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8/11/2019 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
OR operation is Boolean ADDITION
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
Addition
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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Solution