Laboratory II. Exploring Digital Logic

Transcription

1 Laboratory II Exploring Digital Logic Abstract Few devices in the world are as ubiquitous as the seven-segment display. This device, which was invented in the 1960 s, can be found everywhere from digital clocks to pinball machines. Though generally used for numbers, some letters may also be displayed, include forms of a, b, c, d, e, and f. This allows seven-segment displays to be used not only for decimal values, but hexadecimal as well. To use a seven-segment display, however, an engineer must know not only how to get a segment to glow but also which segments should glow for each character. Rules must be generated for each of the seven segments based upon the different possible characters to display. In this lab, students will learn a set of basic rules for converting inputs to outputs. Furthermore, they will learn how the seven-segment display works and how to use the logical rules in conjunction with the micro-controller to display a message. Objectives Students completing this lab will be able to: Construct and minimize up to three-variable logic functions, Design and build logic circuits with up to three input variables and seven output variables with a set number of TTL gates and, Design and build logic circuits to drive a standard 7-segment display. Background Binary Notation In EGR 53, you learned that the way computers store numbers is by keeping track of several 0s and 1s using binary notation. Binary is a number system that uses only the characters 0 and 1, so that each digit ends up representing some power of 2 rather than a power of 10. While this makes numbers take up for space to record, it allows the computer to perform lightening fast calculations. For this lab, you need to be able to understand binary notation as it relates to having a computer count from 0 to some number. The micro-controller for this lab will be used to generate a counter signal made up of either two or three different outputs. Because these outputs can only take on two values - on or off - they will be used to represent a binary digit. Specifically, imagine that you have three output wires connected to lights, and that a light represents a 0 if it is off and a 1 if it is on. You must decide which light is the 1s light, which is the 2s s light, and which is the 4s light. From there, you can interpret the lights to represent a number as follows: 4s Light 2s Light 1s Light Binary Decimal off off off off off on off on off off on on on off off on off on on on off on on on

2 Later you will see how to combine the three outputs with logic to produce different output functions based on what value the counter displays. Logical Operators, Truth Tables, and Gray Code In EGR 53, you also learned how to combine logical and relational operators to ask questions. The answers to these questions generally resulted in True of False (or logical 1s and 0s, respectively) that could be used to select certain pieces of code to run (if trees) or indicate that a certain piece of code was to run again (while loops). You can also use logical operators to write logical output functions based on input conditions. For example, assume that the micro-controller is keeping track of two temperatures. One output - for now call it P12 - is on if the temperature of the room is below 60 o F and off if it is 60 o F or above. This represents a Too Cold condition. Another output - say, P13 - is on if the temperature of the room is above 85 o F and off if it is 85 o F or below. This represents the Too Hot condition. Using these, you can determine a function for the Just Right condition, which basically happens when you are (NOT Too Cold) AND (NOT Too Hot). If some variable L1 is the Just Right function, you can write this as: L1 = ( P12) & ( P13) or using a more common notation for logical statements in ECE, L1 = P12 P13 where the bar over the variable means NOT and having variables multiplied by each other means AND. This latter part makes sense since for the product of two boolean to be 1, both variables must be 1. This exactly parallels the fact that for a logical AND to be true, both parts have to be true. Note that you can rewrite (NOT Too Cold) AND (NOT Too Hot) using DeMorgan s Theorem 1 as NOT (Too Cold OR Too Hot). This gives: or, again using more typical notation, L1 = (P12 P13) L1 = P12 + P13 where summation represents the OR operator. Note in this case that the bar stretches out over the entire term, meaning the NOT happens after the OR is completed. This is fundamentally different from LX = P12 + P13 which would be read as (NOT Too Cold) OR (NOT Too Hot), which will always be true! Once you determine the proper function, it can be used to generate a table that indicates the outputs of the function based upon all possible input combinations. This kind of table is known as a truth table, because it presents the different input combinations that make a function true (as well as how it is false). For example, for L1 above, the truth table would be: P12 P13 L Generally, truth tables are written in such a way that the inputs count up in binary from 0. For the truth table above, assuming P13 is your least significant, or 1s digit and P12 is your 2s digit, the inputs count from 0 to 3. 1 Elements of Digital Logic, Barrowman, p Parallax,

3 Compound Logic and Minimization Sometimes, you will end up writing complicated logical functions with several variables and may want to find the most efficient way to represent the function. For example, assume you are teaching a robot how to maneuver through a room while avoiding running into fires and getting stuck on walls. You have temperature sensors on either side of the front (Tright and Tleft, which indicate 0 if the temperature is loow and 1 if the temperature is high) as well as impact sensors on either side of the front (Iright and Ileft, which indicate 0 if the sensor is not against an object and 1 if the sensor is against an object). Given these inputs, you decide the courses of action given in Table 1. Note that the case number will eventually refer to the binary value obtained by translating the outputs. 0. If your right temperature is low and your left temperature is low and your right impact sensor is off and your left impact sensor is off, move forward one step 1. If your right temperature is low and your left temperature is low and your right impact sensor is off and your left impact sensor is on, back up a little then turn right 2. If your right temperature is low and your left temperature is low and your right impact sensor is on and your left impact sensor is off, back up a little then turn left 3. If your right temperature is low and your left temperature is low and your right impact sensor is on and your left impact sensor is on, back up a little then turn right 4. If your right temperature is low and your left temperature is high and your right impact sensor is off and your left impact sensor is off, turn right 5. If your right temperature is low and your left temperature is high and your right impact sensor is off and your left impact sensor is on, back up a little and turn right 6. If your right temperature is low and your left temperature is high and your right impact sensor is on and your left impact sensor is off, back up a little and turn right 7. If your right temperature is low and your left temperature is high and your right impact sensor is on and your left impact sensor is on, back up a little and turn right 8. If your right temperature is high and your left temperature is low and your right impact sensor is off and your left impact sensor is on, back up a little then turn left 9. If your right temperature is high and your left temperature is low and your right impact sensor is off and your left impact sensor is off, turn left 10. If your right temperature is high and your left temperature is low and your right impact sensor is on and your left impact sensor is off, back up a little then turn left 11. If your right temperature is high and your left temperature is low and your right impact sensor is on and your left impact sensor is on, back up a little then turn left 12. If your right temperature is high and your left temperature is high and your right impact sensor is off and your left impact sensor is off, back up a little turn around 13. If your right temperature is high and your left temperature is high and your right impact sensor is off and your left impact sensor is on, back up a little turn around 14. If your right temperature is high and your left temperature is high and your right impact sensor is on and your left impact sensor is off, back up a little turn around 15. If your right temperature is high and your left temperature is high and your right impact sensor is on and your left impact sensor is on, back up a little turn around Table 1: Decision tree for a hypothetical robot The first step along the path to determining and minimizing the output functions is to give names to the input and output functions. In this case, the input functions already have convenient names - Tright, Tleft, Iright, and Ileft. The output functions can be named based on the actions the robot might take: Forward, BackAndRight, BackAndLeft, UTurn, Right, and Left. Now the decision tree can be recast as a set of truth tables as in Table 2 Given the truth tables, it is somewhat easier to construct logical equations for six different output 3

4 Case Tright Tleft Iright Ileft Forward BackAndRight BackAndLeft UTurn Right Left Table 2: Truth tables for six different actions for hypothetical robot functions. For example, an equation for the BackAndLeft would be: BackAndLeft = Tright Tleft Iright Ileft + Tright Tleft Iright Ileft + Tright Tleft Iright Ileft + Tright Tleft Iright Ileft Similar equations can be developed for each of the other possible robot actions. The issue then becomes: is this the most efficient way to write this function? For BackAndLeft in particular, you may notice that if the temperature on the right is high, and the temperature on the left is low, and the robot has hit something on the right side, then it does not matter what is going on with the left side - the robot will back up and turn left. This demonstrates one of the properties of boolean algebra, which is the combining property: or, in this case, ab + ab = a (Tright Tleft Iright) Ileft + (Tright Tleft Iright) Ileft = Tright Tleft Iright This takes two products of four terms and reduces it to a single product of three terms. The difficulty is in the amount of work required to visualize such combinations, especially for multiple variables. It is for that reason that we use a visual device called a Karnaugh map 2. The Karnaugh map is nothing more than an organizational tool to help visualize possible applications of the combining property. The key is to use order the different variables according to a Gray code, which is a sequence of codes, or calculations, where consecutive codes differ in one variable only. 3. You do this so that elements appearing right next to each other represent changes in only one variable. That way, if there are two 1s next to each other, you know you can combine those elements together to eliminate one of the variables. For multi-variable functions, larger collections can be merged to reduce the number of terms and the size of each term. To create the Karnaugh map of a function, pick the variables and then create a two-dimensional version of the truth table by assigning variables to a particular dimension. For example, a possible two-input function you want to minimize may be: F1 = ab + ab Pick a to go along the top and b to go down the side, then use Gray code for the ordering (which for this example is easy - just go from 0 to 1). Then fill in the appropriate value of the truth table in the proper blocks. The Karnaugh map for F1 would be: 2 Fundamentals of Digital Logic, Brown and Vranesic, p McGraw Hill, Fundamentals of Digital Logic, Brown and Vranesic, p McGraw Hill,

5 b xxx a Now comes the minimization - once you have your 1s and 0s together, look for elements that are right next to each other with the same value. For example, in this case, there are the two 1s that are right on top of each other. What you are looking for are sub-matrices of similar values that have dimensions rxc, where r and c are both integer powers of 2. In this example, the 1s form a sub-matrix that is 2x1, so it is a valid grouping for potential minimization. The number of variables you can eliminate from a grouping is equal to log 2 (r c), because that value tells you how many combinations you have in the sub-matric. In the sub-matrix described above, log 2 (r c) = 1 meaning you should be able to get rid of one variable. The variable you can get rid of is whichever one changes in the adjacent blocks. In this case, b is the one that is changes, so b is irrelevant in that sub-matrix and will be removed from the term, thus leaving only some function of a. From the table headings, you can see that b truly does not matter to F1 - F1 is true if a is true regardless of b, and so we can minimize the function to be F1=a. This process can be extended to three- and four-variable functions quite easily. The major organizational change is that you will have to represent two input variables in a single dimension. This is where the Gray code comes in. For example, if we were to choose to minimize the BackAndLeft function, we could choose to have Tright and Tleft along the side and Iright and Ileft along the top. Using the Gray code to determine how each dimension is established yields the Karnaugh map skeleton below. To clarify what each block means, note that x represents the location where Tright is 0, Tleft is 1 (both of which come from the fact x is in the 01 row), Iright is 1, and Ileft is 0 (the latter two coming from x being in the 10 column). Using the same indexing, you can see that y represents the location where Tright is 1, Tleft is 1, Iright is 0, and Ileft is 0. TrightTleft xxx Iright Ileft x 11 y 10 Once the table is organized, you can fill in the logical values. It is often easier to do this if you consider the binary numbers represented by the four variables. For example, if you assume that Tright represents the most significant, or 8s digit, and Ileft represents the least significant, or 1s digit, you can form four-digit binary numbers by looking first at the row and then the column. x above, for example, is in the or 6 10 block while y is in the or block. The full set of block numbers would give: TrightTleft xxx Iright Ileft Now you can simply take the case numbers from the decision tree and enter them into the Karnaugh map. For instance, if you create the Karnaugh map for BackAndLeft you get: TrightTleft xxx Iright Ileft From this, look for the largest sub-matrices that satisfy the size constraints given above and determine the smallest subset of these matrices that covers every 1 at least once. You are allowed to overlap, but do not overlap needlessly. For example, given the locations above, there are three sub-matrices of size 2, but you only need two of them to completely represent the output of the function. The first sub-matrix contains 0010 and 1011 (first and last entries in the fourth column - so note that you are allowed to wrap), meaning 5

6 you can eliminate one variable. This combination covers two values of Tright, meaning Tright is irrelevant for those two blocks, and the logic of the sub-matrix can be represented by Tleft Iright Ileft. A possible second sub-matrix contains 1001 and 1011, which covers two values of Ileft. The logic for that submatrix is thus Tright Tleft Iright. There is a third sub-matrix of size two that contains 1011 and 1010, but since both of those values have already been represented by other logic, you do not need to include that particular expression. Generally, you try to make the largest groupings possible and then make sure you do not include terms that are wholly included by other terms. When finished, you can combine the logic for each of your sub-matrices with OR statements. For example, the final minimized logic for this function is: BackAndLeft = Tleft Iright Ileft + Tright Tleft Iright which is decidedly shorter than the initial version. The same process could be used to minimize the BackAndRight function. The Karnaugh map for that is: TrightTleft xxx Iright Ileft In this case, you can form a 2x2 sub-matrix with the first and second rows of the second and third columns. Because the sub-matrix has four entries, you can eliminate two variables total. Looking at the specific rows and columns for that sub-matrix, you can see that Tleft and Iright are irrelevant - the logic those four entries have in common involves Tright being false and Ileft begin true. Logically, that portion becomes Tright Ileft. All that remains is the 1 at The largest group it can be a part of is a 1x2 sub-matrix of the terms 0111 and You can eliminate Ileft from this grouping to get Tright Tleft Iright. Now that all the possible 1s are covered, you can write the BackAndRight as a combination of the two sub-matrices: BackAndRight = Tright Ileft + Tright Tleft Iright As a final note on Karnaugh maps, occasionally it is easier to make large groupings of 0s than it is of 1s. For those times, you simply build the reverse logic and throw a NOT at the beginning. Basic Gate Symbols and Operation When planning out a logic circuit, you can either draw it by hand or use a computer aided design tool such as MMLogic to help. Regardless, you need to know what the common symbols for different logic gates are. Figure 1 is a screen-shot from the MMLogic program that displays the symbols for 2, 3, and 4-input versions of the six most common gates: AND, OR, XOR, NAND, NOR, and XNOR. The gates for IS (a buffer) and NOT are also shown. Note that the inverting functions NAND, NOR, XNOR, and NOT have circles at the output node - this indicates you are inverting the typical output of a gate. When you want to represent a logical function, you simply connect wires from the input variables to the input nodes of a gate. The value obtain by the gate s function will obtained at the output node. For example, the two different circuits to represent the Just Right function, L1a = P12 P13 L1b = P12 + P13 are in Figure 2 and there are two different input combinations shown. Note that in the top figure, both switches are set to off, meaning P12 is a 0 and P13 is a 0, leading to L1a being on. For the bottom figure, P12 is on, so the function shows that L1b is off. Seven Segment Displays One version of the Seven-Segment Display is discussed on pp of the Elements of Digital Logic book from Parallax. Basically, a seven-segment display is nothing more that seven (or potentially 6

7 Figure 1: Common Logic Gate Symbols 7

8 Figure 2: Circuits for L1a = P12 P13 and L1b = P12 + P13 eight) light emitting diodes in a specific pattern. The eighth light comes from the decimal point, which is added to some versions of the display. There are two fundamentally different ways you can send inputs to a seven-segment display. In one version, which Parallax uses, you have eight inputs - one for each of the segments and one for the decimal point. Whenever you want to have a particular diode or set of diodes come on, you run current through the corresponding inputs. This input scheme leads to maximum flexibility for the display, as you can choose any 1 of the 256 possible combinations of LEDs to come on. On the other hand, it also means you have to do some translation if you know you want to display a particular number. For example, if simply want the number 2 to show up, you have to use the map on p. 125 of the Elements of Digital Logic book to figure out that you will need to turn diodes A, B, D, E, and G on. To get the number 7, you would need A, B, and C only. Because there are many instances where you simply want to display a number, another version of the seven-segment display is available that is created specifically to convert the binary representation of a number into the number in base 10 on the display. For these displays, there are only five inputs where one input controls the decimal point and the other four are translated from binary into the appropriate signals to the other seven segments to produce the number in decimal form. For these, if you want to display a 2, you input the binary version of 2 into the four numerical inputs - in other words, low-low-high-low. For a 7, you would put in low-high-high-high. Furthermore, the translator built into the chip is generally made to understand hexadecimal as well, so you can get the letters A, b, C, d, E, and F by entering the binary versions of 10, 11, 12, 13, 14, and 15 respectively. The mix of upper and lower case letters comes from the fact that, with seven segments, you cannot represent a, B, or D as distinct from o, 8, and 0. Furthermore, you cannot represent lower case E or F at all. Convention has C displayed upper case. The MMLogic program is capable of using both 8- and 5-input versions of the seven-segment display. The graphical pinout for MMLogic is somewhat different from the actual wiring of the display used by Parallax. It is also somewhat difficult to connect since there are eight leads very close together. To alleviate this problem, you will be given a model for a pre-wired display that uses MMLogic senders and receivers. These two structures in MMLogic make wiring easier, as you can hook some input variable to a sender and then use a receiver somewhere else in the circuit to relay that value along. The contents of the Two7SD.lgi file are shown in Figure 3. Notice that the top-to-bottom order of the MMLogic inputs matches the counter-clockwise order of the Parallax display. If you need more than two displays, you simply need to change the names of the senders and receivers. Note that you can have multiple receivers for a single sender, but you cannot have multiple senders with the same name. To test various combinations of the eight inputs, you will be given the Sample7SD.lgi file shown in Figure 4. When this program is running, you can change any of the eight inputs to see what shows up on the display. In this case, power is being delivered to segments C, E, F, and G and an approximation of the letter h is on the display. 8

9 Figure 3: Two7SD.lgi file showing two sender/receiver enabled seven segment displays Figure 4: Sample7SD.lgi file showing an h Equipment Equipment, including software, needed for this laboratory includes 1. Parallax Digital Logic Trainer Board, including 17 grounded LEDs, breadboard, 4 pushbuttons, and pre-installed Texas Instrument 74-series TTL logic gates, specifically: (2) 74C00 quad NAND (2) 74C02 quad NOR (2) 74C08 quad AND (2) 74C32 quad OR (1) 74C14 hex NOT Parallax Elements of Digital Logic Instruction Booklet Parallax Basic Stamp 2 Micro-controller Parallax PBASIC compiler Softronix Multimedia Logic program (1) Seven-segment display (8) 470 Ω resistors Connecting wires Experiment 1. Parallax Guide Chapter 5 Activities 1 and 2 Activity 1: Guided Tour. Note especially that the trainer board (parallax digital trainer, or PDT) has several built-in components that will prove useful for this lab, including digital logic chips along the top, LEDs at various locations, pushbuttons, and rocker switches. Also notice that the PDT work space allows up to two connections for each input to each gate. If you need more than two connections, you will need to use the breadboard. Activity 2: Exploring Gates. The contact bounce problem is one that can cause problems if you try to use pushbuttons to simulate a crisp change from 5 V to 0 V or vice versa. You will discover a way around it by using the micro-controller outputs to simulate the action of a switch. 9

10 Also, if you discover a problem with a particular gate while running tests, note this in your lab book and inform the instructor. Either the gates will be replaced or, for just this week, you will need to avoid that particular gate. Activities 3 & 4. Though you will not be performing Activities 3-4, you should know the truth table for the NAND, NOR, XOR, and XNOR gates. 2. Parallax Guide Chapter 6 Activities 4 and 5 Activity 4: Building the Multiplexer. A multiplexer is basically an electrical device that allows you to have several input channels and select which one goes to the output. Basically, it is a cable box - your cable box receives multiple inputs all centered on different frequencies, as the channel changers selects the channel (or address) that is allowed to send its information to your single output (the television set). Activity 5: Building the Demultiplexer. A demultiplexer allows you to take a single input signal and decide which among several outputs will receive the signal. An analogy for this would be going to an electronics store and wanting to test out different CD players with a CD that you bring. You get to decide which player (address) gets the output from the single CD (input) that you have. 3. Micro-controller-based counter. Rather than exploring Chapter 7 and beyond of the Parallax guide at this time, you will write a program to make your BS2 count to 7 in binary using PBASIC program for use in later parts of this lab. To do this, you will write a program that counts to 1, then expand to 3, and finally to 7. You will be using outputs P12-P14 on the digital logic trainer as your bits with P12 serving as the least significant bit (2 0, or the 1 s place) and P14 serving as your most significant bit (2 2, or the 4 s place). To visualize the outputs, connect P12 through P14 to LEDs 0 through 2, respectively. That way, the number of the LED is the same as the power of 2 it represents. Note that these LEDs already have their anode connected to the ground of the device - generally, you will need to make sure that the anode is wired to ground. Once you have connected the three wires, you should enter, save, compile, and run the following program, OneBitCounter.bs2: {$STAMP BS2} {$PBASIC 2. 5 } LOW 12 DO PAUSE 500 TOGGLE 12 LOOP This program should toggle LED 0 every half-second. Congratulations - you now have a one bit counter! The other two LEDs remain dark, indicating the the 2 s place and the 4 s place remain at 0. This kind of output would be useful if you wanted to simulate the action of a single pushbutton being switched every half second, because avoids any problems with contact bounce. Next, you expand this counter to two bits so that you can both examine circuits with two inputs and count to 3. Enter, save, compile, and run the following code for TwoBitCounter.bs2: {$STAMP BS2} {$PBASIC 2. 5 } LOW 13 LOW 12 DO PAUSE 500 TOGGLE 12 PAUSE 500 TOGGLE 13 TOGGLE 12 LOOP 10

11 This version should toggle LED 0 every half-second and LED 1 every second. If you look carefully, you are now counting from 0 to 3 with those two lights. Later, you will design a three-bit counter program as part of the Exploration for use with your Assignment. 4. The logic behind ECE. For this, you are going to design the logic necessary to make a seven-segment display write the letters ECE in order with a space after the second E. TRUTH TABLE LISTED FOR OUTPUTS TRUTH TABLE FOR EACH SEGMENT MMLogic Information WIRING DIAGRAM ASSISTANCE Exploration Expand the TwoBitCounter.bs2 program into a ThreeBitCounter.bs2 program by having the microcontroller also toggle P14. When completed, you should run through the numbers 0 through 7 in binary every 4 seconds. Assignment Now that you have a three-bit counter and experience in programming a scrolling message board, design and build the logic for writing SUCCESS with a space after the third S. TRUTH TABLE LISTED FOR OUTPUTS TRUTH TABLE FOR EACH SEGMENT MMLogic information - stop here as a take home assignment? 11