# Needing some sum

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A half adder is used for adding together the two least ificant digits in a binary sum such as the one shown in Figure The four possible combinations of two binary digits A and B are shown in Figure In the sum shown in Figure The carry has rippled through two stages of the addition.

Carry ripple, through many stages, in adder circuits generates unacceptable delays, and methods are now available to eliminate this problem. Figure The half adder is used for adding together the two least ificant bits dotted b The addition of the four possible combinations of two binary digits A and B with a carry to the next most ificant stage of addition c Truth table for the half adder d NAND implementation of the half adder e NOR implementation of the half adder.

The additions shown in Figure The Needing some sum headed A and B display every combination of the two binary digits to be added, while the third and fourth columns are the corresponding tabulations of the sum S and carry Crespectively. The Boolean equations for the sum and carry read directly from the truth table are:. Sarah L. We begin by building a 1-bit half adder.

As shown in Figure 5. If A and B are both 1, S is 2, which cannot be represented with a single binary digit. Instead, it is indicated with a carry out C out in the next column.

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In a multi-bit adder, C out is added or carried in to the next most ificant bit. For example, in Figure 5. However, the half adder lacks a C in input to accept C out of the column. The full adderdescribed in Needing some sum next section, solves this problem. TABLE 3. Example of reduction process.

The resulting array of full-adders and half-adders is shown in Figure 3. For the final 2-to-l reduction, a 7-bit CPA is needed. Reduction by columns of eight 5-bit magnitudes. Cost of reduction: 26 FAs and 4 HAs. The total delay of the scheme consists of the delay of the reduction array and the delay of the final CPA. Since the delay of the CPA depends on the of bits, a more aggressive reduction might be applied to reduce the precision of the final adder at the expense of additional counters Exercise 3.

The full-adder extends the concept of the half-adder by providing an additional carry-in Cin input, as shown in Figure 5. This cell adds the three binary input s to produce sum and carry-out terms. Figure 5. One-bit full-adder cell. The truth table for this de is shown in Table 5. Table 5. One-bit full-adder cell truth table. From viewing the truth table, the Sum output is only a logic 1 when one or three but not two of the inputs is logic 1. The Boolean expression for this is in reduced form :. From viewing the truth table, the Cout output is only a logic 1 when two or three of the inputs is logic 1.

This can be drawn as a circuit schematic as shown in Figure 5. One-bit full-adder circuit schematic. Any of half- and full-adder cells can be connected together to form an n-bit addition. In this de, there is no Cin input. In the full-adder implementation with two half-adders, the load on the carry-in is larger than the load on other als. Since the delay of the carry-out al is affected by this load, it is convenient to reduce it.

Determine Needing some sum effect of this modification on the delay of the carry-out al, using the characteristics of Table 2. TABLE 2. Characteristics of a family of CMOS gates. Determine the delay of a bit adder using the full-adder characteristics of Table 2. Compare delay and size with a 2-bit carry-ripple adder implemented with radix-2 full-adders use average delays.

An important logic de created from the basic logic gates is the half-adder, shown in Figure 6. This cell adds the two binary input s and produces sum and carry-out terms. The truth table for this de is shown in Table 6. Table 6. Half-adder cell truth table. The Sum output is only a logic 1 when either but not both inputs are logic This can be drawn as a logic diagram as shown in Figure 6. Figure 6. Half-adder logic diagram. A dataflow VHDL description for the one-bit half-adder that uses the two logic expressions is shown in Figure 6. Here, two expressions are placed in the architecture body one expression for the Sum output on line 17, the Needing some sum for the Cout output on line VHDL code for a one-bit half-adder.

Clearly such a complexity would be unwieldy were it not for integrated circuits, which are quite capable of implementing complex circuitry in small-scale form. Adders are Needing some sum most used of the more complex operators in a typical de. In certain cases, ASIC deers sometimes employ special versions using combinations of half-adders and full-adders. This may work very efficiently in the case of a gate array device, for example, but it will typically result in a very bad FPGA implementation. When using an adder with constants, a little thought goes a long way.

In fact, a little algebra also goes a long way in FPGAs. The conditional—sum adder generates all possible sums and carries in a manner similar to the carry—select adder [ 22 ]. The conditional-sum adder uses a modified half-adder to generate sums and carries in the first phase. The second phase uses log 2 W d levels of multiplexers to conditionally combine neighboring bits into the final sum [ 36 ]. This adder can easily be pipelined by placing latches after the first phase and after every multiplexer stage. The conditional—sum adder, implemented using dynamic CMOS circuits [ 25 ], is usually faster than a carry—look—ahead adder, but both adders have complex interconnections that require large chip areas. The conditional—sum adder should be considered as a candidate for highspeed CMOS adders.

A full adder circuit is central to most digital circuits that perform addition or subtraction. It is so called because it adds together two binary digits, plus a carry-in digit to produce a sum and carry-out digit. The truth table and Needing some sum Karnaugh maps for it are shown in Table 4. Table 4. The truth table and Karnaugh maps for a full adder. X and Y are the two bits to be added, C in and C out the carry-in and carry-out bits, and S the sum.

Two 1's with no carry-in are added using a full adder. What are the outputs? Adding two 1's in binary gives a result of 0 with a carry-out of 1. Two 1's with a carry-in of 1 are added using a full adder. Using the Karnaugh maps to obtain minimised expressions for S and C outwe notice the chequerboard pattern of an XOR gate in the sum term to give:. Circuit diagram of a full adder. Download as PDF. Set alert. About this. Arithmetic circuits B. View chapter Purchase book. Digital Building Blocks Sarah L. The delay in the critical path is roughly 3. This is actually the Exclusive-OR function, so:. The Cout output is logic 1 only when all two inputs are logic 1 i. The circuit to implement the full adder is shown in Fig. Needing some sum
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