Design of Information and Control SystemsProgrammable Logic Controllers - Fundamentals

Institute of Information Engineering, Automation and Mathematics

22. marca 2020

Contents

What are programmable logic controllers? (PLC)

Implementation of logic using Boolean algebra

Ladder Logic - LAD

Programmable Logic Controller (PLC)

Programmable Logic Controller

Programmable Logic Controller

Programmable Logic Controller

Programmable Logic Controller

Programmable Logic Controller

Programmable Logic Controller

Electric

sourceMain

unit

Input and

output

modules

Programmable Logic Controller

Electric

source

Main

unit

Input and

output

modules

Programmable Logic Controller

Electric

sourceMain

unit

Input and

output

modules

Programmable Logic Controller

Electric

sourceMain

unit

Input and

output

modules

Programmable Logic Controller

Electric

sourceMain

unit

Input and

output

modules

Electric source:

from outlet - 230V

produces 24V forpowering the PLC andmodules

Main unit:

processor

memory

communication

peripherals

Modules:

input

output

digital/analog

Programmable Logic Controller

PLC Siemens S7-300

Programmable Logic Controller

PLC Siemens S7-300

Programmable Logic Controller

PLC Siemens S7-1200

Programmable Logic Controller

PLC Siemens S7-1200

Programmable Logic Controller

Processor and memory

PLC

Logic

inputs outputs

switch

sensor

light

Programmable Logic Controller

light

spınac

sensor

switch

snımac 2

tlacidlo 2

Processor and memory

PLC

Logic

inputs outputs

024V

0V

3.3V

10

1

3.3V24V

3.3V24V

00V

0V

Programmable Logic Controller

light

spınac

sensor

switch

snımac 2

tlacidlo 2

Processor and memory

PLC

Logic

inputs outputs

024V

0V

3.3V

10

1

3.3V24V

1

3.3V24V

13.3V

24V

Programmable Logic Controller

light

spınac

sensor

switch

snımac 2

tlacidlo 2

Processor and memory

PLC

Logic

inputs outputs

024V

0V

3.3V

1 1

3.3V24V

1

3.3V24V

13.3V

24V

Programmable Logic Controller

light

spınac

sensor

switch

snımac 2

tlacidlo 2

Processor and memory

PLC

Logic

inputs outputs

024V

0V

3.3V

1 1

3.3V24V

1

3.3V24V

13.3V

24V

Programmable Logic Controller

PLC performs a set of simple tasks:

reads the state of inputs

executes logical and mathematical instructions

based on results, activates/deactivates outputs

For the purpose of this lecture, PLC will perform only logical operations.

Programmable Logic Controller

PLC performs a set of simple tasks:

reads the state of inputs

executes logical and mathematical instructions

based on results, activates/deactivates outputs

For the purpose of this lecture, PLC will perform only logical operations.

Boolean algebra

Logical operations

operate only with values 0 and 1 (yes/no, on/off, true/false)

use logical functions - values on input are mapped to values on outputcan be written down in multiple ways

logical statement (mathematical form)ladder diagram (graphical form)electrical diagram (graphical form with switches)electronic diagram (graphical form with logic gates)

Logical operations

NOT - logical negationlogical function: Y = A

conjunction: is not

NOT gate

Truth table

A A0

1

1

0

Logical operations

NOT - logical negationlogical function: Y = A

conjunction: is not

NOT gate

Truth table

A A0 11

0

Logical operations

NOT - logical negationlogical function: Y = A

conjunction: is not

NOT gate

Truth table

A A0 11 0

Logical operations

AND - logical conjunctionlogical function: Y = A · B

conjunction: and; at the same time

AND gate

Truth table

A B A · B0 0

0

0 1

0

1 0

0

1 1

1

Logical operations

AND - logical conjunctionlogical function: Y = A · B

conjunction: and; at the same time

AND gate

Truth table

A B A · B0 0 00 1

0

1 0

0

1 1

1

Logical operations

AND - logical conjunctionlogical function: Y = A · B

conjunction: and; at the same time

AND gate

Truth table

A B A · B0 0 00 1 01 0

0

1 1

1

Logical operations

AND - logical conjunctionlogical function: Y = A · B

conjunction: and; at the same time

AND gate

Truth table

A B A · B0 0 00 1 01 0 01 1

1

Logical operations

AND - logical conjunctionlogical function: Y = A · B

conjunction: and; at the same time

AND gate

Truth table

A B A · B0 0 00 1 01 0 01 1 1

Logical operations

OR - logical disjunctionlogical function: Y = A + B

conjunction: or

OR gate

Truth table

A B A + B0 0

0

0 1

1

1 0

1

1 1

1

Logical operations

OR - logical disjunctionlogical function: Y = A + B

conjunction: or

OR gate

Truth table

A B A + B0 0 00 1

1

1 0

1

1 1

1

Logical operations

OR - logical disjunctionlogical function: Y = A + B

conjunction: or

OR gate

Truth table

A B A + B0 0 00 1 11 0

1

1 1

1

Logical operations

OR - logical disjunctionlogical function: Y = A + B

conjunction: or

OR gate

Truth table

A B A + B0 0 00 1 11 0 11 1

1

Logical operations

OR - logical disjunctionlogical function: Y = A + B

conjunction: or

OR gate

Truth table

A B A + B0 0 00 1 11 0 11 1 1

Logical operations

NAND - negated logicalconjunction

logical function: Y = A · B

conjunction: not at the same time

NAND gate

Truth table

A B A · B0 0

1

0 1

1

1 0

1

1 1

0

Logical operations

NAND - negated logicalconjunction

logical function: Y = A · B

conjunction: not at the same time

NAND gate

Truth table

A B A · B0 0 10 1

1

1 0

1

1 1

0

Logical operations

NAND - negated logicalconjunction

logical function: Y = A · B

conjunction: not at the same time

NAND gate

Truth table

A B A · B0 0 10 1 11 0

1

1 1

0

Logical operations

NAND - negated logicalconjunction

logical function: Y = A · B

conjunction: not at the same time

NAND gate

Truth table

A B A · B0 0 10 1 11 0 11 1

0

Logical operations

NAND - negated logicalconjunction

logical function: Y = A · B

conjunction: not at the same time

NAND gate

Truth table

A B A · B0 0 10 1 11 0 11 1 0

Logical operations

NOR - negated logicaldisjunction

logical function: Y = A + B

conjunction: neither

NOR gate

Truth table

A B A + B0 0

1

0 1

0

1 0

0

1 1

0

Logical operations

NOR - negated logicaldisjunction

logical function: Y = A + B

conjunction: neither

NOR gate

Truth table

A B A + B0 0 10 1

0

1 0

0

1 1

0

Logical operations

NOR - negated logicaldisjunction

logical function: Y = A + B

conjunction: neither

NOR gate

Truth table

A B A + B0 0 10 1 01 0

0

1 1

0

Logical operations

NOR - negated logicaldisjunction

logical function: Y = A + B

conjunction: neither

NOR gate

Truth table

A B A + B0 0 10 1 01 0 01 1

0

Logical operations

NOR - negated logicaldisjunction

logical function: Y = A + B

conjunction: neither

NOR gate

Truth table

A B A + B0 0 10 1 01 0 01 1 0

Logical operations

XOR - exclusive logicaldisjunction

logical function: Y = A ⊕ B

conjunction: one or the other but notboth

XOR gate

Truth table

A B A ⊕ B0 0

0

0 1

1

1 0

1

1 1

0

Logical operations

XOR - exclusive logicaldisjunction

logical function: Y = A ⊕ B

conjunction: one or the other but notboth

XOR gate

Truth table

A B A ⊕ B0 0 00 1

1

1 0

1

1 1

0

Logical operations

XOR - exclusive logicaldisjunction

logical function: Y = A ⊕ B

conjunction: one or the other but notboth

XOR gate

Truth table

A B A ⊕ B0 0 00 1 11 0

1

1 1

0

Logical operations

XOR - exclusive logicaldisjunction

logical function: Y = A ⊕ B

conjunction: one or the other but notboth

XOR gate

Truth table

A B A ⊕ B0 0 00 1 11 0 11 1

0

Logical operations

XOR - exclusive logicaldisjunction

logical function: Y = A ⊕ B

conjunction: one or the other but notboth

XOR gate

Truth table

A B A ⊕ B0 0 00 1 11 0 11 1 0

Ladder Logic

Ladder Logic

PLC programming

a way to design logical controllers

program is represented graphically

Ladder Logic

Ladder Logic

Basic elements of ladder logic

Logical network

1

1

1

Normallyopen

contact

Normallyclosedcontact

input

input

0

0

1

1 1

0

Ladder Logic

Basic elements of ladder logic

Logical network

1

1

1

Normallyopen

contact

Normallyclosedcontact

input

input

0

0

1

1 1

0

Ladder Logic

Basic elements of ladder logic

Logical network

1

1

1

Normallyopen

contact

Normallyclosedcontact

input

input

0

0

1

1 1

0

Ladder Logic

Basic elements of ladder logic

Logical network

1

1

1

Normallyopen

contact

Normallyclosedcontact

input

input

1

0

1

1 1

1

Ladder Logic

Basic elements of ladder logic

Logical network

1

1

1

Normallyopen

contact

Normallyclosedcontact

input

input

1

1

1

1 0

1

Ladder Logic

Example: light bulb control

a single room with 3 light switches

pressing of either switch will result in bulb turning on

Ladder Logic

Example: light bulb control

ovl adaci e prvky(vstupy)

ovl adany prvok(vystup)

zdroj el ektri ny

switch 1

switch 2

switch 3

light bulb

Ladder Logic

Example: light bulb control

ovl adaci e prvky(vstupy)

ovl adany prvok(vystup)

zdroj el ektri ny

switch 1

switch 2

switch 3

light bulb

electrical source

Ladder Logic

Example: light bulb control

ovl adaci e prvky(vstupy)

ovl adany prvok(vystup)

zdroj el ektri ny

switch 1

switch 2

switch 3

light bulb

electrical source

control elements(inputs)

Ladder Logic

Example: light bulb control

ovl adaci e prvky(vstupy)

zdroj el ektri ny

switch 1

switch 2

switch 3

light bulb

electrical source

control elements(inputs)

controlled element(output)

Ladder Logic

Example: light bulb control

ovl adaci e prvky(vstupy)

zdroj el ektri ny

switch 1

switch 2

switch 3

light bulb

switch 1 = 0

switch 3 = 0switch 2 = 0

light = 0

Ladder Logic

Example: light bulb control

ovl adaci e prvky(vstupy)

zdroj el ektri ny

switch 1

switch 2

switch 3

light bulb

switch 1 = 1

switch 3 = 0switch 2 = 0

light = 1

Ladder Logic

Example: light bulb control

ovl adaci e prvky(vstupy)

zdroj el ektri ny

switch 1

switch 2

switch 3

light bulb

switch 1 = 1

switch 3 = 0switch 2 = 0

light = 1

Ladder Logic

Example: light bulb control

ovl adaci e prvky(vstupy)

zdroj el ektri ny

switch 1

switch 2

switch 3

light bulb

switch 1 = 0

switch 3 = 0switch 2 = 1

light = 1

Ladder Logic

Example: light bulb control

ovl adaci e prvky(vstupy)

zdroj el ektri ny

switch 1

switch 2

switch 3

light bulb

switch 1 = 0

switch 3 = 0switch 2 = 1

light = 1

Ladder Logic

Example: light bulb control

ovl adaci e prvky(vstupy)

zdroj el ektri ny

switch 1

switch 2

switch 3

light bulb

switch 1 = 0

switch 3 = 1switch 2 = 0

light = 1

Ladder Logic

Example: light bulb control

ovl adaci e prvky(vstupy)

zdroj el ektri ny

switch 1

switch 2

switch 3

light bulb

switch 1 = 0

switch 3 = 1switch 2 = 1

light = 1

Ladder Logic

Example: light bulb control

ovl adaci e prvky(vstupy)

zdroj el ektri ny

switch 1

switch 2

switch 3

light bulb

The bulb is ON ifat least ONE

of the switches ispressed.

Ladder Logic

Example: car engine start

The engine will start only if the key is present, the brake pedal is stepped on,and driver presses start button.

All of these requirements (inputs) must be fulfilled at the same time.

Ladder Logic

Example: car engine start

key break button

start

brake

Ladder Logic

Example: car engine start

key break button

start

brake

Ladder Logic

Example: car engine start

key break button

start

brake

Ladder Logic

Example: car engine start

key break button

start

brake

Ladder Logic

Example: car engine start

key break button

start

brake

Ladder Logic

Example: car engine start

key break button

start

brake

Ladder Logic

Example: car engine start

key break button

start

brake

Ladder Logic

Example: car engine start

key break button

start

brake

Ladder Logic

Example: car engine start

key break button

start

brake

Ladder Logic

Example: car engine start

key break button

start

brake

Ladder Logic

Example: car engine start

key break button

start

All the inputs must be active at the same time.

brake

Ladder Logic

Light bulb example

The light will turn on if:

switch 1 is pressed

OR

switch 2 is pressed

OR

switch 3 is pressed

Logical disjunction: LIGHT = S1 + S2 + S3

Ladder Logic

Light bulb example

The light will turn on if:

switch 1 is pressed

OR

switch 2 is pressed

OR

switch 3 is pressed

Logical disjunction: LIGHT = S1 + S2 + S3

Ladder Logic

Light bulb example

The light will turn on if:

switch 1 is pressed

OR

switch 2 is pressed

OR

switch 3 is pressed

Logical disjunction: LIGHT = S1 + S2 + S3

Ladder Logic

Light bulb example

The light will turn on if:

switch 1 is pressed

OR

switch 2 is pressed

OR

switch 3 is pressed

Logical disjunction: LIGHT = S1 + S2 + S3

Ladder Logic

Light bulb example

The light will turn on if:

switch 1 is pressed

OR

switch 2 is pressed

OR

switch 3 is pressed

Logical disjunction: LIGHT = S1 + S2 + S3

Ladder Logic

Light bulb example

The light will turn on if:

switch 1 is pressed

OR

switch 2 is pressed

OR

switch 3 is pressed

Logical disjunction: LIGHT = S1 + S2 + S3

Ladder Logic

Light bulb example

The light will turn on if:

switch 1 is pressed

OR

switch 2 is pressed

OR

switch 3 is pressed

Logical disjunction: LIGHT = S1 + S2 + S3

Ladder Logic

Engine start example

The engine will start if:

key is present

AND

brake pedal is stepped on

AND

start button is pressed

Logical conjunction: START = KEY · BRAKE · BUTTON

Ladder Logic

Engine start example

The engine will start if:

key is present

AND

brake pedal is stepped on

AND

start button is pressed

Logical conjunction: START = KEY · BRAKE · BUTTON

Ladder Logic

Engine start example

The engine will start if:

key is present

AND

brake pedal is stepped on

AND

start button is pressed

Logical conjunction: START = KEY · BRAKE · BUTTON

Ladder Logic

Engine start example

The engine will start if:

key is present

AND

brake pedal is stepped on

AND

start button is pressed

Logical conjunction: START = KEY · BRAKE · BUTTON

Ladder Logic

Engine start example

The engine will start if:

key is present

AND

brake pedal is stepped on

AND

start button is pressed

Logical conjunction: START = KEY · BRAKE · BUTTON

Ladder Logic

Engine start example

The engine will start if:

key is present

AND

brake pedal is stepped on

AND

start button is pressed

Logical conjunction: START = KEY · BRAKE · BUTTON

Ladder Logic

Engine start example

The engine will start if:

key is present

AND

brake pedal is stepped on

AND

start button is pressed

Logical conjunction: START = KEY · BRAKE · BUTTON

Logical operations in LAD

Logical operations in LAD

NOT - logical negationlogical function: Y = A

conjunction: is not

Y

A

NOT in LAD

Truth table

A A0 11 0

Logical operations in LAD

NOT - logical negationlogical function: Y = A

conjunction: is not

Y

A

NOT in LAD

Truth table

A A0 11 0

Logical operations in LAD

AND - logical conjunctionlogical function: Y = A · B

conjunction: and; at the same time

Y

A B

AND in LAD

Truth table

A B A · B0 0 00 1 01 0 01 1 1

Logical operations in LAD

AND - logical conjunctionlogical function: Y = A · B

conjunction: and; at the same time

Y

A B

AND in LAD

Truth table

A B A · B0 0 00 1 01 0 01 1 1

Logical operations in LAD

OR - logical disjunctionlogical function: Y = A + B

conjunction: or

Y

A

B

OR in LAD

Truth table

A B A + B0 0 00 1 11 0 11 1 1

Logical operations in LAD

OR - logical disjunctionlogical function: Y = A + B

conjunction: or

Y

A

B

OR in LAD

Truth table

A B A + B0 0 00 1 11 0 11 1 1

Logical operations in LAD

NAND - negated logicalconjunction

logical function: Y = A · B

conjunction: not at the same time

Y

A

B

NAND in LAD

Truth table

A B A · B0 0 10 1 11 0 11 1 0

Logical operations in LAD

NAND - negated logicalconjunction

logical function: Y = A · B

conjunction: not at the same time

Y

A

B

NAND in LAD

Truth table

A B A · B0 0 10 1 11 0 11 1 0

Logical operations in LAD

NOR - negated logicaldisjunction

logical function: Y = A + B

conjunction: neither

Y

A B

NOR in LAD

Truth table

A B A + B0 0 10 1 01 0 01 1 0

Logical operations in LAD

NOR - negated logicaldisjunction

logical function: Y = A + B

conjunction: neither

Y

A B

NOR in LAD

Truth table

A B A + B0 0 10 1 01 0 01 1 0

Logical operations in LAD

XOR - exclusive logicaldisjunction

logical function: Y = A ⊕ B

conjunction: one or the other but notboth

Y

A B

A B

XOR in LAD

Truth table

A B A ⊕ B0 0 00 1 11 0 11 1 0

Logical operations in LAD

XOR - exclusive logicaldisjunction

logical function: Y = A ⊕ B

conjunction: one or the other but notboth

Y

A B

A B

XOR in LAD

Truth table

A B A ⊕ B0 0 00 1 11 0 11 1 0

Construction of LAD from logical statements

D = (A · B) + C

C D

A B

Construction of LAD from logical statements

D = (A · B) + C

C D

A B

Construction of LAD from logical statements

D = (A · B) + C

C D

A B

Construction of LAD from logical statements

D = (A · B) + C

C D

A B

Construction of LAD from logical statements

D = (A · B) + C

C D

A B

Construction of LAD from logical statements

D = (A · B) + C

C D

A B

Construction of LAD from logical statements

D = (A · B) + C

C D

A B

Construction of LAD from logical statements

D = (A · B) + C

C D

A B

Construction of LAD from logical statements

X = (A · ∼B) + (B · ∼A)

A BX

A B

Construction of LAD from logical statements

X = (A · ∼B) + (B · ∼A)

A BX

A B

Construction of LAD from logical statements

X = (A · ∼B) + (B · ∼A)

A BX

A B

Construction of LAD from logical statements

X = (A · ∼B) + (B · ∼A)

A BX

A B

Construction of LAD from logical statements

X = (A · ∼B) + (B · ∼A)

A BX

A B

Construction of LAD from logical statements

X = (A · ∼B) + (B · ∼A)

A BX

A B

Construction of LAD from logical statements

X = (A · ∼B) + (B · ∼A)

A BX

A B

Construction of LAD from logical statements

X = (A · ∼B) + (B · ∼A)

X

A B

A B

Construction of LAD from logical statements

X = (A · ∼B) + (B · ∼A)

X

A B

A B

Construction of LAD from logical statements

X = (A · ∼B) + (B · ∼A)

X

A B

A B

Construction of logical statements from LAD

E

A

D C B

F

A = ∼D+ E +∼F( )·C ∼B·

Construction of logical statements from LAD

E

A

D C B

F

A = ∼D+ E +∼F( )·C ∼B·

Construction of logical statements from LAD

E

A

D C B

F

A = ∼D+ E +∼F( )·C ∼B·

Construction of logical statements from LAD

E

A

D C B

F

A = ∼D+ E +∼F( )·C ∼B·

Construction of logical statements from LAD

E

A

D C B

F

A = ∼D+ E +∼F( )·C ∼B·

Construction of logical statements from LAD

E

A

D C B

F

A = ∼D+ E +∼F( )·C ∼B·

Construction of logical statements from LAD

E

A

D C B

F

A = ∼D+ E +∼F( )·C ∼B·

Construction of logical statements from LAD

E

A

D C B

F

A = ∼D+ E +∼F( )·C ∼B·

Construction of logical statements from LAD

E

A

D C B

F

A = ∼D+ E +∼F( )·C ∼B·

Construction of logical statements from LAD

E

A

D C B

F

A = ∼D+ E +∼F( )·C ∼B·

Simplification of logical statements

B

Y

A B D

A

Y = D · (D · (A · B +∼B · A) +∼D · C · A)

D

A

C D

A B

Y

Y = D ·A

Simplification of logical statements

Basic axioms

Idempotency:A + A = A A · A = A

Associativity:(A + B) + C = A + (B + C) (A · B) · C = A · (B · C)

Commutative property:A + B = B + A A · B = B · A

Distributivity:A+(B·C) = (A+B)·(A+C) A · (B+C) = (A ·B)+(A ·C)

Identity:A + 0 = A A + 1 = 1 A · 0 = 0 A · 1 = A

Complement:A +∼A = 1 A · ∼A = 0

Simplification of logical statements

Y = D · (D · (A · B +∼B · A) +∼D · C · A)

Y = D · (D · (A · B +∼B · A) +∼D · C · A)Y = D · (D · A · (B +∼B) +∼D · C · A)Y = D · (D · A · (B +∼B) +∼D · C · A)

Y = D · (D · A · 1 +∼D · C · A)Y = D · (D · A +∼D · C · A)Y = D · (D · A +∼D · C · A)

Y = D · A · (D +∼D · C)Y = D · A · (D +∼D · C)

Y = D · A · D + D · A · ∼D · CY = D · A · D + D · A · ∼D · C

Y = D · A · D + 0 · A · CY = D · A · D

Y = D · A

Simplification of logical statements

Y = D · (D · (A · B +∼B · A) +∼D · C · A)Y = D · (D · (A · B +∼B · A) +∼D · C · A)

Y = D · (D · A · (B +∼B) +∼D · C · A)Y = D · (D · A · (B +∼B) +∼D · C · A)

Y = D · (D · A · 1 +∼D · C · A)Y = D · (D · A +∼D · C · A)Y = D · (D · A +∼D · C · A)

Y = D · A · (D +∼D · C)Y = D · A · (D +∼D · C)

Y = D · A · D + D · A · ∼D · CY = D · A · D + D · A · ∼D · C

Y = D · A · D + 0 · A · CY = D · A · D

Y = D · A

Simplification of logical statements

Y = D · (D · (A · B +∼B · A) +∼D · C · A)Y = D · (D · (A · B +∼B · A) +∼D · C · A)

Y = D · (D · A · (B +∼B) +∼D · C · A)

Y = D · (D · A · (B +∼B) +∼D · C · A)Y = D · (D · A · 1 +∼D · C · A)

Y = D · (D · A +∼D · C · A)Y = D · (D · A +∼D · C · A)

Y = D · A · (D +∼D · C)Y = D · A · (D +∼D · C)

Y = D · A · D + D · A · ∼D · CY = D · A · D + D · A · ∼D · C

Y = D · A · D + 0 · A · CY = D · A · D

Y = D · A

Simplification of logical statements

Y = D · (D · (A · B +∼B · A) +∼D · C · A)Y = D · (D · (A · B +∼B · A) +∼D · C · A)

Y = D · (D · A · (B +∼B) +∼D · C · A)Y = D · (D · A · (B +∼B) +∼D · C · A)

Y = D · (D · A · 1 +∼D · C · A)Y = D · (D · A +∼D · C · A)Y = D · (D · A +∼D · C · A)

Y = D · A · (D +∼D · C)Y = D · A · (D +∼D · C)

Y = D · A · D + D · A · ∼D · CY = D · A · D + D · A · ∼D · C

Y = D · A · D + 0 · A · CY = D · A · D

Y = D · A

Simplification of logical statements

Y = D · (D · (A · B +∼B · A) +∼D · C · A)Y = D · (D · (A · B +∼B · A) +∼D · C · A)

Y = D · (D · A · (B +∼B) +∼D · C · A)Y = D · (D · A · (B +∼B) +∼D · C · A)

Y = D · (D · A · 1 +∼D · C · A)

Y = D · (D · A +∼D · C · A)Y = D · (D · A +∼D · C · A)

Y = D · A · (D +∼D · C)Y = D · A · (D +∼D · C)

Y = D · A · D + D · A · ∼D · CY = D · A · D + D · A · ∼D · C

Y = D · A · D + 0 · A · CY = D · A · D

Y = D · A

Simplification of logical statements

Y = D · (D · (A · B +∼B · A) +∼D · C · A)Y = D · (D · (A · B +∼B · A) +∼D · C · A)

Y = D · (D · A · (B +∼B) +∼D · C · A)Y = D · (D · A · (B +∼B) +∼D · C · A)

Y = D · (D · A · 1 +∼D · C · A)Y = D · (D · A +∼D · C · A)

Y = D · (D · A +∼D · C · A)Y = D · A · (D +∼D · C)Y = D · A · (D +∼D · C)

Y = D · A · D + D · A · ∼D · CY = D · A · D + D · A · ∼D · C

Y = D · A · D + 0 · A · CY = D · A · D

Y = D · A

Simplification of logical statements

Y = D · (D · (A · B +∼B · A) +∼D · C · A)Y = D · (D · (A · B +∼B · A) +∼D · C · A)

Y = D · (D · A · (B +∼B) +∼D · C · A)Y = D · (D · A · (B +∼B) +∼D · C · A)

Y = D · (D · A · 1 +∼D · C · A)Y = D · (D · A +∼D · C · A)Y = D · (D · A +∼D · C · A)

Y = D · A · (D +∼D · C)Y = D · A · (D +∼D · C)

Y = D · A · D + D · A · ∼D · CY = D · A · D + D · A · ∼D · C

Y = D · A · D + 0 · A · CY = D · A · D

Y = D · A

Simplification of logical statements

Y = D · (D · (A · B +∼B · A) +∼D · C · A)Y = D · (D · (A · B +∼B · A) +∼D · C · A)

Y = D · (D · A · (B +∼B) +∼D · C · A)Y = D · (D · A · (B +∼B) +∼D · C · A)

Y = D · (D · A · 1 +∼D · C · A)Y = D · (D · A +∼D · C · A)Y = D · (D · A +∼D · C · A)

Y = D · A · (D +∼D · C)

Y = D · A · (D +∼D · C)Y = D · A · D + D · A · ∼D · CY = D · A · D + D · A · ∼D · C

Y = D · A · D + 0 · A · CY = D · A · D

Y = D · A

Simplification of logical statements

Y = D · (D · (A · B +∼B · A) +∼D · C · A)Y = D · (D · (A · B +∼B · A) +∼D · C · A)

Y = D · (D · A · (B +∼B) +∼D · C · A)Y = D · (D · A · (B +∼B) +∼D · C · A)

Y = D · (D · A · 1 +∼D · C · A)Y = D · (D · A +∼D · C · A)Y = D · (D · A +∼D · C · A)

Y = D · A · (D +∼D · C)Y = D · A · (D +∼D · C)

Y = D · A · D + D · A · ∼D · CY = D · A · D + D · A · ∼D · C

Y = D · A · D + 0 · A · CY = D · A · D

Y = D · A

Simplification of logical statements

Y = D · (D · (A · B +∼B · A) +∼D · C · A)Y = D · (D · (A · B +∼B · A) +∼D · C · A)

Y = D · (D · A · (B +∼B) +∼D · C · A)Y = D · (D · A · (B +∼B) +∼D · C · A)

Y = D · (D · A · 1 +∼D · C · A)Y = D · (D · A +∼D · C · A)Y = D · (D · A +∼D · C · A)

Y = D · A · (D +∼D · C)Y = D · A · (D +∼D · C)

Y = D · A · D + D · A · ∼D · C

Y = D · A · D + D · A · ∼D · CY = D · A · D + 0 · A · C

Y = D · A · DY = D · A

Simplification of logical statements

Y = D · (D · (A · B +∼B · A) +∼D · C · A)Y = D · (D · (A · B +∼B · A) +∼D · C · A)

Y = D · (D · A · (B +∼B) +∼D · C · A)Y = D · (D · A · (B +∼B) +∼D · C · A)

Y = D · (D · A · 1 +∼D · C · A)Y = D · (D · A +∼D · C · A)Y = D · (D · A +∼D · C · A)

Y = D · A · (D +∼D · C)Y = D · A · (D +∼D · C)

Y = D · A · D + D · A · ∼D · CY = D · A · D + D · A · ∼D · C

Y = D · A · D + 0 · A · CY = D · A · D

Y = D · A

Simplification of logical statements

Y = D · (D · (A · B +∼B · A) +∼D · C · A)Y = D · (D · (A · B +∼B · A) +∼D · C · A)

Y = D · (D · A · (B +∼B) +∼D · C · A)Y = D · (D · A · (B +∼B) +∼D · C · A)

Y = D · (D · A · 1 +∼D · C · A)Y = D · (D · A +∼D · C · A)Y = D · (D · A +∼D · C · A)

Y = D · A · (D +∼D · C)Y = D · A · (D +∼D · C)

Y = D · A · D + D · A · ∼D · CY = D · A · D + D · A · ∼D · C

Y = D · A · D + 0 · A · C

Y = D · A · DY = D · A

Simplification of logical statements

Y = D · (D · (A · B +∼B · A) +∼D · C · A)Y = D · (D · (A · B +∼B · A) +∼D · C · A)

Y = D · (D · A · (B +∼B) +∼D · C · A)Y = D · (D · A · (B +∼B) +∼D · C · A)

Y = D · (D · A · 1 +∼D · C · A)Y = D · (D · A +∼D · C · A)Y = D · (D · A +∼D · C · A)

Y = D · A · (D +∼D · C)Y = D · A · (D +∼D · C)

Y = D · A · D + D · A · ∼D · CY = D · A · D + D · A · ∼D · C

Y = D · A · D + 0 · A · CY = D · A · D

Y = D · A

Simplification of logical statements

Y = D · (D · (A · B +∼B · A) +∼D · C · A)Y = D · (D · (A · B +∼B · A) +∼D · C · A)

Y = D · (D · A · (B +∼B) +∼D · C · A)Y = D · (D · A · (B +∼B) +∼D · C · A)

Y = D · (D · A · 1 +∼D · C · A)Y = D · (D · A +∼D · C · A)Y = D · (D · A +∼D · C · A)

Y = D · A · (D +∼D · C)Y = D · A · (D +∼D · C)

Y = D · A · D + D · A · ∼D · CY = D · A · D + D · A · ∼D · C

Y = D · A · D + 0 · A · CY = D · A · D

Y = D · A

Simplification of logical statements

A = ∼B · (C · (∼D + E + C) +∼F · C)

A = ∼B · (C · (∼D + E + C) +∼F · C)A = ∼B · (C · ∼D + C · E + C · C +∼F · C)A = ∼B · (C · ∼D + C · E + C · C +∼F · C)

A = ∼B · (C · ∼D + C · E + C +∼F · C)A = ∼B · (C · ∼D + C · E + C +∼F · C)

A = ∼B · C · (∼D + E + 1 +∼F)A = ∼B · C · (∼D + E + 1 +∼F)

A = ∼B · C · 1A = ∼B · C

Simplification of logical statements

A = ∼B · (C · (∼D + E + C) +∼F · C)A = ∼B · (C · (∼D + E + C) +∼F · C)

A = ∼B · (C · ∼D + C · E + C · C +∼F · C)A = ∼B · (C · ∼D + C · E + C · C +∼F · C)

A = ∼B · (C · ∼D + C · E + C +∼F · C)A = ∼B · (C · ∼D + C · E + C +∼F · C)

A = ∼B · C · (∼D + E + 1 +∼F)A = ∼B · C · (∼D + E + 1 +∼F)

A = ∼B · C · 1A = ∼B · C

Simplification of logical statements

A = ∼B · (C · (∼D + E + C) +∼F · C)A = ∼B · (C · (∼D + E + C) +∼F · C)

A = ∼B · (C · ∼D + C · E + C · C +∼F · C)

A = ∼B · (C · ∼D + C · E + C · C +∼F · C)A = ∼B · (C · ∼D + C · E + C +∼F · C)A = ∼B · (C · ∼D + C · E + C +∼F · C)

A = ∼B · C · (∼D + E + 1 +∼F)A = ∼B · C · (∼D + E + 1 +∼F)

A = ∼B · C · 1A = ∼B · C

Simplification of logical statements

A = ∼B · (C · (∼D + E + C) +∼F · C)A = ∼B · (C · (∼D + E + C) +∼F · C)

A = ∼B · (C · ∼D + C · E + C · C +∼F · C)A = ∼B · (C · ∼D + C · E + C · C +∼F · C)

A = ∼B · (C · ∼D + C · E + C +∼F · C)A = ∼B · (C · ∼D + C · E + C +∼F · C)

A = ∼B · C · (∼D + E + 1 +∼F)A = ∼B · C · (∼D + E + 1 +∼F)

A = ∼B · C · 1A = ∼B · C

Simplification of logical statements

A = ∼B · (C · (∼D + E + C) +∼F · C)A = ∼B · (C · (∼D + E + C) +∼F · C)

A = ∼B · (C · ∼D + C · E + C · C +∼F · C)A = ∼B · (C · ∼D + C · E + C · C +∼F · C)

A = ∼B · (C · ∼D + C · E + C +∼F · C)

A = ∼B · (C · ∼D + C · E + C +∼F · C)A = ∼B · C · (∼D + E + 1 +∼F)A = ∼B · C · (∼D + E + 1 +∼F)

A = ∼B · C · 1A = ∼B · C

Simplification of logical statements

A = ∼B · (C · (∼D + E + C) +∼F · C)A = ∼B · (C · (∼D + E + C) +∼F · C)

A = ∼B · (C · ∼D + C · E + C · C +∼F · C)A = ∼B · (C · ∼D + C · E + C · C +∼F · C)

A = ∼B · (C · ∼D + C · E + C +∼F · C)A = ∼B · (C · ∼D + C · E + C +∼F · C)

A = ∼B · C · (∼D + E + 1 +∼F)A = ∼B · C · (∼D + E + 1 +∼F)

A = ∼B · C · 1A = ∼B · C

Simplification of logical statements

A = ∼B · (C · (∼D + E + C) +∼F · C)A = ∼B · (C · (∼D + E + C) +∼F · C)

A = ∼B · (C · ∼D + C · E + C · C +∼F · C)A = ∼B · (C · ∼D + C · E + C · C +∼F · C)

A = ∼B · (C · ∼D + C · E + C +∼F · C)A = ∼B · (C · ∼D + C · E + C +∼F · C)

A = ∼B · C · (∼D + E + 1 +∼F)

A = ∼B · C · (∼D + E + 1 +∼F)A = ∼B · C · 1

A = ∼B · C

Simplification of logical statements

A = ∼B · (C · (∼D + E + C) +∼F · C)A = ∼B · (C · (∼D + E + C) +∼F · C)

A = ∼B · (C · ∼D + C · E + C · C +∼F · C)A = ∼B · (C · ∼D + C · E + C · C +∼F · C)

A = ∼B · (C · ∼D + C · E + C +∼F · C)A = ∼B · (C · ∼D + C · E + C +∼F · C)

A = ∼B · C · (∼D + E + 1 +∼F)A = ∼B · C · (∼D + E + 1 +∼F)

A = ∼B · C · 1A = ∼B · C

Simplification of logical statements

A = ∼B · (C · (∼D + E + C) +∼F · C)A = ∼B · (C · (∼D + E + C) +∼F · C)

A = ∼B · (C · ∼D + C · E + C · C +∼F · C)A = ∼B · (C · ∼D + C · E + C · C +∼F · C)

A = ∼B · (C · ∼D + C · E + C +∼F · C)A = ∼B · (C · ∼D + C · E + C +∼F · C)

A = ∼B · C · (∼D + E + 1 +∼F)A = ∼B · C · (∼D + E + 1 +∼F)

A = ∼B · C · 1

A = ∼B · C

Simplification of logical statements

A = ∼B · (C · (∼D + E + C) +∼F · C)A = ∼B · (C · (∼D + E + C) +∼F · C)

A = ∼B · (C · ∼D + C · E + C · C +∼F · C)A = ∼B · (C · ∼D + C · E + C · C +∼F · C)

A = ∼B · (C · ∼D + C · E + C +∼F · C)A = ∼B · (C · ∼D + C · E + C +∼F · C)

A = ∼B · C · (∼D + E + 1 +∼F)A = ∼B · C · (∼D + E + 1 +∼F)

A = ∼B · C · 1A = ∼B · C