1. CSE 370 – Winter 2002 - Combinational Implementation - 1
Overview
Last lecture
Examples of two-level combinational logic designs
Implicants, prime implicants, Essential prime implicants
Today
Two-level logic using NANDs and NORs
Multilevel logic
12/5/2018
CSE 370 – Winter Combinational Implementation - 1

2. Combinational logic implementation
Two-level logic
implementations of two-level logic
NAND/NOR
Multi-level logic
factored forms
and-or-invert gates
Time behavior
gate delays
hazards
Regular logic
multiplexors
decoders
PAL/PLAs
ROMs
12/5/2018
CSE 370 – Winter Combinational Implementation - 2

3. Implementations of two-level logic
Sum-of-products
AND gates to form product terms (minterms)
OR gate to form sum
Product-of-sums
OR gates to form sum terms (maxterms)
AND gates to form product
F = AB’C’ + A’B + BC
G = (A+B)(B+C’)(A’+B’+C)
12/5/2018
CSE 370 – Winter Combinational Implementation - 3

4. Two-level logic using NAND gates
Step 1: Replace minterm AND gates with NAND gates
and place compensating inversion at inputs of OR gate
12/5/2018
CSE 370 – Winter Combinational Implementation - 4

5. Two-level logic using NAND gates (cont’d)
Step 2: OR gate with inverted inputs is a NAND gate
de Morgan's: A' + B' = (A • B)'
Two-level NAND-NAND network
inverted inputs are not counted
in a typical circuit, inversion is done once and signal distributed
12/5/2018
CSE 370 – Winter Combinational Implementation - 5

6. Two-level logic using NOR gates
Step 1: Replace maxterm OR gates with NOR gates
and place compensating inversion at inputs of AND gate
12/5/2018
CSE 370 – Winter Combinational Implementation - 6

7. Two-level logic using NOR gates (cont’d)
Step 2: AND gate with inverted inputs is a NOR gate
de Morgan's: A' • B' = (A + B)'
Two-level NOR-NOR network
inverted inputs are not counted
in a typical circuit, inversion is done once and signal distributed
12/5/2018
CSE 370 – Winter Combinational Implementation - 7

8. Two-level logic using NAND and NOR gates
NAND-NAND and NOR-NOR networks
de Morgan's law: (A + B)' = A' • B' (A • B)' = A' + B'
written differently: A + B = (A' • B')' (A • B) = (A' + B')'
In other words ––
OR is the same as NAND with complemented inputs
AND is the same as NOR with complemented inputs
NAND is the same as OR with complemented inputs
NOR is the same as AND with complemented inputs OR OR
AND
AND
NAND
NAND
NOR
NOR
12/5/2018
CSE 370 – Winter Combinational Implementation - 8

9. Conversion between forms
Convert from networks of ANDs and ORs to networks of NANDs and NORs
introduce appropriate inversions ("bubbles")
Each introduced "bubble" must be matched by a corresponding "bubble"
conservation of inversions
do not alter logic function
Example: AND/OR to NAND/NAND A B C D Z A B C D
NAND
OR with complemented inputs is the same as NAND
12/5/2018
CSE 370 – Winter Combinational Implementation - 9

10. Conversion between forms (cont’d)
Example: verify equivalence of two forms A B C D Z A B C D Z
NAND
Z = [ (A • B)' • (C • D)' ]'
= [ (A' + B') • (C' + D') ]'
= [ (A' + B')' + (C' + D')' ]
= (A • B) + (C • D) 
12/5/2018
CSE 370 – Winter Combinational Implementation - 10

11. Conversion between forms (cont’d)
Example: map AND/OR network to NOR/NOR network A B C D Z
NOR A B C D Z
NOR \A \B \C \D Z
Step 1
Step 2
conserve
"bubbles"
conserve
"bubbles"
12/5/2018
CSE 370 – Winter Combinational Implementation - 11

12. Conversion between forms (cont’d)
Example: verify equivalence of two forms
NOR \A \B \C \D Z A B C D Z
Z = { [ (A' + B')' + (C' + D')' ]' }’ /* ((X’ + Y’ )’)’ */
= { (A' + B') • (C' + D') }’ /* (X • Y)’*/
= (A' + B')' + (C' + D')’ /* X’ + Y’ */
= (A • B) + (C • D) 
12/5/2018
CSE 370 – Winter Combinational Implementation - 12

13. CSE 370 – Winter 2002 - Combinational Implementation - 13
Multi-level logic
x = A D F + A E F + B D F + B E F + C D F + C E F + G
reduced sum-of-products form – already simplified
6 x 3-input AND gates + 1 x 7-input OR gate (that may not even exist!)
25 wires (19 literals plus 6 internal wires)
x = (A + B + C) (D + E) F + G
factored form – not written as two-level S-o-P
1 x 3-input OR gate, 2 x 2-input OR gates, 1 x 3-input AND gate
10 wires (7 literals plus 3 internal wires) A
B C
D E F G X
12/5/2018
CSE 370 – Winter Combinational Implementation - 13

14. Choosing the “best” realization
Two-level logic: usually has minimal delay
Usually means faster circuits
But with more gates (more transistors) and more wires (larger circuit area); hence requires more power.
Easy to eliminate static hazards (see later)
Sometimes requires large fan-ins
Multilevel logic usually requires less gates
Usually means smaller circuits
Less gates, less wires, shorter wires
Harder to eliminate hazards
12/5/2018
CSE 370 – Winter Combinational Implementation - 14

15. Issues with multilevel design
No global definition of an “optimal” multilevel circuit
Depends on user-defined goals (number of gates, delay etc.)
Use of synthesis (starting with a formal description and proceeding to a logic diagram) to meet the design goals
Synthesis requires CAD-tool help
No simple hand methods like Karnaugh maps
CAD tools manipulate Boolean expressions with some “heurisitics” to achieve goals
12/5/2018
CSE 370 – Winter Combinational Implementation - 15

16. Conversion of multi-level logic to NAND gates
Level 1 Level 2 Level 3 Level 4
F = A (B + C D) + B C'
original
AND-OR network A C D B \C F
introduction and
conservation of bubbles A C D B \C F
redrawn in terms
of conventional
NAND gates A C D \B B \C F
12/5/2018
CSE 370 – Winter Combinational Implementation - 16

17. Conversion of multi-level logic to NORs
Level 1 Level 2 Level 3 Level 4
F = A (B + C D) + B C' A C D B \C F
original
AND-OR network
introduction and
conservation of bubbles A C D B \C F
redrawn in terms
of conventional
NOR gates \A \C \D B \B C F
12/5/2018
CSE 370 – Winter Combinational Implementation - 17

18. Conversion between forms
Example A X B C D F
(a)
original circuit A X B C D F
(b)
add double bubbles at inputs \D A X B C F \X
(d)
insert inverters to fix mismatches \D A \X B C F
(c)
distribute bubbles
some mismatches
12/5/2018
CSE 370 – Winter Combinational Implementation - 18

19. CSE 370 – Winter 2002 - Combinational Implementation - 19
AND-OR-Invert gates
AOI function: conceptually three stages of logic — AND, OR, Invert
multiple gates "packaged" as a single circuit block
AND
OR Invert
logical concept A B C D Z
NAND NAND
Invert
possible implementation A B C D Z
& +
2x2 AOI gate
symbol
& +
3x2 AOI gate
symbol
3 inputs, 2 stacks
12/5/2018
CSE 370 – Winter Combinational Implementation - 19

20. Conversion to AOI forms
General procedure to place in AOI form
compute the complement of the function in sum-of-products form
by grouping the 0s in the Karnaugh map
Example: XOR implementation –– A xor B = A' B + A B'
AOI form: F = (A' B' + A B)'
0 1
1 0 A B
& + A' B' A B F
12/5/2018
CSE 370 – Winter Combinational Implementation - 20

21. Examples of using AOI gates
F = B C' + A C' + A B
F' = A' B' + A' C + B' C
Implemented by 2-input 3-stack AOI gate
F = (A + B) (A + C') (B + C')
F' = (B' + C) (A' + C) (A' + B')
Implemented by 2-input 3-stack OAI gate
Example: 4-bit equality function
Z = (A0 B0 + A0' B0')(A1 B1 + A1' B1')(A2 B2 + A2' B2')(A3 B3 + A3' B3')
0 1
0 0
1 1
1 0 C B A
each implemented in a single 2x2 AOI gate;
then NOR them together
12/5/2018
CSE 370 – Winter Combinational Implementation - 21

22. Examples of using AOI gates (cont’d)
Example: AOI implementation of 4-bit equality function
high if A0  B0 low if A0 = B0 A0 B0 A1 B1 A2 B2 A3 B3
& +
NOR Z
conservation of bubbles
if all inputs are low
then Ai = Bi, i=0,...,3
output Z is high
12/5/2018
CSE 370 – Winter Combinational Implementation - 22

23. Summary for multi-level logic
Advantages
circuits may be smaller
gates have smaller fan-in
circuits may be faster
Disadvantages
more difficult to design
tools for optimization are not as good as for two-level
analysis is more complex
12/5/2018
CSE 370 – Winter Combinational Implementation - 23