# EE2420 – Digital Logic Summer II 2013 Hassan Salamy Ingram School of Engineering Texas State University Set 2: Boolean Logic.

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EE2420 – Digital Logic Summer II 2013 Hassan Salamy Ingram School of Engineering Texas State University Set 2: Boolean Logic

Boolean Algebra 2  The basis for much of what we will do in this class is grounded in Boolean algebra, named for George Boole.  In the early chapters of the textbook we will study the Theorems and Laws of Boolean Algebra and techniques for implementing Boolean functions using electronic circuits.  In later chapters we will study how Boolean functions can be used to implement arithmetic and other useful functions.  Later still we will adapt these techniques to create components critical to modern computing such as memory elements and sequence logic.

Electrical Circuits (Over)simplified  For those of you who have had no experience with electrical circuits, the following is a vastly oversimplified primer.  The simple model of matter is that it is made up of atoms which have positively-charged nuclei orbited by negatively-charged electrons. Given sufficient energy, an electron can be dislodged from its atom.  Think of electricity as the flow of energy carried by the movement of electrons dislodged from atoms.  An electrical circuit is formed when there is a closed path around which the electrons flow. 3

Circuits Primer part 2  In electrical circuits, we are interested in the movement of energy from one location to another.  Some elements in a circuit provide energy. These are sources.  Some elements in a circuit expend energy. These are sinks.  The total energy sourced must equal the total energy expended.  We typically measure energy somewhat indirectly. Energy is measured in Joules (J). Power is energy per unit time. Power is measured in Watts (W). A Watt is one Joule per second.  1W = 1J/s  A 100W light bulb expends 100 Joules each second. 4

Circuits Primer part 3  We said a 100W light bulb expends 100 Joules each second.  We also said to think of electricity as the flow of energy carried by the movement of electrons dislodged from atoms.  To get energy to the light bulb, we make a circuit from the power producing plant to the light bulb.  In that circuit each moving electron is charged with energy at the source. When an electron passes through the sink (light) that energy is expended.  The amount of power that can be supplied depends on two factors:  The number of electrons that can move through the wires  The amount of energy carried with each electron 5

Circuits Primer part 4  The number of electrons flowing in a circuit is called current. Current is measured in Amperes (A). An Ampere is one Coulomb of electrons per second. A Coulomb is 6,241,507,648,655,549,400 (approx 6.24 x 10 18 ) electrons.  The amount of energy per electron is called voltage. Voltage is measured in Volts (V). One Volt is one Joule per Coulomb.  So to determine power we take the product of the number of electrons flowing (current) and the energy per electron (voltage) and we should get a measure of power (wattage)  Voltage * Current = Power  In units V*A=W or (J/C)*(C/s) = (J/s) 6

Circuits Primer part 5  Most of the devices that we will be dealing with in this class will operate using relatively low power circuits.  However, this does not make them 100% safe. It also does not make it impossible to damage the equipment if used incorrectly.  Electricity flows very easily through most metals, including jewelry. It would be wise when you are dealing with any circuitry to remove any jewelry from your hands (rings, watches, etc. ) or long necklaces that could contact the circuitry.  Be careful how you wire the power supply to your circuits. If you reverse the polarity and try to drive current “backward” through the circuit, you are likely to cause damage.  Ask your lab monitor if you are not sure. 7

The Scope of Boolean Algebra 8  In common arithmetic, we are often concerned with the range of values that a variable may take. This is generally determined by the nature of the problem:  The integers 1-12 – the month of the year  A 7-digit decimal integer – local telephone number  A real number in the interval from 0 to 100 – a test grade  Imaginary numbers.  For Boolean algebra, every variable is constrained to one of two values: True or False  Any convenient two-valued pair can be used to represent the True/False pair: on/off, high/low, clockwise/counter-clockwise, closed/open, loud/soft, 1/0  We will commonly use the 1/0 notation with 1=True and 0=False

Switches 9  A very early electronic implementation of Boolean algebra recognized that the state of a switch (open or closed) could be used to represent a Boolean variable (True or False) and that switches could be wired in combination to represent complex Boolean algebra equations. Open SwitchClosed Switch

Electrically-controlled Switches 10  A simple mechanical switch (like a light switch on a wall) requires physical effort to manually open or close the contact.  An electrically-controlled switch is opened or closed based on the state of an input.  For a normally-open switch, when the input is inactive, the switch is open and when the input is active, the switch closes.  For a normally-closed switch, when the input is inactive, the switch is closed and when the input is active, the switch opens. Normally-openNormally-closed

Types of Switches 11  The earliest devices used for electrically-controlled switching for Boolean algebra purposes were relays.  A relay uses an input to generate an electromagnetic field that physically moves the switch contacts into position.  Because relays depend on physical movement of the contacts, they are too slow for modern digital computing purposes.  Modern systems use transistors to perform the function of switches without requiring mechanical movement.  While transistors are complex devices capable of much more than digital switching, when used for switching they may be modeled as very simple electrically-controlled switches.

Active-high Input Switching Devices 12  BJT – Bipolar Junction Transistor  MOSFET – Metal-Oxide Semiconductor Field Effect Transistor Normally-open Switch NPN BJT N-channel MOSFET input Conducting path NMOS: +V=On 0V=Off Lecture 2: Boolean Logic12/28

Active-low Input Switching Devices 13  BJT – Bipolar Junction Transistor  MOSFET – Metal-Oxide Semiconductor Field Effect Transistor Normally-closed Switch PNP BJT P-channel MOSFET input Conducting path PMOS: +V=Off 0V =On 13/28

Switch Logic 14  In the following circuit, the battery is connected to the lamp if switch A is closed and switch B is closed. If either switch is open, there is not a complete circuit.  This is the Boolean function AND Battery Lamp Switch ASwitch B

Switch Logic 15  In the following circuit, the battery is connected to the lamp if switch A is closed or switch B is closed. If either switch is closed, there is a complete circuit.  This is the Boolean function OR Battery Lamp Switch A Switch B

Switch Logic 16  In the following circuit, the battery is connected to the lamp and the output voltage is high if the input to the normally-closed switch is low.  Conversely, when the input is high, the lamp is disconnected and the output drops to 0.  Therefore, the output is the opposite of the input.  This is the Boolean function NOT Battery Lamp Input Output

Logic Gates 17  Special circuit symbols are used for the Boolean logic functions regardless of the type of switches used to implement the functions. ANDORNOT

Logic circuits 18  Logic circuits perform operations on digital signals  Implemented as electronic circuits where signal values are restricted to a few discrete values  In binary logic circuits there are only two values, 0 and 1  The general form of a logic circuit is a switching network  In the figure to the right, the inputs are labeled X n and the outputs are labeled Y n  Not shown in the figure are the required connections for providing power to the circuit. Switching Network X1X1 X2X2 X3X3 XmXm Y1Y1 Y2Y2 Y3Y3 YnYn discrete values

Boolean algebra 19  Direct application to switching networks  Work with 2-state devices  2-valued Boolean algebra (switching algebra)  Use a Boolean variable (X, Y, etc.) to represent an input or output of a switching network  Boolean variable may take on only two values (True, False)  We will use a notation of 0 for False and 1 for True  So the possibilities for a variable are: X=0, X=1  These symbols are not binary numbers, they simply represent the 2 states of a Boolean variable  They are not voltage levels, although they commonly refer to the low or high voltage input/output of some circuit element

Notation Conventions 20  Boolean AND  The raised dot “· ”  x· y  The intersection symbol “  ”  x  y  No symbol  xy  Boolean OR  Plus sign “+”  x + y  The union symbol “  ”  x  y  Boolean NOT  Single quotation “ ’ ” following  x’  Bar over letter   Exclamation point preceding  !x  Tilde preceding  ~x

Inversion of a function 21  If a function is defined as  f(x 1, x 2 )= x 1 + x 2  Then the complement of f is  f(x 1, x 2 )= x 1 + x 2 = (x 1 + x 2 )’  Similarly, if  f(x 1, x 2 )= x 1 · x 2  Then the complement of f is  f(x 1, x 2 )= x 1 · x 2 = (x 1 · x 2 )’

Truth tables 22  Tabular listing that fully describes a logic function  Output value for all input combinations (valuations) x1x1 x2x2 x 1 · x 2 000 010 100 111 x1x1 x2x2 x 1 + x 2 000 011 101 111 x1x1 x1’x1’ 01 10 AND OR NOT

Truth tables 23  Truth table for AND and OR functions of three variables

Truth tables of functions 24  If L(x,y,z)=x+yz, then the truth table for L is: +

Logic gates and networks 25  A larger circuit is implemented by a network of gates  Called a logic network or logic circuit x 1 x 2 x 3 fx 1 x 2 +  x 3  =

Logic gates and networks 26  Draw the truth table and the logic circuit for the following function  F(a,b,c) = ac+bc’ a c b

Analysis of a logic network 27  To determine the functional behavior of a logic network, we can apply all possible input signals to it x 1 x 2 1100  f 0001  1101  0011  0101  Network that implements fx 1 x 1 x 2  += A B

Analysis of a logic network 28  The function of a logic network can also be described by a timing diagram (gives dynamic behavior of the network) 1 0 1 0 1 0 1 0 1 0 x 1 x 2 A B f Time Timing diagram

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