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Chapter 7Logic Circuits
Goal
1. 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.
IntroductionIntroduction
Why Digital Signal Conditioning?1. Multivariable Process Control2. Easy Linearization of Non-Linear System3. Easy Modification and Solution of Complicated System 4. Fully Integrated System using Networking Technique5. Immune from Noise, Power Drift & Analog Problem
Why Not Digital Signal Conditioning?1. Loss of Accuracy2. A/D Converter and D/A Converter Dependence
Digital Signal Compared to Analog Signal
TTL Level
TTL: Transistor-Transistor Logic
DIGITAL FUNDAMENTALSDIGITAL FUNDAMENTALS
Digital Word : Binary WordDecimal Number : 10 BaseOctal and Hexa Number : 8 and 16 Base
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.
Digital Words
Positive versus Negative Logic
Transmission of Digital Information
Positive Logic : High as 1 Low as 0Negative Logic : High as 0 Low as 1
Binary Numbersm
mbbbN −−− +++= 222 22
1110 L
1 than lessnumber 10 base10 =N1 than lessnumber 2 base... 121 =− mm bbbb
number 2 basein digits ofnumber =m
Gray Code
In Gray Code, Each Word differs in only one bit from each of its adjacent words
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.
Complement Arithmetic
One’s complement : Replacement of 1s by 0s, and vice versa.
0100110110110010 (one’s complement)
Two’s complement : Addition of 1 to the one’s complement,neglecting the carry (if any) out of the most significant bit
0100110110110010 (one’s complement)
Complement Arithmetic
Complements are useful for representing negative numbers and performing subtraction in computers.
Example of Subtraction
AAA =AAI = 00 =ABAAB = ( ) ( ) ABCCABBCA ==
AND
0=AA AA =
Inversion
( ) ( ) CBACBACBA ++=++=++
( ) ACABCBA +=+AA =+ 0
11 =+A 1=+ AAAAA =+
OR
( ) ( ) CBACBACBA ++=++=++
Truth Table for Associate Law
Implementation of Boolean Algebra
( )( )EDDCABCCBAF ++++=
DEEDACF +++= )(
De Morgan’s Laws
CBAABC ++= ( ) FEDFED =++
In a Logic Expression 1) 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
Proof of De Morgan’s Law :
TSTS ∪=∩
TxorSxThenTSxeiTSx ∉∉∩∉∩∈ ..
TSxTxorSxHence ∪∈→∈∈
( ) FEDFED =++
CBAABC ++= TSTS ∩=∪
TxorSxThenTSx ∈∈∪∈
TSxTxorSxHence ∩∈→∉∉
NAND, NOR and XOR Gates
NAND : AND followed by InverterNOR : OR followed by InverterXOR : Exclusively ORBuffer : Same Output with InputEquivalence : True if Inputs are same
XOR followed by Inverter 011110101000
=⊕=⊕=⊕=⊕
XOR
Logical Sufficiency of NAND or NOR
Classical Search Logic Decision
Current Search Logic Decision
Logic Implementation by Boolean AlgebraLogic Implementation by Boolean Algebra
ΒCΑCΑΒΑD__⋅⋅+⋅+⋅=
Alarm Condition1. Low Level with High Pressure2. High Level with High Temperature3. High Level with Low Temperature
and High Pressure
1 2 3
Direct Writing of Required Condition
CBACABA ⋅⋅+⋅+⋅
AND , OR , NOT Gate
( ) ( )CBACABA
CBACABA
⋅⋅⋅⋅+⋅=
⋅⋅+⋅+⋅
NAND , NOR, NOT gates
ΒCΑCΑΒΑD__⋅⋅+⋅+⋅=
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 1. 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 0. The output is the product of these sums.Sum terms that include all of the input variables (or their inverses) are called maxterms.
(SOP) Sum-of-Products Implementation
CBA ⋅⋅
CBA ⋅⋅
CBA ⋅⋅CBA ⋅⋅
(POS) Product-of-Sums Implementation
CBA ++
CBA ++CBA ++CBA ++
Take Inverse of Input!
Example of Logic Circuit DesignResident Heating System ,Day Time, Heating is required if temperature falls below 18oCNight Time, Heating is required if temperature falls below 25oCLogic Signal : D, H, L
D : Daytime =1 , Night Time = 0H : Temperature > 25oC = 1 , < 25oC = 0L : Temperature < 18oC = 1 , < 18oC = 0
Temperature can not beT < 18oC & T > 25oC
at a time
Neglect the Case!!
LHDLHDLHDmF ++==∑ )5,4,0(
LHDHDF +=
))()(()7,3,1( LHDLHDLHDMF ++++++==∏
011110101000
=⊕=⊕=⊕=⊕
Operation
011110101000DBA
Truth TableIn SOP
BABAmF +==∑ )2,1(
))(()4,0( BABAMF ++==∏
In POS
XOR Operation
Display Logic
4 bit logic : 10 case : 7 Output
1
1
1
1
1
1
1
G
1
1
1
1
1
1
F
1
1
1
1
E
111100018
11110019
11111107
1101106
11110105
1100104
111111003
11101002
1110001
111100000
DCBA2 Bit BinaryDecimal
Logic SimplificationSimplification Criterion:
F = (…) + (…) + (…) + … + (…)- with a minimum number of terms- with the fewest possible number of literals
* There may be many, equally good expressions!
With proper duplication of product terms, only the distributive law would do the job.
1. Algebraic Operational SimplificationSimple Algebraic Operation can make Simpler form such as
BCA
ABCA
ABCBABA
ABCBCACBACBACBACBAF
+=
+=
++=
++++=),,(
Use any Boolean Axioms and Theorems, or(1) 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
2. Algebraic Boolean Simplification
But, still we have some difficulties:(1) no specific rules to predict each succeeding step(2) difficult to determine whether the simplest expression has been achieved
ABCBCDDA
ABCBCDBBDA
ABCBCDBDADBACBAF
++=
+++=
+++=
)(
),,(
BCD=1, Only if B=1, C=1, D=1 1111 ==== AorAABCorDA Q
Thus BCD is redundant can be dropped
ABCDACBAF +=),,(
1. 2D Graphical Representation of Truth Table2. 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
Karnaugh Maps (Veitch Diagram)
Cube in K-Map
Products of Variable with Cube
Example of K-Map
- Outputs depend on PAST as well as PRESENT inputs- Simple Memory Logic based on Time Scale- Synchronized by a “Clock”
Sequential Logic Circuits
Flip-Flop
S-R Flip-Flop
S-R Flip-Flop with 2 NOR Gates
S-R Flip-Flop with 2 NAND Gates
Serial-In Parallel-Out Shift Register
Parallel-In Serial-Out Shift Register
Ripple Counter