Introduction to Digital Logic Missouri S&T University CPE 2210 Adders

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1 Introduction to Digital Logic Missouri S&T University CPE 22 Egemen K. Çetinkaya Egemen K. Çetinkaya Department of Electrical & Computer Engineering Missouri University of Science and Technology 23 February 28 rev Egemen K. Çetinkaya

2 Outline Introduction Half-adder Full-adder Carry-ripple adder Incrementer Summary 23 February 28 MST CPE22 2

3 Digital Logic Systems Overview Combinatorial logic circuits for no memory systems Boolean algebra to mathematically design/analyze logic gates are building blocks Sequential logic circuits for memory systems Finite State Machines to mathematically design/analyze flip-flops and latches store memory flip-flops and latches are building blocks of sequential logic Sequential logic circuits (aka ntrollers) mbine mbinatorial circuits storage elements (e.g. registers) 23 February 28 MST CPE22 3

4 Digital Systems Components analog phenomena electric signal A2D digital data digital data sensors and other inputs Digital System D2A electric signal actuators and other outputs digital data digital data Transducer: sensor + actuator Not all sensors/actuators require A2D/D2A nversion Digital system can be implemented: microprocessor readily available, cheap, easy to program, easy to reprogram custom circuit smaller, faster, nsume less power 23 February 28 MST CPE22 4

5 Digital Systems Paths Digital systems have two paths: datapath circuit ntrol circuit Datapath circuit store data manipulate data transfer data from one part to another Control circuit ntrols the operation of datapath circuit 23 February 28 MST CPE22 5

6 Subtractors Comparators Datapath Components Building Block examples ALUs: Arithmetic Logic Units Registers Shifters Multiplexer/demultiplexers Enders/deders Counters/timers 23 February 28 MST CPE22 6

7 Overview add two N-bit numbers E.g.: 2-bit adder adds two 2-bit numbers 4-bit adder adds two 4-bit numbers Types: carry-ripple: there is ripple effect this will be our focus in this class carry-select: uses multiplexers, faster carry-lookahead: for faster add operations 23 February 28 MST CPE22 7

8 Add Operation Compute sum, add carry to next lumn Lets add A= and B= A: B: February 28 MST CPE22 8

9 4-bit Adder Example Compute sum, add carry to next lumn Lets add A= and B= A: + B: b a ci b a ci b a ci b a SUM Full-adders Half-adder 23 February 28 MST CPE22 9

10 Half-Adder Adds two bits, generates sum bit and carry-out bit Lets try to design the circuit: A: + B: b a SUM 23 February 28 MST CPE22

11 Half-Adder Adds two bits, generates sum bit and carry-out bit Lets try to design the circuit: Next steps? A: + B: b a SUM 23 February 28 MST CPE22

12 Half-Adder Adds two bits, generates sum bit and carry-out bit Lets try to design the circuit: Next steps? Capture the function Create equations Implement as a circuit A: + B: b a SUM 23 February 28 MST CPE22 2

13 Half-Adder Adds two bits, generates sum bit and carry-out bit Lets try to design the circuit: Next steps? Capture the function a b A: + B: b a SUM 23 February 28 MST CPE22 3

14 Half-Adder Adds two bits, generates sum bit and carry-out bit Lets try to design the circuit: Next steps? Create equations a b A: + B: b a SUM = ab, s = a b+ab = a b 23 February 28 MST CPE22 4

15 Half-Adder Adds two bits, generates sum bit and carry-out bit Lets try to design the circuit: Next steps? Implement as a circuit = ab, s = a b a b A: + B: b a SUM a b Half-adder (HA) s s 23 February 28 MST CPE22 5

16 Full-Adder Adds three bits, generates sum bit and carry-out bit Lets try to design the circuit: A: + B: b a SUM 23 February 28 MST CPE22 6

17 Full-Adder Adds three bits, generates sum bit and carry-out bit Lets try to design the circuit: Next steps? A: + B: b a SUM 23 February 28 MST CPE22 7

18 Full-Adder Adds three bits, generates sum bit and carry-out bit Lets try to design the circuit: Next steps? Capture the function Create equations Implement as a circuit A: + B: b a SUM 23 February 28 MST CPE22 8

19 Full-Adder Adds three bits, generates sum bit and carry-out bit Lets try to design the circuit: Next steps? Capture the function a b ci 23 February 28 MST CPE22 9

20 Full-Adder Adds three bits, generates sum bit and carry-out bit Lets try to design the circuit: Next steps? Create equations =a bc+ab c+abc +abc =a bc+abc+ab c+abc+abc +abc =bc(a+a )+ac(b+b )+ab(c+c ) =ab+ac+bc Note that treating ci as c a b ci 23 February 28 MST CPE22 2

21 Full-Adder Adds three bits, generates sum bit and carry-out bit Lets try to design the circuit: Next steps? Create equations s=a b c+a bc +ab c +abc s=a (b c+bc )+a(b c +bc) s=a (b c)+a(b c) s=a b c Note that treating ci as c a b ci 23 February 28 MST CPE22 2

22 Full-Adder Adds three bits, generates sum bit and carry-out bit Lets try to design the circuit: Next steps? Implement as a circuit = ab + ac + bc s = a b c a b Full-adder (FA) s ci a b Full adder (FA) ci s 23 February 28 MST CPE22 22

23 Carry-Ripple Adder Carry-ripple adder: Uses half- and full-adders E.g.: 4-bit carry-ripple adder can build any size adder based on full- and half-adders a3 b3 a2 b2 a b a b a b ci a b ci a b ci a b FA FA FA HA a3a2aa b3b2bb s s s s 4-bit adder s3 s2 s s s3 s2 s s 23 February 28 MST CPE22 23

24 Carry-Ripple Adder Carry-ripple adder: Can also use only full-adders E.g.: 4-bit carry-ripple adder a3 b3 a2 b2 a b a b ci a b ci a b ci a b ci a b ci a3a2aa b3b2bb FA FA FA FA 4-bit adder ci s s s s s3 s2 s s s3 s2 s s 23 February 28 MST CPE22 24

25 8-bit Adder How can we build 8-bit adder using 4-bit adders? 23 February 28 MST CPE22 25

26 8-bit Adder How can we build 8-bit adder using 4-bit adders? By serially nnecting 4-bit adders a7a6a5a4 b7b6b5b4 a3a2aa b3b2bb a3a2aa b3b2bb 4-bit adder ci a3a2aa b3b2bb 4-bit adder ci a7.. a b7.. b 8-bit adder ci + s3 s2 s s s3 s2 s s s7.. s s7 s6 s5 s4 s3 s2 s s C 23 February 28 MST CPE22 26

27 Incrementers -bit Incrementer Incrementers might be useful too They use half-adders and one of the input is i.e. incrementing by a3 a2 a a a b HA a b HA a b HA a b HA a3 a2 a a + 3 s2 s s s3 s2 s s 23 February 28 MST CPE22 27

28 Summary add two N-bit numbers Half-adder: adds two bits, generates sum bit and carry-out bit Full-adder: adds three bits, generates sum bit and carry-out bit Types: carry-ripple: carry ripples through adders carry-select: for faster add operation, uses multiplexers carry-lookahead: for faster add operation Incrementer increments by using half-adder 23 February 28 MST CPE22 28

29 References and Further Reading [V2] Frank Vahid, Digital Design with RTL Design, VHDL, and Verilog, 2nd edition, Wiley, 2. [BV29] Stephen Brown and Zvonko Vranesic, Fundamentals of Digital Logic with VHDL Design, 3rd edition, McGraw-Hill, 29. [G23] Donald D. Givone, Digital Principles and Design, McGraw-Hill, 23. [S27] John Seiffertt, Digital Logic for Computing, st edition, Springer, February 28 MST CPE22 29

30 End of Foils 23 February 28 MST CPE22 3

Introduction to Digital Logic Missouri S&T University CPE 2210 Electric Circuits

Introduction to Digital Logic Missouri S&T University CPE 2210 Electric Circuits Egemen K. Çetinkaya Egemen K. Çetinkaya Department of Electrical & Computer Engineering Missouri University of Science and