# Digital System Ch5-1 Chapter 5 Synchronous Sequential Logic Ping-Liang Lai ( 賴秉樑 ) Digital System 數位系統.

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Digital System Ch5-1 Chapter 5 Synchronous Sequential Logic Ping-Liang Lai ( 賴秉樑 ) Digital System 數位系統

Digital System Ch5-2 Outline of Chapter 5 5.1Introduction 5.2Sequential Circuits 5.3 Storage Element: Latches 5.4Storage Element: Flip-Flops 5.5Analysis of Clocked Sequential Circuits 5.7State Reduction and Assignment 5.8Design Procedure

Digital System Ch5-3 5-1Introduction Combinational circuits  Contains no memory elements  The outputs depends on the inputs

Digital System Ch5-4 5-2Sequential 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

Digital System Ch5-5 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.

Digital System Ch5-6 Fig. 5.2 Synchronous clocked sequential circuit

Digital System Ch5-7 5-3Latches ( 栓鎖器 ) 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)

Digital System Ch5-8 SR latch with NAND gates Fig. 5.4 SR latch with NAND gates

Digital System Ch5-9 SR latch with control input  En=0, no change  En=1, 0/1 1/S' 1/R' Fig. 5.5 SR latch with control input

Digital System Ch5-10 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

Digital System Ch5-11 Fig. 5.7 Graphic symbols for latches

Digital System Ch5-12 5-4Flip-Flops ( 正反器 ) 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

Digital System Ch5-13 If level-triggered flip-flops are used ( 準位觸發 )  The feedback path may cause instability problem Edge-triggered flip-flops ( 邊緣觸發 )  The state transition happens only at the edge  Eliminate the multiple-transition problem

Digital System Ch5-14 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

Digital System Ch5-15  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 !

Digital System Ch5-16 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

Digital System Ch5-17  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

Digital System Ch5-18 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 0 1 Holding data Set Reset Clk=0 Clk=1 positive-edge-triggered

Digital System Ch5-19 Setup and Hold Time for FF The setup time  D input must be maintained at a constant value prior to the application of the positive CP pulse.  Data to the internal latches. The hold time  D input must not changes after the application of the positive CP pulse.  Clock to the internal latch. Q D CK D T setup T hold D Q T propagation delay

Digital System Ch5-20 Summary  CP=0: (S, R) = (1, 1), no state change  CP=  ↑  : state change once  CP=1 and ↓ : state holds  Eliminate the feedback problems in sequential circuits All flip-flops must make their transition at the same time

Digital System Ch5-21 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

Digital System Ch5-22 JK FF 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 JKQ(t+1) 00Q(t) (No change) 010 (Reset) 101 (Set) 11Q’(t) (Complement)

Digital System Ch5-23 T FF T (toggle) 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 T Q ( t +1) 0 Q ( t ) (No change) 1 Q’ ( t ) (Complement)

Digital System Ch5-24 Summary Characteristic tables

Digital System Ch5-25 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

Digital System Ch5-26 Direct inputs Asynchronous set and/or asynchronous reset Fig. 5.14 D flip-flop with asynchronous reset

Digital System Ch5-27 5-5Analysis of Clocked Sequential Circuits A sequential circuit  (inputs, current state)  (output, next state).  A state equation (also called transition equation) specified the next state as a function of the present state and inputs.  A state transition table or state transition diagram.

Digital System Ch5-28 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) = A'x The output equation  y(t) = (A(t)+B(t))x'(t)  y = (A+B)x' Fig. 5.15 Example of sequential circuit

Digital System Ch5-29 State Table State transition table  = State equations A ( t + 1) = Ax + Bx B ( t + 1) = Ax y = Ax + Bx

Digital System Ch5-30 State Equation A ( t + 1) = Ax + Bx B ( t + 1) = Ax y = Ax + Bx

Digital System Ch5-31 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

Digital System Ch5-32 Flip-Flop Input Equations The part of circuit that generates the inputs to flip-flops  Also called excitation functions  D A = Ax +Bx  D B = A'x The output equations  To fully describe the sequential circuit  y = (A+B)x' Note  Q(t+1)=D Q DADA DBDB y

Digital System Ch5-33 Analysis with D FFs 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

Digital System Ch5-34 Analysis with JK flip-flops The next-state can be derived by following procedure 1.Determine the flip-flop equations in terms of the present state and input variables ( 用目前狀態和輸入變數的觀點決定正反器的輸入方程式 ); 2.List the binary values of each input equation ( 列出每一個輸入方程式的 二進位值 ); 3.Use the corresponding flip-flop characteristic table to determine the next state values in the state table ( 使用相對應的的正反器特性表，決定狀 態表中的次一狀態值 ).

Digital System Ch5-35 Analysis with JK Flip-Flops An sequential circuit with JK FFs  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 J A =B, K A =Bx’ J B =x’, K B =A’x+Ax’

Digital System Ch5-36  J A = B, K A = Bx'  J B = x', K B = A'x + Ax'  Derive the state table  Or, derive the state equations using characteristic equation. JKQ(t+1) 00Q(t) (No change) 010 (Reset) 101 (Set) 11Q’(t) (Complement)

Digital System Ch5-37 State transition diagram State equation for A and B: Fig. 5.19 State diagram of the circuit of Fig. 5.18 J A =B, K A =Bx’ J B =x’, K B =A’x+Ax’

Digital System Ch5-38 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

Digital System Ch5-39 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

Digital System Ch5-40 State Table T Q ( t +1) 0 Q ( t ) (No change) 1 Q’ ( t ) (Complement)

Digital System Ch5-41 Mealy and Moore Models The Mealy model ( 密利模型 ): 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 ( 莫爾模型 ): the outputs are functions of the present state only (Fig. 5-20).  The outputs are synchronous with the clocks.

Digital System Ch5-42 Mealy and Moore Models Fig. 5.21 Block diagram of Mealy and Moore state machine

Digital System Ch5-43 5-7State Reduction and Assignment State Reduction ( 狀態簡化 )  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.  An example state diagram showing in Fig. 5.25. Fig. 5.25State diagram

Digital System Ch5-44 State Reduction  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. 5.25State diagram State:aabcdeffgfga Input:01010110100 Output:00000110100

Digital System Ch5-45 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.

Digital System Ch5-46 Reducing the state table  e = g (remove g);  d = f (remove f);

Digital System Ch5-47  The reduced finite state machine State:aabcdeddedea Input:01010110100 Output:00000110100

Digital System Ch5-48  The checking of each pair of states for possible equivalence can be done systematically (section 9-5).  The unused states are treated as don't-care condition  fewer combinational gates. Fig. 5.26 Reduced State diagram

Digital System Ch5-49 State Assignment State Assignment ( 狀態指定 )  To minimize the cost of the combinational circuits.  Three possible binary state assignments. (m states need n-bits, where 2 n > m)

Digital System Ch5-50  Any binary number assignment is satisfactory as long as each state is assigned a unique number.  Use binary assignment 1.

Digital System Ch5-51 5-8Design Procedure Design Procedure for sequential circuit ( 設計程序 )  The word description of the circuit behavior to get a state diagram; ( 從文 字敘述及所需的操作規格，獲得電路的狀態圖 )  State reduction if necessary; ( 如果需要，簡化狀態數量 )  Assign binary values to the states; ( 指定狀態的二進位值 )  Obtain the binary-coded state table; ( 獲得二進位編碼的狀態表 )  Choose the type of flip-flops; ( 選擇欲使用的正反器形式 )  Derive the simplified flip-flop input equations and output equations; ( 推導 出已化簡的輸入方程式，及輸出方程式 )  Draw the logic diagram; ( 繪製邏輯圖 )

Digital System Ch5-52 Synthesis using D flip-flops An example state diagram and state table: design a circuit to detect a sequence of three or more consecutive 1’s. Fig. 5.27 State diagram for sequence detector

Digital System Ch5-53 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

Digital System Ch5-54 Fig. 5.28 Maps for sequence detector

Digital System Ch5-55 Sequence Detector The logic diagram Fig. 5.29 Logic diagram of sequence detector

Digital System Ch5-56 Excitation Tables ( 激勵表 ) A state diagram  flip-flop input functions  Straightforward for D flip-flops.  We need excitation tables for JK and T flip- flops. JKQ(t+1) 00 Q(t) (No change) 01 0 (Reset) 10 1 (Set) 11 Q’(t) (Complement) TQ ( t +1) 0 Q ( t ) (No change) 1 Q’ ( t ) (Complement)

Digital System Ch5-57 Synthesis using JK FFs The same procedure with D FFs, but a extra excitation table must be used. The state table and JK flip-flop inputs.

Digital System Ch5-58  J A = Bx'; K A = Bx  J B = x; K B = (A ⊕ x) ‘  y = ? Fig. 5.30 Maps for J and K input equations

Digital System Ch5-59 Fig. 5.31 Logic diagram for sequential circuit with JK flip-flops

Digital System Ch5-60 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

Digital System Ch5-61 The state table and the flip-flop inputs

Digital System Ch5-62 Fig. 5.33 Maps of three-bit binary counter

Digital System Ch5-63 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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