第 4 章 逻辑门和布尔代数第一部分：逻辑门孙卫强1sunwqsjtu.edu.cn内容提要o逻辑门(Logic Gates)n反相器(Inverter)n与门(AND gate)n或门(OR gate)n与非门(NAND gate)n或非门(NOR gate)n异或和同或门(XOR 和XNOR)o需要掌握的内容n逻辑符号和表达式n真值表n波形图2sunwqsjtu.edu.cn反相器InputOutputLOW (0)HIGH (1)HIGH (1)LOW (0)低电平有效的指示符号3sunwqsjtu.edu.cn反相器的波形图InputOutputLOW (0)HIGH (1)HIGH (1)LOW (0)前半部分电路后半部分电路t电压t电压4sunwqsjtu.edu.cn与门(AND gate)InputOutputABLOW (0)LOW (0)LOW (0)LOW (0)HIGH (1)LOW (0)HIGH (1)LOW (0)LOW (0)HIGH (1)HIGH (1)HIGH (1)前半部分电路后半部分电路t电压t电压t电压5sunwqsjtu.edu.cn其它的逻辑门o或门o与非门（非或门）o或非门（非与门）6sunwqsjtu.edu.cn异或门(XOR gate)oXOR: Exclusive-ORo只有当两个输入具有相反的逻辑值时才输出1InputOutputAB000011101110作用：用以产生二进制加法的和7sunwqsjtu.edu.cn同或门(XNOR gate)oNXOR: Exclusive-NORo只有当两个输入具有相同的逻辑值时才输出1InputOutputAB001010100111ABAB =AB =8sunwqsjtu.edu.cn集成电路逻辑门oCMOS vs. TTLo逻辑系列n74HC00o逻辑门的种类（课本129页）n00：4个二输入与非门n02: 4个二输入或非门n04: 十六进制反相器n30: 单个8输入的与非门9sunwqsjtu.edu.cn集成门电路的特性o传输延时(Propagation Delay)o直流输入电压(DC supply Voltage)o功耗(Power Dissipation)o输入和输出逻辑电平(Logic Levels)nVIL,VIHnVOL,VOHo速度功率乘积o扇出和负载10sunwqsjtu.edu.cn逻辑电平和噪声容限高态直流高态直流噪声容限噪声容限低态直流低态直流噪声容限噪声容限11sunwqsjtu.edu.cnCMOS门电路参数的实例12sunwqsjtu.edu.cn第 4 章 逻辑门和布尔代数第二部分：布尔代数孙卫强13sunwqsjtu.edu.cn布尔代数（Boolean algebra）o1854年， George Boole， An Investigation of the Laws of Thought，on which are Founded the Mathematical Theories of Logic and Probabilitiesn提出了布尔代数，“基于人类逻辑思考的本性”，将思想翻译成符号。并且指出，这些符号只需要两个值，即0和1o1938年，Claude E. Shannon, A Symbolic Analysis of Relay and Switching Circuits,硕士论文n提出将布尔代数用于分析和优化继电器逻辑电路14sunwqsjtu.edu.cnGeorge BooleNature and Design of this WorkoThe design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed;to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.o“the operations of the mind are in a certain real sense subject to laws, and that a science of the mind is therefore possible.”15sunwqsjtu.edu.cn公理 (Axiom)o公理(A1) 如果X1，则X=0；(A1)如果X0，则X=1(A2) 如果X=0，则X=1； (A2)如果X=1，则X=0(A3) 00=0 (A3)1+1=1(A4) 11=1 (A4)0+0=0(A5) 01=10=0 (A5)0+1=1+0=1这些公理已经完备地描述了布尔代数。这些公理已经完备地描述了布尔代数。16sunwqsjtu.edu.cn定理 (Theorem)o交换律(Commutative Law)nA+B = B+A, AB = BAo结合律(Associative Law)nA+(B+C) = (A+B)+CnA(BC) = (AB)Co分配律(Distributive Law)nA(B+C) = AB+AC17sunwqsjtu.edu.cn布尔代数常用的公式18sunwqsjtu.edu.cnDe Morgan定理oAugustus De Morgan (1806 - 1871) n和George Boole一起，是符号逻辑(Symbolic Logic)的奠基人oDeMorgan定理n变量乘积取反等于将每个变量取反，然后再求其和n变量求和取反等于将每个变量取反，然后再求其积19sunwqsjtu.edu.cnDe Morgan定理20sunwqsjtu.edu.cnDeMorgan定理的应用=21sunwqsjtu.edu.cn逻辑电路的布尔分析o布尔代数提供了一种描述逻辑电路工作机理的方法ADCBXCDB+CD=A(B+CD)InputOutputABCD0000000010001000011001000010100110001110100001001010100101111100111011111011111122sunwqsjtu.edu.cn本部分小结o逻辑门n基本逻辑门n符号和表达式n真值表和波形图o布尔代数n布尔代数的公理n布尔代数的定理nDeMorgan定理及其应用n布尔代数和逻辑电路分析23sunwqsjtu.edu.cn作业o第三章：2,10,13,34,42,43o第四章：2,3,6e,10bd,13cd24sunwqsjtu.edu.cn