MET 487 Lecture 9
Monday, July 19, 2010
5:30 PM
Chapter 6. Digital Circuits
6.1 Introduction
6.2 Digital Representations
Number Systems
• Decimal Number System
• Binary Number System
• Octal Number System
• Hexadecimal Number System
Binary Number System
• Bit - Binary digit
• Least Significant Bit (LSB)
• Most Significant Bit (MSB)
Example: Decimal to Binary Conversion
123 => 1111011
Method 1:
Table 6.1
Successive Divisions Quotient Remainder
123/2
61
1
61/2
30
1
30/2
15
0
15/2
7
1
7/2
3
1
3/2
1
1
1/2
0
1
Result
LSB
MSB
1111011
Method 2:
Binary Weight (2 ^ 8 = 256; 2 ^ 9 = 512; 2^10 = 1,024 or 1K)
------------------------------------------------2^7 2^6 2^5 2^4 2^3 2^2 2^1 2^0
-----------------------------------------------128 64 32 16 8
4 2
1
------------------------------------------------0 1 1
1 1
0 1
1
128 > 123 => place 0 in bit 7 position
64 < 123 => (123 - 64 = 59); place 1 in bit 6 position
32 > 59 => (59 - 32 = 27); place 1 in bit 5 position
16 < 27 => (27 - 16 = 11); place 1 in bit 4 position
8 < 11 => (11 - 8 = 3); place 1 in bit 3 position
4 > 3 => place 0 in bit 2 position
2 < 3 => (3 - 2 = 1); place 1 in bit 1 position
1 = 1 ; place 1 in bit 0 position
Example 6.1: Binary Arithmetic
MET-487-Lectures Page 1
Example 6.1: Binary Arithmetic
0
+ 0
-----0 (Sum)
1
+0
----1 (Sum)
1
+1
-------1 (Carry bit to the next higher bit position)
0 (Sum)
9 + 3 = 12 = (1100) - base 2
CY 0 1 1
10 0 1
+0 0 1 1
---------11 0 0 (sum bits)
9 x 3 = 27 = (11011) - base 2
1001
x 0011
------------------
Table 6.2 Hexadecimal Symbols and Equivalents
Binary
Hexadecimal Decimal
Octal
0000
0
0
0
0001
1
1
1
0010
2
2
2
0011
3
3
3
0100
4
4
4
0101
5
5
5
0110
6
6
6
0111
7
7
7
1000
8
8
10
MET-487-Lectures Page 2
1000
8
8
10
1001
9
9
11
1010
A (a)
10
12
1011
B (b)
11
13
1100
C (c )
12
14
1101
D
13
15
1110
E
14
16
1111
F
15
17
Examples:
123 (base 10) = 0111 1011 (base 2) = 7B (base 16)
123 (base 8) = 001 111 011 (base 2) = 173 (base 8)
ASCII Codes: 8-bit
A (0100 0001) = 41 Hex = 65 (base 10)
B = (0100 0010) = 42 Hex, 66 (base 10)
..
Z=
90 (base 10)
UNICODE (16-bit): xxxx xxxx ASCII
65536 = 2^8 * 2^8 = 65,536 256 * 256
Binary Coded Decimal (BCD) - use 4 bit-code number to encode each decimal digit
123 (base 10) = 0001 0010 0011 (BCD)
123 (Base 10) = 0111 1011 (base 2)
6.3 Combinational Logic and Logic Classes
NOT
C = A'
AND
C= A ∙ B
OR
C= A +B
NAND
C = (A ∙B)'
NOR
C = (A + B)'
XOR
C = A O B = A ∙ B' + A' ∙ B
Buffer
C= A
MET-487-Lectures Page 3
Example 6.2: A Logic Circuits with
Three Inputs: A, B, C
Three Outputs: D, E, F
Boolean Equations:
D= A B
E = D + C' = (A ∙ B) + C'
F = (E ∙ C')'
Gates:
AND Gate x 1 (7408 x 1)
OR Gate x 1 (7432 x 1)
NOT Gate (7404 x 1)
NAND Gate (7400 x 1)
NOR gate (7402 x 1)
Truth Table
Inputs
Outputs
A
B
C
D
E
F
0
0
0
0
1
0
0
0
1
0
0
1
0
1
0
0
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
1
1
1
0
1
1
0
1
1
1
1
1
1
7404 Inverter
Pasted from <http://www.cs.uiowa.edu/~jones/logicsim/man/node5.html>
MET-487-Lectures Page 4
7400 NAD
Pasted from <http://www.cs.uiowa.edu/~jones/logicsim/man/node5.html>
6.4 Timing Diagrams
6.5 Boolean Algebra
Example 6.3 Simplifying a Boolean Expression
6.6 Design of Logic Networks
6.7 Finding a Boolean Expression Given a Truth Table
Example 6.4 Sum of Products and Products of Sums
LAB 6:
• Build the logic circuit - Example 6.2
• Verify the logic circuits using the Truth Table
6.8 Sequential Logic
6.9 Flip-Flops
6.10 Applications of Flip-Flops
6.11 TTL and CMOS Integrated Circuits
6.12 Special Purpose Digital Integrated Circuits
6.13 Integrated Circuit System Design
MET-487-Lectures Page 5