two values, false and true used in mathematical logic, and to two voltage levels, LOW and HIGH used in switching circuits.

Transcription

1 Digil Logi/Design. L. 3 Mrh 2, 26 3 Logi Ges nd Boolen Alger 3. CMOS Tehnology Digil devises re predominnly mnufured in he Complemenry-Mel-Oide-Semionduor (CMOS) ehnology. Two ypes of swihes, s disussed in se.., re implemened s pir of omplemenry MOS Fields-Effe-Trnsiors FETs: nmos rnsisor normlly open swih, pmos rnsisor normlly losed swih, Suh pir of rnsisors is used o uild n inverer nd onsequenly ll digil devies When he inpu volge V is low, h is, he logi signl =, he pmos rnsisor is losed nd nmos rnsisor is open. Consequenly he oupu volge V y is high, h is, he logi signl y =. The inverse siuion ours when he inpu volge is high. A.P. Ppliński 3 Digil Logi/Design. L. 3 Mrh 2, Boolen Alger nd Logi Ges Operions performed y logi ges n e onvenienly desried in Boolen lger. The (wo-vlued) Boolen lger is defined on se of wo elemens, B = {, }, wo inry operors, OR (+) nd AND ( ), one unry operor, NOT ( ), ( ) Two Boolen vlues nd orrespond o wo vlues, flse nd rue used in mhemil logi, nd o wo volge levels, LOW nd HIGH used in swihing iruis. A.P. Ppliński 3 2

2 Digil Logi/Design. L. 3 Mrh 2, 26 Three si Boolen (logi) operions nme: AND OR NOT symol: operion: y = y = + y = y y The resul of he AND operion is if nd only if oh opernds re. We lso sy h he oupu of he AND ge is HIGH (ssered) if oh inpu signls re HIGH (ssered). y A 2-vrile ruh le liss vlues of he oupu signl y {, } (resuls of logi operions) for ll possile ominions of inpu signls,, {, } (opernds). The AND operion or logi mulipliion is idenil wih rihmei mulipliion. The resul of he OR operion is if les one opernd is. We lso sy h he oupu of he OR ge is HIGH (ssered) if les one inpu signl is HIGH (ssered). The OR operion or logi ddiion differs from rihmei ddiion, euse + = no 2! The NOT operor or logi omplemen n e rihmeilly inerpreed y he following epression: = The NOT ge or INVERTER omplemens inpu signls: =, =. A.P. Ppliński 3 3 Digil Logi/Design. L. 3 Mrh 2, Timing digrms Operions performed y logi ges n lso e desried y mens of iming digrms.. + In he emple he pir of inpu signls, goes hrough ll possile ominions in he following wy: = ( ) 2 {,, 3, 2} Suh ode is known s he Gry ode A.P. Ppliński 3 4

3 Digil Logi/Design. L. 3 Mrh 2, Boolen Epressions nd Logi Digrms Boolen epressions re formed from: wo Boolen onsns, (, ), hree si logi operions, (, +, ), nd prenheses ( ). The order of evluion is: epressions inside prenheses, omplemen (NOT), logi mulipliion (AND), Logi ddiion (OR). Consider Boolen (logi) epression f = + = + where,,, f nd Boolen vriles. The equivlen logi digrm: f A.P. Ppliński 3 5 Digil Logi/Design. L. 3 Mrh 2, 26 In order for he logi ddiion o e performed efore mulipliion we hve o dd prenheses: The equivlen logi digrm: f = ( + ) = ( + ) f The AND operor (he mulipliion sign) my e omied nd we n wrie f = + or f = ( + ) Idenify: inpu-oupu pors, or signls (wires) onneed o he ouside world ges nd signl or nes (wires), h is. signls ineronneing ges inpus nd oupus. A.P. Ppliński 3 6

4 Digil Logi/Design. L. 3 Mrh 2, VHDL Hrdwre Desripion Lnguge emple Digil devies/iruis n e desried/modelled using hrdwre desripion lnguge, VHDL. The desripion onsise of wo min prs: Inpu-oupu pors re speified y he ENTITY The irui sruure or funion is speified y n ARCHITECTURE Emple: The onens of he log ir.vhd file: -- his is ommen -- VHDL is NOT se sensiive -- To emphsize, he key words re pilized -- his is lk-o logi irui ENTITY log_ir_ IS PORT (,, : IN BIT ; f : ou BIT ) ; END [ENTITY] log_ir_ ; -- n rhieure for log_ir_ ARCHITECTURE rh_ OF log_ir_ IS BEGIN f <= ( OR NOT ) AND ; -- SIGNAL ssignmen END [ARCHITECTURE] rh_ ; A.P. Ppliński 3 7 Digil Logi/Design. L. 3 Mrh 2, Truh les nd Krnugh Mps The ehviour of logi irui, h is, he vlues of he oupu signls for ll ominions of inpu signls n e equivlenly desried y: Boolen epression, he ruh le, he Krnugh mp, Consider he following Boolen (logi Funion f = ( + ) The ruh le liss he vlues of he funion f for ll 2 3 = 8 ominions of hree inpu vriles () 2 + f A.P. Ppliński 3 8

5 Digil Logi/Design. L. 3 Mrh 2, A 3-vrile Krnugh mp Krnugh mps re represenions of Boolen hyper-ues. A onep of djen veries A Krnugh mp for f = ( + ) 3.8 Theorems of Boolen Alger nd heir irui inerpreion Trnsformions nd simplifiion of logi iruis re sed on vriey of Boolen lger heorems whih n esily e verified y Doule Complemen Involuion propery = Doule NOT operion n e removed. A.P. Ppliński 3 9 Digil Logi/Design. L. 3 Mrh 2, 26 Operions wih onsns + = + = = = Operions wih repeed rgumens + = + = = = A.P. Ppliński 3

6 Digil Logi/Design. L. 3 Mrh 2, 26 OR nd AND re ommuive operions ll ge inpus re idenil: OR: + = + AND: = OR nd AND re ssoiive operions n-inpu ges eis: OR: + ( + ) = ( + ) + = + + OR operion (logi ddiion) is disriuive ( + ) = + AND operion (logi mulipliion) is lso disriuive! + = ( + ) ( + ) A.P. Ppliński 3 Digil Logi/Design. L. 3 Mrh 2, 26 Verifiion of he disriuive lw for he AND operion using he ruh le mehod: + = ( + ) ( + ) () 2 LHS + + RHS For ll ominions of vriles LHS = RHS, herefore, he disriuive lw for he logi mulipliion is vlid. A.P. Ppliński 3 2

7 Digil Logi/Design. L. 3 Mrh 2, 26 Duliy Priniple Every heorem of he Boolen lger remins vlid if he operors nd onsns re inerhnged, h is: AND OR Emple: If he following equliy ( + ) = ( ) + ( ) is vlid, hen inerhnging + wih we oin he dul equliy: whih is lso vlid. + ( ) = ( + ) ( + ) A.P. Ppliński 3 3 Digil Logi/Design. L. 3 Mrh 2, 26 Asorpion Rules. + = Verifiion y lgeri mnipulion: + = ( + ) =, euse + = 2. ( + ) = he dul equliy 3. + = + Imporn! 4. ( + ) = Asorpion rules re imporn in irui simplifiion. A.P. Ppliński 3 4

8 Digil Logi/Design. L. 3 Mrh 2, 26 De Morgn s Theorems. The omplemen of produ (AND) is equl o he sum (OR) of he omplemens: ( ) = +. (. ) + equivlenly (. ) + This is he NAND ge (NOT AND) 2. The omplemen of sum (OR) is equl o he produ (AND) of he omplemens (he dul heorem): ( + ) = + ( + ). equivlenly ( + ) This is he NOR ge (NOT OR). A.P. Ppliński 3 5 Digil Logi/Design. L. 3 Mrh 2, All wo-vrile funions y = F (, ) A.P. Ppliński 3 6

9 Digil Logi/Design. L. 3 Mrh 2, NAND nd NOR ges he ruh les for he NAND nd NOR ges A.P. Ppliński 3 7