Lecture 16 OUTLINE Logic Binary representations Combinatorial logic circuits Reading Chap 7 7 5 EE40 Summer 2005 Lecture 16 Instructor Octavian Florescu 1 Digital Circuits Introduction Analog signal amplitude is continuous with time Digital signal amplitude is represented by a restricted set of discrete numbers Binary only two values are allowed to represent the signal High or low i e logic 1 or 0 Digital word Each binary digit is called a bit A series of bits form a word Byte is a word consisting of 8 bits Advantages of digital signal Digital signal is more resilient to noise can more easily differentiate high 1 and low 0 Transmission Parallel transmission over a bus containing n wires Faster but short distance internal to a computer or chip Serial transmission transmit bits sequentially Longer distance EE40 Summer 2005 Lecture 16 Instructor Octavian Florescu 2 1 Binary Representation N bit can represent 2N values typically from 0 to 2N 1 3 bit word can represent 8 values e g 0 1 2 3 4 5 6 7 Conversion Integer to binary Fraction to binary 13 510 1101 12 and 0 39210 0 0110012 Octal and hexadecimal EE40 Summer 2005 Lecture 16 Instructor Octavian Florescu 3 Logic Gates and Memories Logic gates Combine several logic variable inputs to produce a logic variable output Memory Memoryless output at a given instant depends the input values of that instant Momory output depends on previous and present input values EE40 Summer 2005 Lecture 16 Instructor Octavian Florescu 4 2 Boolean algebras are algebraic structures which capture the essence of the logical operations AND OR and NOT as well as the corresponding set theoretic operations intersection union and complement They are named after George Boole an English mathematician at University College Cork who first defined them as part of a system of logic in the mid 19th century Specifically Boolean algebra was an attempt to use algebraic techniques to deal with expressions in the propositional calculus Today Boolean algebras find many applications in electronic design They were first applied to switching by Claude Shannon in the 20th century EE40 Summer 2005 Lecture 16 Instructor Octavian Florescu 5 Boolean algebras The operators of Boolean algebra may be represented in various ways Often they are simply written as AND OR and NOT In describing circuits NAND NOT AND NOR NOT OR and XOR eXclusive OR may also be used Mathematicians often use for OR and for AND since in some ways those operations are analogous to addition and multiplication in other algebraic structures and represent NOT by a line drawn above the expression being negated EE40 Summer 2005 Lecture 16 Instructor Octavian Florescu 6 3 Boolean Algebra NOT operation inverter AND operation AgA A Ag1 A AgA 0 A A 1 Ag0 0 AgB BgA AgB gC Ag B gC OR operation A A A A 1 1 A 0 A A B B A A B C A B C EE40 Summer 2005 Lecture 16 Instructor Octavian Florescu 7 Graphic Representation A gA 0 A A 1 A A Full square complete set 1 Yellow part NOT A A White circle A EE40 Summer 2005 Lecture 16 Instructor Octavian Florescu 8 4 Graphic Representation A B A AB B A B AB AB A B g A B AgB A B Exclusive OR yellow and blue part intersection overlap part exactly when only one of the input is true EE40 Summer 2005 Lecture 16 Instructor Octavian Florescu 9 Boolean Algebra Distributive Property Ag B C AgB AgC A B gC A B g A C De Morgan s laws A B AgB AgB A B An excellent web site to visit http en wikipedia org wiki Boolean algebra EE40 Summer 2005 Lecture 16 Instructor Octavian Florescu 10 5 Examples F A B C A B C C D D E F C A D E D E EE40 Summer 2005 Lecture 16 Instructor Octavian Florescu 11 Logic Functions Symbols Notation NAME NOT SYMBOL A OR A B AND A B EE40 Summer 2005 Lecture 16 NOTATION F F F F A TRUTH TABLE A F 0 1 1 0 F A B A B 0 0 0 1 1 0 1 1 F 0 1 1 1 F A B A B 0 0 0 1 1 0 1 1 F 0 0 0 1 Instructor Octavian Florescu 12 6 Logic Functions Symbols Notation 2 A B NOR F A B NAND XOR exclusive OR F A B F EE40 Summer 2005 Lecture 16 F A B A B 0 0 0 1 1 0 1 1 F 1 0 0 0 F A B A B 0 0 0 1 1 0 1 1 F 1 1 1 0 F A B A B 0 0 0 1 1 0 1 1 F 0 1 1 0 Instructor Octavian Florescu 13 Circuit Realization A B AB AB A B g A B A gB A B A A B AB A B B AB EE40 Summer 2005 Lecture 16 Instructor Octavian Florescu 14 7 Fan in Fan out Complex digital operations are formed with a variety of gates interconnected to yield the desired logic function Sometimes a number of inputs are connected to one gate input and output of a gate may be connected to a number of gates Fan in the maximum number of logic gates that can be connected at the input of a gate without altering its performance Fan out the maximum number of logic gates that can be connected to the output of a gate without altering its performance Typical fan in and fan out numbers are 3 EE40 Summer 2005 Lecture 16 Instructor Octavian Florescu 15 Inverter NOT Gate Vin Vout Ideal Transfer Characteristics Vout V 2 EE40 Summer 2005 Lecture 16 V Vin Instructor Octavian Florescu 16 8 NMOS Resistor Pull Up V DD Circuit Voltage Transfer Characteristic vOUT RD iD A iD vIN VDD F vDS vOUT vIN VDD 0 VT vIN VDD VDD RD increasing vGS vIN VT 0 vGS vin VT EE40 Summer 2005 Lecture 16 VDD vDS A F 0 1 1 0 Instructor Octavian Florescu 17 Disadvantages of NMOS Logic Gates Large values of RD are required in order to achieve a low value of VOL keep power consumption low Large resistors are needed but these take up a lot of space One solution is to replace the resistor with an NMOSFET that is always on EE40 Summer 2005 Lecture 16 Instructor Octavian Florescu 18 9 The CMOS Inverter Intuitive Perspective SWITCH MODELS CIRCUIT VDD VDD VDD S G Rp D VOUT VOUT VIN VOL 0 V D G VOUT VOH VDD Rn S Low static power consumption since one MOSFET is always off in steady state EE40 Summer 2005 Lecture 16 VIN VDD VIN 0 V Instructor Octavian Florescu 19 CMOS Inverter Voltage Transfer Characteristic N sat P sat VOUT N off P lin VDD VDD G C S D VOUT VIN N sat P lin D G A B D S E N lin P sat N lin P off 0 0 EE40 Summer 2005 Lecture 16 VDD Instructor Octavian Florescu VIN 20 10 …
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