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4 problems, 4 pages Exam One Solutions 4 February 2009
Problem 1 (2 parts, 20 points) Switch-level Design
Several incomplete circuits are shown below. Complete each circuit by adding the needed switching network so the output is pulled high or low for all combinations of inputs (i.e., no floats or shorts). Complete each circuit (pull down, pull up, or both) and write the expression if one is not given. Assume both inputs and complements are available.
A D E
B C F
OutX OutY
A
B
C
D
E
F
A
B
C
F
E
A
B C
E F
OUTx = A โ
( B + C ) +( D + E ) โ
F
OUTy = ( A + B โ
C )โ
E โ
F
4 problems, 4 pages Exam One Solutions 4 February 2009
Problem 2 (2 parts, 28 points) Mixed Logic Reengineering
For the following expressions, implement the Boolean expression using the specified gate type. Use correct mixed-logic notation. Do not simplify the expression. You may use multi-input gates. Minimize the total transistors (switches) required. When possible, use common subexpressions to reduce gate counts. Also determine the number of switches used in each implementation.
Part A (14 points) Implement A โ
( B + C )โ
(( B + C )+ D + E )using only AND and NOT gates.
A B C
D E
Out
switches = 1 x 8t + 3 x 6t + 5 x 2t = 36t
Part B (14 points) Implement A + ( B โ
C + D )+ E โ
F using only NAND and NOT gates.
A B C D E F
Out
switches = (^) 1 x 6t + 3 x 4t + 4 x 2t = 26t
4 problems, 4 pages Exam One Solutions 4 February 2009
Problem 4 (2 parts, 30 points) Karnaugh Maps
Part A (12 points) Given the following Karnaugh Map, circle and list all the prime implicants for a product-of-sums (POS) expression, indicating which are essential. Derive the simplified POS expression.
1 X 1 0
A
A
B B
C
C
C
D
D D
prime implicants
essential? yes no
0
B + C
B + D
C + D
B + C + D
simplified POS expression ( B + D ) โ
( B + C ) โ
( C + D ) (โ
B + C + D )
Part B (18 points) For the following expression, derive a simplified sum of products expression using a Karnaugh Map. Circle and list all the prime implicants for a sum-of-products (SOP) expression, indicating which are essential.
Out =( B + D )โ
( A + B + D )โ
( A + B + C )
A
A
B B
C
C
C
D
D D
prime implicants
essential? yes no
B C
A B
A C D
A B D
simplified SOP expression (^) A โ
B + B โ
C + A โ
B โ
D
