Arithmetic Circuits

1
Arithmetic Circuits
2
Interest

• Simplified CPU, showing only the arithmetic-logic
  unit (ALU) and how it connects with the registers
  (collectively called the register file).

3
Arithmetic circuits in the computer

• Adders Subtracters Add subtract in binary.
• CPU does this in arithmetic logic unit (ALU)
• ALU
• Add subtract
• Multiply divide
• Complement compare.
• Shift rotate.
• Increment and decrement.
• Perform logic operations (AND, OR, XOR)

4
Binary Addition

• Example

11 carry 11010 1100 100110
26 12 38

• Double check answer in base 10 number

5
Questions

• Q. What is the sum of binary 110001100
  101010101?
• A. 1011100001
• Q. What is the base 10 number of the sum of
  binary 110001100 101010101?
• A. 51212864321737.
• Q. What is the sum of binary 11111 110011?
• A. 1010010
• Q. What is the base 10 number of the sum of
  binary 11111 110011?
• A. 6416282.

6
Half Adders
Block Diagram
Truth Table
Inputs
Outputs
S (Sum)
A
Half Adder
B
Co (Carry out)
Logic Diagram
Inputs
Half Adder
Outputs
A
S (Sum)
XOR
B
AND
Co (Carry out)

• The half adder circuit adds only the LSB (1s)
• A full adder must be used for 2s, 4s, 8s, 16s
  higher places in binary addition.

7
Full Adders
Truth Table
Block Diagram
Inputs
Outputs
Cin
S (Sum)
Full Adder
A
Co (Carry out)
B
Logic Diagram
Inputs
Full Adder
Outputs
Half Adder
Cin
A
S
S
B
Half Adder
A
S
B
Co
Co
Co
8
Three-Bit Adders

• Half and full adders are connected to form adders
  that add several binary digits (bits) at one
  time.
• Organized as you would add and carry by hand.
• In a parallel adder, all bits are applied to the
  inputs at the same time.

9
Parallel Three-Bit Adder
Problem
A1
A2
A3

B1
B2
B3
A
S
Half Adder1s
Co
B
Inputs
Cin
S
Full Adder
A
Co
B
Cin
S
Full Adder
A
Co
B
8s
4s
2s
1s
Sum Output
10
Binary Subtraction

• Example

11010 - 1100 1110
26 - 12 14

• Double check answer in base 10 number

11
Questions

• Q. What is the solution for binary 1100100 -
  1010101?
• A. 1111
• Q. What is the base 10 number of the subtraction
  of binary 1100100 - 1010101?
• A. 842115.
• Q. What is the solution for binary 11001 - 1011?
• A. 1110
• Q. What is the base 10 number of the subtraction
  of binary 11001 - 1011?
• A. 84214.

12
Half Subtractors
Block Diagram
Truth Table
Inputs
Outputs
Di (Difference)
A
Half Subtractor
B
Bo (Borrow)
Logic Diagram
Inputs
Half Subtractor
Outputs
A
Di (Difference)
XOR
B
AND
Bo (Borrow)

• Adders and subtractors are very similar.
• When subtracting several columns of binary
  digits, take into account borrowing.

13
Full Subtractors
Truth Table
Block Diagram
Inputs
Outputs
Bin
Di (Difference)
Full Subtractor
A
Bo (Borrow)
B
Logic Diagram
Inputs
Full Subtractor
Outputs
Half Subtractor
Bin
A
Di
Di
B
Half Subtractor
A
Di
B
Bo
Bo
Bo
14
Parallel Four-Bit Subtractor
Problem
A2
A1
A0
A3
-
B3
B2
B1
B0
Di
Half Subtractor 1s
A
Bo
B
Bin
Di
Full Subtractor 2s
A
Bo
Inputs
B
Bin
Di
Full Subtractor4S
A
Bo
B
Bin
Di
Full Subtractor8S
A
Bo
B
Output Difference
8s
4s
2s
1s
15
Using adders for subtraction
Decimal subtraction
Binary subtraction
Special technique subtraction
1010 1001 10011
1010 - 0110 100
10 - 6 4
1s complement And add
End around carry
1

100
Difference

• 1s complement is illustrated above.
• With 2s complement numbers, you can add
  subtract signed numbers (i.e. -1)

16
IC Adders
Problem
7483 4-bit binary adder
A2
A1
A0
A3
-
B3
B2
B1
B0
A1
Vcc
S1
B1
S2
A2
S3
B2
A3
Inputs
S4
B3
C4
A4
7483
B4
Carry input
C0
GND
4s
2s
8s
1s
16s
Output
17
Binary Multiplication

• Binary multiplication is just repeated addition.

6 x 5 30
6
6
6
6
6





30
Multiplicand
Multiplier
Product
110 x 101 110 000 110 11110
Multiplicand
Multiplier
First partial product 1s
Second partial product 2s
Third partial product 4s
Product
18
Questions

• Q. What is the product of binary 10111 1010?

• Q. What is the product of the above question
  using decimal numbers?

19
Binary Multipliers
Repeated Addition-type Multiplier System Block
Diagram
Disadvantage Slow with large numbers.
110 x 101 11110
Problem
6 x 5 30
20
Add Shift-Type Multiplier Circuit
Operation
Multiplicand Register
Multiplier Register
Accumulator Register
Control Line
Start
1
1
0
1
0
0
1
0
0
0
Clear Load
1 Add 0 Do not add
1s
Parallel Adder
Control 1
ParallelAdder
2s
4s
Multiplier Register
Accumulator Register
Control 1 Add 0110
1s
2s
1
1
0
1
Add
0
0
1
4s
8s
Multiplier Register
Accumulator Register
The 1 is lost.
Multiplier Register
Accumulator Register
0
0
0
1
Shift Right
1
1
0
1
Parallel Adder
Control 0
1
0
0
1
0
0
0
Multiplier Register
Accumulator Register
Control 0 Do not add.
0
0
0
1
1
1
0
Add
Multiplier Register
Accumulator Register
The 0 is lost.
1
1
0
0
0
1
0
Shift Right
0
Control 1
Parallel Adder
110 x 101 11110
Problem
6 x 5 30
Multiplier Register
Accumulator Register
Control 1 Add 0110
1
1
0
1
0
1
1
Add
The 1 is lost.
Multiplier Register
Accumulator Register
Finish
1
0
0
1
1
1
0
Shift Right
1
Final Product
21
2s Complement

• 2s complement is widely used in microprocessors.
• Microprocessors must process both negative and
  positive numbers.
• 2s complement represents
• Sign of the number.
• Magnitude of the number.
• 2s complement procedure
• Convert decimal to binary numbers.
• Change to 1s complement.
• Add 1.

22
2s Complement 4-bit
Sign Bit 1(-) 0)
Accumulator Register
0
0
0
MSB
LSB
23
Questions

• Q. What is 2s complement in a 4 bit register for
  negative 5?

• Q. What is 2s complement in a 4 bit register for
  negative 9?

• A. Need an 8-bit register to do this operation.

24
2s Complement Addition

• 2s complement notation is easily used in addition
  and subtraction of signed numbers.
• 2s complement is in the sum of sample problems.

0001 0101 0110
1 5 6
1110 1101 11011
(-2) (-3) -5
Discard
0110 1101 10011
(6) (-3) 3
(2) (-4) -2
0010 1100 1110
Discard
25
2s Complement Subtraction

• An adder is used for subtraction. This is done by
  converting the subtrahend to its 2s complement
  and adding.

0110 1110 10100
6 - 2 4
1110 1111 11101
(-2) -(1) -3
Discard
Discard
1010 0011 1101
(-6) -(-3) -3
(-4) -(-4) 0
1100 0100 10000
Discard
26
Questions

• Q. Using 2s complement, what is the sum of
  negative 5 and negative 1?

• Q. Using 2s complement, what is the sum of
  negative 1 and positive 1?

27
Questions Continued

• Q. Using 2s complement, what is the difference of
  negative 5 and negative 1?

• Q. Using 2s complement, what is the difference of
  negative 1 and positive 1?

28
2s Complement 8-bit
Sign Bit 1() 0(-)
Accumulator Register
0
0
0
0
0
0
0
2s
64s
1s
32s
16s
8s
4s
MSB
LSB
29
Questions

• Q. Using 2s complement, what is the sum of
  positive 55 and positive 21?

• Q. Using 2s complement, what is the sum of
  negative 71 and positive 63?

30
Questions Continued

• Q. Using 2s complement, what is the difference of
  positive 125 and positive 127?

• Q. Using 2s complement, what is the difference of
  negative 6 and positive 31?

Discard
31
2s Complement Adder/Subtractor
Problem
A2
A1
A0
A3
2s Complement Inputs
B3
B2
B1
B0
Cin
S
/-
A0
Full Adder 1s
A
Co
B
B0
Cin
S
A1
Full Adder 2s
A
Co
B
B1
Cin
S
A2
Full Adder 4S
A
Co
B
B2
Cin
S
A3
Full Adder8S
A
Co
B
Mode Control
2s Complement OutputSum or Difference
B3
sign
4s
2s
1s
Subtract1Add 0
32
Troubleshooting a Full Adder

• Visual Inspection Use all senses.
• Check power supply and ground.
• Check inputs for outputs in truth table using a
  logic probe.
• Isolate to the AND or XOR or OR gate.

33
Conclusions

• Q. What are the logic components that make up a
  half adder?
• A. An Exclusive OR gate and AND gate.