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# plc_bool - 89 6 BOOLEAN LOGIC DESIGN Topics Boolean algebra...

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89 6. BOOLEAN LOGIC DESIGN 6.1 INTRODUCTION The process of converting control objectives into a ladder logic program requires structured thought. Boolean algebra provides the tools needed to analyze and design these systems. 6.2 BOOLEAN ALGEBRA Boolean algebra was developed in the 1800’s by James Bool, an Irish mathematician. It was found to be extremely useful for designing digital circuits, and it is still heavily used by electrical engi- neers and computer scientists. The techniques can model a logical system with a single equation. The equation can then be simplified and/or manipulated into new forms. The same techniques developed for circuit designers adapt very well to ladder logic programming. Boolean equations consist of variables and operations and look very similar to normal algebraic equations. The three basic operators are AND, OR and NOT; more complex operators include exclusive or (EOR), not and (NAND), not or (NOR). Small truth tables for these functions are shown in Figure 62. Each operator is shown in a simple equation with the variables A and B being used to calculate a value for X. Truth tables are a simple (but bulky) method for showing all of the possible combinations that will turn an output on or off. Topics: Objectives: • Be able to simplify designs with Boolean algebra • Boolean algebra • Converting between Boolean algebra and logic gates and ladder logic • Logic examples

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90 Figure 62 Boolean Operations with Truth Tables and Gates In a Boolean equation the operators will be put in a more complex form as shown in Figure 63. The variable for these equations can only have a value of 0 for false, or 1 for true. The solution of the equation follows rules similar to normal algebra. Parts of the equation inside parenthesis are to be solved first. Operations are to be done in the sequence NOT, AND, OR. In the example the NOT func- tion for C is done first, but the NOT over the first set of parentheses must wait until a single value is available. When there is a choice the AND operations are done before the OR operations. For the given AND A 0 0 1 1 B 0 1 0 1 X 0 0 0 1 X A B = OR A 0 0 1 1 B 0 1 0 1 X 0 1 1 1 X A B + = NOT A 0 1 X 1 0 X A = EOR A 0 0 1 1 B 0 1 0 1 X 0 1 1 0 X A B = NAND A 0 0 1 1 B 0 1 0 1 X 1 1 1 0 X A B = NOR A 0 0 1 1 B 0 1 0 1 X 1 0 0 0 X A B + = Note: The symbols used in these equations, such as + for OR are not universal stan- dards and some authors will use different notations. A B A B A B A B A B A X X X X X X Note: The EOR function is available in gate form, but it is more often converted to its equivalent, as shown below. X A B A B A B + = = Note: By convention a false state is also called off or 0 (zero). A true state is also called on or 1.
91 set of variable values the result of the calculation is false. Figure 63 A Boolean Equation The equations can be manipulated using the basic axioms of Boolean shown in Figure 64. A few of the axioms (associative, distributive, commutative) behave like normal algebra, but the other axioms have subtle differences that must not be ignored.

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plc_bool - 89 6 BOOLEAN LOGIC DESIGN Topics Boolean algebra...

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