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SEQUENTIAL CIRCUITS Flip-Flops Analysis

Ciletti, M. D. and M. M. Mano. Digital design, fourth edition. Prentice Hall. NJ.


FLIP-FLOPS Characteristic Tables, Characteristic Equations

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CHARACTERISTIC TABLES

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Characteristic Tables • A characteristic table defines the logical properties of a

flip-flop by describing its operations in tabular form. • For D, JK, and T flip-flops, the characteristic table defines

the next state as a function of the inputs and the present state. • For the characteristic tables, • Q(t) refers to the present state

• Q(t+1) refers to the next state


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Characteristic Tables: D FF • The next state of a D flip-flop is dependent only on the D

input and is independent of the present state. • Q(t+1) = D • Next state value is equal to the D value.

• Notice that there is no “no change” condition.


Characteristic Table: D FF

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Characteristic Tables: JK FF • The next state of a JK flip-flop • Q(t+1) = Q(t) when J and K are both 0 – no change

• Q(t+1) = 0 when J = 0 and K = 1 – reset • Q(t+1) = 1 when J = 1 and K = 0 – set

• Q(t+1) = Q’(t) when J and K are both 1 – output complemented


Characteristic Table: JK FF

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Characteristic Tables: T FF • Characteristic table of a T flip-flop • Q(t+1) = Q(t) when T = 0 – no change

• Q(t+1) = Q’(t) when T = 1 – output complemented

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Characteristic Table: T FF

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CHARACTERISTIC EQUATIONS

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Characteristic Equations • The logical properties of a flip-flop can be expressed

algebraically with a characteristic equation. • For D flip-flop: Q(t+1) = D • For JK flip-flop: Q(t+1) = JQ’ + K’Q • For T flip-flop: Q(t+1) = TQ’ + T’Q = T  Q • Characteristic equations can be obtained from the circuit.


ANALYSIS OF CLOCKED SEQUENTIAL CIRCUITS State Equation, State Table, State Diagram

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Introduction • Analysis describes what a given circuit will do under

certain operating conditions. • Behavior of a clocked sequential circuit is determined

from • the inputs

• the outputs • the state of the flip-flops used

• The outputs and the next state are both a function of the

inputs and the present state.


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Introduction • The analysis of a sequential circuit consists of obtaining a

table or a diagram for the time sequence of inputs, outputs, and internal states. • Boolean expressions, that describe the behavior of the

sequential circuit, can be determined. The expressions

must include the necessary time sequence.


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Introduction • A logic diagram is recognized as a clocked sequential

circuit if it includes flip-flops with clock inputs. • The flip-flops can be of any type (D, JK, T). • The logic diagram may or may not include combinational

circuit gates.


STATE EQUATIONS

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State Equations • The behavior of clocked sequential circuits can be

described algebraically by means of state equations. • A state equation (a.k.a. transition equation) specifies the

next state as a function of the present state and inputs.

Consider the figure in the next slide..


Sample Sequential Circuit

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State Equations • The sample sequential circuit consists of • two D flip-flops A and B • input x • output y

• The state equations • A(t+1) = A(t)x(t) + B(t)x(t)

//at time t

• B(t+1) = A’(t)x(t)

//at time t

• Since all the next state variables are a function of the

present state, the designation (t) can be omitted.


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State Equations • The present-state value of the output can be expressed

algebraically as • y(t) = [A(t) + B(t)]x’(t)

• Since

all variables are in the present state, the

designation (t) can be omitted. The output Boolean

equation now is, • y = (A + B)x’


STATE TABLES

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State Table • The time sequence of inputs, outputs, and flip-flop states

can be enumerated in a state table. • The state table consists of four sections, • present state – state of flip-flop at any given time t • input – value of input for each possible present state

• next state – state of flip-flop after one clock cycle (at time t+1) • output – value of output at time t for each present state and input

condition


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State Table • Derivation of a state table requires listing all possible

binary combinations of present states and inputs. • The next state section values are determined from the

state equations of the flip-flops used. • The output section values are determined from the output

equation.


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State Table

A(t+1) = Ax + Bx B(t+1) = A’x y = Ax’ + Bx’

State table for the sample circuit


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State Table • Given a sequential circuit with m flip-flops and n inputs, 2m+n

rows are needed for the state table. • The number of columns for each section: m for present state

and next state, n for input, the number of output variables. • The binary numbers 0 to 2m+n – 1 are listed under the present

state and input sections. • Values under next state section are derived from the state

equations of the flip-flops. Values under the output section are derived from the output equations.


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State Table (version 2) • It is sometimes convenient to express the state table in a

slightly different form. • A different state table has three sections: present state,

next state, and output. • The input conditions are enumerated under the next state

and output sections.


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State Table (version 2)

A(t+1) = Ax + Bx B(t+1) = A’x y = Ax’ + Bx’

Another form of the state table for the sample circuit


STATE DIAGRAMS

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State Diagram • A state diagram is a graphical representation of the state

table. • A state diagram is derived from the information given in

the state table.


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State Diagram Present State (A,B)

Input (x) Output (y)

Next State (A,B) State diagram for the sample circuit


State Diagram

Draw the state diagram of the sample circuit

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State Diagram

State diagram for the sample circuit


INPUT/OUTPUT EQUATIONS

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Flip-Flop Input/Output Equations • An output equation is a Boolean function describing the

external output of the combinational circuit part of the sequential circuit. • A flip-flop input equation is a Boolean function describing

the part of the circuit that generates the inputs to the flipflops. • With the flip-flop input equations and output equations,

the logic diagram of the sequential circuit can be drawn.


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Flip-Flop Input Equation • Sample Circuit: 2 input equations and 1 output equations

DA = Ax + Bx DB = A’x

y = Ax’ + Bx’ • DA means that we are using a D flip-flop with output A

• DB means that we are using a D flip-flop with output B


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ANALYSIS WITH FLIP-FLOPS More examples: D, JK, T

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Analysis: D Flip-Flop • Consider the following input equation for the circuit to be

analyzed,

• DA = A  x  y • Description of the circuit, • the DA symbol implies the use of a D flip-flop with output A • no output equations are given – output of the circuit comes from the

output of the flip-flop used • What to do: a) logic diagram, b) next state equations, c)

state table, and d) state diagram


Analysis: D Flip-Flop a. Logic Diagram of DA = A  x  y

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Analysis: D Flip-Flop b. Next State Equation of DA = A  x  y

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Analysis: D Flip-Flop c. State Table of DA = A  x  y

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Analysis: D Flip-Flop c. State Diagram of DA = A  x  y

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