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**L14: Boolean Logic and Basic Gates**

Quick announcements Deadline for HW3 extended to 11/14/2005 Grading formula in “grade so far” does not include extra credit. Used for grade promotion. Grades histogram. AVG(grades so far = 33.04) L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

LECTURE 14: Transistors to logic gates Boolean Logic Basic logic gates Introduction to switching logic Turning logic circuits into switching expressions Simplifying switching expressions, Karnaugh Maps (no time) L14: Boolean Logic and Basic Gates Fall 2005

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**From Transistors to Computer Chips**

L14: Boolean Logic and Basic Gates Fall 2005

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**Transistors - Where it all starts**

Transistors are physical structures Bipolar Junction Transistors Field Effect Transistors For digital logic, you can think of a transistor as a switch that has two states: “ON” and “OFF” Wiring transistors together, we can create logic functions and storage elements L14: Boolean Logic and Basic Gates Fall 2005

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**Example: MOSFET (Metal Oxide Semiconductor Field Effect Transistor)**

Source Gate Drain Two different types of MOSFET p channel n channel L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

Transistor as a Switch Drain Drain Gate Gate Source Source The voltage on the gate decides if the switch will be open (i.e., “OFF”) or closed (i.e., “ON”) L14: Boolean Logic and Basic Gates Fall 2005

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**CMOS - Complementary Metal Oxide Semiconductors**

Today, CMOS is the most common technology used for manufacturing digital computer chips Combines p-channel and n-channel MOSFETS High levels of integration are possible Low power requirements L14: Boolean Logic and Basic Gates Fall 2005

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**when Gate voltage is high, when Gate voltage is low,**

MOSFETs n-Channel p-Channel Source Drain Gate Gate Source Drain Designed so that when Gate voltage is high, switch is “ON” Designed so that when Gate voltage is low, switch is “ON” L14: Boolean Logic and Basic Gates Fall 2005

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**Transistors and Logic Gates**

Transistors can be connected together with wires Connect transistors to create simple, logical functions - these functions are called logic gates In modern computer chips, the transistors and wires are very, very small L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

CMOS Inverter Vdd = “High” Input Output Input Output Low High High Low Ground = “Low” = 0V L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

NOT Logic Gate Input A Output Input A Output 1 1 The NOT logic gate is an inverter Logical function: A’ or A L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

CMOS NAND Logic Gate Vdd = “High” Output Input A Input A Input B Output Input B 1 1 1 1 1 1 1 L14: Boolean Logic and Basic Gates Ground = “Low” = 0V Fall 2005

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**L14: Boolean Logic and Basic Gates**

NAND Logic Gate Input A Input B Output Input A Output 1 1 1 Input B 1 1 1 1 Logical function “NAND” AB alternatively (AB)’ L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

AND Logic Gate Input A Input B Output Input A Output 1 Input B 1 1 1 1 Logical function “AND” AB L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

OR Logic Gate Input A Input B Output Input A Output 1 1 Input B 1 1 1 1 1 Logical function “OR” A + B L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

NOR Logic Gate Input A Input B Output Input A Output 1 1 Input B 1 1 1 Logical function “NOT OR” (A + B) alternatively (A + B)’ L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

XOR Logic Gate Input A Input B Output Input A Output 1 1 Input B 1 1 1 1 Logical function “Exclusive OR” A B + A B alternatively A’B + AB’ L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

Truth Tables Truth Tables are used to define the output for any given combination of inputs Input A Input B Input A Input B Input C Output Output 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 L14: Boolean Logic and Basic Gates Fall 2005

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**Using Truth Tables to Create Digital Logic Functions**

Input x Input y Input z Output 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

A Generalized Procedure for Creating a Digital Logic Design from a Truth Table Canonical Sum of Products form Create an input line for each input variable, and branch off another line with a NOT gate to form the complement of the variable. You will need as many AND gates as there are “1”s in the truth table. The inputs to each AND gate are the input variables or their complements - as indicated by the truth table entries. For example if x=0, y=1, and z=0 and OUTPUT=1, you will need x’ , y, and z’ as inputs into one of the AND gates You will need one OR gate. All of the outputs of the AND gates will go into the OR gate. The output of the OR gate is the output function for the circuit L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

A Generalized Procedure for Creating a Digital Logic Design from a Truth Table Canonical Product of Sums form Create an input line for each input variable, and branch off another line with a NOT gate to form the complement of the variable. You will need as many OR gates as there are “0”s in the truth table. The inputs to each OR gate are the input variables or their complements - as indicated by the truth table entries. For example if x=0, y=0, and z=1 and OUTPUT=0, you will need x , y, and z’ as inputs into one of the OR gates You will need one AND gate. All of the outputs of the OR gates will go into the AND gate. The output of the AND gate is the output function for the circuit L14: Boolean Logic and Basic Gates Fall 2005

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**Canonical Sum of Products**

Using sum of products is one way to get a correct circuit design from a truth table Approach doesn’t always give you the most efficient circuit design! L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

Boolean Algebra George Boole, 19th Century mathematician Developed a mathematical system (algebra) involving logic later known as “Boolean Algebra” Primitive functions: AND, OR and NOT The power of BA is there’s a one-to-one correspondence between circuits made up of AND, OR and NOT gates and equations in BA + means OR,• means AND, x means NOT L14: Boolean Logic and Basic Gates Fall 2005

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**Switching Algebra Identities**

Identity Law 1 A = A 0 + A = A Null Law 0 A = A = 1 Idempotent Law AA = A A + A = A Inverse Law AA = 0 A + A = 1 L14: Boolean Logic and Basic Gates Fall 2005

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**More Switching Algebra Identities**

Commutative Law AB = BA A + B = B + A Associative Law (AB)C = A(BC) (A + B) + C = A + (B + C) Absorption Law A(A + B) = A A + AB = A Distributive Law A + BC = (A + B)(A + C) A(B + C) = AB + AC L14: Boolean Logic and Basic Gates Fall 2005

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**One Other Theorem: Involution**

A = A (A’)’ = A L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

DeMorgan’s Law A B = A + B alternatively (AB)’ = A’ + B’ A + B = A B alternatively (A + B)’ = A’ B’ Very useful for simplifying functions! L14: Boolean Logic and Basic Gates Fall 2005

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**Turning a Switching Expression into A Digital Logic Circuit**

The expression has the directions “and’s” turn into “and” gates; “or’s” turn into “or” gate, complements can be “not” gates…. Example: xy + z’ = output x y output z L14: Boolean Logic and Basic Gates Fall 2005

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**Working with Switching Algebra Quickie Quiz 1 (Draw this circuit)**

f (A,B,C) = A’ + B’ + A’ B C A B C A’ B’ A’ B C F(A,B,C) L14: Boolean Logic and Basic Gates Fall 2005

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**Working with Switching Algebra**

Complicated functions can be simplified using Boolean Algebra L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

Another Example Input A Output Input B What’s the logic function? ((A + B’)’ B’)’ Simplify using DeMorgan’s Law and identities = (A + B’) + B = A + (B’ + B) = A + 1 = 1 L14: Boolean Logic and Basic Gates Fall 2005

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**Working with Switching Algebra QUICKIE QUIZ (Simplify this function)**

f (w,x,y,z) = x y + w’ x y z’ + x’ y = y (x + x’ + w’ x z’) = y ( 1) = y L14: Boolean Logic and Basic Gates Fall 2005

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**Simplified solutions aren’t always unique**

f(x,y,z) = x’ y’ z’ + x’ y z’ + x’ y z + x y’ z’ + x y’ z + x y z = x’ z’ + x y’ + y z = x’ y + y’ z’ + x z = x’ z’ + y’ z’ + y z + x z L14: Boolean Logic and Basic Gates Fall 2005

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**Karnaugh Maps A Tool for Simplifying Logic Functions**

x y 0 0 0 1 1 1 1 0 z 1 Karnaugh map for three variables – a truth table turned on its side L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

Karnaugh Maps Karnaugh maps are tools to help you visually/graphically group the “1s” in a truth table to develop simplified expressions From the logic expression, you can develop a logic- gate-level design implementing the function L14: Boolean Logic and Basic Gates Fall 2005

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**Representing Functions Using a Karnaugh Map**

x y 0 0 0 1 1 1 1 0 z 1 L14: Boolean Logic and Basic Gates Fall 2005

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**Karnaugh Map Guidelines (1 of 3)**

Prepare the map using the truth table inputs Each “1” on the Karnaugh map represents a min-term for a “sum of products” expression of the truth table’s function You can simplify the logic expression by combining the product terms (the “min terms”) that are grouped on the Karnaugh map Groups of two, four, eight, (any power of two) “1” ‘s on the map can be used to develop a simpler expression of fewer variables L14: Boolean Logic and Basic Gates Fall 2005

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**Karnaugh Map Guidelines (2 of 3)**

Groups can “wrap around” the sides of the Karnaugh map Top and bottom rows Right-most and left-most columns Determine which variables can be eliminated from the min-terms using the row and column entries of the Karnaugh map Any variable that appears as both a “1” and a “0” in the grouping can be eliminated from the product term For a complete expression, you need to have expressions for every “1” on the Karnaugh map and combine the terms in a “sum of products” expression L14: Boolean Logic and Basic Gates Fall 2005

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**Karnaugh Map Guidelines (3 of 3)**

You can have a “1” included in more than one simplified expression Redundancy is okay if you end up with the same or fewer terms in your sum of products expression Combine the simplified expressions into a sum of products form Translate the sum of products form into a logic gate design L14: Boolean Logic and Basic Gates Fall 2005

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**L14: Boolean Logic and Basic Gates**

Another Example Input x Input y Input z Output 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 L14: Boolean Logic and Basic Gates Fall 2005

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**Another Example with Three Variables**

x y z 1 0 0 1 0 1 1 0 1 f(x,y,z) = z’ + x y z L14: Boolean Logic and Basic Gates Fall 2005

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