1.10 (a) Effects of logic gates AND, OR, NOT on binary signals in a processor

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1 Chapter 1.10 Logic Gates 1.10 (a) Effects of logic gates AND, OR, NOT on binary signals in a processor Microprocessors are the central hardware that runs computers. There are several components that make a processor. The first is the transistor. Next, are logic gates where you put more than one transistor to work with others. The transistor is fundamental because it carries an electrical charge and can direct the current flow through the logic gates. The microprocessor works with binary arithmetic by using binary math to perform operations. When a microprocessor is designed, along with other design focus areas "Logic gate cell library (a library is collection of all low level logic functions like AND, OR and NOT etc.), which is used to implement the logic" is also deeply planned and developed. By themselves, transistors are not very functional. However, if you combine them with other transistors, you get logic gates. Logic gates carry out the instructions mathematical or otherwise that a processor performs, for example a logic gate performs a logical operation on one or more logic inputs and produces a single logic output. When you connect a variety of logic gates together, the results are interesting circuits. The logic is called Boolean logic and is most commonly found in digital circuits. As mentioned logic gates are primarily implemented electronically using transistors in microprocessors, but for other devices can also be constructed using electromagnetic relays (relay logic), fluidic logic, pneumatic logic, optics, molecules, or even mechanical elements. Following three logic gates are part of our syllabus. 1. AND gate, 2. OR gate, and 3. NOT gate. AND gate: AND gate symbol The AND gate is a digital logic gate that implements logical conjunction - it behaves according to the table on your right. A HIGH output (1) results only if both the inputs to the AND gate are HIGH (1). If neither or only one input to the AND gate is HIGH, a LOW output results. In another sense, the function of AND effectively finds the minimum between two binary digits. INPUT OUTPUT A B A AND B (Q)

2 OR Gate: OR gate symbol The OR gate is a digital logic gate that implements logical disjunction - it behaves according to the table on your right. A HIGH output (1) results if one or both the inputs to the gate are HIGH (1). If neither input is HIGH, a LOW output (0) results. In another sense, the function of OR effectively finds the maximum between two binary digits. NPUT A B OUTPUT A + B (Q) NOT gate (Inverter): NOT gate symbol In digital logic, an inverter or NOT gate is a logic gate which implements logical negation. Not gate represents perfect switching behavior. NPUT A OUTPUT NOT A

3 1.10 (b) Calculate outcome from a set of logic gates given input In this part 1.10 (b) we will explore the application of Boolean algebra in the design of electronic circuits. The basic elements of circuits are gates. Each type of gate implements a Boolean operation. We will study combinational circuits, which means the circuits whose output depends only on the input and not on any memory. Consider Boolean expression ; i.e., is the complement of. Now and. This Boolean operation, i.e., compliment can be implemented using a device called NOT gate or the Inverter. It can be expressed as below: NOT gate (INVERTER) Now consider Boolean expression ; i.e., a is the Boolean product of &. As we know that,, and. This Boolean operation, i.e., product can be implemented using a device called AND gate. It can be expressed as below: AND gate Next consider Boolean expression ; i.e., a is the Boolean sum of &. As we know that and. This Boolean operation, i.e., sum can be implemented using a device called OR gate. It can be expressed as below: OR gate In circuitry theory, NOT, AND, and OR gates are the basic gates. Any circuit can be designed using these gates. The circuits designed depend only on the inputs, not on the output. In other words, these circuits have no memory. Also these circuits are called combinatorial circuits. The symbols NOT gate, AND gate, and OR gate are also considered as basic circuit symbols, which are used to build general circuits. The circuits for expressions x y and xy are shown below in figures (a) and (b), respectively:

4 The circuits for expressions x +y and x+y are shown below in figures (a) and (b), respectively: The circuit for x y and x +y are shown below in figures (a) and (b), respectively: (a) (b) Let a be a Boolean expression, then its circuit diagram is defined as follows: 1. If a is x, then it circuit diagram is given by a NOT gate, 2. If a is x+y, then it circuit diagram is given by a OR gate, 3. If a is xy, then it circuit diagram is given by a AND gate, 4. If a is B, then it circuit diagram is shown below, 5. If a is Bx, then it circuit diagram is shown below, 6. If a is B+x, then it circuit diagram is shown below, 7. If a is By, then it circuit diagram is shown below, 8. If a is B+y, then it circuit diagram is shown below,

5 9. If a is xy +x y, then it circuit diagram is shown below, We now break xy and x y and include their own circuits in diagram above and create a new circuit with four gates. In order to simplify the circuit above we can split x and y inputs half way and use it for two or more gates. So the above diagram can be simplified and presented as: Truth Table: A truth table is a mathematical table used in logic specifically in connection with Boolean algebra and Boolean functions to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their logical variables. In particular, truth tables can be used to tell whether a propositional expression is true for all legitimate input values, that is, logically valid. Practically, a truth table is composed of one column for each input variable (for example, A and B), and one final column for all of the possible results of the logical operation that the table is meant to represent (for example, A OR B). Each row of the truth table therefore contains one possible configuration of the input variables (for instance, A=true B=false), and the result of the operation for those values. Number of possible rows in a truth table is directly dependant on the number of inputs and can be easily find out by applying 2 n, where n is the number of inputs mentioned in truth table.

6 Here is a truth table giving definitions of the most commonly used 3 out of the 16 possible truth functions of 2 binary inputs (P,Q are thus boolean variables): P Q P AND Q P OR Q NOT P NOT Q T T T T F F T F F T F T F T F T T F F F F F T T Note that total number of rows is, 2 2 = 4. Applications of truth tables: In computer science, truth tables can be used to reduce basic Boolean operations to simple correlations of inputs to outputs, without the use of logic gates or code. For example, a binary addition can be represented with the truth table: A B C R where A = First Operand B = Second Operand C = Carry R = Result This truth table is read left to right: Value pair (A,B) equals value pair (C,R). Or for this example, A plus B equal result R, with the Carry C. Note that this table does not describe the logic operations necessary to implement this operation; rather it simply specifies the function of inputs to output values.

7 In this case it can only be used for very simple inputs and outputs, such as 1's and 0's, however if the number inputs increases, the size of the truth table will increase. For instance, in an addition operation, one needs two operands, A and B. Each can have one of two values, zero or one. The number of combinations of these two values is 2x2, or four. So the result is four possible outputs of C and R. The first "addition" example above is called a half-adder. A full-adder (not part of 2010 computing syllabus) is when the carry from the previous operation is provided as input to the next adder. Thus, a truth table of eight rows would be needed to describe a full adder's logic: A B C* C R Same as previous, but C* = Carry from previous adder The next level after logic gates in microprocessor gets even more complicated. First, we combined transistors together to make logic gates. Now, in combining a whole lot of logic gates together in clever ways, we can make complex circuits that do things like adding two numbers together, multiplying two numbers together, moving data from one place to another, and so on. Adders: In electronics, an adder or summer is a digital circuit that performs addition of numbers. In modern computers adders reside in the arithmetic logic unit (ALU) where other operations are performed. The most common adders operate on binary numbers. There are two types of adders 1. Half adder, which adds only two values and 2. Full adder (Not part of this syllabus), which adds more than two values.

8 Half adder: A half adder is a logical circuit that performs an addition operation on two one-bit binary numbers often written as A and B. The half adder output is a sum of the two inputs usually represented with the signals and S where. Following is the logic table for a half adder: Inputs Outputs A B C S As an example, a Half Adder can be built with an OR gate, two AND gates and one Inverter. Other forms of half adder using OR, AND and Not gates could be: Can you device Boolean equations for all above four circuits?

9 1.10(c) Logic gates as a form of refreshable memory and as an accumulator. In a computer's central processing unit (CPU), an accumulator is a register in which intermediate arithmetic and logic results are stored. Without a register like an accumulator, it would be necessary to write the result of each calculation (addition, multiplication, etc.) to main memory, perhaps only to be read right back again for use in the next operation. Access to main memory is slower than access to a register like the accumulator because the technology used for the large main memory is slower (but cheaper) than that used for a register. The example for accumulator use is summing a list of numbers. The accumulator is initially set to zero, and then each number in turn is added to the value in the accumulator. Only when all numbers have been added is the result held in the accumulator written to main memory An accumulator is build with an adder whose sum can be loaded into a register as shown in figure below. Both adder and register in an accumulator are made up of logic gate circuits. Accumulators are a basic building block of most large digital logic projects. As an analogy, you can think of an up accumulator as a file cabinet. It starts out empty. If you add two, it now holds the value of two. If you add three more it now holds five. Ad der R e gi st er One of the more interesting things that you can do with Boolean gates is to create memory with them. If you arrange the gates correctly, they will remember an input value. This simple concept is the basis of RAM (random access memory) in computers, and also makes it possible to create a wide variety of other useful circuits like CMOS (BIOS). Memory relies on a concept called feedback. That is, the output of a gate is fed back into the input. The simplest possible feedback circuits using two inverters are shown below: If you follow the feedback path, you can see that if output happens to be 1, it will always be 1. If it happens to be 0, it will always be 0. Since it's nice to be able to control the circuits we create, this one does let us see how feedback works. It turns out that in "real" circuits, you can actually use this sort of simple inverter feedback approach to hold bits as long as required.

10 This type of circuit is commonly known as Flip-Flop circuits. A flip-flop holds a single bit of memory. In reality, flip-flops are a bit more complicated and have 5 (or so) logic gates (transistors) per flip-flop. Transistors are all enclosed in an IC, or integrated circuit. Consider a 1 GB memory chip (DIMM), which is 1 GB = 8,589,934,592 bits of memory. That s about 43 million transistors. In reality, those transistors are split into 9 ICs of about 5 million transistors each. The Intel Pentium IV processors have 55 million transistors.

11 Example Questions: 1. Draw a circuit diagram for = (xy' + x'y)z 2. Device a suitable Boolean expression for the circuit below: 3. Determine what the following logic circuit does Hint: Construct a truth table with 8 rows (2 3 ), 3 "input" columns (A, B, C) and 5 "output" columns (D, E, F, G, H) Determine the "outputs" D, E, F, G, H one at a time and in that order. 4. Create a truth table for an OR gate with two inputs A and B. 5. (i) Copy and complete this table for the circuit. (ii) Give a possible use for this circuit in a processor, explaining your answer.

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