modified from john wakerly lecture #2 and #3 cmos gates at the transistor level boolean algebra...
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![Page 1: Modified from John Wakerly Lecture #2 and #3 CMOS gates at the transistor level Boolean algebra Combinational-circuit analysis](https://reader033.vdocument.in/reader033/viewer/2022061614/56649d795503460f94a5d045/html5/thumbnails/1.jpg)
Modified from John Wakerly Lecture #2 and #3
CMOS gates at the transistor level
Boolean algebraCombinational-circuit analysis
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CMOS NAND Gates
• Use 2n transistors for n-input gate
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• CMOS NAND -- switch model
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• CMOS NAND -- more inputs (3)
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• Inherent inversion. • Non-inverting buffer:
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• 2-input AND gate:
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CMOS NOR Gates
• Like NAND -- 2n transistors for n-input gate
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NAND vs. NOR
• For a given silicon area, PMOS transistors are “weaker” than NMOS transistors.
NAND NOR
• Result: NAND gates are preferred in CMOS.
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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.)
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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.
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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
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Logic symbols
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Theorems
• Proofs by perfect induction
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More Theorems
• N.B. T8, T10, T11
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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!
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N-variable Theorems
• Prove using finite induction• Most important: DeMorgan theorems
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DeMorgan Symbol Equivalence
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Likewise for OR
• (be sure to check errata!)
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DeMorgan Symbols
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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)
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Truth table vs. minterms & maxterms
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Combinational analysis
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Signal expressions
• Multiply out:
F = ((X + Y) Z) + (X Y Z)
= (X Z) + (Y Z) + (X Y Z)
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New circuit, same function
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“Add out” logic function
• Circuit:
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Shortcut: Symbol substitution
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Different circuit, same function
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Another example
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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”.
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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