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Ece421) Digital Electronics Fundamental (Ece422 DIGITAL SYSTEM FUNDAMENTALS (ECE421) COURSE / CODE DIGITAL ELECTRONICS FUNDAMENTAL (ECE422) Medium Scale Integrated (MSI) devices It’s sometimes not reasonable to do all the design work at the gate-level. For example, it is a common practice to use full-adders as the building blocks in building a ripple-carry adder. When designing circuits, there are certain building blocks that are used over and over again. Thus, being familiar with a set of building blocks that are commonly used can make designing digital devices easier. Building blocks that are more complex than gates are sometimes called medium-scale integrated devices, or MSI devices for short. Exactly what makes a device “medium” in scale rather than “large” or “very large” (as in VLSI) is not always agreed on, but generally it’s all right stick with devices that are fairly simple and where small versions of them can be implemented in a couple dozen gates or so. Special Combinational Logic Circuits 1. Adders A half adder can add two bits. It has two inputs, generally labeled A and B, and two outputs, the sum S and carry C. S is the two-bit XOR of A and B, and C is the AND of A and B. Essentially the output of a half adder is the sum of two one-bit numbers, with C being the most significant of these two outputs. A half adder is a logical circuit that performs an addition operation on two binary digits. The half adder produces a sum and a carry value which are both binary digits. The drawback of this circuit is that in case of a multi-bit addition, it cannot include a carry. Sum(S) = A B and Carry(C) = AB A full adder is capable of adding three bits: two bits and one carry bit of earlier calculation. It has three inputs - A, B, and carry Cin, such that multiple full adders can be used to add larger numbers. Hence, a full adder can perform an addition operation on three binary digits. The full adder produces a sum and carry value, which are both binary digits. Exercises: Q1. Draw the half-adder circuit using NAND gates only. Q2. Draw the truth table for a full adder and write down the output expressions. Q2. Implement the operation of a full adder using two half-adders and an OR gate. Q3. What are the advantages of full-adder over half-adder? Mohd Uzir Kamaluddin / July 2019 page 1 Parallel Binary Adders As we discussed that a single full adder performs the addition of two one bit numbers and an input carry. For performing the addition of binary numbers with more than one bit, more than one full adder is required depends on the number bits. Thus, a parallel adder is used for adding all bits of the two numbers simultaneously. By connecting a number of full adders in parallel, n-bit parallel adder is constructed. From the below figure, it is to be noted that there is no carry at the least significant position, hence we can use either a half adder or made the carry input of full adder to zero at this position. The figure below shows a parallel 4-bit binary adder which has three full adders and one half-adder. The two binary numbers to be added are A3A2A1A0 and B3B2B1B0 which are applied to the corresponding inputs of full adders. This parallel adder produces their sum as C4S3S2S1S0 where C4 is the final carry. In the 4-bit adder, first block is a half-adder that has two inputs as A0B0 and produces their sum S0 and a carry bit C1. Next block should be full adder as there are three inputs applied to it. Hence this full adder produces their sum S1 and a carry C2. This will be followed by other two full adders and thus the final sum is C4S3S2S1S0. Most commonly Full Adders designed in dual in-line package integrated circuits. A typical 74LS83 is a 4-bit full adder. Arithmetic and Logic Unit of a unit computer consist of these parallel adders to perform the addition of binary numbers. Mohd Uzir Kamaluddin / July 2019 page 2 2. Subtractor Subtractor is a circuit which is used to subtract two binary number (digit) and provides Difference and Borrow as output. Half Subtractor is used for subtracting one single bit binary digit from another single bit binary digit. A logic circuit which is used for subtracting three single bit binary digit is known as Full Subtractor. Parallel Binary Subtractors To perform the subtraction of binary numbers with more than one bit is performed through the parallel subtractors. This parallel subtractor can be designed in several ways, including combination of half and full subtractors, all full subtractors, all full adders with subtrahend complement input, etc. The below figure shows a 4-bit parallel binary subtractor formed by connecting one half subtractor and three full subtractors. Mohd Uzir Kamaluddin / July 2019 page 3 In this subtractor, 4-bit minuend A3A2A1A0 is subtracted by 4-bit subtrahend B3B2B1B0 and gives the difference output D3D2D1D0. The borrow output of each subtractor is connected as the borrow input to the next preceding subtractor. It is also possible to design a 4 bit parallel subtractor 4 full adders as shown in the below figure. This circuit performs the subtraction operation by considering the principle that the addition of minuend and the complement of the subtrahend is equivalent to the subtraction process. We know that the subtraction of A by B is obtained by taking 2’s complement of B and adding it to A. The 2’s complement of B is obtained by taking 1’s complement and adding 1 to the least significant pair of bits. Hence, in this circuit 1’s complement of B is obtained with the inverters (NOT gate) and a 1 can be added to the sum through the input carry. Parallel Adder / Subtractor The operations of both addition and subtraction can be performed by a one common binary adder. Such binary circuit can be designed by adding an Ex-OR gate with each full adder as shown in below figure. The figure below shows the 4-bit parallel binary adder/subtractor which has two 4 bit inputs as A3A2A1A0 and B3B2B1B0. The mode input control line M is connected with carry input of the least significant bit of the full adder. This control line decides the type of operation, whether addition or subtraction. Mohd Uzir Kamaluddin / July 2019 page 4 When M = 1, the circuit is a subtractor and when M = 0, the circuit becomes adder. The Ex-OR gate consists of two inputs to which one is connected to the B and other to input M. When M = 0, B Ex-OR of 0 produce B. Then full adders add the B with A with carry input zero and hence an addition operation is performed. When M = 1, B Ex-OR of 0 produce B complement and also carry input is 1. Hence the complemented B inputs are added to A and 1 is added through the input carry, nothing but a 2’s complement operation. Therefore, the subtraction operation is performed. Note: B 0 B and B 1 B Exercises: Q1. Show on the parallel adder/subtractor circuit the operation of 10012 + 01112. Q2. Show on the parallel adder/subtractor circuit the operation of 11012 - 10012. 3. Multiplier A binary multiplier is an electronic circuit used in digital electronics, such as a computer, to multiply two binary numbers. It is built using binary adders. A simple 2-bit multiplier, A2A1 x B2B1 can be designed using logic gates as shown in the truth table below. Another method commonly used is to do multiplication by repeated addition. The multiplicand is added by itself the number of times as indicated by the multiplier. See the block diagram. To increase the speed of the multiplication process, other method of multiplication is used such as: Shift and add multiplier Serial multiplier Mohd Uzir Kamaluddin / July 2019 page 5 Serial/Parallel multiplier Array multiplier Booth multiplier Exercise: The multiplication of a 3-bit number B by a 4- bit number A is illustrated below. i) Based on the multiplication process shown, implementation a 4-bit x 3-bit multiplier using adders. ii) Show the multiplication of 10112 by 1002 by clearly indicating the values at the inputs and outputs. 4. Comparator A digital comparator or magnitude comparator is a hardware electronic device that takes two numbers as input in binary form and determines whether one number is greater than, less than or equal to the other number. Comparators are used in central processing units (CPUs) and microcontrollers (MCUs). An XNOR gate is a basic comparator, because its output is "1" only if its two input bits are equal. The truth table shows a 1-bit comparator and its logic circuit is given below. Figure below shows the logic diagram and truth table for a 2-bit comparator. Mohd Uzir Kamaluddin / July 2019 page 6 A 4-bit magnitude comparator TTL 74LS85. An 8-bit word comparator. For comparators that compares 3 bits or more, the truth table method of designing comparators will result in a huge truth table. Design of a 4-bit comparator using the deduction method. Mohd Uzir Kamaluddin / July 2019 page 7 Exercises: a) What does a comparator circuit do? What are the output of a comparator? b) A comparator determines when two binary numbers are equal, true or false? c) If a comparator compares two 2-bit numbers A, B and produces the output D1 = 1 if A<B and D2 = 1 if A=B Derive the expressions for D1 and D2 and implement the comparator using logic gates.
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