Sequential Circuits

2

Sequential vs. Combinational

• Combinational Logic: Output depends only on current input

− TV channel selector (0-9)

• Sequential Logic: Output depends not only on current input but also on current

state of the system (which depends on past input values)− TV channel selector (up-down)

Need some type of memory to remember the current state

inputs outputssystem

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Sequential Logic

• Sequential Logic circuits Remember past inputs and past circuit state.Outputs from the system are “fed back” as new inputs.The storage elements are circuits that are capable of

storing binary information: memory.

4

Feedback Loop• Feedback:

A signal s1 depends on another signal whose value depends on s1 − (perhaps with several intermediate signals).

s1

5

Base of MemoryConsider the following circuit:

It can differentiate between two different states as it has only one feedback line that can keep one of two values, 0 or 1.

A circuit with n feedback lines has 2n potential states, and that the memory of our circuit depends on the number of its feedback lines:

PI

PI

01

n

6

Synchronous vs. Asynch.

• Synchronous sequential circuit: the behavior can be defined from knowledge of its

signal at discrete instants of time. achieves synchronization by using a timing signal

called the clock.

• Asynchronous sequential circuit: the behavior is dependent on the order of input

signal changes over continuous time, and output can change at any time (clockless).

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Clock Signal

Different duty cycles

Clock generator: Periodic train of clock pulses

Rising Clock Edge

Falling Clock Edge

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Clock Signal

Clock is distributed throughout the whole design

All components synchronizes itself with it.

9

Synchronous Circuits

Combinational

Logic

time

clk

10

SR latch (NOR version)

RS

R

S

Q

Q’

Q

hold 0 1

unstable

S

0 0 1 1

R

0 1 0 1

Truth Table:Next State = F(S, R, Current State)

S(t) R(t) Q(t) Q(t+)

0 0 0 0 (hold)0 0 1 1 (Hold)0 1 0 0 (reset)0 1 1 0 (reset)1 0 0 1 (set)1 0 1 1 (set)1 1 0 Not allowed1 1 1 Not allowed

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SR LatchTruth Table:Next State = F(S, R, Current State)

S

R

Q

R-SLatch Q+

Derived K-Map:

Characteristic Equation:

Q+ = S + R Q t

R

SR 00 01 11 10

0 0 X 1

1 0 X 1

0

1

Q ( t )

S

S(t) R(t) Q(t) Q(t+)

0 0 0 0 (hold)0 0 1 1 (Hold)0 1 0 0 (reset)0 1 1 0 (reset)1 0 0 1 (set)1 0 1 1 (set)1 1 0 Not allowed1 1 1 Not allowed

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R=S=1 ??

• Illegal output, becauseWhen S=R=1, both outputs go to zero. If both inputs now go to 0, the state of the SR flip

flop is depends on which input remains a 1 longer before making transition to 0.

Hence, “undefined” state. MUST be avoided.

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Timing Diagram

100

R

S

Q

Q’

Reset Hold Set

ForbiddenState

Reset Set

ForbiddenState

Race

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Timing Diagram of SR Latch

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SR Latch State Diagram

• Theoretical State Diagram

Q Q Q Q

Q Q

0 1 1 0

0 0

SR = 1 0

SR = 0 1

SR = 0 1

SR = 1 1

SR = 1 0

SR = 1 1

SR = 00, 01 SR = 00, 10

Q Q1 1

SR = 0 0

SR = 0 0, 11

SR = 11

SR = 1 0SR = 0 1

Q Q Q Q

Q Q

0 1 1 0

0 0

Q Q1 1

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SR Latch State Diagram

Very difficult to observe R-S Latch in the 1-1 state Ambiguously returns to state 0-1 or 1-0

Q Q Q Q

Q Q

0 1 1 0

0 0

Q Q Q Q

Q Q

0 1 1 0

0 0

SR = 1 0

SR = 0 1

SR = 0 1

SR = 1 1

SR = 1 0

SR = 1 1

SR = 00, 01 SR = 00, 10

SR = 0 0

SR = 11

SR = 0 0

• Observed State Diagram

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S’R’ Latch (NAND version)

S’

R’

Q

Q’

0 00 11 01 1

S’ R’ Q Q’0

1

1

0

1 0 Set

0 0 1 0 1 1 1 0 1 1 1 0

X Y NANDS

R

R-SLatch

Q

Q’

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S’

R’

Q

Q’

0 00 11 01 1

S’ R’ Q Q’1

1

1

0 1 0 Hold

0 0 1 0 1 1 1 0 1 1 1 0

X Y NAND

1 0 Set

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S’

R’

Q

Q’

0 00 11 01 1

S’ R’ Q Q’1

0

0

1

0 0 1 0 1 1 1 0 1 1 1 0

X Y NAND

1 0 Hold

1 0 Set0 1 Reset

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S’

R’

Q

Q’

0 00 11 01 1

S’ R’ Q Q’1

1

0

1

0 0 1 0 1 1 1 0 1 1 1 0

X Y NAND

0 1 Hold

1 0 Set0 1 Reset1 0 Hold

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S’

R’

Q

Q’

0 00 11 01 1

S’ R’ Q Q’0

0

1

1

0 0 1 0 1 1 1 0 1 1 1 0

X Y NAND

0 1 Hold

1 0 Set0 1 Reset1 0 Hold

1 1 Disallowed

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SR Latch with Clock signal

Latch is sensitive to input changes ONLY when C=1Latch is sensitive to input changes ONLY when C=1

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SR Latch with Clock signal

S’

R’

Q

Q’

S

R

CLK

S R CLK S’ R’ Q Q’

0 0 1 1 1 Q0 Q0’ Store 0 1 1 1 0 0 1 Reset1 0 1 0 1 1 0 Set1 1 1 0 0 1 1 DisallowedX X 0 1 1 Q0 Q0’ Store

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D LatchOne way to eliminate the undesirable

indeterminate state in the RS flip flop:− ensure that inputs S and R are never 1

simultaneously. This is done in the D latch:

D

C

DLatch

Q

Q’

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D Latch (cont.)D

S R CLK Q Q’

0 0 1 Q0 Q0’ Store 0 1 1 0 1 Reset1 0 1 1 0 Set1 1 1 1 1 DisallowedX X 0 Q0 Q0’ Store

0 1 0 11 1 1 0X 0 Q0 Q0’

D CLK Q Q’

S’

R’

Q

Q’

S

R

CLK

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D Latch Timing Diagram

D

C

DLatch

Q

Q’

C

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D-Latch Circuit

G D Q Q+

0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1

01010011

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D Latch with Transmission Gates

C=1 C=1 TG1 closes and TG2 opens TG1 closes and TG2 opens Q’=D’ and Q’=D’ and Q=DQ=D C=0 C=0 TG1 opens and TG2 closes TG1 opens and TG2 closes Hold Q and Hold Q and Q’Q’

2

1

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JK Latch

J, K both one yields toggle


Q+ = Q K’ + Q’ J

K

J Q

Q’

J-K Latch

J(t) K(t) Q(t) Q(t+)

0 0 0 0 (hold)0 0 1 1 (Hold)0 1 0 0 (reset)0 1 1 0 (reset)1 0 0 1 (set)1 0 1 1 (set)1 1 0 1 (toggle)1 1 1 0 (toggle)

Derived K-Map:

K

JK 00 01 11 10

0 0 1 1

1 0 0 1

0

1

Q ( t )

J

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JK Latch Using SR LatchHow to eliminate the forbidden state in SR?

Idea: use output feedback to guarantee that R and S are never both one

J, K both one yields toggle


Q+ = Q K + Q J

J(t) K(t) Q(t) Q(t+)

00

00

01

0 HOLD

1

00

11

01

0 RESET

0

11

00

01

1 SET

1

11

11

01

1 TOGGLE

0

R-S latch

K

J S

R

Q

Q’ Q’

Q

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JK Lacth Race Condition

J

K

Q

\ Q

100 Set Reset Toggle

Toggle Correctness: Single State change per clocking event

Solution: Master/Slave Flipflop

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Flip-Flops

Latches are “transparent” (= any change on the inputs is seen at the outputs immediately).

This causes synchronization problems!

Solution: use latches to create flip-flops that can respond (update) ONLY on SPECIFIC times (instead of ANY time).

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Alternatives in FF choice

• Type of FFRSDJK

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D-FF

0 00 11 01 1

0011

D Q Q

c Truth table

DQ

Timing for D Flip-Flop (Falling-Edge Trigger)

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Rising Edge D-FF

Falling-Edge Circuit?

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Symbols

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Compare 3 Types

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Setup & Hold Times

Setup and Hold Times for an Edge-Triggered D Flip-Flop

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Edge-Triggered D Flip-Flop

Figure 11-17. Determination of Minimum Clock Period

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Master-Slave FF configuration using SR latches

– Enables level-triggered behaviorEnables level-triggered behavior

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S R CLK Q Q’

0 0 1 Q0 Q0’ Store 0 1 1 0 1 Reset1 0 1 1 0 Set1 1 1 1 1 DisallowedX X 0 Q0 Q0’ Store

Master-Slave FF configuration using SR latches (cont.)

•When C=1, master is enabled and stores new data, slave stores old data.•When C=0, master’s state passes to enabled slave (Q=Y), master not sensitive to new data (disabled).

Master Slave

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Master-Slave J-K Flip-Flop

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Master-Slave J-K Flip-Flop

Sample inputs while clock high Sample inputs while clock low

Uses time to break feedback path from outputs to inputs!Uses time to break feedback path from outputs to inputs!

Correct ToggleOperation

P

P’

Master outputs

Slave outputs

Set Reset T oggle 1's

Catch 100

J

K

C

P

P‘‘

Q

Q’

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Edge-Triggered FF1's Catching: a 0-1-0 glitch on the J or K inputs leads to a state change! forces designer to use hazard-free logic

Solution: edge-triggered logic

Negative Edge-TriggeredD flipflop

4-5 gate delays

setup, hold timesnecessary to successfully

latch the input

Characteristic Equation:Q+ = D

Q

Q

D

Clk=1

R

S

0

0

D

DD

Holds D when clock goes low

Holds D when clock goes low

Negative edge-triggered FFwhen clock is high

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T Flip-Flop

T Flip-Flop

0 00 11 01 1

0110

T Q Q

b

QTQTQTQ

Timing Diagram for T Flip-Flop (Falling-Edge Trigger)

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Implementation of T-FF

Implementation of T Flip-Flop

QTQTQKQJQ

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FFs with Additional Inputs

Figure 11-27. D Flip-Flop with Clock Enable

CEDECQQ

CEDECQDQ in The MUX output :

The characteristic equation :

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Asynchronous Preset/Clear

Many times it is desirable to asynchronously (i.e., independent of the clock) set or reset FFs.

Example: At power-up to that we can start from a known state.

Asynchronous set == direct set == Preset

Asynchronous reset == direct reset == Clear

There may be “synchronous” preset and clear.

D

C

S

R

Q

Q

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Asynchronous Set/Reset

S

C1

1J

1K

R

IEEE standard graphics symbol for JK-FF with direct set & reset

Cn indicates that Cn controls all other inputs whose label starts with n. In this case, C1 controls J1 and K1.

S R C1 1J 1K Q(t+1)

0 1 X X X 1 – Preset

1 0 X X X 0 – Clear

0 0 X X X Undefined

1 1 0 0 Q(t) – Hold

1 1 0 1 0 – Reset

1 1 1 0 1 – Set

1 1 1 1 Q(t)’ -- Complement

Function Table

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Asynchronous Inputs

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Synchronous Reset

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Timing Parameters

• ts - setup time• th - hold time• tw - clock

pulse width• tpx - propa-

gation delay tPHL - High-to-

Low tPLH - Low-to-

High tpd - max (tPHL,

tPLH)

Sequential Circuits

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