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

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

CMOS NAND -- switch model

CMOS NAND -- more inputs (3)

Inherent inversion. Non-inverting buffer:

2-input AND gate:

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

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

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.)

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.

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¢

Logic symbols

Theorems Proofs by perfect induction

More Theorems N.B. T8¢, T10, T11

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!

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

DeMorgan Symbol Equivalence

Likewise for OR (be sure to check errata!)

DeMorgan Symbols

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)

Truth table vs. minterms & maxterms

Combinational analysis

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

New circuit, same function

“Add out” logic function Circuit:

Shortcut: Symbol substitution

Different circuit, same function

Another example

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”.

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