Adders and Combinational Circuits: Half Adder and Full Adder

Hello students, welcome to Kwickprep. Your calculator adds numbers in a blink, but inside it there are only logic gates. So how do gates add one plus one? Today we build a half adder and a full adder, with truth tables and circuits. Then we meet encoders, decoders and multiplexers.

First, a new term. A combinational circuit is one whose output depends only on the inputs at this moment. It has no memory, so it does not care what the inputs were a second ago. It is built only from logic gates joined together. Adders, encoders, decoders and multiplexers are all combinational circuits.

Let us add two single bits, the way you add in class. Zero plus zero is zero, with no carry. Zero plus one is one, with no carry. One plus zero is also one, with no carry. One plus one is two, which is one zero in binary, so we write zero and carry one. So adding two bits always gives two outputs: a sum bit and a carry bit.

A half adder is the circuit that adds two bits, A and B. Look at the sum column: zero, one, one, zero. That is one only when the inputs are different, which is XOR. Now look at the carry column: zero, zero, zero, one. That is one only when both inputs are one, which is AND.

So the half adder circuit needs just two gates, with A and B going into both. The sum S equals A XOR B, from one XOR gate. The carry C equals A dot B, from one AND gate. It is called half because it cannot accept a carry coming in from the previous column.

To see the problem, add two numbers with more than one bit. Take binary one one plus zero one. In the right column, one plus one gives zero, carry one. In the left column, we must add one plus zero plus that carry, which is three bits. A half adder has only two inputs, so we need a full adder, which takes A, B and a carry in.

Let us build the full adder truth table row by row, counting the ones in each row. Row one has no ones, so sum zero, carry out zero. Row two has one one, so sum one, carry zero. Row three also has one one, so sum one, carry zero. Row four has two ones, which is binary one zero, so sum zero, carry one.

Now the last four rows. Row five has one one, so sum one, carry zero. Row six has two ones, so sum zero, carry one. Row seven has two ones, so sum zero, carry one. Row eight has three ones, which is binary one one, so sum one and carry one.

From the table we get two expressions. The sum S is A XOR B XOR carry in, which is one when the number of ones is odd. The carry out is A dot B, plus carry in dot, A XOR B, which is one when at least two inputs are one. Another correct form is A dot B plus B dot carry in plus A dot carry in.

Here is how a full adder is built from two half adders. First, half adder one adds A and B, giving sum S one and carry C one. Next, half adder two adds S one and the carry in, giving the final sum S two and a carry C two. Then an OR gate joins C one and C two to give the carry out. We use OR, because both carries can never be one together.

Pause the video and work this out. A full adder gets A equal to one, B equal to zero, and carry in equal to one. What are the sum and the carry out? There are two ones, which is binary one zero, so the sum is zero and the carry out is one.

Now three more combinational circuits. An encoder turns one active input line into a binary code. This is a four to two encoder, with inputs D zero to D three and two outputs. When D one is active, the output is zero one. When D two is active, it is one zero, and when D three is active, it is one one. A keyboard uses this idea, turning the key you press into a code. An eight to three encoder, called octal to binary, works the same way.

A decoder does the reverse: a binary code switches on exactly one of its outputs. This is a two to four decoder. Inputs zero zero switch on D zero, which is not A dot not B. Inputs zero one switch on D one. Inputs one zero switch on D two. Inputs one one switch on D three, which is A dot B. Notice each output is simply one minterm. Computers use decoders to pick one memory location from its address.

A multiplexer, or MUX, picks one of many data inputs and sends it to a single output. The choice is made by select lines, like pressing a channel number on a TV remote. A four to one MUX has data inputs D zero to D three and two select lines. Select zero zero sends D zero to the output. Select zero one sends D one. Select one zero sends D two, and one one sends D three. A two to one MUX has the expression Y equals not S dot D zero plus S dot D one.

Let us compare them, since exams ask for the difference. A four to two encoder has four input lines and gives a two bit code. A two to four decoder takes a two bit code and switches on one of four lines. A four to one multiplexer has four data inputs plus two select lines, and just one output. For bigger circuits, every extra code bit doubles the number of lines.

Let us revise what we learned today. A half adder gives sum A XOR B and carry A dot B. A full adder also adds a carry in, so it can add multi-bit numbers column by column. A full adder is two half adders plus one OR gate. An encoder turns an active line into a code, and a decoder turns a code into one active line. A multiplexer uses select lines to choose one input for its single output.