Digital Electronics 1 Logic Gates Digital Electronics 2 Interest Sensor Technology in digital...

Digital Electronics1

Logic Gates


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.


Using Logic Circuits

A . Boolean Algebra1 . Developed by George Boolean, a 19th century

mathematician.a . His theories were used to develop an assembly of

gears and pulleys to be used to drive a grain elevator.

b . A Boolean expression is nothing more than a description of the input conditions necessary to get a desired output.

CENT-113 Digital Electronics4

Tracing Logic Gates

A

B

C C

A

B

C C

A + B + C(A + B + C)(C)


Tracing Logic Gates

AB

C

B

B

C

A AB

B + C

(AB)(B + C)

(AB)(B + C)


Questions• Q. Draw the logic circuit for AB + CD + ABC = Y

•A.

Y

A

B

C

D


Karnaugh Maps• In 1953, Maurice Karnaugh published a paper about his

system of mapping & simplifying Boolean expressions.• Example:

Inputs Output

B A Y

0 0 0

0 1 1

1 0 1

1 1 1

OR Truth Table Karnaugh Map

x

x xA

B

A

B

Boolean Expression

AB + AB + AB = Y

x

x xA

A

B B

SimplifiedBoolean Expression

A + B = Y

2 Terms:1 Variable = 2 Blocks2 Variables = 1 Block


Karnaugh MapsExample using 3 variables:

Inputs Output

C B A Y

0 0 0 0

0 0 1 1

0 1 0 1

1 0 0 1

1 1 0 1

1 0 1 1

0 1 1 1

1 1 1 1


x x x

x x x xA

B

A

B

Boolean Expression

ABC + ABC + ABC +ABC + ABC + ABC +ABC = Y

A

A

B BSimplified

Boolean Expression

A + B + C = Y

3 Terms:1 Variable = 4 Blocks

2 Variables = 2 Blocks3 Variables = 1 Block

C CC

x x x

x x x x

CC C


Questions• Q. Simplify AB + CBA + BAC + CB + AC.

x

x

x xx

xB

C CC

A

A

B

•A. A + C


Questions• Q. Simplify ABC + CB + AC + BCA + CA.

xx Xx

xxB

C CC

A

A

B

•A. A + BC + BC



Inputs Output

D C B A Y

0 0 0 0 0

0 0 0 1 1

0 0 1 0 1

0 1 0 0 1

1 0 0 0 1

1 1 0 0 1

1 0 1 0 1

1 0 0 1 1

0 1 1 0 1

0 1 0 1 1

0 0 1 1 1

1 1 1 0 1

0 1 1 1 1

1 0 1 1 1

1 1 0 1 1

1 1 1 1 1


x x x

x x x x

x x x x

x x x xA

B

A

BBoolean Expression

ABCD + ABCD + ABCD +ABCD + ABCD + ABCD +ABCD + ABCD + ABCD +ABCD + ABCD + ABCD +ABCD + ABCD + ABCD= Y

A

A

B BSimplified

Boolean Expression

A + B + C + D = Y4 Terms:

1 Variable = 8 Blocks 2 Variables = 4 Blocks3 Variables = 2 Blocks4 Variables = 1 Block

C CC

CC C

D

D

D

x x x

x x x x

x x x x

x x x x D

D

D

4 corner rule


Questions• Q. Simplify ABCD + DC + ABD + DAC.

•A. AB + AD + DC

A

B

A

B

C CC

D

D

D

xx

x

xx

x

xx


Questions• Q. Simplify ABCD + DC + ABC + BC + ABD.

A

B

A

B

C CC

D

D

Dxxxx

xx

x

x

x

•A. BC + AD + AC + CD (be aware of 4 corners)


Questions• Q. Simplify ABCD + ABCD + ACB + CB

+ AB + ACD + ABCD + ABCD + ABC + ACBD.

A

B

A

B

C CC

D

D

Dxxx

x

x

x

xx

xx

•A. CD + BC + AB + AC + BD



Karnaugh Map Boolean Expression

ABCDE + ABCDE + ABDE +ABD + ABCDE + ABCDE = Y

Simplified Boolean Expression

BD = Y

5 Terms:1 Variable = 16 Blocks 2 Variables = 8 Blocks3 Variables = 4 Blocks4 Variables = 2 Blocks5 Variables = 1 Block

4 corner rule

A

BA

B

CC

C

DD

D

A

BA

B

CC

C

DD

D

E

E

XX

XX

XX

XX


Questions• Q. Simplify ABCDE + ABCDE + ABCDE +

ABCDE + ABCD + ABCE + ABCDE = Y.

A

BA

B

CC

C

DD

D

EA

BA

B

CC

C

DD

D

E

XX

XX

X

X

•A. ABD + ABC


Questions• Q. Simplify ABCDE + ABCDE + ABD + ABC

ABCDE + ABCD + ABCD + CD = Y.

A

BA

B

CC

C

DD

D

EA

BA

B

CC

C

DD

D

E

X

XX

X

X

X

X

X

X

X

X

X

X

X

XX

XX

•A. AC + CD + BD = Y


Questions• Q. Simplify ABCDE + ABCD + ACD + ABCDE

+ ABD + ABD + ABCDE = Y.

A

BA

B

CC

C

DD

D

EA

BA

B

CC

C

DD

D

E

X

X

X

X

XX

X

X

X

•A. AD + ABCE


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.


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

W Output1

0

1234

567C B A

DataInputs

1

DataSelectorInputs 0 1 1

01

234567

11

Output

Rotary Switch


Data Selector IC Logic ProblemsInputs Output

D C B A Y

0 0 0 0 0

0 0 0 1 1

0 0 1 0 1

0 0 1 1 1

1 0 0 0 1

1 1 0 0 1

1 0 1 0 1

1 0 0 1 1

0 1 1 0 1

0 1 0 1 1

0 0 1 1 1

1 1 1 0 1

0 1 1 1 1

1 0 1 1 1

1 1 1 0 0

1 0 1 1 1

1 1 1 1 1

01234567

8910111213141516 C B AD

Data Inputs

Data Select Inputs

OutputW


Data Selector Folding Technique• Demonstrate using page 76.


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


De Morgan’s Theorems• First Theorem

• Second Theorem

BA

Y =AB = Y A + B = Y

AB

Y

=

AB

Y

A + B = Y

BA

Y

AB = Y


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).– (A + B) + (BC) = Y

• Step 3: Complement the entire function (long overbar).– (A + B) + (BC) = Y

• Step 4: Eliminate all groups of double overbars.– (A + B) + (BC) = Y

– Solution: (A + B) + (BC) = Y


Conclusion• Q. What is the minterm of A + B?•A. AB

• Q. What is the chip number for a data selector chip?

•A. 74151

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