Binary Numbers and Buffer Gate

Binary Numbers

• Binary numbers use base two, employing only 0 and 1
• Circuits use 0 to indicate an off state, and 1 to indicate an on state
• The off state is equivalent to "false," while the on state is equivalent to "true"
• An off state has 0 volts; an on state typically has 5 volts (or a non-zero voltage)

Buffer Gate

• A triangle pointing right symbolizes the buffer gate, with input on the left and output on the right
• Output of a buffer gate mirrors its input
• Input (A) of 1 (on state) results in an output of 1
• Input (A) of 0 (off state) results in an output of 0
• A buffer gate uses an NPN transistor, a voltage source, and an LED
• Applying voltage to the transistor's base (input A on) activates the transistor, allowing current flow and turning the LED on (output is 1)
• Without voltage at the transistor's base (input A off), the transistor stays off, preventing current flow and the LED remains off (output is 0)

NOT Gate

• The NOT gate symbol resembles a buffer gate but includes a circle in the front
• Output of a NOT gate is the complement of its input (A or A bar)
• Input of 0 yields an output of 1
• Input of 1 yields an output of 0
• A NOT gate uses a similar transistor circuit, but the LED is connected across the collector and emitter
• Applying voltage to the base (input A on) activates the transistor, allowing current to flow from collector to emitter
• Current takes the path of least resistance (through the transistor), so no current flows through the LED, turning the LED off; hence, a 1 input gives a 0 output
• When input A is off, the transistor is off, so current cannot flow through it
• Instead, current flows through the LED, turning it on; hence, a 0 input yields a 1 output

NOT Gate Function

• Transforms an off state into an on state
• Transforms an on state into an off state
• Changes A into A prime

AND Gate

• Its symbol has two inputs, A and B, with multiplication between them
• The output is A times B

AND Gate Circuit and Truth Table

• Requires two transistors connected in series
• If both inputs A and B are off (0), the LED is off (0)
• If A is on (1) and B is off (0), the LED is off (0)
• If A is off (0) and B is on (1), the LED is off (0)
• If both A and B are on (1), the LED is on (1)
• The LED illuminates only when both A and B are on

OR Gate

• Its symbol has two inputs, A and B, with addition between them
• Output is A plus B

OR Gate Circuit and Truth Table

• Two NPN transistors connected in parallel are used
• If both transistors A and B are off (0), the LED is off (0)
• If transistor A is on (1) and B is off (0), the LED is on (1)
• If transistor B is on (1) and A is off (0), the LED is on (1)
• If both transistors are on (1), the LED is on (1)
• If A or B is on, the output is one

NAND Gate

• It is the complement of the AND gate
• Its symbol resembles an AND gate but with a circle at the end
• The output is the complement of A times B

NAND Gate Truth Table

• When both inputs A and B are on (1), the output is off (0)
• Output column is the complement of the AND Gate's output column

NAND Gate Construction

• An AND gate followed by a NOT gate creates a NAND gate
• The complement of AB is A*B prime

NAND and NOT Gate Equivalence

• A NAND gate with both inputs connected together acts as a NOT gate
• The truth table for the combined NAND gate has identical input columns
• There are only two possibilities: Both inputs are 0 or both are 1
• With a 0 input, the output is 1; with a 1 input, the output is 0, which is what defines a NOT gate

NOR Gate

• It is the complement of the OR gate
• Its symbol looks like the OR gate with a circle at the end
• Output is A plus B, complemented

NOR Gate Truth Table

• If either A or B is on, the output is off

Logic Gate Functions

• Understanding logic functions requires knowing how to write a function given a block diagram
• Essential to identify logic gates
• AND gate signifies multiplication
• OR gate indicates addition

Function example 1

• F = A * B * C when all inputs feed into AND gates

Function example 2

• F = XY + XY' when one input goes into an AND gate, then a second AND gate, and they are connected with an OR gate

Function Example 3

• F = (A+B)(AB')

Function Example 4

• F = (X+Y)(X'+Z)

Challenge Problem example function

• F = X'(XY + Z) + W + XY'Z

Drawing a Block Diagram From a Function

• Function: F = AB + C
• Employ an AND gate for A and B, and connect it to an OR gate with input C

Block Diagram Example 2

• F = XY + X'Y + Y'Z
• Utilize AND gates for the products (XY, X'Y, Y'Z), then link to an OR gate
• This expression is called the sum of products

Sum of Products (SOP)

• SOP expression is the sum of product terms
• A literal is each variable, whether complemented or uncomplemented
• Example: the expression currently uses 6 literals

Block Diagram Example 3

• F = (A + B')(A + B + C)(B' + C')
• Use OR gates for the sum terms, then combine them using an AND gate
• This equation is a product of sums

Product of Sums (POS)

• POS expression signifies the product (multiplication) of multiple sum terms, such as (A + B) * (C + D)
• Example above has 7 literals in the function listed

SOP vs POS Expression

• You must be able to identify functions as SOP or POS (or both, or neither)

Expression Example 1

• Expression: XY' + XYZ' + X'YZW + XY'Z'
• It's a SOP expression because it is the sum of 4 product terms

Expression Example 1 Details

• Has four variables (W, X, Y, Z)
• Totals 12 literals
• Contains a min term

Min Term

• A standard product term includes each variable in the specific problem
• Must include all four variables either complemented or uncomplemented
• The min term in example one is: X'YZW

Expression Example 2

• Expression: (X + Y)(X' + Z)(Y + Z')(X + Y + Z')
• It's a POS expression because it is a product of four sum terms

Expression Example 2 Details

• Has three variables: X, Y, and Z
• Includes nine literals
• Contains a max term

Max Term

• A sum term that includes each variable in the specific problem, complemented or uncomplemented
• The max term for Expression Example 2 is: (X + Y + Z')

Expression Example 3

• Expression: XY'ZW
• It can be defined as both SOP and POS because it's one term
• From a POS perspective, terms X, Y', Z, and W are all added to 0
• Has 4 variables: W, X, Y, Z
• Has 4 literals

Terms

• A PRODUCT TERM can be a single variable x, or y prime
• A SUM TERM can be an individual literal x, or x + y

Expression Example 4

• Expression: A + B + C' + D + E'
• Can be represented as both SOP and POS expression
• SOP is a sum of five product terms
• POS is a product of one sum term, which can be assumed
• Has 5 variables: A, B, C, D, E
• Has 5 literals: A, B, C', D, E'

Expression Example 5

• A(B + CD')
• It is neither a SOP or POS Expression
• Has 4 variables
• Has 4 literals

Rules of Boolean Algebra

• Includes the commutative, associative, identity, and null property
• OR logic gate = plus sign
• AND logic gate = multiplication
• In Boolean Algebra, 1 + 1 = 1; but in old school algebra 1 + 1 = 2

OR Gate Logic

• With an OR gate, an output turns on if one or both inputs are on
• The output is one if both A and B are in the on state
• "One plus one equals one" in the context of logic gates
• A+B will be on if A is on and B is on

Complement Property

• A + A' = 1
• A * A' = 0
• If A is zero, A' (A prime) must be one and vice-versa
• 0 + 1 = 1, meaning an OR logic gate outputs "on" if one input is off and the other is on

AND Gate Logic with Complements

• With an AND gate, output switches on only when both inputs are on
• A' will be off if A is on
• A and A' cannot be on simultaneously because of the nature of the AND gate
• The output is always zero (off) when using an AND logic gate with A and A' as the inputs

Explanation of binary numbers, which are based on the number two, using only 0 and 1, and how they relate to circuits where 0 represents the off state and 1 represents the on state. Description of the buffer gate symbol and its functionality.