# Boolean Algebra for Ladder Logic

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Boolean Algebra

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Boolean Algebra

after mathematician George Boole

A means of reducing the number of logic

input needed to realize the desired output.

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Drill Motor Boolean Equations

CR1

CR1

CR3

CR1

CR3

CR2

CR3

CR3

CR3

CR2

CR1=Stop (Start + CR1)

CR2=CR1*LS1*LS2

CR3=LS2 * (CR2+CR3)

Extend = CR3

Retract = CR3

Motor = CR3

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Boolean Algebra

X = A + B

AND operation is BooleanMULTIPLICATION

X = AB

These operations follow some of the samelaws as standard algebra.

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Associative Laws

Associate: group together

X = A + (B + C) = (A + B) + C

Multiplication

X = A(BC) = (AB)C

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Distributive Laws

Distribute: divide into groups

X = A(B + C) = AB + AC

X = (A + B) (C + D) = AC + AD + BC + BD changes logic from

OR then AND to AND then OR

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Boolean Theorems

Boolean Rules

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Operations with 1 and 0

A0 = 0 (A and Zero is Zero)

any 0 input gives 0 output with AND

A1 = A (A and One is A) A + 0 = A

A + 1 = 1

A 1 X

0 1 0

1 1 1

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Idempotent Laws

Idem = same

AA = A

0 and 0 = 0

1 and 1 = 1

A + A = A

0 or 0 = 0 1 or 1 = 1

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Define Compliment

Compliment means inverse.

Put A into a NOT logic and the result is A-

compliment

Sometimes written: A

rather than A with a bar over it

apostrophe is easier to type.

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Involution Law

A double negative is a positive.

Compliment the input twice and its back to

the original.

(A)=A

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Laws of Complimentarity

Combine input and inputs compliment.

AA = 0.

Either the number or its compliment is alwayszero.

A + A = 1.

One term will always be zero. The other term will always be one.

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Simplification Theorems

A + AB = A + B

A + AB = A + B

Cant derive from

previous theorems, but

can demonstrate the

theorems validity with

a truth table.

Last two columns are

the same. Thus thesolution is the same for

all possible A, B.

A B A A

B

A+

A

B

A+

B

0 0 1 0 0 00 1 1 1 1 1

1 0 0 0 1 1

1 1 0 0 1 1

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More Simplification Theorems

Can derive these from previous theorems.

AB + AB = A

(A + B) (A + B) = A A + AB = A

A(A+B) = A

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Proof of A + AB = A

A B AB A+AB

0 0 0 0

0 1 0 0

1 0 0 1

1 1 1 1

A+AB = A(1+B)

distributive

A(1+B) = A(1)

operations with 1, 0.

A(1) = A

operations with 1, 0.

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Example

Convert the given ladder diagram into a

Boolean equations which contains a control

relay, a solenoid, two push buttons and three

limit switches.

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Solution

2111

3))21(1(1

LSLSCRSOL

LSPBPBCRCR

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Example

Two push buttons, PB1 and PB2 are used to

control four lamps LAMP1, Lamp2 and LAMP

3 using the set of equations below:

2)14(4

213

212

211

PBPBLAMPLAMP

LAMPLAMPLAMP

PBPBLAMP

PBPBLAMP

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