Combinational Circuits – The Half Adder

COMBINATIONAL CIRCUIT

• A combinational circuit is a connected arrangement of logic gates with the set of inputs and outputs.
• A combinational circuit can be described by a truth table showing the binary relationship between n input variables and m output variables.
• Two examples: half adder, full adder
• These circuits serve as basic building blocks for the construction of more complicated arithmetic circuits.

BINARY ADDITION

• Consider: 0 1 0 1 1 1 0 + 0 0 1 1 1 0 0 0 1 1 1 1 0 0 carry

_______________________

1 0 0 1 0 1 0

• We need 1-bit adders…

HALF-ADDER

• The most basic arithmetic circuit is the addition of two binary digits.
• A combinational circuit that performs the arithmetic addition of two bits is called a half-adder.
• It consists of two inputs and two outputs:
• Two inputs get summed
• Two outputs: sum and carry bits + 1 is binary 10, which has two digits.

A half-adder

Input X Sum S

Half adder Input Y Carry out Co

X Y S Co The plus-circle 0 0 0 0 (modulo addition 0 1 1 0 of 1-bit values) 1 0 1 0 means XOR, 1 1 0 1 the dot means AND

S = X ⊕ Y Co = X •Y

Half-adder

The XOR gate calculates the sum

The AND gate calculates the carry

FULL-ADDER

• A full-adder is a combination circuit that forms the arithmetic sum of three input bits.
• It consists of three inputs and two outputs.
• Two of the inputs represent the two bits to be added.
• The third input represents the carry from the previous lower significant position.
• One of the outputs will be the carry into the next higher significant position, i.e. into the next full adder.
• Of course two output bits are necessary because the arithmetic sum of three binary digits ranges from 0 to 3, and 2 and 3 need two bits.

A full adder

Input X Sum S

Input Y Full adder Carry in Ci Carry out Co

Ci X Y S Co

a 0 0 0 0 0
b 0 0 1 1 S = X ⊕ Y 0 Co = X • Y
c 0 1 0 1 0
d 0 1 1 0 1
e 1 0 0 1 0
f 1 0 1 0 S = X ⊕ Y 1 Co = X + Y
g 1 1 0 0 1
h 1 1 1 1 1

A full adder

• We could make this from two half-adders:

Counts how many buttons are on in binary!

0 + 1 + 1 = 10

Time to Imagine

• A full-adder is a combination circuit that forms the arithmetic sum of three input bits.
• Suppose we added a second full adder (A2) to the output of the first adder (A1), and connected the Ci pin of A2 to Co of A1?
• Wouldn't we be able to add two 2-bit numbers?
• A 3rd adder?
• A 4th....

Multi bit addition – combine 8 adders with carry to get 8-bit addition (of two 8-bit

numbers).

Sn S2 S1 S0

C0 C0

Carry

full full full ½ adder adder adder adder

Y X Ci Ci Ci

n n Y2 X2 Y1 X1 Y0 X0

In the Lab...

• Play with Gates (in a “sim”) and construct truth tables.
• Design a half adder
• I made some video tutorials that might also help:
• Logisim introduction: https://youtu.be/fwPw28O5ORI
• Half-adder tutorial: https://youtu.be/BpvlMgb40U4
• Full-adder tutorial: https://youtu.be/1qLAk0AsH40
• Computer Systems:
• We’re building a computer up conceptually!
• Get your Pi and hardware
• Check Canvas regularly
• Information and bits
• Digital Circuit Design:
• Gates, Half-Adders and Full-Adders
• Next week: more complex circuits and Flip Flops (memory)!