3. PRINCIPLES OF COMBINATIONAL LOGIC
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1 Principle of ombinational Logic -. PRINIPLES OF OMINTIONL LOGI Objectives. Understand the design & analysis procedure of combinational logic.. Understand the optimization of combinational logic.. efinitions of ombinational & Sequential Logic ombinational Logic: ircuits without a Memory Switch controlled house lamp, full adder n input variables x x n ombinational Logic ircuit/functions (F) y y m m output variables Sequential Logic: ircuits with a Memory Three-position lamp, traffic light controller Inputs ombinational Logic ircuit/functions (F) Memeory Element Outputs General logic design sequence Problem Statement Truth Table onstruction Switching Equations Equation Simplification Logic iagram rawn Logic ircuit uilt
2 Principle of ombinational Logic - () Problem Statements to Truth Tables Ex.: Make the electric motor move. Operating condition:. One of two operators is in position. if material is present to be moved. the protective interlock switch is closed. Step. Set I/O variables Variable I/O escription Variable I/O escription a In operator is in the interlock switch is s In position closed b In operator is in position m In material is present M Out the signal to turn the motor on and off Step. Truth table a b m s M a b m s M Step. Switching equation: M abms abms abms
3 Principle of ombinational Logic - Ex. conveyor system brings raw material in from three different sources. different speed conveyors: sensors If any source has a product, then M4 can be turned on. If S detects a material, only M can be turned on. If S is empty, then either or or both can be turned on. Step. Set I/O variables Variable I/O escription Variable I/O escription S I Sensor M O Motor S I Sensor M O Motor S I Sensor M O Motor M4 O Motor 4 Step. Truth table S S S M M M M4 S S S M M M M4
4 Principle of ombinational Logic -4 () Terminology Literal: a oolean variable or its complement Minterm Maxterm Sum term Product term Sum of product (SOP) Product of sum (POS) anonical forms: canonical sum of products, canonical product of sums. Generation of Switching Equations from Truth Tables Ex. Input variables Minterm Maxterm Output a b c Term esignation Term esignation F abc m abc M abc m abc M abc m a b c M abc m a b c M ab c m 4 a b c M 4 abc m 5 a b c M 5 abc m a b c M abc m a b c M,,,4,5,,,4,5,, F f a b c m a bc ab c ab c abc abc,,,,,, F m a b c a b c a bc F a b c a b c a bc a b c a b c a b c M MM M
5 Principle of ombinational Logic -5. Two-Level Simplification on t forget that tools are written by mere mortals and do not always do the correct things! You must still be able to check the output of the tool. The essence of simplification is repeatedly to find two-element subsets of the onset in which only one variable changes its value while the other variables do not., F f a b ab ab a b b a c.f. F f b, a () oolean ubes X -cube Y X -cube XYZ Y Z X -cube WXYZ Y Z W X 4-cube () K-Map Method Karnaugh map or K-map : n alternative reformulation of the truth table MS LS -variable K-map ecimal value 4 5 -variable K-map
6 Principle of ombinational Logic variable K-map Ex. F 4 5, F f,, F f F 4 5 F f,, m,5,,
7 Principle of ombinational Logic ,, F f,, F f ,,, F f,,, F f Finding the Minimum Product of Sums From ,,, F f F
8 Principle of ombinational Logic -8 Implicant: a single element of the on-set or any group of elements that can be combined together in a K-map. Prime implicant: an implicant that cannot be combined with another one to eliminate a literal. Each prime implicant is an implicant with as few literals as possible. Essential prime implicant: a particular element of the on-set which is covered by a single prime implicant Prime implicants:,,,,, Essential prime implicants:,,,, F Essential prime implicants must be part of the minimized expression. () K-Maps Revisited: 5- and -Variable Functions Five-Variable K-Map = E = = E = E
9 Principle of ombinational Logic -9 E E,,,, m,5,, 8,,,5,,9,,, 4, 9, f E E E E E Six-Variable K-Map E = = = = EF EF EF EF = EF EF 9 8 = EF = EF 5 4 =
10 Principle of ombinational Logic -,,,,, m, 8,,8, 4,,4,, 4, 45,5,5,58, f E F EF EF F EF E E F F (4) Incompletely Specified Functions (on t are Terms) on t care: minterms or maxterms that are not used as part of the output Ex: inary to EX- code conversion inary EX- W X Y Z X X X X X X X X X X X X X X X X X X X X X X X X
11 Principle of ombinational Logic -,,, 5 8 9,,,,4,5,,,,,, 4,9,,,,4,5,,,,, 4,,8,,,,4,5,,,,, 4,,8,,,,4,5 f W X Y Z m d f W X Y Z m d f W X Y Z m d f W X Y Z m d wx yz y wx yz y w 4 5 X X 8 9 X X 5 4 X X x w 4 5 X X 8 9 X X 5 4 X X x z W XZ XY wx yz y wx yz z XY XZ XYZ y w 4 5 X X 8 9 X X 5 4 X X x w 4 5 X X 8 9 X X 5 4 X X x z Y Z YZ z Z
12 Principle of ombinational Logic -.4 Quine-Mcluskey Method (Tabulation Method) eveloped in the mid 95s. systematic procedure for generating all prime implicants and extracting a minimum set of primes covering the on-set. Ex. F f,,,,,,8,,,4,5 Step : Group binary representation of the minterms according to the number of s contained. Step : ny two minterms which differ from each other by only one variable can be combined, and the unmatched variable removed. The minterms in one section are compared with those of the next section down only, because two terms differing by more than one bit cannot match. Step : Repeat step. Step 4: The unchecked terms in the table form the prime-implicants. Step 5: Prime-implicant table etermination of prime-implicant Step Step Step, -,,8, - -, -,8,, - -,8 -,,4, , -,4,, , -, - 4,4-5,5-4,5 - Prime-implicant table Minterms 8 4 5,(-) X X,,8,(--) X X X X,,4,5(--) X X X X F
13 Principle of ombinational Logic - Ex. F,,, m,,,,5 d,,5 etermination of prime-implicant Step Step Step,(),,,(,) - -,(),,,(,) - -,(),,5,(,4) - -,5(4),,5,(,4) - - 5,(),,,5(4,8) - -,(4),,,5(4,8) - -,(8) 5 5,(),5(8),5(4) Prime-implicant table Minterms 5,,,(--) X X,,5,(--) X X X,,,5(--) X X X X F or F Ex. F,,, m,4,,,8,9,,,5 etermination of prime-implicant Step Step Step,9(8) 8,9,,(,) ,() 8,9,,(,) ,9() 8,() 9,() 9,(),(),5(8) 5,5(4)
14 Principle of ombinational Logic -4 Prime-implicant table Minterms ,9(-) X X 4,(-) X X,(-) X X,5(-) X X,5(-) X X 8,9,,(--) X X X X F.5 Positive Versus Negative Logic F Voltage Truth Table F Low Low Low Low High Low High Low Low High High High Positive Logic F Negative Logic F Ex. esign the logic so an active low output is generated when power is on (an active high signal), the system is not reset (an active low signal), an interlock is closed (an active low signal), a run signal is present (active low), and data are ready (active high) Input variables Power on = PWR.H Reset = RST.L Interlock = ITL.L Run = RUN.L ata Ready = RY.H Output variables Out = OUT.L PWR.H RST.L ITL.L RUN.L RY.H OUT.L
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