Logic Gates Digital Electronics

Interest Sensor Technology in digital cameras use CCD and CMOS technology to convert light into electrons. CMOS has many TTL logic gates that you can read a schematic and determine simplified output. Digital Electronics

Using Logic Circuits . Boolean Algebra . Developed by George Boolean, a 19th century mathematician. . His theories were used to develop an assembly of gears and pulleys to be used to drive a grain elevator. . A Boolean expression is nothing more than a description of the input conditions necessary to get a desired output. Digital Electronics

CENT-113 Digital Electronics Tracing Logic Gates A A A + B + C B B (A + B + C)(C) C C C C CENT-113 Digital Electronics

Tracing Logic Gates A AB A (AB)(B + C) B B B (AB)(B + C) C B + C C Digital Electronics

Questions Q. Draw the logic circuit for AB + CD + ABC = Y A. A B C Y D Digital Electronics

Simplified Boolean Expression Karnaugh Maps In 1953, Maurice Karnaugh published a paper about his system of mapping & simplifying Boolean expressions. Example: OR Truth Table Karnaugh Map Boolean Expression Inputs Output B A Y 1 B B AB + AB + AB = Y x A Simplified Boolean Expression A B B A + B = Y x A 2 Terms: 1 Variable = 2 Blocks 2 Variables = 1 Block A Digital Electronics

Simplified Boolean Expression Karnaugh Maps Example using 3 variables: OR Truth Table Karnaugh Map Boolean Expression Inputs Output C B A Y 1 B B ABC + ABC + ABC + ABC = Y x A A C C C Simplified Boolean Expression B B x A + B + C = Y A 3 Terms: 1 Variable = 4 Blocks 2 Variables = 2 Blocks 3 Variables = 1 Block A C C C Digital Electronics

Questions Q. Simplify AB + CBA + BAC + CB + AC. A. A + C x B A C Digital Electronics

Questions Q. Simplify ABC + CB + AC + BCA + CA. A. A + BC + BC x X B A Digital Electronics

Simplified Boolean Expression Karnaugh Maps Example using 4 variables: OR Truth Table Karnaugh Map Boolean Expression Inputs Output D C B A Y 1 B B ABCD + ABCD + ABCD + ABCD + ABCD + ABCD = Y x D A D A D C C C B B Simplified Boolean Expression x D A A + B + C + D = Y D 4 Terms: 1 Variable = 8 Blocks 2 Variables = 4 Blocks 3 Variables = 2 Blocks 4 Variables = 1 Block A D C C C Digital Electronics 4 corner rule

Questions Q. Simplify ABCD + DC + ABD + DAC. A. AB + AD + DC x B D A C Digital Electronics

Questions Q. Simplify ABCD + DC + ABC + BC + ABD. x A. BC + AD + AC + CD (be aware of 4 corners) Digital Electronics

Questions Q. Simplify ABCD + ABCD + ACB + CB + AB + ACD + ABCD + ABCD + ABC + ACBD. A B C D x A. CD + BC + AB + AC + BD Digital Electronics

Karnaugh Maps Example using 5 variables: BD = Y ABCDE + ABCDE + ABDE + Boolean Expression A B C D ABCDE + ABCDE + ABDE + ABD + ABCDE + ABCDE = Y X X X E X A B C D X X Simplified Boolean Expression X E X BD = Y 5 Terms: 1 Variable = 16 Blocks 2 Variables = 8 Blocks 3 Variables = 4 Blocks 4 Variables = 2 Blocks 5 Variables = 1 Block Digital Electronics 4 corner rule

Questions Q. Simplify ABCDE + ABCDE + ABCDE + ABCDE + ABCD + ABCE + ABCDE = Y. A B C D E X A. ABD + ABC Digital Electronics

Questions Q. Simplify ABCDE + ABCDE + ABD + ABC ABCDE + ABCD + ABCD + CD = Y. A B C D E X A. AC + CD + BD = Y Digital Electronics

Questions Q. Simplify ABCDE + ABCD + ACD + ABCDE + ABD + ABD + ABCDE = Y. A B C D E X A. AD + ABCE Digital Electronics

Using NAND Logic A NAND TTL chip is the Quad 2 input 7400. NAND gates are used in many logic circuits because they solve many logic problems with fewer chips. A two input inverted OR gate is the same as a NAND gate. Digital Electronics

Data Selectors (multiplexers) A single pole 8 position rotary switch works similar to a data selector chip. Logic Symbol for a 1 of 8 data selector. Data Selector 1 2 3 4 5 6 7 Rotary Switch 1 Data Inputs 1 1 2 W Output 3 1 1 4 Output 5 6 7 C B A Data Selector Inputs 1 1 Digital Electronics

Data Selector IC Logic Problems Inputs Output D C B A Y 1 Data Inputs 1 2 3 4 5 6 7 W Output 8 9 10 11 12 13 14 15 16 C B A D Data Select Inputs Digital Electronics

Data Selector Folding Technique Demonstrate using page 76. Digital Electronics

More Complex Logic Problems Programmable array of logic gates that within a single IC with multiple outputs: PAL: Programmable Array Logic: 16 inputs, 8 outputs. ROMs: Read-only memories or gate arrays. PROMs: Programmable ROMs: User programmable gate arrays. EPROMs: Erasable PROMs: User programmable gate arrays. Programmable gate arrays Digital Electronics

De Morgan’s Theorems = = First Theorem Second Theorem A A Y Y B B AB = Y A + B = Y Second Theorem A A Y Y B B = AB = Y A + B = Y Digital Electronics

De Morgan’s Theorems Maxterm to Minterm. Given: (AB) (B + C) = Y Step 1: Change all ORs to ANDs and all ANDs to ORs. (A + B) + (BC) = Y Step 2: Complement each individual variable (short overbar). Step 3: Complement the entire function (long overbar). Step 4: Eliminate all groups of double overbars. Solution: (A + B) + (BC) = Y Digital Electronics

Conclusion Q. What is the minterm of A + B? A. AB Q. What is the chip number for a data selector chip? A. 74151 Digital Electronics