DESIGN PROBLEM : 4-bit increment by 2 circuit

Problem: Derive the logical expression for a 4-bit increment by 2 circuit and draw the architecture of it.

Solution: The task here is to design a circuit that increments its count by two. Since, it is a 4-bit circuit, the total number of possible states is 16. Each state transitions to the state which has a binary value two greater than it. Now, there are two possible scenarios based upon the initial state that the counter gets into:

1. It can count 0 -> 2 -> 4 -> 6 -> 8 -> 10 -> 12 -> 14 -> 0 (their binary equivalents)

2. It can count 1 -> 3 -> 5 -> 7 -> 9 -> 11 -> 13 -> 15 -> 1 (their binary equivalents)

The state transition table can be represented as shown below:

We can find the expression for outputs using K-maps as below.

Expression for D3(next): Let us first derive the expression for D3(next). The K-map can be represented as below:

The expression for D3(next) as derived from K-map is:
D3(next) = D3.D2' + D3.D1' + D3'.D2.D1
D3(next) = D3.(D2'+D1') + D3'.D2.D1.
D3(next) = D3.(D2.D1)'+D3'.(D2.D1)
D3(next) = D3 (exor) (D2.D1) 

Expression for D2(next): Given below is the K-map derived from state transition table for D2(next).

The expression for D2(next) as derived from K-map is:
D2(next) = D2'.D1 + D2.D1' = D2 (exor) D1

Expression for D1(next):  Given below is the K-map derived from state transition table for D1(next).

The expression for D1(next) is derived from K-map as:
D1(next) = D1'

Expression for D0(next): Given below is the K-map for D0(next).

The expression for D0(next) is:
D0(next) = D0

Combining all the expressions, the circuit is as given below:

Can you come up with a better solution for this problem? Let us know your views in comments.

This question was asked by one Himadri Roy Pramanik on our post your query page. You can also post your queries there. We will try to answer using our limited knowledge.

Clock gating checks in case of mux select transition when both clocks are running

PROBLEM: In the following figure, it is desired to toggle the select of the mux from CLOCK_DIV to CLOCK and both the clocks are running. What are the architectural and STA considerations for the same?

This is a very good example to understand how clock gating checks work, although you may/may not find any practical application for the same. We have to toggle the select of the multiplexer such that there is no glitch at the output. Let us consider architectural considerations first:

Architectural considerations:

Launching flip-flop of 'select' signal: In the post clock gating checks at a multiplexer, we discussed that if there is a mux getting clock at its inputs and select as data, then, there are two possible scenarios:

  • If the other clock is at state "0", then AND type check is formed and select has to launch from negative edge-triggered flip-flop
  • If the other clock is at state "1", then OR type check is formed and select has to launch from positive edge-triggered flip-flop
Now, since both the clocks are running simultaneously, both with act as "other clock" for each other. Let us choose to keep both the clocks at state "0" when select toggles. The same discussion holds true for the other scenario as well, just that appropriate values will hold. Thus,

(i) Both clocks required to be at state '0' when clock toggles
(ii) There is AND-type clock gating check formed between 'select' and both clocks 
(iii) 'select' launches from negative edge-triggered flip-flop.

Valid negative edges when 'select' can toggle: Now, as mentioned above both the clocks should be zero when select toggles. Figure below shows the valid and invalid edges where 'select' can toggle. As it turns out, select can toggle only on edges labelled "VALID" as both "CLOCK" and "DIV_CLOCK" will be zero then.

 So, to ensure that "SEL" toggles only when DIV_CLOCK is "0", we can add logic to the input of the flip-flop launching "SEL" such that it allows to propagate "SEL" only when DIV_CLOCK is "0".

In the above diagram, flip-flop launching "SEL" will hold its value when DIV_CLOCK = 0. We have to keep in mind that this implementation is just a representation of what needs to be done. the actual implementation may be more complex than this depending upon the requirements.

Timing considerations: Now coming to the timing considerations, we need to ensure that the setup and hold conditions are met, which are as shown in the figure below: