal mode
(eq. S’R’=0)
should be carefully
checked first, so it is obtained
Y = S’ + Ry
VIDYA SAGAR P82