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**Synchronous Sequential Logic**

Chapter 5

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**5-1 Introduction Combinational circuits contains no memory elements**

the outputs depends on the inputs

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**5-2 Sequential Circuits ■ Sequential circuits a feedback path**

the state of the sequential circuit (inputs, current state) Þ (outputs, next state) synchronous: the transition happens at discrete instants of time asynchronous: at any instant of time

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**Synchronous sequential circuits**

a master-clock generator to generate a periodic train of clock pulses the clock pulses are distributed throughout the system clocked sequential circuits most commonly used no instability problems the memory elements: flip-flops binary cells capable of storing one bit of information two outputs: one for the normal value and one for the complement value maintain a binary state indefinitely until directed by an input signal to switch states

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Fig. 5.2 Synchronous clocked sequential circuit

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**5-3 Latches Basic flip-flop circuit two NOR gates**

more complicated types can be built upon it directed-coupled RS flip-flop: the cross-coupled connection an asynchronous sequential circuit (S,R)= (0,0): no operation (S,R)=(0,1): reset (Q=0, the clear state) (S,R)=(1,0): set (Q=1, the set state) (S,R)=(1,1): indeterminate state (Q=Q'=0) consider (S,R) = (1,1) Þ (0,0)

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**SR latch with NAND gates**

Fig. 5.4 SR latch with NAND gates

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**SR latch with control input**

En=0, no change En=1, output depends inputs S, R S_ 1/S' 0/1 R_ 1/R' Fig. 5.5 SR latch with control input

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D Latch eliminate the undesirable conditions of the indeterminate state in the RS flip-flop D: data gated D-latch D Þ Q when En=1; no change when En=0 S_ 1/D' 0/1 R_ 1/D Fig. 5.6 D latch

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Fig. 5.7 Graphic symbols for latches

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**5-4 Flip-Flops A trigger Level triggered – latches**

The state of a latch or flip-flop is switched by a change of the control input Level triggered – latches Edge triggered – flip-flops Fig. 5.8 Clock response in latch and flip-flop

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**If level-triggered flip-flops are used Edge-triggered flip-flops**

the feedback path may cause instability problem Edge-triggered flip-flops the state transition happens only at the edge eliminate the multiple-transition problem

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**Edge-triggered D flip-flop**

Master-slave D flip-flop two separate flip-flops a master flip-flop (positive-level triggered) a slave flip-flop (negative-level triggered) Fig. 5.9 Master-slave D flip-flop

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**CP = 1: (S,R) Þ (Y,Y'); (Q,Q') holds**

CP = 0: (Y,Y') holds; (Y,Y') Þ (Q,Q') (S,R) could not affect (Q,Q') directly the state changes coincide with the negative-edge transition of CP 第三版內容，參考用!

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**Edge-triggered flip-flops A D-type positive-edge-triggered flip-flop**

the state changes during a clock-pulse transition A D-type positive-edge-triggered flip-flop Fig. 5.10 D-type positive-edge-triggered flip-flop

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**three basic flip-flops (S,R) = (0,1): Q = 1 (S,R) = (1,0): Q = 0 **

(S,R) = (1,1): no operation (S,R) = (0,0): should be avoided Fig. 5.10 D-type positive-edge-triggered flip-flop

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第三版內容，參考用!

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**The propagation delay time**

The setup time D input must be maintained at a constant value prior to the application of the positive CP pulse The hold time D input must not changes after the application of the positive CP pulse The propagation delay time The interval between the trigger edge and the stabilization of the output to a new state 50% VH

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**Summary CP=0: (S,R) = (1,1), no state change CP=: state change once**

CP=1: state holds

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**Other Flip-Flops The edge-triggered D flip-flops**

The most economical and efficient Positive-edge and negative-edge Fig. 5.11 Graphic symbols for edge-triggered D flip-flop

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**JK flip-flop D=JQ'+K'Q J=0, K=0: D=Q, no change J=0, K=1: D=0 Þ Q =0**

J=1, K=1: D=Q' Þ Q =Q' Fig. 5.12 JK flip-flop

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**T flip-flop D = T⊕Q = TQ'+T'Q T=0: D=Q, no change T=1: D=Q' Þ Q=Q'**

Fig. 5.13 T flip-flop

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**Characteristic tables**

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**Characteristic equations**

D flip-flop Q(t+1) = D JK flip-flop Q(t+1) = JQ'+K'Q T flop-flop Q(t+1) = T⊕Q

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**Direct inputs asynchronous set and/or asynchronous reset Fig. 5.14**

D flip-flop with asynchronous reset

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**5-5 Analysis of Clocked Sequential Ckts**

A sequential circuit (inputs, current state) Þ (output, next state) a state transition table or state transition diagram Fig. 5.15 Example of sequential circuit

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**State equations A compact form The output equation**

A(t+1) = A(t)x(t) + B(t)x(t) B(t+1) = A'(t)x(t) A compact form A(t+1) = Ax + Bx B(t+1) = Ax The output equation y(t) = (A(t)+B(t))x'(t) y = (A+B)x'

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State table State transition table = state equations

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State equation A(t + 1) =Ax + Bx B(t + 1) = Ax y = Ax + Bx

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**State diagram State transition diagram a circle: a state**

a directed lines connecting the circles: the transition between the states Each directed line is labeled 'inputs/outputs‘ a logic diagram Û a state table Û a state diagram Fig. 5.16 State diagram of the circuit of Fig. 5.15

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**Flip-flop input equations**

The part of circuit that generates the inputs to flip-flops Also called excitation functions DA = Ax +Bx DB = A'x The output equations to fully describe the sequential circuit y = (A+B)x'

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**Analysis with D flip-flops**

The input equation DA=A⊕x⊕y The state equation A(t+1)=A⊕x⊕y Fig. 5.17 Sequential circuit with D flip-flop

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**Analysis with JK flip-flops**

Determine the flip-flop input function in terms of the present state and input variables Used the corresponding flip-flop characteristic table to determine the next state Fig. 5.18 Sequential circuit with JK flip-flop

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JA = B, KA= Bx' JB = x', KB = A'x + Ax‘ derive the state table Or, derive the state equations using characteristic eq.

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**State transition diagram**

State equation for A and B: Fig. 5.19 State diagram of the circuit of Fig. 5.18

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**Analysis with T flip-flops**

The characteristic equation Q(t+1)= T⊕Q = TQ'+T'Q Fig. 5.20 Sequential circuit with T flip-flop

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**The input and output functions**

TA=Bx TB= x y = AB The state equations A(t+1) = (Bx)'A+(Bx)A' =AB'+Ax'+A'Bx B(t+1) = x⊕B

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State Table

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Mealy and Moore models the Mealy model: the outputs are functions of both the present state and inputs (Fig. 5-15) the outputs may change if the inputs change during the clock pulse period the outputs may have momentary false values unless the inputs are synchronized with the clocks The Moore model: the outputs are functions of the present state only (Fig. 5-20) The outputs are synchronous with the clocks

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Fig. 5.21 Block diagram of Mealy and Moore state machine

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**5-7 Synthesizable HDL Models of Sequential Circuits**

Behavioral Modeling Example: Two ways to provide free-running clock Example: Another way to describe free-running clock

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**Behavioral Modeling always statement Examples:**

Two procedural blocking assignments: Two nonblocking assignments:

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**Flip-Flops and Latches**

■ HDL Example 5.1

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**Flip-Flops and Latches**

■ HDL Example 5.2

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**Characteristic Equation**

Q(t + 1) = Q ⊕ T Q(t + 1) = JQ + KQ For a T flip-flop For a JK flip-flop ■ HDL Example 5.3

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**HDL Example 5-3 (Continued)**

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HDL Example 5-4 Functional description of JK flip-flop

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State Diagram ■ HDL Example 5.5: Mealy HDL model

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**HDL Example 5-5 (Continued)**

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**HDL Example 5-5 (Continued)**

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Mealy_Zero_Detector Fig Simulation output of Mealy_Zero_Detector

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**HDL Example 5-6: Moore Model FSM**

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**Simulation Output of HDL Example 5-6**

Fig Simulation output of HDL Example 5.6

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**Structural Description of Clocked Sequential Circuits**

■ HDL Example 5.7: State-diagram-based model

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**HDL Example 5-7 (Continued)**

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**HDL Example 5-7 (Continued)**

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**HDL Example 5-7 (Continued)**

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**HDL Example 5-7 (Continued)**

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**Simulation Output of HDL Example 5-7**

Fig Simulation output of HDL Example 5.7

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**5-7 State Reduction and Assignment**

reductions on the number of flip-flops and the number of gates a reduction in the number of states may result in a reduction in the number of flip-flops a example state diagram Fig State diagram

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**only the input-output sequences are important **

state a a b c d e f f g f g a input output only the input-output sequences are important two circuits are equivalent have identical outputs for all input sequences the number of states is not important Fig State diagram

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**Equivalent states two states are said to be equivalent**

for each member of the set of inputs, they give exactly the same output and send the circuit to the same state or to an equivalent state one of them can be removed

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**Reducing the state table**

e=f d=?

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**the reduced finite state machine**

state a a b c d e d d e d e a input output

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**the checking of each pair of states for possible equivalence can be done systematically (9-5)**

the unused states are treated as don't-care condition Þ fewer combinational gates Fig. 5.26 Reduced State diagram

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**State assignment to minimize the cost of the combinational circuits**

three possible binary state assignments

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**any binary number assignment is satisfactory as long as each state is assigned a unique number**

use binary assignment 1

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5-8 Design Procedure the word description of the circuit behavior (a state diagram) state reduction if necessary assign binary values to the states obtain the binary-coded state table choose the type of flip-flops derive the simplified flip-flop input equations and output equations draw the logic diagram

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**Synthesis using D flip-flops**

An example state diagram and state table Fig. 5.27 State diagram for sequence detector

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**The flip-flop input equations**

A(t+1) = DA(A,B,x) = S(3,5,7) B(t+1) = DB(A,B,x) = S(1,5,7) The output equation y(A,B,x) = S(6,7) Logic minimization using the K map DA= Ax + Bx DB= Ax + B'x y = AB

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Fig. 5.28 Maps for sequence detector

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**Sequence detector The logic diagram Fig. 5.29**

Logic diagram of sequence detector

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**Excitation tables A state diagram Þ flip-flop input functions**

straightforward for D flip-flops we need excitation tables for JK and T flip-flops

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**Synthesis using JK flip-flops**

The same example The state table and JK flip-flop inputs

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JA = Bx'; KA = Bx JB = x; KB = (A⊕x)‘ y = ? Fig. 5.30 Maps for J and K input equations

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Fig. 5.31 Logic diagram for sequential circuit with JK flip-flops

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**Synthesis using T flip-flops**

A n-bit binary counter the state diagram no inputs (except for the clock input) Fig. 5.32 State diagram of three-bit binary counter

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**The state table and the flip-flop inputs**

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Fig. 5.33 Maps of three-bit binary counter

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**Logic simplification using the K map**

TA2 = A1A2 TA1 = A0 TA0 = 1 The logic diagram Fig. 5.34 Logic diagram of three-bit binary counter

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