T Flip-Flop

A T (toggle) flip-flop is a complementing flip-flop and can beobtained from a JK flip-flop when the two inputs are tied together.

When T = 0D = Q and no change in output

QQTD

When T = 1D = Q’ and the output complements

'QQTD

Characteristic Tables and Equations

J K Q(t+1)0 0 Q(t) No change0 1 0 Reset1 0 1 Set1 1 Q’(t) Complement

D Q(t+1)0 0 Reset1 1 Set

T Q(t+1)0 Q(t) No change1 Q’(t) Complement

DtQ )1(

QTTQQTtQ '')1(

QKJQtQ '')1(

Q(t) = present stateQ(t+1) = next state after one clock period

Analysis of Clocked Sequential Circuits

The behavior of a clocked sequential circuit is determined from theinputs, outputs, and the state of its flip-flops.

State EquationA state equation (transition equation) specifies the next state as a function of the present state and inputs.

State TableA state table (transition table) consists of: present state, inputnext state and output.

State DiagramThe information in a state table can be represented graphically ina state diagram. The state is represented by a circle and the transitionsbetween states are indicated by directed lines connecting the circles.

1. Determine the flip-Flop input equations in terms of the presentstate and input variables.

2. Substitute the input equations into the flip-flop characteristicequation to obtain the state equations.

3. Use the corresponding state equations to determine the next state values in the state table.

Analysis Procedure

Analysis of Clocked Sequential Circuits

Example of a Sequential Circuit

)(')]()([

)()(')1(

)()()()()1(

txtBtAy

txtAtB

txtBtxtAtA

State Equations

note mistake in Fig. 5-15 p. 181

(t+1) next state of the flip-flopone clock edge later.

xAtB

BxAxtA

')1(

)1(

')( xBAy

Flip-flop input equations(excitation equations)

xAD

BxAxD

B

A

'

Example of a Sequential Circuit (continued)

Present Next State Input State OutputA B x A B y0 0 0 0 0 00 0 1 0 1 00 1 0 0 0 1 0 1 1 1 1 01 0 0 0 0 11 0 1 1 0 01 1 0 0 0 11 1 1 1 0 1

xAtB

BxAxtA

')1(

)1(

')( xBAy

Present Next State Input State OutputA B x A B y0 0 0 0 0 0

0 0 1 0 1 0 a0 1 0 0 0 1

0 1 1 1 1 0 b1 0 0 0 0 1

1 0 1 1 0 0 c1 1 0 0 0 11 1 1 1 0 1

a

a: When the sequential circuit is in present state 00 and the input is 1, the output is 0.After the next clock cycle, the circuit goes to the next state 01.b: When the sequential circuit is in present state 01 and the input is 1, the output is 0.After the next clock cycle, the circuit goes to the next state 11.c: No change in state.

c

b

Example of a Sequential Circuit (continued)

Sequential Circuit Analysis with D Flip-Flops

yxADA

a

b

( 1)A t A x y

xAKxJ

BxKBJ

BB

AA

'

'

1. Flip-Flop input equations:

Example of Sequential Circuit with JK Flip-Flops

2. Substitute the input equations into the flip-flop characteristicequation to obtain the state equations.

'''')'( '')1(

'')''( ')1(

'')1(

'')1(

BxAABxxBBxABxtB

AxABBAABxBAtA

BKBJtB

AKAJtA

BB

AA

3. Use the corresponding state equations to determine the next state values in the state table.

xAKxJ

BxKBJ

BB

AA

'

'

JK Flip-Flop characteristic equation Flip-Flop input equations

Sequential Circuit state equations

Example of Sequential Circuit with JK FF (continued)

Present Next Flip-Flop State Input State InputsA B x A B 0 0 0 0 1 0 0 1 00 0 1 0 0 0 0 0 10 1 0 1 1 1 1 1 00 1 1 1 0 1 0 0 11 0 0 1 1 0 0 1 11 0 1 1 0 0 0 0 01 1 0 0 0 1 1 1 11 1 1 1 1 1 0 0 0

BBAA KJKJ

'''')'( '')1(

'')''( ')1(

BxAABxxBBxABxtB

AxABBAABxBAtA

Example of Sequential Circuit with JK FF (continued)

Example of Sequential Circuit with T Flip-Flops

ABy

xT

BxT

B

A

1. Flip-Flop input equations:

2. Substitute the input equations into the flip-flop characteristicequation to obtain the state equations.

ABy

xT

BxT

B

A

BxtB

BxAAxABABxABxtA

)1(

' ' ')'()'()1(

TQQTQTtQ '')1(

Flip-Flop input equations

T Flip-Flop characteristic equation

Sequential Circuitstate equations

Example: T Flip-Flops circuit (continued)

BxtB

BxAAxABABxABxtA

)1(

' ' ')'()'()1(

Present Next State Input State OutputA B x A B y0 0 0 0 0 00 0 1 0 1 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 0 01 0 1 1 1 0 1 1 0 1 1 11 1 1 0 0 1

Example: T Flip-Flops circuit (continued)