DESIGN OF A FULL ADDER CIRCUIT

CHAPTER ONE

1.0 INTRODUCTION

Adder is one of the most important components of a CPU (central processing unit). Arithmetic logic unit (ALU), floating-point unit and address generation like cache or memory access unit use it. In addition, Full adders are important components in other applications such as digital signal processors (DSP) architectures and microprocessors [1-5]. Arithmetic functions such as ‘addition’, ‘subtraction’, ‘multiplication’ and ‘division’ are some examples, which use ‘adder’ as a main building block. As a result, design of a high-performance full-adder is very useful and important [2, 3, 6-8]. On the other hand, increasing demand for portable equipments such as cellular phones, personal digital assistant (PDA), and notebook personal computer, arises the need of using area and power efficient VLSI circuits[2, 9-12]. Low-power and high-speed adder cells are used in battery-operation based devices [13-15].

1.1   Half adder

            The half adder is an example of a simple, functional digital circuit built from two logic gates. A half adder adds two one-bit binary numbers A and B. It has two outputs, S and C (the value theoretically carried on to the next addition); the final sum is 2C + S. The simplest half-adder design, pictured on the right, incorporates an XOR gate for S and an AND gate for C. Half adders cannot be used compositely, given their incapacity for a carry-in bit.

Example half adder logic diagram

1.2   Full adder

            A full adder adds binary numbers and accounts for values carried in as well as out. A one-bit full adder adds three one-bit numbers, often written as A, B, and Cin; A and B are the operands, and Cin is a bit carried in (in theory from a past addition). The full-adder is usually a component in a cascade of adders, which add 8, 16, 32, etc. binary numbers. The circuit produces a two-bit output sum typically represented by the signals Cout and S, where .

Schematic symbol for a 1-bit full adder with Cin and Cout drawn on sides of block to emphasize their use in a multi-bit adder

Full adder adds binary numbers and accounts for values carried in as well as out. A one-bit full adder adds three one-bit numbers, often written as A, B, and Cin; A and B are the operands, and Cin is a bit carried in (in theory from a past addition). The full-adder is usually a component in a cascade of adders, which add 8, 16, 32, etc. binary numbers. The circuit produces a two-bit output sum typically represented by the signals Cout and S, where

 The one-bit full adder's truth table is:

Example full adder logic diagram; the AND gates and the OR gate can be replaced with NAND gates for the same results

A full adder can be implemented in many different ways such as with a custom transistor-level circuit or composed of other gates. One example implementation is with  and .

In this implementation, the final OR gate before the carry-out output may be replaced by an XOR gate without altering the resulting logic. Using only two types of gates is convenient if the circuit is being implemented using simple IC chips which contain only one gate type per chip. In this light, Cout can be implemented as .

A full adder can be constructed from two half adders by connecting A and B to the input of one half adder, connecting the sum from that to an input to the second adder, connecting Ci to the other input and OR the two carry outputs. Equivalently, S could be made the three-bit XOR of A, B, and Ci, and Cout could be made the three-bit majority function of A, B, and Ci.

1.3   MORE COMPLEX ADDERS

1.3.1 RIPPLE CARRY ADDER

4-bit adder with logic gates shown

It is possible to create a logical circuit using multiple full adders to add N-bit numbers. Each full adder inputs a Cin, which is the Cout of the previous adder. This kind of adder is a ripple carry adder, since each carry bit "ripples" to the next full adder. Note that the first (and only the first) full adder may be replaced by a half adder.

The layout of a ripple carry adder is simple, which allows for fast design time; however, the ripple carry adder is relatively slow, since each full adder must wait for the carry bit to be calculated from the previous full adder. The gate delay can easily be calculated by inspection of the full adder circuit. Each full adder requires three levels of logic. In a 32-bit [ripple carry] adder, there are 32 full adders, so the critical path (worst case) delay is 3 (for carry propagation in first adder) + 31 * 2 (for carry propagation in later adders) = 65 gate delays.

1.3.2   CARRY-LOOKAHEAD ADDERS

4-bit adder with carry lookahead

To reduce the computation time, engineers devised faster ways to add two binary numbers by using carry-lookahead adders. They work by creating two signals (P and G) for each bit position, based on if a carry is propagated through from a less significant bit position (at least one input is a '1'), a carry is generated in that bit position (both inputs are '1'), or if a carry is killed in that bit position (both inputs are '0'). In most cases, P is simply the sum output of a half-adder and G is the carry output of the same adder. After P and G are generated the carries for every bit position are created. Some advanced carry-lookahead architectures are the Manchester carry chain, Brent–Kung adder, and the Kogge–Stone adder.

Some other multi-bit adder architectures break the adder into blocks. It is possible to vary the length of these blocks based on the propagation delay of the circuits to optimize computation time. These block based adders include the carry bypass adder which will determine P and G values for each block rather than each bit, and the carry select adder which pre-generates sum and carry values for either possible carry input to the block.

Other adder designs include the carry-save adder, carry-select adder, conditional-sum adder, carry-skip adder, and carry-complete adder.

 1.3.3   LOOKAHEAD CARRY UNIT

A 64-bit adder

By combining multiple carry lookahead adders even larger adders can be created. This can be used at multiple levels to make even larger adders. For example, the following adder is a 64-bit adder that uses four 16-bit CLAs with two levels of LCUs.

 1.4    COMPRESSORS

We can view a full adder as a 3:2 lossy compressor: it sums three one-bit inputs, and returns the result as a single two-bit number; that is, it maps 8 input values to 4 output values. Thus, for example, a binary input of 101 results in an output of 1+0+1=10 (decimal number '2'). The carry-out represents bit one of the result, while the sum represents bit zero. Likewise, a half adder can be used as a 2:2 lossy compressor, compressing four possible inputs into three possible outputs.

Such compressors can be used to speed up the summation of three or more addends. If the addends are exactly three, the layout is known as the carry-save adder. If the addends are four or more, more than one layer of compressors is necessary and there are various possible design for the circuit: the most common are Dadda