April2000 LD201 : Logic Design (Question 3)Suggested Solutions

A beer-bottling factory contains a production line for putting caps on bottles. The production line consists of a bottle-capper and three bottle fillers, you have to design a warning system to indicate whether the bottle-capper is being overloaded by the bottle-fillers.

The bottle-capper C has two different operation states: normal, in which it can handle 60 bottles per minute, and super, in which it can handle 75 bottles per minute. The bottle-capper is fed from three bottle-fillers F₁, F₂, F₃, which fill 50,30 and 20 bottles per minute respectively. The factory operates on a 24-hour basis, so at any one time at least one bottle-filler is in operation. The warning device W should light up when the output of the bottle-fillers exceeds the capacity of the bottle-capper.

(a) Draw a truth table describing the behavior of W in terms of C, F₁, F₂, F₃. For C, use ‘1’ to mean the bottle-capper is running in super mode, ‘0’ for normal mode. [3 marks]

Truth table:

C	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1
F₁	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
F₂	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
F₃	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
W	´	0	0	0	0	1	1	1	´	0	0	0	0	0	1	1

Here, a ‘1’ for C means that the bottle-capper is running in super mode, a ‘1’ for an F₁ means that the bottle-filler is in operation, and a ‘1’ for W means that the warning light should be lit; ´ denotes ‘don’t care’, as that situation cannot happen. (1 mark for layout of the table, including a key explaining the interpretations of the ‘1’s. 2 marks for correct entries in the last row; 1 mark if there is a single mistake and no marks if there is more than one mistake.) [3 marks]

(b) Give an unsimplified sum-of-products expression for when W should light up. [2 marks]
Unsimplified SOP expression for W:

C’F₁F’₂F₃+C’F₁F₂F’₃ + C’F₁F₂F₃ + CF₁F₂F’₃ + CF₁F₂F₃

(2 marks for a correct answer; 1 mark if at most one term is wrong; no marks otherwise.) [2 marks]

(c) Draw a Karnaugh map for this expression, and hence derive simplified sum-of-products and product-of-sum forms of the expression. [4 marks]
The Karnaugh map is

The 1s are grouped with dashes and the 0s with dots – note that one grouping wraps around (1 mark for correct grouping of 1s, 1 mark for 0s), so the SOP form is

C’F₁F₃+ F₁F₂(1 mark) and the POS form is F₁(F₂+F₃)( C’ + F₂) (1 mark) [4 marks]

(d) Using the laws of boolean algebra, show that these two expressions are logically equivalent. Identify the laws used. [3 marks]
Equivalence of SOP and POS:

F₁(F₂ + F₃)(C’ + F₂)

= distribute product over sum, twice

C’ F₁ F₂ + F₁ F₂F2 + C’ F₁ F₃ + F₁ F₂ F₃

= idempotence of conjunction

C’ F₁ F₂ + F₁ F₂ + C’ F₁ F₃ + F₁ F₂ F₃

= distribute product over sum

F₁ F₂ (C’ + 1 + F₃) + C’ F₁ F₃

= dominance

F₁ F₂ + C’ F₁ F₃

(2 marks for a completely correct calculation, or failing complete corrections, 1 mark for at least two correct steps. Final 1 mark for a reasonable attempt at identifying the laws used.) [3 marks]

(e) Draw a circuit showing an implemention of the simplified sum-of-product expression, using AND, OR and NOT gates. [3 marks]
Circuit for SOP expression:

(1 mark for correct AND layer, 1 mark for correct OR layer, 1 mark for labeling and clarity.) [3 marks]