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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)
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