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Figure 4.9: True table to 2-input car horn circuit
Figure 4.10a: NOT gate
Figure 4.10b: NOT gate
Figure 4.11: A normally closed relay contact as an inverter
Figure 4.12: NOT function with AND and OR functions
4.2.3The Not Function
Logically, the NOT function causes the output state to be the opposite state of the input. Because of this, it is often called the inverter. The NOT output is False (“0”) if the input is True (“1”) and, inversely, the output becomes True (“1”) if the input is False (“0”). The NOT function has only one input. It is graphically shown in Figure 4.10a and it has two states in its true table, as in Figure 4.10b.
In relay logic applications, a normally closed switch or a normally closed contact has the function of NOT gate (Figure 4.11). The normally closed push button in the first rung serves as an inverter function,
The NOT function of A is called “NOT A” and is represented as an A with a bar on top,
4.3 Boolean Algebra
4.3.1Three Logic Functions
Boolean algebra is the fundamental mathematical expression of logic circuits. The algebraic symbols indicate the logical relationships between groups of inputs and outputs. Often they are called Boolean expressions. Series and parallel combinations are built with AND, OR, and NOT logical operators. Use a dot (•) between inputs to represent the AND operator, a (+) between inputs to represent the OR operator, and a bar over the letter (
Table 4.4: Logic operations using Boolean algebra
Boolean algebra can be used to express any complex logic relations by using a combination of the three basic logic functions. Two examples follow:
4.3.2Order of Boolean Algebra Operations
The order of Boolean algebra operations follows two rules:
1.Grouping signs have the highest precedence. When grouping items, the order is parentheses ( ) first, brackets [ ] second, and braces { } third.
2.Besides grouping signs, the order of priority is the NOT function first, the AND function second, and the OR function last.
Example 4.1: Priority of operations
The priority of operations in this Boolean equation
Y = (A + B)C + (B + C)D
means that equations are to be done in the demonstrated order.
| 1st precedence (parentheses): | (A + B) and (B + C) |
| 2nd precedence (AND function): | (A + B)C and (B + C)D |
| 3rd precedence (OR function): | (A + B)C + (B + C)D |
4.4 Converting Logic Gates to Boolean Equations
Many logic control circuits are given in the logic gate form. It is often desirable to convert logic gate circuits to their corresponding Boolean equations. The converting procedure follows the associated level from left to right. The output of the logic gate in the lower (left) level becomes one of input of the next (right) level. Two examples are given in this section to convert the logic circuits to their Boolean equations.
Example 4.2: Using Boolean equation to express a circuit
The logic circuit shown in Figure 4.13 consists of one OR and one AND function.
The Boolean equation can be derived as follows. From the OR function we can find,
| X = A + B | (1) |
From the AND function we have,
| Y = XC | (2) |
Substitute (1) into (2) to obtain,
Y = (A + B)C
Example 4.3: Convert a gate circuit to its Boolean equation
The gate logic circuit in Figure 4.14 has two AND functions, one NOT function, and one OR function.
The Boolean equations for the three basic logic functions can be derived as below:
| (1) | |
| (2) | |
| (3) |
Combine (1), (2), and (3) to yield the Boolean equation as:
Figure 4.13: Sample logic circuit
Figure 4.14: Gate logic circuit
4.5 Input Elements
Boolean equations are often converted to ladder diagrams in PLC applications. Understanding the connecting relationship between input elements for the three basic logic functions (AND, OR, and NOT) in the ladder diagrams is the key to constructing ladder diagrams from Boolean equations.
4.5.1The NOT Function
The NOT function of an element is translated into either a normally closed (NC) contact or switch. Inversely,
