Chapter 7 Logic Circuits

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1 Chapter 7 Logic Circuits

2 Goal. Advantages of digital technology compared to analog technology. 2. Terminology of Digital Circuits. 3. Convert Numbers between Decimal, Binary and Other forms. 5. Binary Arithmetic Operations in digital systems. 6. Interconnect Logic Gates to Implement a given logic function. 7. Karnaugh Maps to Minimize the Number of gates to implement a logic function.

3 Introduction Why Digital Signal Conditioning?. Multivariable Process Control 2. Easy Linearization of Non-Linear System 3. Easy Modification and Solution of Complicated System 4. Fully Integrated System using Networking Technique 5. Immune from Noise, Power Drift & Analog Problem Why Not Digital Signal Conditioning?. Loss of Accuracy 2. A/D Converter and D/A Converter Dependence

4 Digital Signal Compared to Analog Signal TTL Level TTL: Transistor-Transistor Logic

5 Digital Words DIGITAL FUNDAMENTALS Digital Word Decimal Number Octal and Hexa Number Positive versus Negative Logic : Binary Word : Base : 8 and 6 Base Positive Logic : High as Low as Negative Logic : High as Low as Transmission of Digital Information In parallel transmission, an n-bit word is transferred on n wires, one wire for each bit, plus a common or ground wire. In serial transmission, the successive bits of the word are transferred one after the other with a single pair of wires.

6 Binary Numbers N 2 L 2 = b 2 + b N = base number less than b b2... b m bm = base 2 number less than m = number of digits in base 2 number b m m

7 Gray Code In Gray Code, Each Word differs in only one bit from each of its adjacent words

8 Binary Addition Overflow and Underflow We must be aware of the possibility of overflow in which the result exceeds the maximum value that can be represented by the word length in use.

9 Complement Arithmetic One s complement : Replacement of s by s, and vice versa. (one s complement) Two s complement : Addition of to the one s complement, neglecting the carry (if any) out of the most significant bit (one s complement)

10 Complement Arithmetic Complements are useful for representing negative numbers and performing subtraction in computers. Example of Subtraction

11 AND AI = A A = AA = A AB = BA A ( BC) = ( AB) C = ABC

12 Inversion A A = A = A

13 OR A + A = A A + = A + = A + A = A ( B + C) = AB AC A + ( A + B) + C = A + ( B + C) = A + B + C

14 Truth Table for Associate Law ( A + B) + C = A + ( B + C) = A + B + C

15 Implementation of Boolean Algebra ( )( ) C D D E F = ABC + ABC F = C( A + D + E) + DE

16 De Morgan s Laws ABC = A + B + C ( D + E + F ) = D E F In a Logic Expression ) the Variables are replaced by their Inverses, 2) AND Operation is replaced by OR, 3) OR operation is replaced by AND 4) the entire expression is inverted, Resulting expression yields the same as before Any combinatorial logic function can be implemented solely with AND gate & Inverters. Similarly Any combinatorial logic function can be implemented solely with OR gate & Inverters

17 Proof of De Morgan s Law : T S T S = T x or S x Then T S x e i T S x.. T S x T x or S x Hence ( ) F D E F E D = + + C B A ABC + + = T S T S = T x or S x Then T S x T S x T x or S x Hence

18 NAND, NOR and XOR Gates NAND : AND followed by Inverter NOR : OR followed by Inverter XOR = XOR : Exclusively OR = Buffer : Same Output with Input Equivalence : True if Inputs are same = XOR followed by Inverter =

19 Logical Sufficiency of NAND or NOR

20 Classical Search Logic Decision

21 Current Search Logic Decision

22 Logic Implementation by Boolean Algebra Alarm Condition. Low Level with High Pressure 2. High Level with High Temperature 3. High Level with Low Temperature and High Pressure Direct Writing of Required Condition D = _ Α Β + Α C + _ Α C 2 3 Β

23 C B A C A B A + + AND, OR, NOT Gate ( ) ( ) C B A C A B A C B A C A B A + = + + NAND, NOR, NOT gates Β C Α C Α Β Α D + + =

24 Synthesis of Logic Circuits (SOP) Sum-of-Products Implementation In SOP, we form a product of all the input variables (or their inverses) for each row of the truth table for which the result is logic. The output is the sum of these products. Product terms that include all of the input variables (or their inverses) are called minterms. (POS) Product-of-Sums Implementation In POS, we form a sum of all the input variables (or their inverses) for each row of the truth table for which the result is logic. The output is the product of these sums. Sum terms that include all of the input variables (or their inverses) are called maxterms.

25 (SOP) Sum-of-Products Implementation A A B B C C A B A B C C

26

27 (POS) Product-of-Sums Implementation Take Inverse of Input! A + B + A + B + A + B + A + B + C C C C

28

29 Example of Logic Circuit Design Resident Heating System, Day Time, Heating is required if temperature falls below 8 o C Night Time, Heating is required if temperature falls below 25 o C Logic Signal : D, H, L D : Daytime =, Night Time = H : Temperature > 25 o C =, < 25 o C = L : Temperature < 8 o C =, < 8 o C = Temperature can not be T < 8 o C & T > 25 o C at a time Neglect the Case!!

30 F = m(,4,5) = DH L + DH L + DHL F = M (,3,7) = ( D + H + L)( D + H + L)( D + H + L) F = DH + DH L

31 XOR Operation Operation = = = = Truth Table A B D In SOP F = m(,2 ) = AB + AB In POS F = M (,4) = ( A + B)( A + B)

32 Display Logic 4 bit logic : case : 7 Output G F E D C B A 2 Bit Binary Decimal

33 Simplification Criterion: Logic Simplification F = ( ) + ( ) + ( ) + + ( ) - with a minimum number of terms - with the fewest possible number of literals * There may be many, equally good expressions!. Algebraic Operational Simplification Simple Algebraic Operation can make Simpler form such as F( A, B, C) = ABC + ABC + ABC + ABC + ABC = AB + AB + ABC = A + ABC = A + BC With proper duplication of product terms, only the distributive law would do the job.

34 2. Algebraic Boolean Simplification Use any Boolean Axioms and Theorems, or () two adjacent product terms into one bigger product term (2) a term may be used more than once during combination (3) a term may be partitioned into smaller ones F( A, B, C) = ABD + = AD ( B + ABD + B) BCD + = AD + BCD + BCD=, Only if B=, C=, D= AD ABC + BCD + ABC ABC = or ABC = Q A = or A = Thus BCD is redundant can be dropped F ( A, B, C) = AD + ABC But, still we have some difficulties: () no specific rules to predict each succeeding step (2) difficult to determine whether the simplest expression has been achieved

35 Karnaugh Maps (Veitch Diagram). 2D Graphical Representation of Truth Table 2. Simplification by Manual Operation (not for CAD) 3. Simplification of SOP (minterms) 4. Procedure - Draw K-Map - Write Logic Equation in K-Map - Select True Cube

36 Cube in K-Map Products of Variable with Cube

37 Example of K-Map

38 Sequential Logic Circuits - Outputs depend on PAST as well as PRESENT inputs - Simple Memory Logic based on Time Scale - Synchronized by a Clock Flip-Flop

39 S-R Flip-Flop S-R Flip-Flop with 2 NOR Gates S-R Flip-Flop with 2 NAND Gates

40

41

42

43 Serial-In Parallel-Out Shift Register

44 Parallel-In Serial-Out Shift Register

45 Ripple Counter

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