Introduction to Industrial Automation

Figure 4.4  Basic postulates and theorems of Boolean algebra.

---

Logical Design of Automation Circuits 

◾

 

121

In such an empirical design procedure, multiple automation drawings are created by erasing or 

adding additional open or closed contacts or even whole automation branches, until they conclude 

at the correct automation circuit. In this approach, it is very easy to make design mistakes, which 

is the case in an automation circuit that contains unnecessary logical elements, like the ones pre-

sented in Figure 4.4. For this reason, knowledge of Boolean postulates and theorems will enable 

the easy spotting of mistakes even in an empirical design procedure.

Finally, the De Morgan Theorem provides the equivalent complement of a complex logi-

cal function by replacing each of the containing variables with its complement, and each of the 

operations with the corresponding dual one. The most simplified applications of the De Morgan 

Theorem are as follows:

 

A B A B A B A B

+ = ⋅

⋅ = +

In general, when the logical function has a complex form, simplification using the algebraic form 

is quite difficult and time consuming, while in many cases it is not obvious if this function can 

be further simplified. For this reason, various methods of simplification have been developed, the 

Karnaugh method being the most popular. In this book, none of these methods will be examined 

further, since in automation circuits the existence of additional and unnecessary contacts or sec-

ondary relays do not dramatically increase the implementation cost, which is happening in digital 

circuits. However, if for any reason it is absolutely necessary that an automation circuit with mini-

mum realization be implemented, the designer should apply methodologies like the Karnaugh 

method, which will not be analyzed in the contents of this book.