Engr354: Digital Logic Circuits
Engr354: Digital Logic CircuitsChapter 2: Introduction to Logic CircuitsDr. Curtis NelsonChapter 2 Objectives Engr354 Chapter 2Define and illustrate basic logic functions and circuits;Present Boolean algebra for dealing with logic functions;Illustrate logic gates and synthesis of simple circuits;Review CAD tools and the VHDL hardware descriptionlanguage.1
Binary Logic Circuits Logic circuits perform operations on digital signals; These circuits are implemented using electroniccomponents; Binary logic circuits can be found in one of two states– 0 or 1;– off or on;– down or up;– not asserted or asserted;– etc.Switch Representationx 0x 1(a) Two states of a switchSx(b) Symbol for a switchEngr354 Chapter 22
Switch ExampleL(x) xSBatteryLxLight(a) Simple connection to a batterySPowersupplyLx(b) Using a ground connection as the return pathTwo Basic FunctionsPowersupplySSx1x2L(x1, x2) x1 x2LLightL 1 if x1 1 and x2 1L 0 otherwise(a) The logical AND function (series connection)Sx1PowersupplySL(x1, x2) x1 x2LLightx2L 1 if x1 1 or x2 1L 0 otherwise(b) The logical OR function (parallel connection)Engr354 Chapter 23
A Series-Parallel ExampleSPowersupplyx1SSx3LLightx2L(x1, x2, x3) (x1 x2) x3Truth Tables All combinations of inputs on the left; Outputs on the right; 2-input AND and OR functions shown below.Engr354 Chapter 24
3-Input And and Or FunctionsBasic Gatesx1x2x1x2x1 x2 xnx1 x2xn(a) AND gatesx1x2x1x2x1 x2 xnx1 x2xn(b) OR gatesxx(c) NOT gateEngr354 Chapter 25
Example Using Basic Gatesxxx12f (x x ) x1233Sequencing of Inputsxx0 0 1 11 1 0 01A0 0 0 10 1 0 11 1 0 1fB2Network that implementsf x x1 x21xxf (x , x )0010111001111212Truth table for fEngr354 Chapter 26
Timing Diagramxx0 0 1 11 1 0 01A0 1 0 10 0 0 11 1 0 1fB211 0x12 0xA10B10f10TimeTiming diagramExamplexx0 0 1 11 1 0 011 1 0 10 1 0 1g2Network that implements g x x12 Draw a timing diagram below:Engr354 Chapter 27
Boolean Algebra In 1849, George Boole published a scheme for describinglogical thought and reasoning; In 1930s, Claude Shannon applied Boolean algebra todescribe circuits built with switches; Boolean algebra provides the mathematical foundation fordigital design.Notation INVERSION: x x’ !x NOT xf(x1,x2) x1 x2 (x1 x2)’ !(x1 x2) AND: OR:Engr354 Chapter 2 NOT(x1 x2)x1 x2 x 1 Ù x2 x 1 x2x1 x 2 x 1 Ú x28
Precedence of Operations In the absence of parentheses, operations are performed inthis order: NOT, AND, ORx1 x2 x1’ x2’ (x1 x2) ((x1’) (x2’))Principle of Duality On the following pages, axioms and theorems are listed inpairs to show the principle of duality; Given a logic expression, its dual is found by exchanging and operators and 0’s and 1’s; The dual of any true statement is always true.Engr354 Chapter 29
Axioms of Boolean Algebra1.2.3.4.0 0 01 1 10 1 1 0 0if x 0 then x 11 1 10 0 01 0 0 1 1if x 1 then x 0Single-Variable Theorems5.6.7.8.9.Engr354 Chapter 2x 0 0x 1 xx x xx x 0x xx 1 1x 0 xx x xx x 110
2- and 3-Variable Properties10a. x y y x10b. x y y xCommutative11a. x (y z) (x y) z11b. x (y z) (x y) z12a. x (y z) x y x z12b. x y z (x y) (x z)AssociativeDistributive2- and 3-Variable Properties13a. x x y x13b. x (x y) x14a. x y x y x14b. (x y) (x y) x15a. x y x y15b. x y x y16. x x y x yEngr354 Chapter 2AbsorptionCombiningDeMorgan’s Thmx (x y) x y11
Truth Table Proof of DeMorgan’s Theorem15a. x y x yDeMorgan’s TheoremBoolean Algebra Examples1) (a b)(a’ b’) 2) a’b’c’ a’b’c ab’c’ ab’c 3) a’b’c’ a’b’c a’bc’ abc’ ab’c’ ab’c 1) ab’ ba’2) b’3) b’ c’Engr354 Chapter 212
Synthesis Synthesis is the process of creating a circuit from aspecification, i.e. a truth table, schematic, VHDL code, etc.Two Implementations of the Function fx1x2f(a) Sum-of-products formx1x2f(b) Minimal-cost realizationEngr354 Chapter 213
Minterms and Maxterms Minterms– For a function of n variables, a product term in which each of the nvariables appears once is called a minterm Variables may appear in complemented or uncomplementedform; A given minterm is formed by including xi if xi 1 and byincluding not(xi) if xi 0. Maxterms (Principle of duality applies)– For a function of n variables, a sum term in which each of the nvariables appears once is called a maxterm Variables may appear in complemented or uncomplementedform. A given maxterm is formed by including xi if xi 0 and byincluding not(xi) if xi 1.Three-Variable Minterm and Maxterm TableEngr354 Chapter 214
Sum-of-Products (SOP) Form A function f can be represented by an expression that is a sum of minterms.For example, the function below could be represented asf(x1,x2,x3) !x1!x2x3 x1!x2!x3 x1!x2 x3 x1x2!x3 Using short hand notationf(x1,x2,x3) m1 m4 m5 m6 orf(x1,x2,x3) S m(1,4,5,6)Product-of-Sums (POS) Form The same function f can be represented by an expression that is aproduct of sums of maxterms (this follows from DeMorgan’s laws).For example, the function below could also be represented asf(x1,x2,x3) (x1 x2 x3)(x1 !x2 x3)(x1 !x2 !x3)(!x1 !x2 !x3)Using short hand notation,f(x1,x2,x3) M0M2M3M7 orf(x1,x2,x3) P M(0,2,3,7)Engr354 Chapter 215
Canonical A logic expression is said to be canonical if each termcontains all variables, either complemented oruncomplemented.2-Variable Example Synthesize the following function using four different, butequivalent, expressions Engr354 Chapter 2Minterm form;Maxterm form;Canonical form;Minimum form.16
NAND and NOR Gatesxxxx1x1 x12x21 x2 xn2xn(a) NAND gatesxxxx1x1 x12x21 x2 xn2xn(b) NOR gatesDeMorgan’s Theorems in Terms of Logic Gatesx1x2x1x1x2x2(a)x1x2x1 x2 x1 x2x1x1x2x2(b)x1 x2 x1 x2 Function vs. GateEngr354 Chapter 217
SOP Implementation Using NAND Gatesx1x2x1x2x3x4x5x3x4x5x1x2x3x4x5POS Implementation Using NOR Gatesx1x2x1x2x3x4x5x3x4x5x1x2x3x4x5Engr354 Chapter 218
Example Synthesize the following function using four different, butequivalent, expressions Minterm form;Maxterm form;Canonical form;Minimum form.Canonical SOP Implementationfx1x2x3(a) Sum-of-products realizationEngr354 Chapter 219
Canonical POS Implementationx3x2x1f(b) Product-of-sums realizationMinimal ImplementationEngr354 Chapter 220
Textbook Problem 2.28 Brown 3rd Ed. Design the simplest circuit that has three inputs, a, b, and c,which produces an output value of 1 whenever two ormore of the input variables have the value of 1; otherwise,the output has to be 0. Implement the circuit three different ways– With minimum and/or logic;– With nand logic gates only;– With nor logic gates only.Design Entry Truth tables– Practical only for small circuits. Schematic capture– Interconnect symbols in some library;– Facilitates hierarchical design;– Good for medium-to-large circuits;– Difficult to use for very large circuits.Engr354 Chapter 221
Design Entry - Continued Hardware description languages (HDL’s)––––––Similar to a programming language;VHDL and Verilog HDL are IEEE standards;Provide design portability;Allow for sharing and design reuse;Support hierarchical design;Can be combined with schematics.Logic Synthesis Logic synthesis, or logic optimization, is the process oftranslating a truth table, schematic, or VHDL code into anetwork of logic gates; What makes a circuit good depends on the application – doyou want to optimize for speed, area, power?Engr354 Chapter 222
Simulation A functional simulator is used to determine if a designedcircuit operates correctly from a logic perspective. Circuit verification– User provides input values to the circuit;– Simulator determines the circuit response;– User checks responses against desired outputs. A timing simulator is used to check correctness byincorporating the electrical characteristics of a logic designin addition to the logical performance. This simulationrequires– Technology mapping;– Layout synthesis.A Typical CAD SystemDesign conceptionDESIGN ENTRYSchematic captureVHDLSynthesisFunctional simulationNoDesign correct?YesPhysical designTiming simulationNoTiming requirements met?Chip configurationEngr354 Chapter 223
VHDL VHDL - Very high speed integrated circuit hardwaredescription language; Original IEEE standard adopted in 1987; Revised standard in 1993; Originally used for documentation and simulation; Now, it is also used for synthesis; Very complex language, but only a subset is needed todesign a wide range of circuits.Representing Digital Signals in VHDL Each logic signal in a circuit is a data object in VHDL code; Data objects in VHDL are assigned types; A simple type is BIT which is used for objects that can takeonly 2 values: 0 or 1.Engr354 Chapter 224
A Simple Logic Function Examplex1x2fx3Chapter 2 Summary Engr354 Chapter 2Defined and illustrated basic logic functions and circuits;Presented Boolean algebra for dealing with logic functions;Illustrated logic gates and synthesis of simple circuits;Reviewed CAD tools and the VHDL hardware descriptionlanguage.25
Engr354: Digital Logic Circuits Chapter 2: Introduction to Logic Circuits Dr. Curtis Nelson Chapter 2 Objectives Define and illustrate basic logic functions and circuits; Present Boolean algebra for dealing with logic functions; Illustrate logic gates and synthesis of simple circuits
ECE380 Digital Logic Introduction to Logic Circuits: Variables, functions, truth tables, gates and networks Electrical & Computer Engineering Dr. D. J. Jackson Lecture 2-2 Logic circuits Logic circuits perform operations on digital signals – Implemented as electronic circuits wh
Dynamic Logic Dynamic Circuits will be introduced and their performance in terms of power, area, delay, energy and AT2 will be reviewed. We will review the following logic families: Domino logic P-E logic NORA logic 2-phase logic Multiple O/P domino logic Cascode logic
Digital electronic circuits operate with voltages of two logic levels namely Logic Low and Logic High. The range of voltages corresponding to Logic Low is represented with ‘0’. Similarly, the range of voltages corresponding to Logic High is represented with ‘1’. The basic digital elec
3.5 Combinational Circuits 138 Digital logic chips are combined to give us useful circuits. These logic circuits can be categorized as either combinational logic (Section 3.5) or sequential logic (Sec. 3.6). 3.5.1 Basic Concepts 138 The key concept in recognizing a combinational circu
Logic Design Chapter 2: Introduction to Logic Circuits Introduction Logic circuits operate on digital signals Unlike continuous analog signals that have an infinite number of possible values, digital signals are restricted to a few discrete values In particular for binary logic circuits, signals can have only two values: 0 and 1.
Digital Design is concerned with the design of digital electronic circuits. The subject is also known by other names such as logic design, digital logic, switching circuits, and digital systems. Digital circuits are employed in the design of systems such as digital computers, control syst
Digital Logic Fundamentals Unit 1 – Introduction to the Circuit Board 2 LOGIC STATES The output logic state (level) of a gate depends on the logic state of the input(s). There are two logic states: logic 1, or high, and logic 0, or low. The output of some gates can also be in a high-Z (high impedance) state, which is neither a high
Austin, Oscar Palmer Nacogdoches, TX Vietnam War Austin, William . Lopez, Jose Mendoze Mission, TX (Santiago Huitlan, Mexico) World War II (Most sources say that Lopez was born in Texas but he later stated in multiple interviews and his funeral program recorded that he was born in Mexico) Lummus, Jack Ennis, TX World War II Martinez, Benito Fort Hancock, TX Korean War . Compiled by Gayle .
