Counters

1
???? ????

• ????? 2?????

2
Counters
3
Edge Triggering

• Important Issue
• The T Flip-Flop and the JK Flip-Flop 1-1 option
  are not stable since they are toggle
  operations.
• From now all FFs are assumed negative
  edge-triggered Values are changed not when the
  clock is up, but at the exact moment the clock
  goes down notice the circle notation
• Edge Triggering mechanism can be seen in the
  lecture.

4
New Approach To Flip-Flops (1)

• Formerly, we presented each flip-flops Truth
  Table, i.e. the output as a function of former
  state (former output) and the inputs.

Former State
Input
Output
T Flip-Flop Example
5
New Approach To Flip-Flops (2)

• A new Approach For each former output and
  desired output combination, what inputs do we
  need to transform from current to desired?
• This approach is more useful when creating
  counters.

6
New Approach To Flip-Flops (3)
7
Flip-Flop Concatenation

• T Flip-Flop has an interesting Attribute

1
1
10
10
10
(In red, is the clock down-edge)
8
Flip-Flop Concatenation (2)

• But doesnt that resemble counting in binary?

9
Asynchronic Counter

• So we can use this attribute to create a simple
  n-digit binary counter.
• (shown here with the equivalent 1-1 state JK
  Flip-Flop)

10
BCD Counter

• Can we count in a radix which is not ?
• Yes, we can even create a binary-coded decimal
  (BCD) Counter

11
BCD Counter (2)

• To create it we used the diagram

12
BCD Counter Concatenation

• If we can count to 10 (exclusive) we can count to
  100, 1000 and so on

13
BCD Counter

• Two problems
• Developing a custom-made, efficient, asynchronic
  counter, such as the BCD counter is difficult!
• What if we want to create a non-consecutive
  counter easily?

14
Counting Sequence

• A cyclic sequence of binary numbers.
• For instance 0,1,2,4,5,6,0,1,2,4,

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Synchronic Counter

• We want to implement a mechanism for counting in
  a given counting sequence.
• Well focus independently on each digit and the
  changes it undergoes.

16
Building A Sync. Counter

• Algorithm
• Choose a Flip-Flop type to use.
• To each digit (columns), allocate a Flip-flop.
• For each FF input create a truth table in the
  following manner
• For each state (line) do
• Mark state in current line as
• Mark state in next line as
• Determine the desired FFs inputs by the tables.
  Those inputs are actually function results in the
  truth table.
• Simplify the truth table and implement the
  function

17
3-Bit Sync. Counter using T Flip-Flops (2)
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3-Bit Sync. Counter using T Flip-Flops (1)
19
3-Bit Sync. Counter using T Flip-Flops (3)
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3-Bit Sync. Counter using T Flip-Flops (3)
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3-Bit Sync. Counter using T Flip-Flops (4)
22
Example using JK Flip-Flops (2)

• Shown for the 0,1,2,4,5,6,0,1,2,4,5,6, example
• Well do C as an example.

23
Example using JK Flip-Flops (1)
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Example using JK Flip-Flops (3)

• JC
• KC

00
01
11
10
0
1
00
01
11
10
0
1
25
Example using JK Flip-Flops (4)