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Synchronous Sequential Logic Chapter 5

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Digital Circuits 5-2 5-1 Introduction Combinational circuits contains no memory elements the outputs depends on the inputs

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Digital Circuits 5-3 5-2 Sequential Circuits ■ Sequential circuits a feedback path the state of the sequential circuit (inputs, current state) (outputs, next state) synchronous: the transition happens at discrete instants of time asynchronous: at any instant of time

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Digital Circuits 5-4 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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Digital Circuits 5-5 Fig. 5.2 Synchronous clocked sequential circuit

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Digital Circuits 5-6 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) (0,0)

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Digital Circuits 5-7 SR latch with NAND gates Fig. 5.4 SR latch with NAND gates

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Digital Circuits 5-8 SR latch with control input En=0, no change En=1, output depends inputs S, R S_ R_ 0/1 1/S' 1/R' Fig. 5.5 SR latch with control input

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Digital Circuits 5-9 D Latch eliminate the undesirable conditions of the indeterminate state in the RS flip-flop D: data gated D-latch D Q when En=1; no change when En=0 Fig. 5.6 D latch S_ R_ 0/1 1/D' 1/D

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Digital Circuits 5-10 Fig. 5.7 Graphic symbols for latches

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Digital Circuits 5-11 5-4 Flip-Flops A trigger 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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Digital Circuits 5-12 If level-triggered flip-flops are used 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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Digital Circuits 5-13 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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Digital Circuits 5-14 CP = 1: (S,R) (Y,Y'); (Q,Q') holds CP = 0: (Y,Y') holds; (Y,Y') (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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Digital Circuits 5-15 Edge-triggered flip-flops 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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Digital Circuits 5-16 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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Digital Circuits 5-17 第三版內容， 參考用 !

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Digital Circuits 5-18 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% V H

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Digital Circuits 5-19 Summary CP=0: (S,R) = (1,1), no state change CP= : state change once CP=1: state holds

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Digital Circuits 5-20 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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Digital Circuits 5-21 JK flip-flop D=JQ'+K'Q J=0, K=0: D=Q, no change J=0, K=1: D=0 Q =0 J=1, K=0: D=1 Q =1 J=1, K=1: D=Q' Q =Q' Fig. 5.12 JK flip-flop

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Digital Circuits 5-22 T flip-flop D = T ⊕ Q = TQ'+T'Q T=0: D=Q, no change T=1: D=Q' Q=Q' Fig. 5.13 T flip-flop

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Digital Circuits 5-23 Characteristic tables

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Digital Circuits 5-24 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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Digital Circuits 5-25 Direct inputs asynchronous set and/or asynchronous reset Fig. 5.14 D flip-flop with asynchronous reset

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Digital Circuits 5-26 5-5 Analysis of Clocked Sequential Ckts A sequential circuit (inputs, current state) (output, next state) a state transition table or state transition diagram Fig. 5.15 Example of sequential circuit

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Digital Circuits 5-27 State equations 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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Digital Circuits 5-28 State table State transition table = state equations

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Digital Circuits 5-29 State equation A(t + 1) =Ax + Bx B(t + 1) = Ax y = Ax + Bx

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Digital Circuits 5-30 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 a state table a state diagram Fig. 5.16 State diagram of the circuit of Fig. 5.15

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Digital Circuits 5-31 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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Digital Circuits 5-32 Analysis with D flip-flops The input equation D A =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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Digital Circuits 5-33 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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Digital Circuits 5-34 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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Digital Circuits 5-35 State transition diagram Fig. 5.19 State diagram of the circuit of Fig. 5.18 State equation for A and B:

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Digital Circuits 5-36 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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Digital Circuits 5-37 The input and output functions T A =Bx T B = 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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Digital Circuits 5-38 State Table

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Digital Circuits 5-39 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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Digital Circuits 5-40 Fig. 5.21 Block diagram of Mealy and Moore state machine

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Digital Circuits 5-41 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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Digital Circuits 5-42 Behavioral Modeling always statement Examples: Two procedural blocking assignments:Two nonblocking assignments:

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Digital Circuits 5-43 Flip-Flops and Latches ■ HDL Example 5.1

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Digital Circuits 5-44 Flip-Flops and Latches ■ HDL Example 5.2

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Digital Circuits 5-45 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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Digital Circuits 5-46 HDL Example 5-3 (Continued)

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Digital Circuits 5-47 HDL Example 5-4 Functional description of JK flip-flop

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Digital Circuits 5-48 State Diagram ■ HDL Example 5.5: Mealy HDL model

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Digital Circuits 5-49 HDL Example 5-5 (Continued)

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Digital Circuits 5-50 HDL Example 5-5 (Continued)

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Digital Circuits 5-51 Mealy_Zero_Detector Fig. 5.22 Simulation output of Mealy_Zero_Detector

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Digital Circuits 5-52 HDL Example 5-6: Moore Model FSM

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Digital Circuits 5-53 Simulation Output of HDL Example 5-6 Fig. 5.23 Simulation output of HDL Example 5.6

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Digital Circuits 5-54 Structural Description of Clocked Sequential Circuits ■ HDL Example 5.7: State-diagram-based model

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Digital Circuits 5-55 HDL Example 5-7 (Continued)

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Digital Circuits 5-56 HDL Example 5-7 (Continued)

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Digital Circuits 5-57 HDL Example 5-7 (Continued)

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Digital Circuits 5-58 HDL Example 5-7 (Continued)

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Digital Circuits 5-59 Simulation Output of HDL Example 5-7 Fig. 5.24 Simulation output of HDL Example 5.7

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Digital Circuits 5-60 5-7 State Reduction and Assignment State Reduction 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. 5.25 State diagram

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Digital Circuits 5-61 statea a b c d e f f g f g a input0 1 0 1 0 1 1 0 1 0 0 output0 0 0 0 0 1 1 0 1 0 0 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. 5.25 State diagram

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Digital Circuits 5-62 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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Digital Circuits 5-63 Reducing the state table e=f d=?

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Digital Circuits 5-64 the reduced finite state machine statea a b c d e d d e d e a input0 1 0 1 0 1 1 0 1 0 0 output0 0 0 0 0 1 1 0 1 0 0

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Digital Circuits 5-65 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 fewer combinational gates Fig. 5.26 Reduced State diagram

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Digital Circuits 5-66 State assignment to minimize the cost of the combinational circuits three possible binary state assignments

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Digital Circuits 5-67 any binary number assignment is satisfactory as long as each state is assigned a unique number use binary assignment 1

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Digital Circuits 5-68 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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Digital Circuits 5-69 Synthesis using D flip-flops An example state diagram and state table Fig. 5.27 State diagram for sequence detector

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Digital Circuits 5-70 The flip-flop input equations A(t+1) = D A (A,B,x) = (3,5,7) B(t+1) = D B (A,B,x) = (1,5,7) The output equation y(A,B,x) = (6,7) Logic minimization using the K map D A = Ax + Bx D B = Ax + B'x y = AB

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Digital Circuits 5-71 Fig. 5.28 Maps for sequence detector

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Digital Circuits 5-72 Sequence detector The logic diagram Fig. 5.29 Logic diagram of sequence detector

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Digital Circuits 5-73 Excitation tables A state diagram flip-flop input functions straightforward for D flip-flops we need excitation tables for JK and T flip-flops

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Digital Circuits 5-74 Synthesis using JK flip-flops The same example The state table and JK flip-flop inputs

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Digital Circuits 5-75 J A = Bx'; K A = Bx J B = x; K B = (A ⊕ x)‘ y = ? Fig. 5.30 Maps for J and K input equations

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Digital Circuits 5-76 Fig. 5.31 Logic diagram for sequential circuit with JK flip-flops

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Digital Circuits 5-77 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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Digital Circuits 5-78 The state table and the flip-flop inputs

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Digital Circuits 5-79 Fig. 5.33 Maps of three-bit binary counter

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Digital Circuits 5-80 Logic simplification using the K map T A2 = A 1 A 2 T A1 = A 0 T A0 = 1 The logic diagram Fig. 5.34 Logic diagram of three-bit binary counter

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