Digital Electronics Basics: Combinational Logic

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1 Digitl Eletronis Bsis: for Bsi Eletronis by Prof. Mihel Tse Jnury 25

2 Digitl versus nlog So fr, our disussion bout eletronis hs been predominntly nlog, whih is onerned with ontinuously hnging signls signls whose vlues t different times re useful informtion. There is nother form of signls whih is nowdys very importnt nd is being used in n overwhelming number of pplitions DIGITAL. A digitl signl ssumes just few nlog vlues. For exmple, binry signls hve two vlues, e.g., V nd 5V. They trnsmit informtion in the form of sequene of V nd 5V segments. 5V V 2

3 Digitl signls Of ourse, in prtil genertion nd detetion of digitl signls, ext vlues of the ssumed voltge levels re not possible. Noise Mrgins: V OH V OL V 5V V IH V IL Genertion: Detetion: Any voltge higher thn V OH is HIGH. (V OH = 2.4V*) Any voltge lower thn V OL is LOW. (V OL =.4V) Any voltge higher thn V IH is HIGH. (V IH = 2.V) Any voltge lower thn V IL is LOW. (V IL =.8V) * for TTL logi whih is prtiulr kind of logi iruit fmily. 3

4 Logi Wht n digitl iruit do? The simplest tsk we n think of is ombintionl type of logi deision. For exmple, we n design digitl eletroni iruit to mke n instnt deision bsed on some informtion. Here we emphsize instnt in the deision mking proess. Tht mens, the proess hs no time dely. = tody s wether is good = tody is holidy deision = go to pini Suppose our rule is = nd The iruit is simple AND gte. This kind of logi, involving no time dely, is ombintionl logi. 4

5 Truth tbles A esy wy to represent ombintionl logi result is to tbulte ll possible inputs. Truth tble of AND Truth tble of OR Truth tble of NOT Truth tble of NAND Truth tble of NOR Truth tble of OR (Exlusive OR) 5

6 Boolen lgebr Logi n lso be expressed in lgebri form. =. AND gte Truth tble of AND Truth tble of NAND =. NAND gte 6

7 Boolen lgebr Logi n lso be expressed in lgebri form. = + OR gte Truth tble of OR Truth tble of NOR = + NOR gte 7

8 Finding expression from truth tble One we hve the truth tble, we n find the output expression by dding up ll min-terms. Min-term orresponds to the produt term tht give in the output. For exmple, here the min-terms re.,., nd. The expression for is = Question: This is n OR gte! Cn it be simplified down to = +? How n we do it? 8

9 More exmples The expression for is = Truth tble of NOR The expression for is =. Question: These re NAND nd NOR gtes! Cn they be simplified or onverted bk to the originlly derived forms? How n we do it? 9

10 Ciruit reliztion = Note: This is sme s NAND gte (see the truth tble), nd hene should be the sme s The question is HOW TO SIMPLIF A MIN-TERM EPRESSION!

11 Exmple: binry ode to Gry ode onversion Binry Gry ode b x y z Wht re the expressions for x, y nd z? Then, n we design iruit to onverter binry ode to Gry ode? x = b + b + b + b y = b + b + b + b z = b + b + b + b So, for eh of x, y nd z, we need number of inverters, plus 4 AND gtes nd multi-input OR gtes.

12 Boolen lgebr simplifition Bsi Lws: Commuttive: + = +. =. Assoitive: ++ = (+)+ = +(+).. = (.). =.(.) Distributive:.(+) =. +. De Morgn s: + =.. = + 2

13 Exmple: = =.( + ) +. =.+. = +. = (.( + )) = (. +. ) =. + =. = De Morgn lw Distributive. = Tht s right! It s NAND gte! 3

14 Exmple: = = ( + ). +. = +. =.(. ) =.( + ) =. + =. = + De Morgn lw De Morgn lw. = De Morgn lw Tht s right! It s OR gte! 4

15 Boolen lgebr n be tedious. Is there ny esier method to simplify the min-term expression? 5

16 Krnugh Mps A very useful design tool for simplifying ombintionl logi. The expression for is = Therefore = + or =. 6

17 Krnugh mp (with 3 inputs) Suppose we hve three inputs, b nd. Output x is x =.b. +.b. +.b. +.b. b b b Proedure:. Cirle lusters of. 2. Determine the logi expressions for eh luster. 3. Add them up. b b b Hene, we get x =.b. + b. +.b 7

18 x Exmple: Gry ode Binry Gry ode b x y z b b b x = y z b b b b b b y = b + b z = b + b 8

19 Exmple: Gry ode Binry Gry ode b x y z b x = y = b + b z = b + b x y z 9

20 Krnugh mp (with 4 inputs) b d b b b d d d 2

21 Exmple Proedure:. Cirle lusters of. 2. Determine the logi expressions for eh luster. 3. Add them up. b d b b b b d d d x = + b + 2

22 Exmple Derive the iruit for this ombintionl logi. x = + b + b x 22

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