ECE 3110: Introduction to Digital Systems Chapter 6 Combinational Logic Design Practices

1. ECE 3110: Introduction to Digital SystemsChapter 6Combinational Logic Design Practices Adders, subtractors, ALUs

2. Prev… • XOR (2-level, 3-level) • Equivalent symbols • XNOR • Parity Circuits (Odd, even) • Daisy chain • Tree • Comparators • Iterative • Parallel

3. Adders/Subtractors • Half Adder • Full Adder • Ripple Adder • Full Subtractor • Ripple Subtractor • Adder/ Subtractor Circuit

4. Half Adder: adds two 1-bit operands • Truth table :X Y HS=(X+Y) CO0 0 0 00 1 1 01 0 1 01 1 0 1 X H S Y CO

5. Full Adders: provide for carries between bit positions • Basic building block is “full adder” • 1-bit-wide adder, produces sum and carry outputs • Truth table:

6. X Y Cin S Cout 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 Full Adders: provide for carries between bit positions • Basic building block is “full adder” • 1-bit-wide adder, produces sum and carry outputs • Truth table:

7. X Y Cin S Cout 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 Full Adders: provide for carries between bit positions • Basic building block is “full adder” • 1-bit-wide adder, produces sum and carry outputs • Truth table: S is 1 if an odd number of inputs are 1. COUT is 1 if two or more of the inputs are 1. Recall: Table 2-3, pp32

8. Full-adder circuit

9. Full-adder circuit

10. Full-adder circuit

11. Ripple adder • Speed limited by carry chain • Faster adders eliminate or limit carry chain • 2-level AND-OR logic ==> 2n product terms • 3 or 4 levels of logic, carry look-ahead

12. 74x2834-bit adder • Uses carry look-ahead internally

13. 16-bit group-ripple adder

14. Subtraction • Subtraction is the same as addition of the two’s complement. • The two’s complement is the bit-by-bit complement plus 1. • Therefore, X – Y = X + Y’ + 1

15. Full Subtractor = full adder, almost • X,Y are n-bit unsigned binary numbers • Addition : S = X + Y • Subtraction : D = X - Y = X + (-Y) = = X+ (Two’s Complement of Y) = X+ (One’s Complement of Y) + 1 = X+ Y’+ 1

16. Full Subtractor = full adder, almost • X,Y are n-bit unsigned binary numbers • Addition : S = X + Y • Subtraction : D = X - Y = X + (-Y) = = X+ (Two’s Complement of Y) = X+ (One’s Complement of Y) + 1 = X+ Y’+ 1

17. Using Adder as a Subtractor • Ripple Adder can be used as a Subtractor by inverting Y and setting the initial carry ( CIN ) to 1

18. Using Adder as a Subtractor • Ripple Adder can be used as a Subtractor by inverting Y and setting the initial carry ( CIN ) to 1

19. MSI Arithmetic Logic Units (ALU ) • ALU performs Arithmetic and Logical Functions- A , B : 4 bits inputs- S3,S2,S1,S0 : Function select- M=0 : Arithmetic operations +=Plus , - = Minus M=1 : Logical operations : += OR , . =AND • Example : Inputs FunctionsS3 S2 S1 S0 M=0 M=10 0 0 0 F= A-1+CIN F=A’0 1 1 0 F= A-B-1+CIN F=A XOR B’1 0 0 1 F= A+B+CIN F=A XOR B1 0 1 1 F=(A OR B)+ CIN F=A+B1 1 0 0 F= A+A+CIN F= 00001 1 1 1 F=A+CIN F=A 74x181 S0 S1 G S2 P S3 M A=B CIN F0 A0 F1 B0 F2 A1 F3 B1 A2 COUT B2 A3 B3

20. Chapter Summary • Documentation Standards:- Gate symbols, Signals Active Levels, Bubble to Bubble Logic- Block diagram, Schematic Diagram, Timing Diagram. • Combinational Logic design Structures:1-Decoders : Binary Decoders, Cascading decoders • 2-Encoders : Binary Encoder, Priority Encoder, Cascading Encoders, Encoder applications.3-Three State Buffers : SSI buffers, MSI Octal Buffer , Octal Three-state Transceiver

21. Chapter Summary • 4- Multiplexers : MUX operation, Single/Multiple outputs MUX, Expanding MUXs5- Demultiplexers : MUX/DMUX operation, Using Decoders as Demultiplexers.6- XOR and XNOR Gates: Logic Symbols, Equivalent Symbols, Parity Circuits, Parity Circuit application ( memory unit checking )7- Comparators : Parallel Comparators, Iterative Comparators, Cascading Comparators8-Adders : Half Adder, Full Adder, Ripple Adder, Subtractor, Ripple Adder / Subtractor Unit, • 9- Arithmetic Logic Units

22. Next… • Project • Reading Wakerly CH-7