Department of EECS EE100/42-43 Spring 2007 Rev. 1 Introduction to Digital Systems 0. Acknowledgments Many thanks to Prof. Bernhard Boser and National Instruments for funding this project in the Summer of 2007. Ferenc Kovac has been (and will continue to be) an excellent mentor. Winthrop Williams designed the strain gauge lab (a paradigm of the K.I.S.S – Keep it Simple Stupid – philosophy). We are using this as a sensor for interfacing to our digital system (a PIC microcontroller). Tho Nguyen and Karen Tran were the brave souls who first tried out the PIC microcontroller. Kevin Mullaly’s staff installed the microcontroller development tools. Last but not the least, a shout-out to the authors on the internet who made this document possible: Tony R. Kuphaldt’s free description of digital electronics (http://www.ibiblio.org/obp/electricCircuits/Digital/index.html) forms the crux of this document. Combined with Richard Bowles’ excellent description of feedback in digital systems (http://richardbowles.tripod.com/dig_elec/chapter1/chapter1.htm), it is hoped that this document will serve as a self-contained introduction to digital systems. Go Bears! 1. Introduction Logic circuits are the basis for modern digital computer systems. To appreciate how computer systems operate you will need to understand digital logic and Boolean algebra. This chapter provides only a basic introduction to Boolean algebra – describing it in its entirety would take up an entire textbook. I chose to concentrate on the basics of Boolean algebra, rather than on optimizing concepts like Karnaugh Maps. First we start out with the concept of digital vs. analog. 2. Digital vs. Analog [2] The term digital refers to the fact that the signal is limited to only a few possible values. In general, digits signals are represented by only two possible voltages on a wire - 0 volts (which we called "binary 0", or just "0") and 5 volts (which we call "binary 1", or just "1"). We sometimes call these values "low" and "high", or "false" and "true". More complicated signals can be constructed from 1s and 0s by stringing them end-to-end, like a necklace. If we put three binary digits end-to-end, we have eight possible combinations: 000, 001, 010, 011, 100, 101, 110 and 111. In principle, there is no limit to how many binary digits we can use in a signal, so signals can be as complicated as you like. The figure below shows a typical digital signal, firstly represented as a series of voltage levels that change as time goes on, and then as a series of 1s and 0s.Department of EECS EE100/42-43 Spring 2007 Rev. 1 Figure 1. A digital signal Analog electronics uses voltages that can be any value (within limits, of course - it's difficult to imagine a radio with voltages of a million volts!) The voltages often change smoothly from one value to the next, like gradually turning a light dimmer switch up or down. The figure below shows an analog signal that changes with time. Figure 2. An analog signal 3. Number Systems [1] a. The Binary Number System The binary number system is a natural choice for representing the behavior of circuits that operate in one of two states (on or off, 1 or 0). For instance, we studied a diode logic gate (refer to the Diodes and Transistors handout online) when we discussed diode circuits. But before we study logic gates, you need to be intimately familiar with the binary number system – the system used by computers for counting. Lets count from zero to twenty using the decimal number system and the binary number system. Decimal Binary ------- ---------- 0 0 1 1 2 10 3 11 4 100 5 101 6 110 7 111Department of EECS EE100/42-43 Spring 2007 Rev. 1 8 1000 9 1001 10 1010 11 1011 12 1100 13 1101 14 1110 15 1111 16 10000 17 10001 18 10010 19 10011 20 10100 Notice, though, how much shorter decimal notation is over binary notation, for the same number of quantities. What takes five bits in binary notation only takes two digits in decimal notation. An interesting footnote for this topic is the one of the first electronic digital computers, the ENIAC. The designers of the ENIAC chose to represent numbers in decimal form, digitally, using a series of circuits called "ring counters" instead of just going with the binary numeration system, in an effort to minimize the number of circuits required to represent and calculate very large numbers. This approach turned out to be counter-productive, and virtually all digital computers since then have been purely binary in design. This is intuitively due to that fact that a binary number directly maps to the “on” and “off” state in digital systems. Notice that the binary number system and digital logic are actually two different concepts. A binary number is a number in base-2, it is independent of the concept of digital logic. However, the computer revolution is attributed to the very simple fact that mathematics in digital electronics can be represented by binary numbers. This is the number system that we will primarily study, along with the hexadecimal (base-16) system for convenience of representing large digits. To convert a number in binary numeration to its equivalent in decimal form, all you have to do is calculate the sum of all the products of bits with their respective place-weight constants. To illustrate: Convert 110011012 to decimal form: bits = 1 1 0 0 1 1 0 1 . - - - - - - - - weight = 1 6 3 1 8 4 2 1 (in decimal 2 4 2 6 notation) 8 The bit on the far right side is called the Least Significant Bit (LSB), because it stands in the place of the lowest weight (the one's place). The bit on the far left side is called the Most Significant Bit (MSB), because it stands in the place of the highest weight (the one hundred twenty-eight's place). Remember, a bit value of "1" means that the respective place weight gets added to the total value, and a bit value of "0" means that the respective place weight does not get added to the total value. With the above example, we have: 12810 + 6410 + 810 + 410 + 110 = 20510 If we encounter a binary number with a dot (.), called a "binary point" instead of a decimal point, we follow the same
View Full Document
Unlocking...