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6-1 Sequential Circuit Definitions

- sequential circuit
- combinational circuit + storage elements
- storage elements
- store binary information state of the sequential circuit at given state
- outputs are a function of the inputs & present state of the storage elements
- next state of storage elements is also a function of the inputs & the present state

Figure 6-1 Block Diagram of a Sequential Circuit

6-1 Sequential Circuit Definitions

Figure 6-2 logic Structures for Strong Information

6-1 Sequential Circuit Definitions

- two types
- synchronous sequential circuit
- behavior is defined from the knowledge of its signals at discrete instants of time
- asynchronous sequential circuit
- behavior depends on the inputs at any instance of time & the order in continuous time in which the inputs change,
- clock generator
- synchronous sequential circuit has a timing device
- produce a periodic train of clock pulses
- storage elements are affected only upon the arrival of each pulse
- clock pulses are applied with other signals
- the outputs can change their value only in the presence of clock pulses
- clocked sequential circuits

6-1 Sequential Circuit Definitions

- flip-flop
- storage elements employed in clocked sequential circuits
- a binary storage device capable of storing one bit of info
- Normally, a sequential circuit uses many flip-flops
- the transition from one state to the other occurs only at predetermined time intervals dictated by the clock pulses
- two outputs: normal & complemented values

Figure 6-3 Synchronous Clocked Sequential Circuit

6-2 Latches

- A storage element can maintain a binary state indefinitely until directed by an input signal to switch states
- Latch
- most basic types of flip-flops
- simple & most often used within flip-flops
- used with more complex clocking methods to implement sequential circuits
- SR Latch
- a circuit with 2 cross-coupled NOR (or NAND) gates
- 2 inputs: S (set) & R (reset)

6-2 Latches

if S=1, Q=1 (Q\'=0);

if R=1, Q=0 (Q`=1)

if S=R=0, keep previous state (hold)

if S=R=1, undefined state

Figure 6-4 SR Latch

with NOR Gates

Figure 6-5

Logic Simulatiom

Of

SR Latch Behavior

6-2 Latches

- S\'R\' latch with two cross-coupled NAND gates
- the input signals for the NAND require the complement of those values used for the NOR

Figure 6-6 S’R’ Latch with NAND Gates

6-2 Latches

- SR latch with a control input
- a basic S\'R\' latch with 2 NAND gates
- C (control input) acts as an enable signal for the other 2 inputs if C=0, no action; if C=1, act as SR f-f
- the indeterminate condition (S=R=1) ==> seldom used in practice
- but important, all others are constructed from it
- SR latch with control input is called SR (or RS) f-f

Figure 6-7 SR Latch with Control Input

6-2 Latches

- D Latch
- eliminate the undesirable condition of the indeterminate state
- make S & R never equal to 1 at the same time ==> include an inverter
- 2 inputs: D (data) & C (control) D goes to S; D\' goes to R
- act as a temporary storage
- constructed with transmission gates

Figure 6-8 D Latch

Figure 6-9 D Latch with

Transmission Gates

6-3 Flip-Flops

- the state of a latch is allowed to switch by a momentary change of the control unit
- a momentary change is called a trigger
- a sequential circuit has a feedback path
- control pulse goes to logic-1
- the new state of a latch may appear
- the output is connected to the input
- ...…

Form a reliable flip-flop

- master-slave flip-flop & edge-triggered flip-flop

6-3 Flip-Flops

- Master-Slave Flip-Flop

Figure 6-10 SR Master-Slave Flip-Flop

Figure 6-11 Logic Simulation of an SR Master-Slave Flip-Flop

6-3 Flip-Flops

Figure 6-12 Negative Edge-Triggered D Flip-Flop

6-3 Flip-Flops

- Edge-Triggered Flip-flop
- ignore the pulse while it is at a constant level, but triggers only during the transition of the clock signal

Figure 6-13 Positive-Edge-Triggered D Flip-Flop

6-3 Flip-Flops

Standard Graphics Symbols

Figure 6-14 Standard Graphics Symbols for Latches and Flip-Flops

6-3 Flip-Flops

- Direct Inputs
- Preset and Clear inputs highly desirable !!

Figure 6-15 D Flip-Flop with Direct Set and Reset

6-4 Sequential Circuit Analysis

- behavior of a sequential circuit is determined from inputs, outputs, & present state of the circuit
- outputs & the next state are function of inputs & present state
- Input Equations
- a logic diagram of sequential circuit includes F-Fs (any type), or combinational circuit
- the part of the combinational circuit can be described by a set of Boolean functions, called input equations

6-4 Sequential Circuit Analysis

Sequential Circuit Analysis

- Logic diagram
- Input equations and output equations
- State table
- State diagram

6-4 Sequential Circuit Analysis

DA = AX + BX, DB = A\'X, Y = (A+B) X\'

(input equations for F-F) (equations for output Y)

Figure 6-17 Example of a Sequential Circuit

6-4 Sequential Circuit Analysis

- State Table
- functional relationship between inputs, outputs, & flip-flop state
- consist of 4 sections: present state, input, next state, output
- list all possible combinations of present state and inputs
- next state shows states of F-F one clock period later at time t+1
- State table example

Table 6-1 State Table for Circuit of Figure 6-17

6-4 Sequential Circuit Analysis

- State relationship

A(t+1) = DA = AX + BX; B(t+1) = DB = A\'X;

Y = AX\' + BX’

- Two-dimensional state table

Table 6-2 Two-Dimensional State Table for the Circuit in Figure 6-17

6-4 Sequential Circuit Analysis

- Mealy model
- the outputs depend on the inputs and the states
- Moore model
- outputs depend only on the states (a 1-D column suffices)

a Moore model circuit

DA = A X Y, Z = A

Figure 6-18 Logic Diagram and State Table for DA = A X Y

6-4 Sequential Circuit Analysis

- State Diagram
- The information (in a state table) may be represented graphically
- state by a circle & transition between state by directed lines

Figure 6-19 State Diagram

- # sequential circuit of Fig 6-17
- binary number inside circle
- = state of F-F
- directed lines are labeled with (input/output) value

- # sequential circuit of Fig 6-18
- one F-F with 2 states, 2 inputs, no output
- directed lines are labeled w/ (input/output) value

6-4 Sequential Circuit Analysis

- Sequential Circuit Timing

Figure 6-20 Sequential Circuit Timing Parameters

6-4 Sequential Circuit Analysis

- Sequential Circuit Timing

tp= tslack + (tpd,FF + tpd,COMB + ts)

tp≥ max (tpd,FF + tpd,COMB + ts) = tp ,min

Figure 6-21 Sequential Circuit Timing Parths

6-4 Sequential Circuit Analysis

- Ex6-1) Clock Period and Frequency Calculations

1.5ns= tslack + 0.2 +1.3 + 0.1 = tslack + 1.6ns

- Simulation

Figure 6-22 Simulation Timing

6-5 Sequential Circuit Design

- combinational circuit: fully specified by a truth table
- sequential circuit requires a state table for its specification
- first step is to obtain a state table (or state diagram)
- No. of F-F is determined from the no of states (up to 2n)
- Design Procedure

1) Obtain the state diagram

(from problem statement, or state diagram)

2) Obtain the state table

3) Assign binary codes to the states

4) Derive F-F input equations from next state conditions in table

5) Derive the output functions if needed

6) Simplify the input equations & output functions

7) Draw the logic diagram with D F-Fs & combinational gates

6-5 Sequential Circuit Design

- Finding State Diagrams and State Tables

Figure 6-23 Asynchronous and Synchronous Reset for D Flip-flops

6-5 Sequential Circuit Design

- Finding State Diagram and State Tables

Table6-3 State Table for State

Diagram In Figure 6-21

Figure 6-24 Construction of a State Diagram for Example 6.2

6-5 Sequential Circuit Design

- Ex6-3) Finding a State Diagram for a BCD-to-Excess-3 Decoder

Table 6-4 Sequence Tables for Code Converter Example

6-5 Sequential Circuit Design

Figure 6-25 Construction of a State Diagram for Example 6.3

6-5 Sequential Circuit Design

Design Procedure

1) state diagram

2) state table

3) F-F input equationsandoutput functions

4) Simplify the input equations and output functions

5) logic diagram

6-5 Sequential Circuit Design

- Sequence recognizer, 1101

Gray code를 할당함

A, B, C, D: 00,01,11,10

Figure 6-24 Construction of a State Diagram for Example 6.2

6-5 Sequential Circuit Design

- Designing with D Flip-Flops(ABX로 truth table작성)

Table 6-5 Table 6-3 with Names Replaced by Binary Codes

6-5 Sequential Circuit Design

- A(t+1) = DA(A,B,X) = m(3,6,7)
- B(t+1) = DB(A,B,X) = m(1,3,5,7)
- Y(A,B,X) = m(5)

Figure 6-26 Maps for Input Equations and Output Z

6-5 Sequential Circuit Design

Figure 6-27 Logic Diagram for Sequential Circuit with D Flip-Flops

6-5 Sequential Circuit Design D Flip-Flops

- Designing with Unused States
- # A circuit with n F-F has 2n binary states
- # unused states can be treated as don\'t care conditions

Table 6-6 State Table for Designing with Unused States

6-5 Sequential Circuit Design D Flip-Flops

Figure 6-28 Maps for Optimizing Input Equations

6-5 Sequential Circuit Design D Flip-Flops

- Verification

Figure 6-29 Test Sequence Generation for Simulation in Example 6.5

6-5 Sequential Circuit Design D Flip-Flops

Figure 6-30 Simulation for Example 6.5

6-6 Other Flip-Flop Type

- JK and T Flip-Flops

Table 6-7 Flip-Flop Logic, Characteristic Tables and Equations,

and Excitation Tables

6-7 HDL Representation for Sequential Circuits-VHDL

- Ex6-5) VHDL for Positive-Edge-Triggered D Flip-Flop with Reset

Figure 6-31 VHDL Process

Description of Positive-Edge-

Triggered Flip-Flop with Reset

6-7 HDL Representation for Sequential Circuits-VHDL

- Ex6-6) VHDL for the Sequence Recognizer

Figure 6-32 VHDL Process

Description of a Sequence

Recognizer

6-7 HDL Representation for Sequential Circuits-VHDL

Figure 6-33 VHDL Process

Description of a Sequence

Recognizer (continued)

6-7 HDL Representation for Sequential Circuits-VHDL

Table 6-8 Illustration of generation of storage in VHDL

6-8 HDL Representation for Sequential Circuits-Verilog

- Ex6-7) Verilog for Positive-Edge-Triggered D Flip-Flop with Reset

Figure 6-34 Verilog Process Description of Positive-Edge-Triggered

Flip-Flop with Reset

6-8 HDL Representation for Sequential Circuits-Verilog

- Ex6-8) Verilog for Sequence Recognizer

Figure 6-35 Verilog Process

Description of a Sequence

Recognizer

6-8 HDL Representation for Sequential Circuits-Verilog

Table 6-9 Illustration of generation of storage in Verilog

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