tCh10.ppt
- 1. Digital Electronics Principles & Applications Fifth Edition Chapter 10 Arithmetic Circuits ©1999 Glencoe/McGraw-Hill Roger L. Tokheim
- 2. CHAPTER 10 PREVIEW • Binary Addition • Half & Full Adders • Binary Subtraction • Half & Full Subtractors • Parallel Adders and Subtractors • Using Adders for Subtraction • Binary Multiplication • Binary Multipliers • Half & Full Adders • 2s Complement
- 3. • Conceptually similar to decimal addition • Example: Add the binary numbers 1010 and 11 BINARY ADDITION 1 0 1 0 + 1 1 (carry) 1 1 0 1 1
- 4. Add the Binary numbers 11010 and 1100 TEST 1 1 0 1 0 + 1 1 0 0 (carry) 1 0 1 1 0 0 1 (carry) 1
- 5. • 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) Half Adder Input Output Logic Symbol: Logic Diagram:
- 6. Used for adding binary place values other than the 1s place FULL ADDER Logic Symbol: Logic Diagram: A B (sum) C0 (carry out) Full Adder Input Output Cin
- 7. Example: Subtract binary number 101 from 1011 BINARY SUBTRACTION 1 0 1 1 - 1 0 1 (borrow) 0 1 1 0 1 0
- 8. Subtract binary number 11 from 1010 TEST 1 0 1 0 - 1 1 1 1 1 0 1 10 0 0 01 1
- 9. Subtracts LSD column in binary subtraction HALF SUBTRACTOR A B Di (difference) B0 (borrow out) Half Subtractor Input Output Logic Symbol: Logic Diagram:
- 10. Used for subtracting binary place values other than the 1s place FULL SUBTRACTOR Logic Symbol: Logic Diagram: A B Di (difference) B0 (borrow out) Full Subtractor Input Output Bin A B Di B0 H. S. H. S. Bin
- 11. • Use half adder for LSD • Use full adder for other digits PARALLEL ADDING A2 A1 A0 + B2 B1 B0
- 12. PARALLELADDER 1s place uses half-adder 2s, 4s, 8s places use full adders SUM appears here 0 0 1 1 + 1 0 0 0 0 1 0 1 1 1 1 0 0 + 1 1 1 1 1 1 0 1 1 1 0 1 0 0 1 1 1 0 + 0 1 1 0 Parallel adders are available in IC form.
- 13. 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 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
- 14. Example: Multiply the binary numbers 111 and 101. BINARY MULTIPLICATION 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 x 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
- 15. Multiply the binary numbers 101 and 100. 1 0 1 x 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 TEST
- 16. Binary multiplier circuits - utilize repeated addition. BINARY MULTIPLIERS Block Diagram: Multiplicand register Multiplier down counter Product register Adder
- 17. • 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 = -)
- 18. • 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 decimal to binary 1s complement Add 1 2s Complement to Binary 1 1 0 0 (2s C) 0 0 1 1 0 1 0 0 (Binary) 1s complement Add 1
- 19. 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 0 1 0 1 1 1 Discard (+1) + (- 3) (- 2) 0 0 0 1 + 1 1 0 1 0 1 1 1 (Decimal) 2s Complement 2s complement 2s complement
- 20. Add the following 2s complement numbers: 0 1 0 1 + 1 1 0 0 1 0 0 TEST 0 1 (+5) + (- 4) (+1) Discard
- 21. 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.