1. Gates and Circuits Monday/Wednesday Week 7

2. Electronic Circuits  Two types of circuit diagrams See pp. 238 in The Analytical Engine by Decker and Hirshfield for the "mechanical switch" type of diagram. Second style (which we will call "symbolic diagrams") uses pictures for the boolean logic operators. These pictures are called gates.

3. Three Main Gates  AND  OR  NOT

4. Gate Diagrams  Example 1: (MR) + S

5. Gate Diagrams  Example: What does it represent?

6. Gate Diagrams  Example: ((MR) + S) (RS)’

7. Truth Table to Gates  First, build the Boolean algebra expression that gives Z Z = AB + A’B’ Z = (A AND B) OR (NOT A AND NOT B) ABZ TTT TFF FTF FFT

8. Truth Table to Gates  Z = AB + A’B’  Next, build the circuit that goes with the Boolean algebra expression Z ABZ TTT TFF FTF FFT

9. Z = AB + A’B’

10. Binary Arithmetic  We can add binary numbers just like decimal numbers only using base two arithmetic.  For example: 5101 1110 101 + 7+ 111 121100

11. Binary Addition  Notice in addition: 0011 + 0+ 1+ 0 + 1 01110 FalseTrue False True Sum Carry ABSum (1) T (0) F (1) T(0) F(1) T (0) F(1) T (0) F

12. Sum and Carry ABCarry 111 100 010 000 ABSum 110 101 011 000

13. Sum Circuit ABSum 110 101 011 000 Sum = AB’ + A’B

14. Carry Circuit ABCarry 111 100 010 000 Carry = AB

15. Half Adder - Sum and Carry

16. Half Adder  Handles the case where we add two binary digits with no inward carry.

17. Full Adder  Takes a carry in and produces the result and carry out.  So, we have 3 inputs and two outputs.  Combine two half-adders together with an or gate to get a full adder for each binary digit.  How many half adders would we need to add two 8-digit binary numbers? How many gates?

18. Full Adder

19. Binary Subtraction  We do binary subtraction like decimal subtraction only the borrowing is done in 2’s instead of 10’s. 1221111010 - 7- 111 1151110011

20. Subtraction ABSub 110 101 011 000 ABBorrow 110 100 011 000

21. Binary Multiplication  Again, just like decimal except we add and multiply in binary. *01 000 101 5101 x 7x 111 35100011

22. Exercises  1 - From the book, p 266, Construct circuits with the following properties (using AND, OR or NOT gates):

23. Exercises A)B) ABZ 110 101 011 001 ABCZ 1111 1100 1011 1000 0110 0101 0010 0001

24. NAND Gates and NOT  This gate represents (A NAND NOT B).

25. NAND Truth Table ABA NAND B TTF TFT FTT FFT

26. Exercises  2 - Fill in a truth table and give a Boolean expression for the following circuits.

27. Exercises  3 - How would you create a one binary digit multiplier? A two-digit by one-digit multiplier? A two-digit by two-digit multiplier? *01 000 101