1. Elementary Combinational Circuits Introduction Combinational circuits are built from logic gates Can realize arbitrary logical functions Goal is to design efficient circuits Also must keep in mind “extra-logical” properties

2. Elementary Combinational Circuits Gates and corresponding truth tables

3. Elementary Combinational Circuits Gates perform the indicated logical transformation But, can also look at gates as filters acting on data streams If control signal is 1, then AND gate will let signal pass through, If control signal is 0, then output is always 0 If control signal is 1, then OR gate produces 1 If control signal is 0, then output is signal ?

4. Elementary Combinational Circuits Circuits to functions Circuit equivalent to: (() + ()) ((P NAND Q) + ()) ((P NAND Q) + (R'))

5. Elementary Combinational Circuits Circuits to functions (cont.) 2) ((() () ())' XOR ()) 1) (() XOR ()) 3) (((A + B) (C) (D'))' XOR ()) 4) (((A + B) (C) (D'))' XOR (CD))

6. Elementary Combinational Circuits Circuits to truth tables (directly)

7. Elementary Combinational Circuits Functions to circuits (direct) [(A + B)(C)(D')]' XOR [CD]

8. Elementary Combinational Circuits Functions to circuits (through minterms) (P'Q'R' + P'Q'R + P'QR' + P'QR + PQ'R' + PQ'R + PQR')

9. Elementary Combinational Circuits Functions to circuits (through maxterms) (P' + Q' + R')

10. Elementary Combinational Circuits NAND and NOR representations of SOP and POS circuits 1) step 1 justification bubbles cancel 2) step 2 justification generalized DeMorgan

11. Elementary Combinational Circuits Realizing minimal circuits [(A + B)(C)(D')]' XOR [CD]  ABCD (0,1,2,4,5,8,9,12,13)  ABCD (3,6,7,10,11,14,15)

12. Elementary Combinational Circuits Gates and Integrated Circuits (IC’s) in practice Logic families TechnologyDate Relays 1930 ’ s Vacuum Tubes 1940 ’ s-1950 ’ s TTL IC ’ s1960 ’ s-1990 ’ s CMOS IC ’ s1990 ’ s-present

13. Elementary Combinational Circuits Gates and Integrated Circuits (IC’s) in practice Values and voltages BinaryTernary

14. Elementary Combinational Circuits Gates and Integrated Circuits (IC’s) in practice Fan-in and fan-out Fan in is limited for CMOS gates Workaround Propagation time proportional to fan-out soft and hard constraints (ABC)' + (DEF)' ≡ (A' + B' + C' + D' + E' + F') ≡ (ABCDEF)'

15. Elementary Combinational Circuits Gates and Integrated Circuits (IC’s) in practice Gate delays i) rise and fall times not instantaneous ii) outputs lag inputs ii) t pLH not in general equal to t pHL

16. Elementary Combinational Circuits Gates and Integrated Circuits (IC’s) in practice Transistor implementation of gates NAND gateAND gate

17. Elementary Combinational Circuits Summary of topics Gates Gates as filters Circuits to functions Circuits to truth tables Functions to circuits (direct) Functions to circuits through minterms and maxterms NAND and NOR realizations of SOP and POS functions Gates and circuits in practice Logic families Values and voltages Fan-in and fan-out Gate delays Transistor implementations of gates