Journal of Computers - Academy Publisher

1972 JOURNAL OF COMPUTERS, VOL. 6, NO. 9, SEPTEMBER 2011
&
&
CLK
J
K
Q
Q
Q0
&
K
J
Q
Q
Q1
Fig 1. The logic diagram
&
K
J
Q
Q
Q2
In order to obtain the modulus value of the counter and
to know whether the counter can self-starting, a general
procedure is applied in three steps when analyzing this
counter.
Step 1: Obtaining the state transition equations
From the logic diagram of Fig .1, we can obtain the
following expressions for the J and K inputs of each FFs:
J = Q + Q, K = Q + Q
0 2 1 0 2 1
J = QQ , K = Q
1 2 0 1 2
J = QQ , K = Q
2 1 0 2 1
According to the characteristic equation of J-K FFs
[16,17], which is shown as equation (2).
(1)
Q′ = J Q + k Q 0 ≤ i ≤ n−
1 (2)
i i i i i
where i Q′ denotes next state, Q denotes present state,
i
n is the number of FFs. The expressions of J and i
K ( 0≤i≤ 2)
are separately substituted into equation(2),
i
we can obtain the state transition equations as shown in
equations(3):
⎧Q′
= QQQ + Q Q
2 2 1 0 2 1
⎪
⎨Q′
= QQQ + QQ
1 2 1 0 2 1
⎪
⎩ Q′ = QQQ + QQ + QQ
0 2 1 0 2 0 1 0
Step 2: The state truth table
We develop a truth table for the equations(3) as shown
in Tab.1, the present state includes three variables
Q2、 Q and 1 Q in the domain, so there are eight
0
possible combinations of binary values of the variables as
listed in medium columns, which are 000, 001, 010, 011,
100, 101, 110 and 111, the decimal digit corresponding to
each binary value is listed in left column, which are from
0 to 7, the next state is listed in right columns.
After a CLK pulse input, the FFs enter a new
state(from present state to next state), assuming initial
present state is 000, that is,
Q = 0, Q = 0, Q = 0 ( 2 1 0 000
QQQ = ), substituting it
2 1 0
into equation 2 Q′ of equations(3), we can get
corresponding value of next state, the process is as follow:
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(3)
Tab. 1 The truth table
Q2 Q1 Q0
' ' '
Q 2 Q 1 Q 0
Q′ = 0⋅0⋅ 0+ 0⋅ 0= 0
(4)
2
Here, we define such process of obtaining value of i Q′
as one time next state equation calculation, obviously, it
perhaps includes NOT, AND and OR operations. Next,
we can calculate the values of 1 Q′ and 0 Q′.
Q′
= 0⋅0⋅ 0+ 0⋅ 0= 0
1
Q′
= 0⋅0⋅ 0+ 0⋅ 0+ 0⋅ 0= 1
(5)
0
So, after three times calculations, we obtain the next
state QQQ ′ ′ ′ = 001,
and fill the values in correspondence
2 1 0
positions in right-hand column of the truth table, which is
shown in Tab.1.
Regarding a new present state is 001 and substituting it
into equations(3), the new next state can be obtained,
which is 011. Regarding the third present state is 010 and
substituting it into equations(3) again, the new next state
can be obtained, which is 110. This process is continued,
when the next state corresponding to the present state 111
is calculated, the process is ended.
It is clearly to know, in order to complete the truth
table(Tab.1), we need accomplish 24(8*3) times next
state equation calculations.
Step 3: The state transition table and the state
transition diagram
After obtaining the truth table, the next step is to
construct the state transition table and the state transition
diagram.
Firstly, we construct the state transition table. The table
is constructed with three columns, which are comments,
present state and nest state, which is shown in Tab.2.
Regarding initial present state is 000, observing Tab.1,
the next state is 001, filling 000 and 001 to corresponging
column of Tab.2. Regarding the next state 001 is as a new
present state, the new next state is 011. Regarding the
third present state is 011 and the third next state is 010.
This process is continued, in the end, we can observe 101
is as a present state, which next state is 000, the counting
cycle is finished, there are six counting states, which are
000, 001, 011, 010, 110 and 101, we call these states as
counting states. For next two states, which are 100 and