Sistemas Digitais Captulo II Fundamentos do Projecto Digital 1.

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  • Sistemas DigitaisCaptulo IIFundamentos do Projecto Digital1

  • Sistemas DigitaisAlgebra de BooleVarivel LgicaVarivel que tem por domnio dois valores lgicos distintos, normalmente representados por 0 e 1.Funo LgicaFuno que tem por contradomnio o conjunto lgico [0 1].Expresso LgicaConjunto de variveis e constantes lgicas ligadas entre si pelos sinais das funes lgicas elementares.2

  • Sistemas DigitaisFunes ElementaresOperadores Lgicos ElementaresComplementaoIntercepoReunio3

  • Sistemas DigitaisTabelas de Verdade Vantagens

    Facilidade com que se obtm a partir da formulao verbal da funo a implementar; Facilidade na obteno das expresses algbricas nas formas cannicas; Possibilidade de passar directamente implementao de funes lgicas com certos componentes digitais; Constituem o ponto de partida para mtodos grficos e tabulares de simplificao de funes (Mapas de Karnaugh). Representam o valor de determinada funo lgica mediante todas as suas combinaes.11

  • Sistemas DigitaisAxiomas8

  • Sistemas DigitaisAxiomas8

  • Sistemas DigitaisTeoremas da lgebra de Boole9

  • Sistemas DigitaisTeoremas da lgebra de Boole9(OBVIO)

  • Sistemas DigitaisTeoremas da lgebra de Boole9

  • Sistemas DigitaisTeoremas da lgebra de Boole9

  • Sistemas DigitaisFormas Algbricas das Expresses LgicasSoma de ProdutosProdutos de SomasA expresso lgica constituda por somas lgicas de produtos lgicos.A expresso lgica constituda por produtos lgicos de somas lgicas.Formas CannicasConstituem o ponto de partida para alguns mtodos para anlise de circuitos lgicos. a partir delas que se procede implementao de circuitos.Formas Mnimas normalmente a partir das formas mnimas que se procede implementao de circuitos lgicos com portas lgicas discretas. (circuitos mais simples)4

  • Sistemas DigitaisFormas Algbricas das Expresses LgicasForma Mnima Produto de SomasForma Mnima Soma de ProdutosForma Cannica Soma de ProdutosForma Cannica Produtos de SomasFormas CannicasFormas MnimasSomas lgicas de produtos lgicos que contm todas as variveis da funo.Produtos lgicos de Somas lgicas que contm todas as variveis da funo.A expresso lgica constituda por uma soma de produtos tal que o somatrio do numero de produtos e literais mnimo.A expresso lgica constituda por um produto de somas tal que o somatrio do numero de somas e literais mnimo.5

  • Sistemas DigitaisDefinies Mintermo - Produto em que cada uma das variveis aparece apenas uma vez, na forma complementada ou no complementada. A cada combinao de valores das variveis de entrada est associado um mintermo. Maxtermo - Soma em que cada uma das variveis aparece apenas uma vez, na forma complementada ou no complementada. A cada combinao de valores das variveis de entrada est associado um maxtermo. Forma Cannica Produto de Somas - Quando a expresso constituda pelo produto lgico dos maxtermos para os quais a funo toma o valor 0. Forma Cannica Soma de Produtos - Quando a expresso constituda pela soma lgica dos mintermos para os quais a funo toma o valor 1.6

  • Sistemas DigitaisExemplosForma Mnima Produto de SomasForma Mnima Soma de ProdutosForma Cannica Soma de ProdutosForma Cannica Produtos de Somas7

  • Sistemas DigitaisFormas Cannicas a Partir da Tabela de Verdade Dada a Tabela de Verdade da Funo F(A,B,C):- F(A,B,C) uma funo lgica de 3 variveis A,B e C, assumindo o valor 1 para as combinaes 001, 010 e 110 e o valor 0 para as combinaes 000, 011, 100, 101 e 111.- Recorrendo definio de mintermo e maxtermo, facilmente se obtm da tabela a expresso lgica dafuno na forma cannica soma de produtos:ou na forma cannica produto de somas:12

  • Sistemas DigitaisPortas Lgicas13

  • Sistemas DigitaisPortas Lgicas14

  • Sistemas DigitaisPortas Lgicas15

  • Sistemas DigitaisPortas Lgicas16

  • Sistemas DigitaisSntese de Circuitos Lgicos Expresso Lgica Circuito Lgico Correspondente17

  • Sistemas DigitaisSntese de Circuitos Lgicos (Estruturas a dois Nveis) Expresso Lgica (Forma Cannica Soma de Produtos / Estrutura AND-OR) Circuito Lgico Correspondente18

  • Sistemas DigitaisSntese de Circuitos Lgicos (Estruturas a dois Nveis) Expresso Lgica- Forma Cannica Soma de Produtos - Estrutura NAND-NAND Circuito Lgico Correspondente19

  • Sistemas DigitaisSntese de Circuitos Lgicos (Estruturas a dois Nveis) Expresso Lgica- Forma Mnima Soma de Produtos- Estrutura AND-OR Circuito Lgico Correspondente20

  • Sistemas DigitaisSntese de Circuitos Lgicos (Estruturas a dois Nveis) Expresso Lgica Forma Mnima Soma de Produtos Estrutura NAND-NAND Circuito Lgico Correspondente21

  • Sistemas DigitaisSntese de Circuitos Lgicos (Estruturas a dois Nveis) Expresso Lgica Forma Cannica Produto de Somas Estrutura OR-AND Circuito Lgico Correspondente22

  • Sistemas DigitaisSntese de Circuitos Lgicos (Estruturas a dois Nveis) Circuito Lgico Correspondente Expresso Lgica Forma Cannica Produto de Somas Estrutura NOR-NOR23

  • Sistemas DigitaisSntese de Circuitos Lgicos (Estruturas a dois Nveis) Expresso Lgica Forma Mnima Produto de Somas Estrutura OR-AND Circuito Lgico Correspondente24

  • Sistemas DigitaisSntese de Circuitos Lgicos (Estruturas a dois Nveis) Circuito Lgico Correspondente Expresso Lgica Forma Mnima Produto de Somas Estrutura NOR-NOR25

  • Sistemas DigitaisMapas de Karnaugh Representao grfica das tabelas de verdade, servindo de ponto de partida para a minimizao de circuitos combinacionais;

    O mapa, para uma funo lgica de n-variveis, um conjunto de 2n clulas, uma para cada combinao das entradas;26

  • Sistemas DigitaisMapas de Karnaugh Cada clula do Mapa de Karnaugh contm informao sobre a funo. A clula contm 0 ou 1, se ao termo mnimo correspondente na tabela de verdade for respectivamente 0 ou 1;

    Cada clula corresponde a uma combinao das entradas que difere das clulas adjacentes numa s varivel.27

  • Sistemas DigitaisMapas de Karnaugh Para obter uma expresso sob a forma mnima soma de produtos realizamos os seguintes passos:

    1 Realizar agrupamentos de 1s. Agrupar o maior nmero de 1s possvel;. A quantidade de 1s em cada grupo tem que ser da ordem 2n;. Apenas podemos agrupar 1s adjacentes;. 1s em ambos os extremos da tabela so adjacentes;. Todos os 1s tm que ser agrupados.28

  • Sistemas DigitaisMapas de Karnaugh

    2 Obter as expresses lgicas correspondentes a cada agrupamento:. Retirar sob a forma de produto as variveis que correspondem a uma variao no agrupamento.

    3 A expresso final corresponde soma dos diversos produtos obtidos.29

  • Sistemas Digitais30Mapas de Karnaugh

    - Para obter a expresso lgica sob a forma mnima produto de somas procederamos do mesmo modo, no entanto, agrupando 0s ao invs de 1s e negando as respectivas variveis.

  • Sistemas Digitais31Portas e Famlias LgicasCircuito Integrado Componentes electrnicos, integrados num nico dispositivo, que implementam funes lgicas elementares ou outras funes mais complexas.Classificao quanto ao Nvel de IntegraoCada fabricante disponibiliza as especificaes tcnicas e funcionais dos CIs (Datasheets).

    Tipo de Circuito IntegradoN. de Portas LgicasSSI (Small-Scale Integrated)< 10MSI (Medium-Scale Integrated)De 10 a 100LSI (Large-Scale Integrated)De 100 a algumas centenasVLSI (Very Large-Scale Integrated)At 100 milhes

  • Sistemas Digitais32Portas e Famlias LgicasFamlia Lgica Conjunto de circuitos integrados que tm em comum a mesma tecnologia de fabrico.Principais FamliasTTL(Transistor Transistor Logic)CMOS(Complementary Metal Oxid Semiconductor)- Integra transstores bipolares para permitir maior rapidez, embora custa de maior consumo de energia.- Utiliza transstores de efeito de campo, sendo preferida para a implementao de circuitos mais complexos (menos espao em CI) e apresenta um consumo de energia bastante inferior.

  • Sistemas Digitais33Outras Famlias Lgicas

    ECL Emitter Coupled LogicElevada rapidez de funcionamento embora com consumos demasiado elevados. Apenas utilizada quando a velocidade de funcionamento constitua um requisito predominante.

    RTL Resistor-Transistor Logic

    DTL Diode-Transistor Logic

    I2L Integrated-Injection LogicPortas e Famlias LgicasRAPIDAMENTE TORNADAS OBSOLETAS COM A EVOLUO DA ACTUAL TECNOLOGIA

  • Sistemas Digitais34Portas e Famlias LgicasSub-Famlias Lgicas (TTL)

    L (Low Power) Menor consumo, mais lentos;

    S (Schottky) Maior rapidez, maior consumo;

    LS (Low Power Schottky) Circuitos rpidos e com consumos reduzidos;

    AS (Advanced Schottky) Nova gerao com melhores caractersticas relativamente sub-famlia S embora com maiores custos;

    ALS (Advanced Low Schottky) Nova gerao com melhores caractersticas relativamente famlia LS embora com maiores custos.

  • Sistemas Digitais35Sub-Famlias Lgicas (CMOS)

    HC (High-Speed CMOS)

    HCT (High-Speed CMOS/TTL Interface)

    AHC (Advanced High-Speed CMOS)

    AHCT (Advanced High-Speed CMOS/TTL Interface)

    As sub-famlias AHC e AHCT encontram-se disponveis nas verses de 5V e 3V. Portas e Famlias Lgicas

  • Sistemas Digitais36Caractersticas ElctricasTenso de Alimentao A tenso nominal de alimentao de um CI TTL de 5V;

    A tenso nominal de alimentao de um CI CMOS mais flexvel. Um CI CMOS opera com tenses de alimentao at 10V, no entanto usual utilizar-se 5V para que haja compatibilidade com circuitos digitais TTL.

    Para a categoria Low Voltage CMOS, tenso nominal de alimentao de 3V.

  • Sistemas Digitais37Nveis Lgicos para um CI CMOS (5V)Caractersticas Elctricas

  • Sistemas Digitais38Nveis Lgicos para um CI CMOS (3V)Caractersticas Elctricas

  • Sistemas Digitais39Nveis Lgicos para um CI TTLCaractersticas Elctricas

  • Sistemas Digitais40Imunidade a RudoCaractersticas Elctricas

  • Sistemas Digitais41Margem de RudoCaractersticas ElctricasUma medida de imunidade a rudo de uma famlia lgica a denominada Margem de Rudo, definida pelos parmetros VNH e VNL , dados por:

  • Sistemas Digitais42Consumo de Energia (Potncia)Caractersticas ElctricasOs circuitos TTL tm o seu consumo independente da frequncia de operao, enquanto os CIs CMOS tm um consumo que varia de modo aproximadamente linear com a frequncia.

  • Sistemas Digitais43Sempre que um impulso se propaga atravs de uma porta lgica, sofre um atraso no tempo (time delay).Tempo de Propagao tPHL Intervalo de tempo decorrido entre o instante especificado no impulso de entrada da porta e o instante correspondente ao impulso de sada, durante uma transio do estado HIGH para LOW.

    tPLH Intervalo de tempo decorrido entre o instante especificado no impulso de entrada da porta e o instante correspondente ao impulso de sada, durante uma transio do estado LOW para HIGH.

  • Sistemas Digitais44Existe um limite mximo de entradas de portas lgicas que podem ser ligadas simultaneamente sada de uma porta lgica. Este limite designado por Fan-Out.Exemplo: O Fan-Out tpico de uma porta TTL padro de 10. No caso de uma porta Low Power Schottky (LS) o Fan-Out de 20.Fan-OutCaractersticas Elctricas

  • Sistemas Digitais45O elemento bsico de uma porta CMOS o MOSFET (Metal-Oxid Semiconductor Field Effect Transistor)Portas da Famlia CMOS

  • Sistemas Digitais46Circuitos interno tpico de uma porta NOT da famlia CMOS.Portas da Famlia CMOS

  • Sistemas Digitais47Operao de uma porta NOT da famlia CMOS.Portas da Famlia CMOS

  • Sistemas Digitais48Precaues no uso de CIs CMOS.Todos os dispositivos CMOS so sujeitos a serem danificados por electricidade esttica. Em funo disto recomendado que:

    Todos os CIs CMOS sejam envolvidos em espuma plstica condutora para efeito de transporte. Quando removidos da espuma protectora, os pinos do Cl no devem ser tocados;

    Os CIs sejam depositados com os pinos para baixo sobre um superfcie metlica aterrada, quando removidos da espuma protectora. No depositar CIs CMOS sobre qualquer material plstico devido s cargas elctricas estticas que inevitavelmente so geradas por este material;Portas da Famlia CMOS

  • Sistemas Digitais49 Todas as ferramentas, equipamentos de teste e bancadas de metal sejam aterradas. Quando o ambiente de trabalho tal que o ar atmosfrico seja muito seco, somente manusear os CIs CMOS com uma pulseira presa ao pulso ou antebrao, pulseira que deve ser ligada terra atravs de uma resistncia de aproximadamente 1OOK. A resistncia evita choque maior caso a pessoa toque numa fonte de tenso e, ao mesmo tempo, descarrega as cargas elctricas acumuladas durante o manuseamento;

    Os CIs devem ser inseridos nos seus sockets somente quando a tenso de alimentao estiver desligada;Precaues no uso de CIs CMOS.Portas da Famlia CMOS

  • Sistemas Digitais50 Todos os terminais de entrada que no forem utilizados devem ser ligados alimentao ou terra. Se deixados abertos sero influenciados por cargas elctricas estticas na vizinhana, o que provocar um funcionamento errado do circuito.Precaues no uso de CIs CMOS.Portas da Famlia CMOS

  • Sistemas Digitais51O elemento bsico de uma porta TTL o Transstor BipolarPortas da Famlia TTL

  • Sistemas Digitais52Sadas em Totem-PoleA maioria das portas lgicas apresentam-se numa configurao totem-pole, designao que resulta da existncia de dois dispositivos de comutao (Sa, Sb) colocados conforme se indica na figura.

    Quando se encontram ligadas duas sadas em simultneo, pode ocorrer o curto-circuito da fonte de alimentao.

    Devem ser tomados cuidados para que isso nunca acontea, ou utilizar circuitos com outro tipo de sadas.Configuraes de SadaInternamente, o controlo dos dois comutadores realizado de modo a que esteja fechado um e s um dos comutadores.

  • Sistemas Digitais53A interligao de duas ou mais sadas em totem-pole no praticvel. Como se verifica na figura, quando a sada Y1 colocada no estado alto(H) e a sada Y2 colocada no estado baixo (L), existe um curto-circuito entre Vcc e GND atravs dos comutadores fechados Sa do CI1 e Sb do CI2. Alm disso, o estado da sada imprevisvel. Dependendo das caractersticas dos circuitos de sada, a tenso pode estar no intervalo de variao para o estado alto, como para o estado baixo, ou ainda no intervalo de indefinio.Sadas em Totem-Pole

  • Sistemas Digitais54Sadas em Colector Aberto (Open Colector)Configuraes de SadaNeste tipo de sada, em termos da representao simplificada com comutadores, apenas existe o comutador Sb que quando fechado coloca a sada no estado baixo. No circuito externo sada em colector aberto deve existir uma resistncia ligada a Vcc (+5V) resistncia de pull-up - para que a sada apresente um valor aceitvel no intervalo de variao para o estado alto quando Sb est aberto.

  • Sistemas Digitais55As sadas dos circuitos em colector aberto tm a possibilidade de interligao a um ponto comum.Conforme se pode verificar na figura, basta que uma das sadas esteja no estado baixo (Sb fechado), para que a sada comum esteja no estado baixo. Por outro lado, a sada comum est ao nvel alto se todas as sadas esto ao nvel alto (Sb aberto). Resulta, para a sada comum, um comportamento funcional equivalente a uma porta AND designada por porta Wired-AND.Sadas em Colector AbertoConfiguraes de Sada

  • Sistemas Digitais56As sadas de trs estados so idnticas s sadas em totem-pole, existindo, no entanto, uma entrada adicional (enable) que permite que os dois comutadores (Sa e Sb) estejam abertos simultaneamente.Nestas condies diz-se que a sada est num estado de alta impedncia (Hi-Z), como se estivesse desligada do circuito.Sadas em Trs Estados (Tri-State)Configuraes de SadaEste tipo de sada tem particular aplicabilidade quando se pretende, atravs de um linha comum, enviar dados de diferentes dispositivos de origem para um nico dispositivo de destino.

  • Sistemas Digitais57Em algumas situaes, nem todas as entradas de uma porta lgica so necessrias. Por exemplo, quando necessria uma porta AND de duas entradas e apenas se tem disponvel uma porta AND de 3 entradas. Nos CIs TTL, uma entrada deixada desligada comporta-se com estando ao nvel alto (HIGH-H), enquanto que nos CIs CMOS, uma entrada deixada desligada comporta-se como estando ao nvel baixo (LOW-L). Assim sendo, no caso de uma porta AND TTL, como o valor 1 o elemento neutro do produto lgico, poderamos no ligar esta entrada. No entanto, esta uma prtica no aconselhvel, j que as entradas desligadas so muito sensveis ao rudo elctrico. De modo a garantir a fiabilidade do sistema digital, todas as entradas no utilizadas devem ser ligadas a um nvel adequado. Consideraes Prticas

  • Sistemas Digitais58O valor da resistncia R calculado em funo das correntes mximas admissveis na(s) entrada(s), de modo a no provocar incompatibilidade nos nveis de tenso.Em funo do tipo de porta lgica, as entradas no utilizadas devem ser ligadas ao nvel alto ou baixo conforme se exemplifica na figura.Consideraes Prticas

  • Sistemas Digitais59Sistemas de NumeraoSistema Decimal123 = Cento e Vinte e Trs!

    Algoritmo de Clculo de um Nmero123 = 1*102 + 2*101 + 3*100

    Smbolos Vlidos0, 1, 2, 3, 4, 5, 6, 7, 8, 9 BASE 10Sistema Binrio

    Algoritmo de Clculo de um Nmero101b = 1*22 + 0*21 + 1*20 = 1*4 + 0*2 + 1*1 = 5

    Smbolos VlidosMSB Bit mais significativo LSB Bit menos significativo0, 1, BASE 2

  • Sistemas Digitais60Sistemas de NumeraoOutros Sistemas

    Octal- Oito smbolos: 0, 1, 2, 3, 4, 5, 6, 7;

    - Pesos: 1, 8, 64, 512, 4096;

    - Base 8;

    - Exemplos:17o = 1*8 + 7 = 15124o = 1*64 + 2*8 + 4*1 = 84

    - Um digito de um nmero Octal representa 3 dgitos (bits) deum nmero binrio.

  • Sistemas Digitais61Sistemas de NumeraoOutros Sistemas

    Hexadecimal- 16 smbolos: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F;

    - Pesos: 1, 16, 256, 4096, 65536;

    - Base 16;

    - Exemplos:A7h = 10*16 + 7 = 1671DEh = 1*256 + 13*8 + 14*1 = 478

    - Um digito de um nmero Hexadecimal representa 4 dgitos (bits) de um nmero binrio.

  • Sistemas Digitais62Sistemas de NumeraoTabela de Converso

  • Sistemas Digitais63ConversesDecimalExemplo Binrio108 = 1101100b108 / 2 = 54 Resto 0 (LSB)/2 = 27 Resto 0 /2 = 13 Resto 1 /2 = 6 Resto 1 /2 = 3 Resto 0 /2 = 1 Resto 1 /2 = 0 Resto 1 (MSB)

    OctalExemplo Binrio108 = 154o108 / 8 = 13 Resto 4 (LSB) /8 = 1 Resto 5 /8 = 0 Resto 1 (MSB)

  • Sistemas Digitais64Hexadecimal DecimalExemplo: 10AFh

    1 0 A F(Hexadecimal) 0001 0000 1010 1111 (Binrio)1*1 + 1*2 + 1*4 + 1*8 + 1*32 + 1*128 + 1*4096 = = 4271 (Decimal)

    A passagem intermdia por outros sistemas de numerao pode tornar mais simples a converso. Isto acontece devido correspondncia directa entre o sistema binrio e os sistemas hexadecimal e octal.Sistemas de NumeraoTcnicas de Converso

  • Sistemas Digitais65Soma e Subtraco Binria 1 1 1 1X1901 0 1 1 1 1 1 0Y + 141 +1 0 0 0 1 1 0 1 X + Y331 1 0 1 0 0 1 0 1 1Soma de dois nmerosOverflow

    CarryOut Sinal gerado quando aps o resultado da soma sobra um algarismo esquerda (transporte).CarryIn Sinal gerado na operao anterior (transporte), considerado para efectuar a nova operao.

  • Sistemas Digitais66Representao de Nmeros Negativos Complemento de DoisSoma e Subtraco Binria Realiza-se inicialmente o Complemento de Um Invertem-se todos os bits do nmero, troca-se 0s por 1s e vice-versa.

    Cada nmero obtido do anterior somando 1 e ignorando o transporte (carry) que surja na ltima posio.

    Como a adio apenas um extenso da contagem, os nmeros em complemento de dois podem ser somados atravs da adio binria normal, ignorando qualquer carry a partir do MSB.

    O resultado ser sempre correcto desde que o intervalo do sistema de numerao seja respeitado.

  • Sistemas Digitais671 1 1 1 1 1 X1901 0 1 1 1 1 1 0Y - 141 -0 1 1 1 0 0 1 1 X - Y 49 1 0 0 1 1 0 0 0 1Representao de Nmeros Negativos (Exemplo)Soma e Subtraco BinriaIgnorado

  • Sistemas Digitais68OverflowSoma e Subtraco Binria Quando a operao de adio produz um resultado que excede um dos limites do sistema numrico, diz-se que ocorre um overflow.

    A adio de dois nmeros com sinais diferentes nunca produz um overflow. A soma de dois nmeros com sinais idnticos pode provocar um overflow.

    Existe uma regra simples para detectar um overflow na adio: Quando os nmeros a somar tm o mesmo sinal e o resultado tem sinal diferente; Se os bits CIN (Carry In) e COUT (Carry Out) da posio do sinal forem diferentes.

  • Sistemas Digitais69BCD Binary Coded DecimalSoma e Subtraco Binria Codifica os dgitos 0 a 9 atravs de representaes binrias, sem sinal, de 4-bit (0000b a 1001b). As representaes 1010b at 1111b no so utilizadas;

    As converses entre BCD e representaes digitais so simplesmente uma substituio directa de 4-bit por cada dgito decimal;

    Alguns programas colocam dois dgitos num byte (Packed BCD). O objectivo minimizar a memria ocupada. Um byte representa assim nmeros entre 0 e 99.

  • Sistemas Digitais70Adio em BCDSoma e Subtraco Binria Idntica operao de adio de nmeros de 4 bits sem sinal. No entanto, se o resultado exceder 1001, tem que ser somado 6 (0110) ao resultado de modo a obter o valor correcto;

    O BCD um cdigo onde cada dgito decimal pode ser obtido a partir da sua palavra de cdigo atribuindo a cada bit da palavra um peso especfico (8, 4, 2 e 1). Por essa razo o cdigo BCD por vezes referido como cdigo 8421.

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