Presentation on theme: "Digital System Ch5-1 Chapter 5 Synchronous Sequential Logic Ping-Liang Lai ( 賴秉樑 ) Digital System 數位系統."— Presentation transcript:
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 Ch Introduction Combinational circuits Contains no memory elements The outputs depends on the inputs
Digital System Ch 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
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 Ch Latches ( 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 Ch Flip-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 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 D-type positive-edge-triggered flip-flop
Digital System Ch 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 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 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 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 D flip-flop with asynchronous reset
Digital System Ch Analysis 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 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 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 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 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 State diagram of the circuit of Fig 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 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 Block diagram of Mealy and Moore state machine
Digital System Ch State 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 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: Output:
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: Output:
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 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 Ch Design 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 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 Maps for sequence detector
Digital System Ch5-55 Sequence Detector The logic diagram Fig 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 Maps for J and K input equations
Digital System Ch5-59 Fig 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 State diagram of three-bit binary counter
Digital System Ch5-61 The state table and the flip-flop inputs
Digital System Ch5-62 Fig 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 Logic diagram of three-bit binary counter