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Transcript of ©2008 The McGraw-Hill Companies, Inc. All rights reserved. Digital Electronics Principles &...
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Digital ElectronicsDigital Electronics
Principles & ApplicationsPrinciples & ApplicationsSeventh EditionSeventh Edition
Chapter 10Arithmetic Circuits
Roger L. Tokheim
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
INTRODUCTION• Binary Addition
• Half & Full Adders
• Binary Subtraction
• Half & Full Subtractors
• Parallel Adders and Subtractors
• Using Adders for Subtraction
• Binary Multiplication
• Binary Multipliers
• 2s Complement Notation
• 2s Complement Adding/Subtracting
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
• Conceptually similar to decimal addition
• Example: Add the binary numbers 1010 and 11
Binary Addition
1 0 1 0+ 1 1
(carry)
1
1011
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Add the Binary numbers 11010 and 1100
QUIZ
1 1 0 1 0+ 1 1 0 0
(carry)
1
011001
(carry)
1
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
• Logic device that adds two binary numbers
• Only adds Least Significant Digit (LSD) column (1s column) in binary addition
Half Adder
A
B
(sum)
C0 (carry out)
HalfAdder
Input Output
Logic Symbol:
Logic Diagram:
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
?
?
QUIZ
Q#1- This circuit is called a ___.
ANS: Half-adder
Q#2- What are the sum and carry out outputsfrom the half adder circuit?
A = 0
B = 0
ANS: Sum=0, Carry out=0
Q#3- What are the sum and carry out outputsfrom the half adder circuit?
A = 0
B = 1
ANS: Sum=1, Carry out=0
Q#4- What are the sum and carry out outputsfrom the half adder circuit?
A = 1
B = 1
ANS: Sum=0, Carry out=1
Q#5- What are the sum and carry out outputsfrom the half adder circuit?
A = 1
B = 0
ANS: Sum=1, Carry out=0
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Used for adding binary place values other than the 1s place
Full Adder
Logic Symbol:
Logic Diagram:
AB
(sum)
C0 (carry out)
FullAdder
Input Output
Cin
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
?
?
Q#1- This combinational logic circuit is described as a(n) ___.
ANS: full-adder
Q#2- What are the sum and carry out outputs of thisfull-adder circuit?
Cin = 0
A = 0
B = 0
ANS: Sum=0, Carry out=0
Q#3- What are the sum and carry out outputs of thisfull-adder circuit?
Cin = 0
A = 0
B = 1
ANS: Sum=1, Carry out=0
Q#4- What are the sum and carry out outputs of thisfull-adder circuit?
Cin = 1
A = 0
B = 1
ANS: Sum=0, Carry out=1
Q#5- What are the sum and carry out outputs of thisfull-adder circuit?
Cin = 1
A = 1
B = 1
ANS: Sum=1, Carry out=1
Q#6- What are the sum and carry out outputs of thisfull-adder circuit?
Cin = 0
A = 1
B = 1
ANS: Sum=0, Carry out=1
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Example: Subtract binary number 101 from 1011
Binary Subtraction
1 0 1 1- 1 0
1
(borrow)
0110
10
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Subtract binary number 11 from 1010
QUIZ
1 0 1 0- 1 1
1110
1100 001
1
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Subtracts LSD column in binary subtraction
Half Subtractor
A
B
Di (difference)
B0 (borrow out)
HalfSubtractor
Input Output
Logic Symbol:
Logic Diagram:
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
?
?
Q#1- What is the difference and borrow outputs fromThis half-subtractor circuit?
(A – B)
A = 0
B = 0
ANS: Di= 0, Bo= 0
Q#2- What is the difference and borrow outputs fromthis half-subtractor circuit?
(A – B)
A = 1
B = 0
ANS: Di= 1, Bo= 0
Q#3- What is the difference and borrow outputs fromthis half-subtractor circuit?
(A – B)
A = 1
B = 1
ANS: Di= 0, Bo= 0
Q#4- What is the difference and borrow outputs fromthis half-subtractor circuit?
(A – B)
A = 0
B = 1
ANS: Di= 1, Bo= 1
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Used for subtracting binary place values other than the 1s place
Full Subtractor
Logic Symbol:
Logic Diagram:
AB
Di (difference)
B0 (borrow out)
FullSubtractor
Input Output
Bin
AB
DiB0H. S.
H. S.Bin
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QUIZ
?
?
Q#1- What are the Difference and Borrow out output fromthis full-subtractor circuit?
(A – B - Bin)Bin = 0
A = 0
B = 0
HINT: truth table from textbook (Fig. 10-10) is helpfulANSWER: Di = 0, Bo = 0
Q#2- What are the Difference and Borrow out output fromthis full-subtractor circuit?
(A – B - Bin)Bin = 1
A = 0
B = 0
ANSWER: Di = 1, Bo = 1
Q#3- What are the Difference and Borrow out output fromthis full-subtractor circuit?
(A – B - Bin)Bin = 1
A = 1
B = 1
ANSWER: Di = 1, Bo = 1
Q#4- What are the Difference and Borrow out output fromthis full-subtractor circuit?
(A – B - Bin)Bin = 1
A = 0
B = 1
ANSWER: Di = 0, Bo = 1
Q#5- What are the Difference and Borrow out output fromthis full-subtractor circuit?
(A – B - Bin)Bin = 0
A = 1
B = 0
ANSWER: Di = 1, Bo = 0
Q#6- What are the Difference and Borrow out output fromthis full-subtractor circuit?
(A – B - Bin)Bin = 0
A = 1
B = 1
ANSWER: Di = 0, Bo = 0
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
• Use half adder for LSD
• Use full adder for other digits
Parallel Adding
A2 A1 A0
+ B2 B1 B0
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Parallel Adder
1s place uses half-adder
2s, 4s, 8s places use full adders
SUM appears here
Enter binary
numbersto be added here0 0 1 1 + 1 0 0
0
0 1 0 1 1
1 1 0 0 + 1 1 1 1
1 1 0 1 11 0 1 0 0
1 1 1 0 + 0 1 1 0
Parallel adders are available in IC form.
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZQ#1- With the top full adder’s Cin input LOW, this operatesas a half-adder while the bottom three are full adders. (T or F)
4-bit number 4-bit number
Q#2- When the 4-bit parallel adder adds binary 1100 and 0011the sum appearing at the lower right will be ___. Q#3- When the 4-bit parallel adder adds binary 0111 and 1001the sum appearing at the lower right will be ___. Q#4- When the 4-bit parallel adder adds binary 1101 and 1001the sum appearing at the lower right will be ___. Q#5- When the 4-bit parallel adder adds binary 0011 and 0110the sum appearing at the lower right will be ___. Q#6- When the 4-bit parallel adder adds binary 1111 and 1111the sum appearing at the lower right will be ___. Q#7- When the 4-bit parallel adder adds binary 1010 and 1001the sum appearing at the lower right will be ___.
ANS: TrueANS: 01111ANS: 10000ANS: 10110ANS: 01001ANS: 11110ANS: 10011
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Parallel Subtractor Using Full Adders
Note the use of four full adder circuits
Also notice the addition of four inverters on the B inputs to the
FAs
Inverters
HIGH at Carry
in input
makes this perfo
rm as a
full adder c
ircuitBinary numbers to be
subtracted are input here 1 1 1 1 - 0 0 1 1
The result (difference) of the subtraction problem will appear
here.
1 1 0 0
1 0 0 1 - 0 1 1 1
0 0 1 0
HIGH at Carry
in input acts
like adding +1 to a 1s C
number to fo
rm th
e 2s
complement.
1sC is form
ed by four
inverters.
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Example: Multiply the binary numbers 111 and 101.
Binary Multiplication
111000
111110001
1 1 1x 1 0 1
Multiplicand
Multiplier
1st partial product
2nd partial product
3rd partial product
Product
111 x 101 can also be calculated: 111 + 111 + 111 + 111 + 111
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Multiply the binary numbers 101 and 100.
1 0 1x 1 0 0
000000
10100101
QUIZ
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Binary multiplier circuits – utilize repeated addition.
Binary Multipliers
Block Diagram:
Multiplicand
register
Multiplierdown counter
Productregister
Adder
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• 2s complement representation - widely used in microprocessors.
• Represents sign and magnitude
2s Complement Notation
Decimal: +7 +4 +1 0 -1 -4 -7
2s Complement: 0111 0100 0001 0000 1111 1100 1001
MSB LSB
Sign bit (0 = + ; 1 = -)
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• Converting positive numbers to 2s complement:
• Same as converting to binary
• Converting negative numbers to 2s complement:
2s Complement - Conversions
- 4 (decimal)
0 1 0 0
1 0 1 1
- 4 = 1 1 0 0 (2s Complement)
Decimal to 2s Complement
Convert decimalto binary
1s complement
Add 1
2s Complement toBinary
1 1 0 0 (2s C)
0 0 1 1
0 1 0 0 (Binary)
1s complement
Add 1
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
Q#1- Convert the decimal number –4 to 2s complement.
4 (decimal)
0100
1011
Step 1: convert decimal to binary
Step 2: convert to 1s complement
Step 3: add +1 equaling 2s complement
1100 2s C
ANS: -4 = 1100 2sC
Q#2- Convert the decimal number –-7 to 2s complement.
7 (decimal)
0111
1000
1001 2s C
ANS: -7 = 1001 2sC
Q#3- Convert the decimal number –1 to 2s complement.
1 (decimal)
0001
1110
1111 2s C
ANS: -1 = 1111 2sC
Q#4- Convert the decimal number +6 to 2s complement.
6 (decimal)
0110
Not needed
Not needed
ANS: +6 = 0110 2sC
Q#5- Convert the decimal number -8 to 2s complement.
8 (decimal)
1000
0111
1000 2sC
ANS: -8 = 1000 2sC
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
Q#1- Convert the 2s complement number 1110 to decimal.
0001
0010 binary
Step 1: convert to 1s complement
Step 2: Add +1
Step 3: convert binary to decimal
2 decimal
ANS: 1110 2sC = -2 decimal
1110
Q#2- Convert the 2s complement number 0101 to decimal.
Not needed
Not needed
0101 binary
5 decimal
ANS: 0101 2sC = +5 decimal
Q#3- Convert the 2s complement number 1001 to decimal.
1001
0110
0111 binary
7 decimal
ANS: 1001 2sC = -7 decimal
Q#4- Convert the 2s complement number 1111 to decimal.
1111
0000
0001 binary
1 decimal
ANS: 1111 2sC = -1 decimal
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
2s complement notation makes it possible to add and subtract signed numbers
Adding/Subtracting in 2s Complement
(- 1)+ (- 2)
(- 3)
1 1 1 1+ 1 1 1
010111
Discard
(+1)+ (- 3)
(- 2)
0 0 0 1+ 1 1 0
10111
(Decimal)
2s Complement
2s complement
2s complement
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Add the following 2s complement numbers:
0 1 0 1+ 1 1 0
0100
QUIZ
01
(+5)+ (-
4)(+1)Discard
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Practical Suggestion for Binary Math
• Use a scientific calculator.
• Most scientific calculators have DEC, BIN, OCT, and HEX modes and can either convert between codes or perform arithmetic in different number systems.
• Most scientific calculators also have other functions that are valuable in digital electronics such as AND, OR, NOT, XOR, and XNOR logic functions.
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
REVIEW• Binary Addition
• Half & Full Adders
• Binary Subtraction
• Half & Full Subtractors
• Parallel Adders and Subtractors
• Using Adders for Subtraction
• Binary Multiplication
• Binary Multipliers
• 2s Complement Notation
• 2s Complement Adding/Subtracting