Describing Logic Circuits Algebraically
Any logic circuit, no matter how complex, may be completely described using the Boolean operations, because the OR gate, AND gate, and NOT circuit are the basic building blocks of digital systems.
This is an example of the circuit using Boolean expression:
If an expression contains both AND and OR operations, the AND operations are performed first (X=AB+C: AB is performed first), unless there are parentheses in the expression, in which case the operation inside the parentheses is to be performed first (X= (A+B) +C: A+B is performed first).
Circuits containing Inverters
Whenever an INVERTER is present in a logic-circuit diagram, its output expression is simply equal to the input expression with a prime (') over it.
Evaluating Logic Circuit Outputs
Once the Boolean expression for a circuit output has been obtained, the output logic level can be determined for any set of input levels.
These are two examples of the evaluating logic circuit output:
Let A=0, B=1, C=1, D=1
= A'BC (A+D)'
= 0'*1*1* (0+1)'
= 1 *1*1* (1)'
= 1 *1*1* 0
Let A=0, B=0, C=1, D=1, E=1
= [D+ ((A+B)C)'] * E
= [1 + ((0+0)1 )'] * 1
= [1 + (0*1)'] * 1
= [1+ 0'] *1
= [1+ 1 ] * 1
In general, the following rules must always be followed when evaluating a Boolean expression:
1. First, perform all inversions of single terms; that is, 0 = 1 or 1 = 0.
2. Then perform all operations within parentheses.
3. Perform an AND operation before an OR operation unless parentheses indicate otherwise.
4. If an expression has a bar over it, perform the operations of the expression first and then invert the result.
Determining Output Level from a Diagram
The output logic level for given input levels can also be determined directly from the circuit diagram without using the Boolean expression.
Implementing Circuits from Boolean Expression
If the operation of a circuit is defined by a Boolean expression, a logic-circuit diagram can he implemented directly from that expression.
Suppose that we wanted to construct a circuit whose output is y = AC+BC' + A'BC. This Boolean expression contains three terms (AC, BC', A'BC), which are ORed together. This tells us that a three-input OR gate is required with inputs that are equal to AC, BC', and A'BC, respectively.
Each OR-gate input is an AND product term, which means that an AND gate with appropriate inputs can be used to generate each of these terms. Note the use of INVERTERs to produce the A' and C' terms required in the expression.