Sequential Circuits - cs.princeton.edu · 1 1 Sequential Circuits CS 217 2 Combinational circuit...
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1 1 Sequential Circuits CS 217 2 Combinational circuit • Directed acyclic graph (no loops) • Outputs, at any given time, dependent only on inputs at that time (after signal propagation) • Equivalent to one boolean formula per output x 2 y 2 z 2 c 1 c 2
Transcript of Sequential Circuits - cs.princeton.edu · 1 1 Sequential Circuits CS 217 2 Combinational circuit...
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Sequential Circuits
CS 217
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Combinational circuit• Directed acyclic graph (no loops)
• Outputs, at any given time, dependent only on inputs at that time (after signal propagation)
• Equivalent to one boolean formula per output
x2 y2
z2
c1c2
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Cycles in the circuit• What happens if there are cycles?
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Cycles in the circuit• Simulate . . .
0
3
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Cycles in the circuit• Simulate . . .
0 1
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Cycles in the circuit• Simulate . . .
0 1 0
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Cycles in the circuit• Simulate . . .
1 1 0
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Cycles in the circuit• Simulate . . .
1 0 0
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Cycles in the circuit• Simulate . . .
1 0 1
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Cycles in the circuit• Simulate . . .
0 0 1
0
1
time
Outputs, at any given time, dependent not only on inputs at that time;
also dependent on history. A “sequential” circuit.
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Another circuit with cyclesThree inverters:
astable 0 0 1
0
1
time
0 1
Two inverters:
bistable 1 0
0
1
time
0
1
time
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R-S Latch
= NOR gate
Reset
Set
Q
Q
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R-S Latch
= NOR gate
Reset
Set
Q
Q
0
0
0
1
R
S
Q
Q
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R-S Latch
= NOR gate
Reset
Set
Q
Q
0
1
1
0
R
S
Q
Q
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R-S Latch
= NOR gate
Reset
Set
Q
Q
0
0
1
0
R
S
Q
Q
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R-S Latch
= NOR gate
Reset
Set
Q
Q
1
0
0
1
R
S
Q
Q
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R-S Latch
= NOR gate
Reset
Set
Q
Q
0
0
0
1
R
S
Q
Q
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R-S Latch
= NOR gate
Reset
Set
Q
Q
1
0
0
1
R
S
Q
Q
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R-S Latch
= NOR gate
Reset
Set
Q
Q
0
0
0
1
R
S
Q
Q
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Clocked flipflop
Q
Clock
D
Q
D
ClockClock high:
copy D to Q
Clock low:
ignore D, remember Q
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Master/Slave flipflop
Clock
D
Q
D
ClockClock high:
copy D to X; keep Q
Clock low:
copy X to Q; keep X
Q
X
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Master/Slave flipflop
D
Clock
Q
X
D QCircuit symbol:
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Synchronous sequential circuits
D Q
D Q
D Q D0
D1
D2
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Clock
Q0
Q1
Q2
• Flipflops all clocked simultaneously
• Combinational circuit determines next flipflop values (calculates D’s from Q’s).
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Analysis of sequential circuits
D Q
D Q
D Q D0
D1
D2
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Clock
Q0
Q1
Q2
Q2 Q1 Q0 D2 D1 D0
0 0 0 0 0 1
0 0 1 0 1 0
0 1 0 0 1 1
0 1 1 1 0 0
1 0 0 1 0 1
1 0 1 1 1 0
1 1 0 1 1 1
1 1 1 0 0 0
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Input / Output
D Q
D Q
D Q
Q2Q1Q0 I 2I 1 D2D1D0 O0
0 0 0 0 0 0 1 0 10 0 0 0 1 0 0 0 00 0 0 1 0 0 0 0 0
.
.
.
1 1 1 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0
combinational
circuit
outputs
next state
inputs
State Inputs NextS Outputs
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Circuit with I/O
D Q
D Q D0
D1
I0
Clock
Q0
Q1
O0
Q1 Q0 I 0 D1 D0 O0
0 0 0 0 0 0
0 0 1 0 1 0
0 1 0 0 1 0
0 1 1 1 0 0
1 0 0 1 0 0
1 0 1 1 1 0
1 1 0 1 1 0
1 1 1 0 0 1
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State Machine
Q1 Q0 I 0 D1 D0 O0
0 0 0 0 0 0
0 0 1 0 1 0
0 1 0 0 1 0
0 1 1 1 0 0
1 0 0 1 0 0
1 0 1 1 1 0
1 1 0 1 1 0
1 1 1 0 0 1
00 01
11 10
1/0
0/0 0/0
1/0
1/0
1/1
0/0 0/0
input output
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What does it do?
00 01
11 10
1/0
0/0 0/0
1/0
1/0
1/1
0/0 0/0
input outputCounts up if input=1; stays stationary if input=0.
Output is “carry” when counter wraps around.
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Synthesis procedure1. State purpose of circuit in words
2. Make state machine
3. Make truth tables
4. Apply combinational-circuit synthesis procedure:
• Identify “true” rows
• Construct sum-of-products expression
• Construct circuit
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Vending machine1. State purpose of circuit in words
Accept nickels and dimes
Candy costs 15¢
Dispense candy bar and appropriate change
Inputs: D (dime), N (nickel) Outputs: C (candy), O (nickel change)
2. Make state machine
00 01
10
01/0000/0000/00
01/0001/10
00/00
10/11
10/10States:
00: 0¢ credit
01: 5¢ credit
10: 10¢ credit10/00
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Vending machine3. Make truth table
Assume: D&N impossible
“x” means don’t-care
00 01
10
01/0000/0000/00
01/0001/10
00/00
10/11
10/10
10/00
Q1Q0D N D1D0 C O
0 0 0 0 0 0 0 00 0 0 1 0 1 0 00 0 1 0 1 0 0 00 0 1 1 x x x x0 1 0 0 0 1 0 00 1 0 1 1 0 0 00 1 1 0 0 0 1 00 1 1 1 x x x x1 0 0 0 1 0 0 01 0 0 1 0 0 1 01 0 1 0 0 0 1 11 0 1 1 x x x x1 1 0 0 x x x x1 1 0 1 x x x x1 1 1 0 x x x x1 1 1 1 x x x x
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Vending machine4. Make sum-of-products
expressions
D1= Q1Q0DN + Q1Q0DN + Q1Q0DN
Q1Q0D N D1D0 C O
0 0 0 0 0 0 0 00 0 0 1 0 1 0 00 0 1 0 1 0 0 00 0 1 1 x x x x0 1 0 0 0 1 0 00 1 0 1 1 0 0 00 1 1 0 0 0 1 00 1 1 1 x x x x1 0 0 0 1 0 0 01 0 0 1 0 0 1 01 0 1 0 0 0 1 11 0 1 1 x x x x1 1 0 0 x x x x1 1 0 1 x x x x1 1 1 0 x x x x1 1 1 1 x x x x
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Vending machine4. Make sum-of-products
expressions
D1= Q1Q0DN + Q1Q0DN + Q1Q0DN
D0= Q1Q0DN + Q1Q0DN
C = Q1Q0DN + Q1Q0DN + Q1Q0DN
O = Q1Q0DN
Q1Q0D N D1D0 C O
0 0 0 0 0 0 0 00 0 0 1 0 1 0 00 0 1 0 1 0 0 00 0 1 1 x x x x0 1 0 0 0 1 0 00 1 0 1 1 0 0 00 1 1 0 0 0 1 00 1 1 1 x x x x1 0 0 0 1 0 0 01 0 0 1 0 0 1 01 0 1 0 0 0 1 11 0 1 1 x x x x1 1 0 0 x x x x1 1 0 1 x x x x1 1 1 0 x x x x1 1 1 1 x x x x
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Vending machine4. Make gates Q1Q0D N D1D0 C O
0 0 0 0 0 0 0 00 0 0 1 0 1 0 00 0 1 0 1 0 0 00 0 1 1 x x x x0 1 0 0 0 1 0 00 1 0 1 1 0 0 00 1 1 0 0 0 1 00 1 1 1 x x x x1 0 0 0 1 0 0 01 0 0 1 0 0 1 01 0 1 0 0 0 1 11 0 1 1 x x x x1 1 0 0 x x x x1 1 0 1 x x x x1 1 1 0 x x x x1 1 1 1 x x x x
Q1
Q0
D
N
D1
D0
C
O
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Vending machine5. Hook up flipflops, clocks, inputs, outputs
Q1
Q0
D
N
D1
D0
C
O
D Q
D Q
Clock
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Summary• Sequential circuits
� Store state in latches� Connect latches with combinational circuits
• Next time� Building a computer out of sequential circuits
