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Physics 343Advanced Electronics
Engineering 343Digital Systems
Electronics Courses
Chapter 2
Switching Algebra and Logic Circuits
7400 Datasheethttp://www.ti.com/lit/ds/symlink/sn74ls00.pdf
Logic Gates
Logic Gates Circuit Symbols Truth Table Boolean Expression Daisy Chain Multiple Inputs
Figure 2.1 Symbols for OR and AND gates.
Figure 2.3 A NOT gate.
Figure 2.2 AND gate implementation of Property 2b.
OR (written as +)1
a + b (read a OR b) is 1 if and only if a = 1 or b = 1 or both
AND (written as or simply two variables concatenated)
a b = ab (read a AND b) is 1 if and only if a = 1 and b = 1.
NOT (written)
a (read NOT a) is 1 if and only if a = 0
A water sprinkler should turn on between 4am and 5am in the morning only when a rain sensor indicates that it has not rained in the last 48 hours. T = 1 if the time is between 4am and 5amR = 1 if it has rained in the last 48 hours
Create a truth table and design a circuit that would perform this task.
The Book
Inside the cover…
Boolean Theorems
aa = ?a1 = ?a0 = ?aa’ = ?a’’ = ?a(b + c) = ?
Boolean Theorems
a + a = ?a + 1 = ?a + 0 = ?a + a’ = ?
Proof
Proof of theorem P8b.
De Morgan’s Theorems
(AB)’ = ?(A + B) = ?
Exclusive Or
A + B = ? (A + B)’ = ?
DEFINITIONS
Example: f = a’b’c + acd + a’b’c’d
A literal is the appearance of a variable or its complement.
A product term is one or more literals connected by AND operators.
A standard product term, also minterm is a product term that includes each variable of the problem, either uncomplemented or complemented.
A sum of products expression (often abbreviated SOP) is one or more product terms connected by OR operators.
A minimum sum of products expression is one of those SOP expressions for a function that has the fewest number of product terms. If there is more than one expression with the fewest number of terms, then minimum is defined as one or more of those expressions with the fewest number of literals.
(1) xyz + xyz + xyz + xyz + xyz 5 terms, 15 literals
(2) xy + xy + xyz 3 terms, 7 literals
(3) xy + xy + xz 3 terms, 6 literals
(4) xy + xy + yz 3 terms, 6 literals
DEFINITIONS
Example: f = (a’+b’+c)(a+c+d)(a’+b’+c’+d)
A sum term is one or more literals connected by OR operators.
A standard sum term, also called a maxterm, is a sum term that includes each variable of the problem, either uncomplemented or complemented.
A product of sums expression (POS) is one or more sum terms connected by AND operators.
Announcements
The 12:30pm lab should first go to the Linc Lab on the 1st floor of the library on 2/3/2015. The 4:00pm lab should first go to the Info Lab 1 on the 2nd floor of the library on 2/3/2015. After you pick up your tablet from the library at the times above, then you go to room 315 of the science building to begin lab.
Announcements
Work in groups of 2 in lab. All 27 students in the first lab section can begin lab at 12:30pm.Bring your book and lab handout to lab on Tuesday.The due date for Homework 2 has been moved to Thursday, February, 5th.
Chapter 2 Problems
Homework 2 (Page 100) Problems 2(f), 3(a), etc.
Figure 2.5 Minimum sum of product implementation of f.
Figure 2.6 Circuit with only uncomplemented inputs.
Digital Logic Design
Construct a circuit that will add two 2-digit binary numbers together (A1A0 and B1B0) and display the result as a binary number (Z3Z2Z1Z0).
The input numbers A1A0 and B1B0 can be handled with four data switches and the output Z3Z2Z1Z0 can be displayed using four LED’s.
f is 1 if a = 0 AND b = 1 ORif a = 1 AND b = 0 ORif a = 1 AND b = 1
f is 1 if a= 1 AND b = 1 ORif a = 1 AND b = 1 ORif a = 1 AND b = 1
f is 1 if ab = 1 OR if ab = 1 OR if ab = 1
f = ab + ab + ab
P11a. (a + b) = ab P11b. (ab) = a + b
Figure 2.9 NAND gates.
Figure 2.10 Alternate symbol for NAND.
Figure 2.11 Symbols for NOR gate.
Figure 2.13 NAND gate implementation.
= ab + ab
Figure 2.17 An Exclusive OR gate.
For any two product terms where exactly one variable appears uncomplemented in one and complemented in the other, the consensus is defined as the product of the remaining literals. If no such variable exists or if more than one such variable exists, then the consensus is undefined. If we write one term as at1 and the second as at2 (where t1 and t2 represent product terms), then, if the consensus is defined.
at1 ¢ at2 = t1t2
P13a. at1 + at2 + t1t2 = at1 + at2
P13b. (a + t1)(a + t2)(t1 + t2)= (a + t1)(a + t2)
G = DE + ABC + CDE + ABCE
G = C (AB + ABE) + DE + CDE
G = C (B + AE)(B + A) + DE + CDE
G = (C + D E)[C + (B + AE)(B + A)] + DE
Counting
Binary Decimal Hexadecimal0000 0 0
To the chalk board...
More ExamplesConvert 34310 to binary and hexadecimal and octal.
Convert 1101.12 to decimal an octal.
Convert 0.39210 to binary.
Convert 317.28 to binary.
Exercises
Add these binary numbers: 1000.111 + 1100.011
What 2 kinds of logic gates are needed for computer addition?
Desire2Learn
Course Home Pagewww.danbruton.com/EGR343www.danbruton.com/PHY343
Chapter 2
Switching Algebra and Logic Circuits
Remember to bring your book to class on Tuesday for Lab.
Remember Homework 1 & 2.
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