Adders & Circuits (Cambridge International A-Level Computer Science)

Half adders

What is a half adder circuit?

• A half adder circuit is a basic digital circuit used in computation to perform the **addition **of two single bit numbers
• Has** two inputs**, usually labelled as **A **and B
• Produces** two outputs** labelled Carry out (C_out) and Sum(s)

• Remember that you are adding together the binary numbers represented by A and B

• Create the C_outcolumn first then for each row you can just add A and B together and write the answer in 2 bits in the C_out and S columns

  • For example in row 2:

    • A is 0 and B is 1 and 0+1=1
    • 1 = 01 in 2 bits (C_out 0 and Sum 1)

  • In the last row:

    • A is 1 and B is 1 and 1+1 = 2
    • 2 = 10 in 2 bit binary (C_out 1 and Sum 0)

Drawing a half adder circuit

• A half adder circuit has two inputs, typically labelled as A and B, and two outputs: the Sum(S) and **Carry **(C_out)

• This circuit can be created using an** XOR gate for the Sum output** and an AND gate for the Carry output

• Label Inputs:

  • Begin by drawing two lines on the left side of your paper or drawing space
  • Label the top line as 'A' and the bottom line as 'B'
  • These represent your inputs

• XOR Gate (Sum):

  • Draw an XOR gate (often a shape like a curved 'D' or a shape similar to an OR gate but with an additional curved line on the input side) in the middle of the paper or drawing space
  • Connect the A and B lines to the two inputs of the XOR gate
  • The output from the XOR gate is the 'Sum'
  • Draw a line from the output of the XOR gate to the right side of your paper and label it as 'S'

• AND Gate (Carry):

  • Draw an AND gate (typically a D-shaped symbol) above the XOR gate
  • Again, connect the A and B lines to the two inputs of the AND gate
  • The output from the AND gate is the 'Carry'
  • Draw a line from the output of the AND gate to the right side of your paper and label it as 'C_out'

Half Adder Logic Gates

Full adders

What is a full adder circuit?

• A full adder circuit extends the half adder to handle the addition of three bits
• Has three inputs: A, B, and an input carry (C_in)
• Produces two outputs: carry (C_out) and sum (S)

• To easily reproduce this Truth Table, remember:

  • The full adder adds up three binary inputs A,B and C
  • So the answer can be 0,1,2 or 3
  • For each row, add up A, B and C and the write the answer as a 2 bit binary number in the last 2 columns (C_out and Sum)
  • For example in row 4, A=0, B=1 and C=1 -0+1+1=2 which is 10 in binary, so C_outis 1 and Sum is 0
  • In the last row, A=1, B=1 and C=1, 1+1+1=3 which is 11 in binary so C_outis 1 and Sum is 1

Operation

• The "Sum" output provides the XOR of the inputs A, B, and C_in
• The "Carry" output is TRUE if at least two of the inputs A, B, and C_in are TRUE

Drawing a full adder circuit

• A full adder circuit consists of three inputs: A, B, and Carry (C_in), and two outputs: Sum (S) and Carry (C_out)

• It can be designed using two half adders and an OR gate

• Label Inputs:

  • Start by drawing three lines on the left side of your paper or drawing space
  • Label the top line as 'A', the middle line as 'B', and the bottom line as 'C_in'
  • These represent your inputs

• First Half Adder:

  • Draw a half adder with A and B as inputs
  • This consists of an XOR gate (for the Sum) and an AND gate (for the Carry)
  • Label the output of the XOR gate as 'Sum1' and the output of the AND gate as 'Carry1'

• Second Half Adder:

  • Draw a second half adder underneath the first, using Sum1 and C_in as inputs
  • Again, it consists of an XOR gate (for the Sum) and an AND gate (for the Carry)
  • Label the output of the XOR gate as 'S' (final Sum) and the output of the AND gate as 'Carry2'

• OR Gate:

  • Draw an OR gate to the right of the half adders
  • Connect Carry1 and Carry2 to the inputs of the OR gate
  • The output of the OR gate is the final Carry (C_out)

Full Adder Logic Gates