Modified from John Wakerly Lecture #2 and #3

CMOS gates at the transistor level

Boolean algebraCombinational-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”– 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, describedfunctionally 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– 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.Problem 1. Express

function AB+CD+A’C using only EXORs and AND gates

• Problem 2Problem 2. Prove that

A+B = A B AB• Problem 3Problem 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