Binary counting, Boolean Algebra

1
Lecture 5

• Binary counting, Boolean Algebra

2
Digital Electronics

• Digital electronics
• Number Theory, BCD, ASCII
• Parallel vs Serial Transmission
• AND, OR, NOT, NAND Gates
• Boolean Algebra/Combinational Logic
• Sequential Logic
• Flip Flops
• Counter

3
Schematic

• Sensor and Trigger Control Circuit

4
Counting in decimal and binary

• Digital circuits mathematically only have two
  states
• 0 and 1
• FALSE and TRUE
• Binary system (0 and 1) or base 2 number system
  is used

5
Counting in decimal and binary

• Counting in binary
• 0 0
• 1 01
• 2 10
• 3 11
• 4 100
• 5 101
• 6 110
• 7 111
• 8 1000
• 9 1001
• 10 1010

6
Counting in decimal and binary

• To avoid confusion
• For binary 10 say " one zero" not ten
• Mini quiz
• What is decimal 8 equal to in binary (1000)
• What is binary 0110 in decimal (6)
• What is binary 0111 in decimal (7)

7
Place value (1s, 2s, etc)

• The decimal number 3572 is
• 3000 500 70 2
• 3 -Thousands Place
• 5 - Hundreds Place
• 7 - Tens Place
• 2 - Units

8
Place value

• Decimal Place values
• Powers of ten
• Binary Place values
• Powers of two

9
Binary to decimal conversion

• Examples 1111 is
• 8 4 2 1 16
• Example 11011011 is
• 128 64 0 16 8 0 2 1 219

10
Decimal to binary conversion

• Primarily a division problem - For Example 15
• Divide 15 by 2 7 with a remainder of 1 1s
• 7 by 2 3 with a remainder of 1 2s
• 3 by 2 1 with a remainder of 1 4s
• 1 by 2 0 with a remainder of 1 8s
• Therefore 15 decimal is 1111 binary

11
Decimal to binary conversion

• Example 275
• Divide 275 by 2 137 with a remainder of 1 1s
• Divide 137 by 2 68 with a remainder of 1 2s
• Divide 68 by 2 34 with a remainder of 0 4s
• Divide 34 by 2 17 with a remainder of 0 8s
• Divide 17 by 2 8 with a remainder of 1 16s
• Divide 8 by 2 4 with a remainder of 0 32s
• Divide 4 by 2 2 with a remainder of 0 64s
• Divide 2 by 2 1 with a remainder of 0
  128s
• Divide 1 by 2 0 with a remainder of 1 256s
• Therefore 275 decimal is 100010011

12
Decimal to binary conversion

• Check using place value scale
• Computers require conversion from decimal to
  binary for the calculations and back for display
• Encoder and decoder

13
Hexadecimal numbers

• Dec Binary Hex Dec Binary Hex
• 0 0000 0 11 1011 B
• 1 0001 1 12 1100 C
• 2 0010 2 13 1101 D
• 3 0011 3 14 1110 E
• 4 0100 4 15 1111 F
• 5 0101 5 16 10000 10
• 6 0110 6 17 10001 11
• 7 0111 7 18 10010 12
• 8 1000 8 19 10011 13
• 9 1001 9 20 10100 14
• 10 1010 A 21 10101 15

14
Hexadecimal numbers

• Base 16 systems
• In hexidecimal, "10" decimal A, "11" B, "12"
  C, "15" F, and "16" 10
• This system permits the immediate conversion of
  a 4 bit binary number to a "hexadecimal" number
• For example - 1011 B (hex) - use subscript to
  denote base system of number

15
Hex to Binary

• F5 (hex) is F concatenated with 5 which is
• 1111 concatenated with 0101
• or 11110101

16
Binary to Hex

• Separate the binary into groups of 4 and change
  to hex
• 110111101011 is 1101 concat 1110 concat 1011
• D E B
• DEB (hex)

17
Hex to Decimal

• Use place values
• Assume A52 (hex)
• 16 to the 2nd 16 to the 1st 16 to the zero
• 256s 16s 1s
• Ax256 5x16 2x1
• 10x256 5x16 2x1
• 2642

18
Decimal to Hex

• Use the division and remainder rule
• 59 decimal is 59 divided by 16 3 remainder 11
  which is B 1s
• 3 divided by 16 0 remainder 3 16s
• therefore 59 decimal is 3F (hex)

19
Octal numbers

• Older computers use octal numbers which
  characterize
• Conversion is accomplished using the same
  technique
• You should be able to convert
• Octal to binary and binary to octal (groups of 3)
• Octal to decimal and decimal to octal

20
Base Nomenclature

• 1010 1016 102 108
• 1010 (not ) 1016 (not ) 102 (not ) 108
• Subscript to denote base

21
AND gate

• "All or nothing gate
• Gate is a circuit
• Operation can be illustrated with switches
• Series circuit with two switches
• Both switches must be closed to get the lamp to
  light
• AND logic symbol

22
AND gate

• Logic gate is a circuit that can decide what
  output to produce
• See Fig 3.3 for typical "real" circuit
• Output drives an LED
• Develop truth table
• Truth Table gives all possible combinations of A
  and B and the resulting outputs

23
AND Gate

• Boolean expression
• A B Y
• Four ways of expressing the same thing
• Words
• Logic Symbol
• Boolean Expression
• Truth Table

24
OR gate

• Any or all"
• Parallel circuit
• Truth table
• Logic symbol
• Boolean expression

25
Invertor and buffer (NOT gate)

• Only one input
• Produces an output that is opposite the input
• Symbol
• Expression
• Truth table
• Stated "not A"

26
NOT Gate

• "double not A" - referred to as a buffer
• Output equals input
• Buffers/drivers are used to drive lamps, LEDs,
  etc

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NAND gate

• NOT AND or inverted AND
• Symbol and Expression
• Truth Table

28
NAND gate as a Universal gate

• See Figure 3-21 page 36
• Substitute NANDs for other gates

29
Gates with more than two inputs

• Multiple input Gates can be made from
  combinations of other gates
• 3 input AND
• 4 input OR
• The truth table determines the configuration

30
Flip-Flop (4013)

• Fundamental element of sequential logic
• Several types of flip-flop
• R-S
• JK
• D
• All work similarly

31
R-S flip flop
32
R-S flip flop
33
R-S flip flop
34
Clocked R-S flip-Flop
35
Clocked R-S flip-Flop
36
Clocked R-S flip-Flop
37
Flip Flop (Latching)

• Provides the latched version of sensor trigger
• Held until reset by the next clock pulse