1. Modified from John Wakerly Lecture #2 and #3
CMOS gates at the transistor level
Boolean algebra Combinational-circuit analysis

2. CMOS NAND Gates
Use 2n transistors for n-input gate

3. CMOS NAND -- switch model

4. CMOS NAND -- more inputs (3)

5. Inherent inversion.
Non-inverting buffer:

6. 2-input AND gate:

7. CMOS NOR Gates
Like NAND -- 2n transistors for n-input gate

8. NAND vs. NOR
For a given silicon area, PMOS transistors are “weaker” than NMOS transistors.
NAND
NOR
Result: NAND gates are preferred in CMOS.

9. Boolean algebra a.k.a. “switching algebra” Positive-logic convention
deals with boolean values -- 0, 1
Positive-logic convention
analog voltages LOW, HIGH --> 0, 1
Negative logic -- seldom used
Signal values denoted by variables (X, Y, FRED, etc.)

10. Boolean operators Complement: X¢ (opposite of X) AND: X × Y OR: X + Y
Axiomatic definition: A1-A5, A1¢-A5¢
binary operators, described functionally by truth table.

11. More definitions Literal: a variable or its complement
X, X¢, FRED¢, CS_L
Expression: literals combined by AND, OR, parentheses, complementation
X+Y
P × Q × R
A + B × C
((FRED × Z¢) + CS_L × A × B¢ × C + Q5) × RESET¢
Equation: Variable = expression
P = ((FRED × Z¢) + CS_L × A × B¢ × C + Q5) × RESET¢

12. Logic symbols

13. Theorems
Proofs by perfect induction

14. More Theorems
N.B. T8¢, T10, T11

15. Duality Swap 0 & 1, AND & OR Why?
Result: Theorems still true
Why?
Each axiom (A1-A5) has a dual (A1¢-A5¢)
Counterexample: X + X × Y = X (T9) X × X + Y = X (dual) X + Y = X (T3¢) ????????????
X + (X × Y) = X (T9) X × (X + Y) = X (dual) (X × X) + (X × Y) = X (T8) X + (X × Y) = X (T3¢)
parentheses, operator precedence!

16. N-variable Theorems Prove using finite induction
Most important: DeMorgan theorems

17. DeMorgan Symbol Equivalence

18. Likewise for OR
(be sure to check errata!)

19. DeMorgan Symbols

20. Even more definitions (Sec. 4.1.6)
Product term
Sum-of-products expression
Sum term
Product-of-sums expression
Normal term
Minterm (n variables)
Maxterm (n variables)

21. Truth table vs. minterms & maxterms

22. Combinational analysis

23. Signal expressions
Multiply out: F = ((X + Y¢) × Z) + (X¢ × Y × Z¢) = (X × Z) + (Y¢ × Z) + (X¢ × Y × Z¢)

24. New circuit, same function

25. “Add out” logic function
Circuit:

26. Shortcut: Symbol substitution

27. Different circuit, same function

28. Another example

29. Short Review of Exor Logic
A  A = 0
A  A’ = 1
A  1=A’
A’  1=A
A  0=A
A  B= B  A
A B = B A
A(B  C) = AB  AC
A+B = A  B  AB
A+B = A  B when AB = 0
A  (B  C) = (A  B)  C
(A B) C = A (B C)
A+B = A  B  AB =
A B(1  A) = A  BA’
These rules are sufficient to minimize Exclusive Sum of Product expression for small number of variables
We will use these rules in the class for all kinds of reversible, quantum, optical, etc. logic. Try to remember them or put them to your “creepsheet”.

30. Challenge Problems for ambitious students
Problem 1. Express function AB+CD+A’C using only EXORs and AND gates
Problem 2. Prove that
A+B = A  B  AB
Problem 3 . Prove that A+B = A  B when AB = 0
Problem 4. Given are three functions of three inputs:
A = NOT(a), B = NOT(b), C = NOT(c).
You have only two inverters. You can have an arbitrary large set of two-input AND and OR gates.
Realize these three functions with the gates that you have at your disposal. You cannot use other gates. You can use only two inverters. Draw the schematic of the solution