4.4 Combinational Logic Circuits

Digital circuits are created using gates as building blocks. Gates can be combined in different ways to produce different results. Since the output of the gate is defined by a logical function, the effect of gate combinations can be determined using truth tables. Gate circuits that are simple combinations are called combinational logic circuits. In combinational logic circuits the output of the circuit is determined by the input(s) to the circuit at any given moment. The following examples show the output states for various input conditions.

Example I: Determine the output of the following digital circuit when the inputs x = +5V (logic state 1) and y = 0V (logic state 0).

Solution:  

In this case the output is going to be determined by both the AND and the OR gate. When the inputs to the AND gate are 1 and 0, the output from the AND gate will be 0. The circuit output is determined by the OR, which will have input of 0 from the AND gate and 0 from the y input. If the inputs to the OR gates are both 0's, then the output will be 0 for the circuit. Note: electrically, the above circuit is the same as the following circuit.

This reasoning is shown graphically on the circuit diagram as:

The above logic conditions will be valid only if the input logic states remain constant. If the input logic state changes, then the output can change.

Example II:       For the following circuit create a truth table that shows all possible outputs for the various input combinations.

Solution:           Since there are two inputs to this circuit, there are a total of 4 possible input combinations. The result for each specific combination is arrived at by following the analysis described in the previous example. The process is repeated for all 4 possible input combinations to produce the truth table. To facilitate the analysis each of the gates is labeled and the labels in the truth table indicates the output of the gate.

In most situations, we are interested in knowing the performance of a circuit under all the various input conditions versus a single input condition. The results represented in the truth table gives the circuit designer a quick idea of the performance of the circuit under all the different conditions. In a design situation, the input - output relationship of the circuit is already known, and a circuit capable of performing that desired function needs to be created. This process is a little bit more challenging than determining the output of a known circuit. At the present designing a circuit needs creative thinking to arrive at the desired circuit. Later in chapter 6 a systematic method will be explored. The following example shows how a circuit is designed to perform a given function.

Example III:     

Design a digital circuit that can be used in a car to perform the following function. A beep signal is to be sounded in a car if the door is not closed properly, or if the keys are left in the ignition, or if the lights are left on. These requirements could be used to create a truth table, and a logical process determined. The first step is to represent the requirement in symbolic form, with each having one of two possibilities. Yes to a requirement will be labeled 1, and no to a requirement will be labeled 0.

Solution:       

Let each logic condition be represented using a variable as shown  below.

Door open (x)                   Keys left behind (y)         

Lights on (z)                      Beep (result r)

From the truth table it can be seen that the needed result can be achieved by using two OR gates. In this case all arrangements (x+y)+z, (x+z)+y and x+(y+z) will produce the desired result. The desired result can also be achieved using other gates and other arrangements.

Now that the logical operation has been determined using the truth table, a circuit could be designed and built to perform the function. The circuit will be as shown above. Imagine that x, y, and z are input from sensors that detect the state of each action, and the result is connected to an alarm. The alarm would sound as required, and the circuit is able to perform the task of making decisions based on the input signals. The validity of the final circuit could be verified by creating a truth table for it, and then comparing the truth table of the circuit with the truth table of the specifications.            

2.   Determine the output state for the circuits in question 1 when the input state for x and y are 1 and 1.

3.     Create truth tables for the following dual input circuits.

4.     Create truth tables for the following triple input circuits

5.      Design circuits using AND, OR and NOT gates that will produce the following truth table results.