A journey into Digital Logic with Josh and Eric Discovering Digital So what exactly is Digital Electronics? Basic Logic Breadboarding Next comes Inputs and Outputs Its definition - The implementation of two-valued logic through electronic logic gates. In circuits the logical values true and false are represented by two different voltages; 0V for false and +5V for true. Similarly, the binary system uses these different voltages to represent zero and one. But what does this mean? To fully understand this overly complex definition, we must dive into the basics. Number systems They are more complex than you might think... Every number system represents the number of digits it has with its name. Decimal means 10, therefore it has 10 different numbers (0 through 9), meanwhile binary means two therefore it has two unique numbers, 0 and 1. Not only does its name represent its digits, but it also denotes its base. Decimal has a base of 10, binary a base of 2. As you know, for decimal numbers, the first number from right to left represents the ones place. Then the next digit represents the 10s place and so on. You take the number's base and plug it in for X on this chart, thus for binary you would get something like this: Place Value These charts represent place value in a number system. In decimal, in order to count to 10, we first go to 9 then we make it a 0 and the next place a 1. This same setup can be used for any number system This same concept goes for all number systems based on their bases. For binary, we count to 1 then in order to get to 2, we make the first place a 0 and the next a 1. This number 101011 represents our number 43, but in the binary number system. As you can see, there is one 32, one 8, one 2, and one 1. That means (1*32)+(1*8)+(1*2)+(1*1) giving us 43. 2 An output is the result produced by a computer. Once again it can only have a value of 0 or 1 in binary. Outputs An input is any signal or data that is sent to a computer to be processed - Simply stated - It is a switch that can be turned on or off, a binary value of 0 or 1. Inputs This is a simple circuit with one side of a switch connected to power and the other connected to ground. This circuit has one input and one output; The input is the switch which is labeled X and the output is the light, labeled 0 . Every number system is set up like this: Here is an example The number 43 has a 4 in the 10s place and a 3 in the ones place. This means there are 4 tens and 3 ones. This gives us 4*10 and 3*1, which when added gives us our number 43. Truth Tables Truth tables are a way to represent a circuit or to create a circuit to meet specific requirements. When setting up a truth table, you start with 0 and count, in binary, up to two less then the number of inputs squared. In this instance we have 3 inputs, thus we will be counting up to 7 in binary. First column alternates between 0 and 1 2 alternates 00,11 3 alternates 0000,1111 Input X 1 Output 0 1 Furthermore, every number can be represented with every number system. In order to tell these number systems apart, we use the base number as a subscript. For binary we use a subscript of 2 and for decimal, a subscript of 10, but because we assume numbers are in decimal, a subscript for decimal is not required. So how do we make a decimal number a binary one? Well, lets take the number 25. We then make one of these charts with the number system needed. We then take the largest possible number out of 25 from our corresponding values on this chart; in this case its 16. Subtract it from 25 to get 9 and place a one below the 16. Then take the next largest number out (8) and subtract it from 9 to get 1 and place another 1 below the 8. Then the last number we can take out of 1 is 1, so we place another one below the 1 on the chart. This gives us the number 11001 in binary which is equivalent to 25 in decimal. (Don't forget your subscript!) 2 nd rd AOI logic Testing and trouble shooting Maxterms and Minterms Minterm A minterm is the product of the inputs that result in an output of 1. The sum of the minterms is called the sum of products (SOP). A maxterm is the sum of the inputs that result in an output of 0. The product of the maxterms is also called the product of sums (POS). Maxterm An example Minterms XYZ

XYZ

XYZ

XYZ X+Y+Z

X+Y+Z

X+Y+Z

X+Y+Z Maxterms An input is either a 0 or a 1, and these inputs are labeled X,Y, and Z. However, when an input is 0, it is representing its opposite, so we call it not X, shown as X _ To represent the unsimplified version of the output with algebra, we take the sum of the minterms or the product of the maxterms Both the POS and the SOP represent the output in the form of an unsimplified Boolean expression. XYZ+XYZ+XYZ+XYZ=O (X+Y+Z)(X+Y+Z)(X+Y+Z)(X+Y+Z)=O To fully reduce these expressions, one can opt to either Boolean algebra or karnaugh mapping (K-mapping) Lets stick with minterms from now on In order to K-map, we use a chart like this: You can only switch one not at a time! Its also traditional to start with all nots in upper left Now we plug in the minterms into the chart - remember minterms represent 1s. XYZ+XYZ+XYZ+XYZ=O 1 Now what do you do with this? Well, you take and group the 1s into the largest groups possible. However, only groups of 1, 2, 4, 8, etc work. You also want to make sure that you have the fewest groups possible. It is also OK to overlap your groups. Here is what this one looks like. The object is to circle all the ones in the fewest, but biggest groups. Additional Rules You can also have a group of 4, by grouping the corners. You can group 1s across the chart as well. Additional Rules Now that we have a truth table and our simplified Boolean expression, we can now create our circuit. Their are many different ways to make our circuit. AOI stands for the three types of gates it contains; The And gate, the Or gate, and the Inverter gate. These expressions that we obtain represents our output. When two inputs are next to each other (AB) it shows us A and B. A+B represents A or B just as X represents not X. Expressions The and gate The and gate represents two inputs that are paired such as AB. When these two inputs are anded, they only output a one when both A AND B are 1s. The or gate An or gate represent the Boolean expression of the sum of two inputs (A+B). An or gate outputs a 1 when either input is a 1 or when both inputs are 1s. The inverter (not) gate The not gate represents the opposite of an input; it represents not X (X). Nor only logic Nor only logic means that only nor gates are used to make the circuit. A nor gate represents a not or gate. So every output made by an or gate is inverted. Nand only logic Nand only logic means that only nand gates are used to make the circuit. A nand gate is simply a not and gate. So every output from an and gate is inverted. Breadboarding is another process used to simulate a circuit. Like multisim, a breadboard uses logic gates and wires to create a circuit. These columns are for power and ground Each row is connected but not to each other Once finished creating the simulated circuit in either multisim or on a breadboard, you must test it to see if it works. You would go through and test every possibility and compare the results to your truth table. Testing If the truth tables match, you have successfully created your circuit... However, if your truth table is misrepresented, then you messed up somewhere :( This is where troubleshooting comes in Now to simulate our circuit Once circled, we look at one group (orange) and see that we have XYZ and XYZ so the Y and Y cross out and become XZ. The other group, the X and X cross out becoming YZ. Then you add your new terms to get XZ+YZ as our simplified Boolean expression Here is our simplified Boolean expression - XZ+YZ. Now we have to create a simulation and one possible way to do this is on a program called multisim First we would add our power and ground as well as our inputs X Y Z Next, we look at our equation, XZ+YZ and see that we need two and gates, an or gate, and a not gate. You then just simply just connect these gates with their corresponding inputs YZ X Z XZ YZ+XZ 74LS08 (And Chip) 74LS32 (Or Chip) 74LS04 (Not Chip) Chips AOI gates are contained within chips. These chips each have 14 pins labeled 1-14. The 1st pin is located right of the notch, then is numbered counter-clockwise from there. A TTL 74LS08 chip contains 4 And gates A TTL 74LS04 chip contains 4 Not gates A TTL 74LS32 chip contains 4 Or Gates X Y Z First we would connect power and ground to the breadboard. +5 Volts Ground Depending on the breadboard, the switches may already be connected to power and ground Then we insert our chips And connect our ground and power through each of our chips Then we take a look at our equation XZ + YZ and connect our switches accordingly. YZ XZ Z XZ+YZ And finally, we connect our final output into an LED to check if it works Our Example 1 1 1 1 1 now fill in the rest with zeroes 0 0 0 0 1 Y Z YZ If your truth tables failed to match, then you must troubleshoot. You check the row of your truth table that doesn't match and make sure that part of your circuit is correct, although it shouldn't be or else you wouldn't be troubleshooting. There are also tools that tell you if a wire has a binary value of 1 or 0 that may help you troubleshoot if you can't find your problem. The end of our long journey, deep into digital electronics

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# Discovering Digital

The basics of digital logic and electronics.

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