Combinational Circuits & Functions: Construction & Conversion

Instructor: Martin Gibbs

Martin has 16 years experience in Human Resources Information Systems and has a PhD in Information Technology Management. He is an adjunct professor of computer science and computer programming.

In this lesson, you will learn the basics of combinational circuits, and the logic behind them. You will learn how to represent them with various methods, and to convert between presentation methods.

Combinational Logic Circuits

It's become a tired cliche by now, but everything in computers is really 1s and 0s, True and False, ON and OFF. Even the most complex circuits are constructed of switches that are either on or off.

Given this fact, we can build combinational functions by using Boolean logic.

In electronics, a circuit is a set of inputs and outputs. Think of a basic electrical circuit in your residence: wires are connected from the main power supply to switches, lights, and outlets. Lights should only turn on when the corresponding switch is flipped to ON (TRUE). The lights don't have to remember anything; all they care about is if the switch was flipped.

A combinational circuit, and combinational logic is based on Boolean logic, but it has no memory and acts only on the current input. That is: if switch A is ON, light B is ON.

Figure 1 shows a high-level example of the flow of a combinational circuit.


Figure 1: Combinational Logic
Combinational logic


Creating Combinational Logic and Functions

Since everything is 1s and 0s, ON/OFF or TRUE/FALSE, we can then use Boolean logic methods to design our combinational circuits and their functions. There are several representations we can use, the three main methods being the logic diagrams, Boolean expressions, and truth tables.

Figure 2 shows a sample circuit with three inputs (A, B, C) and a single output (Q). The connections and wiring are also displayed. Don't worry too much about the shapes of each figure in the circuit; the lesson dives into the nitty gritty later. Right now we need to focus on the representations and how each can be created from the other.


Figure 2: Combinational Circuit Example
Combinational circuit


You can only build a combinational circuit from NAND and NOR gates. NAND means NOT-AND; the top-most gate is a NAND gate. The bottom symbol is a NOR gate mean a NOT-OR. Don't worry, next we are going to break down each gate with its truth tables.

NAND Gate

Let's look at the outputs of a NAND gate. This means A NOT-AND B, or A.B (A dot B). Graphically this is represented by a line over the A.B in the diagram.

Logic Gate Circuit Diagram

The NAND gate is a NOT-AND. Figure 3 showcases the NAND gate's logic diagram


Figure 3: NAND Gate
NAND gate


It sounds a bit backwards, but remember it is a NAND gate. Therefore, if any of the inputs are OFF, then the output of the gate is ON.

Truth Table

Knowing that any OFF input will result in an ON state of A.B, we can build a truth table.

A B A.B
0 0 1
0 1 1
1 0 1
1 1 0

Having covered the NAND gate, let's take a look at the NOR gate.

The NOR Gate

The NOR, or NOT-OR gate means that if ANY of the inputs are ON, then the gate is OFF. In other words, if both are off, then the output is ON. This is indicated by an A + B notation, with the line above (as in the NAND gate). Let's look at the diagram first.

Logic Diagram

Figure 3 shows the logic diagram of the NOR gate.


Figure 3: NOR Logic Gate
NOR Logic Gate


Truth Table

The truth table for the NOR gate is as follows:

A B A+B
0 0 1
0 1 0
1 0 0
1 1 0

Converting From One to the Other

Let's put it all together and look at our full circuit as shown in Figure 4. We still have the NAND gate and the NOR gate, but now we have the input C.


Figure 4: Combinational Logic Circuit
Combinational Logic Circuit


But, based on the diagram, we know that we have a NAND and a NOR. Look at input C: there is no gate, so if C is ON, and none of the other switches are on, then Q is ON.

Given the diagram, we can build a Boolean Expression as seen in Figure 5. Each dot (.) means an AND condition. It's a NAND gate AND a NOR gate AND gate C.


Figure 5: Combinational Boolean Expression
Combinational Boolean Expression


The symbols A.B and A+B (with the lines above), AND C tell us the valid combinations that we can have. We have three inputs, but the only way that Q is going to be on is when C is ON and the other inputs are OFF.

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