Digital-electronics 简明教程

Full Adder in Digital Electronics

What is a Full Adder?

能够对两个二进制位（位）和进位位进行加法，并产生和位和进位位作为输出的组合逻辑电路称为 full-adder 。

A combinational logic circuit that can add two binary digits (bits) and a carry bit, and produces a sum bit and a carry bit as output is known as a full-adder.

换句话说，设计用于对三个二进制位进行加法并产生两个输出（和和进位）的组合电路称为全加器。因此，全加器电路对三个二进制位进行加法，其中两个是输入，一个是前一次加法产生的进位。全加器的框图和电路图如图 1 所示。

In other words, a combinational circuit which is designed to add three binary digits and produces two outputs (sum and carry) is known as a full adder. Thus, a full adder circuit adds three binary digits, where two are the inputs and one is the carry forwarded from the previous addition. The block diagram and circuit diagram of the full adder are shown in Figure-1.

因此，全加器的电路由一个异或门、三个与门和一个或门组成，它们连接在一起，如图 1 中的全加器电路所示。

Hence, the circuit of the full adder consists of one EX-OR gate, three AND gates and one OR gate, which are connected together as shown in the full adder circuit in Figure-1.

Operation of Full Adder

全加器采用三个输入，即 A、B 和 Cin。其中，A 和 B 是两个二进制位，Cin 是二进制加法的上一个阶段的进位位。全加器的和输出通过将位 A、B 和 Cin 异或获得。而进输出位 (Cout) 则通过与和或操作获得。

Full adder takes three inputs namely A, B, and Cin. Where, A and B are the two binary digits, and Cin is the carry bit from the previous stage of binary addition. The sum output of the full adder is obtained by XORing the bits A, B, and Cin. While the carry output bit (Cout) is obtained using AND and OR operations.

Truth Table of Full Adder

真值表是指示逻辑电路的输入和输出变量之间的关系并解释逻辑电路操作的一种方法。以下是全加器电路的真值表 −

Truth table is one that indicates the relationship between input and output variables of a logic circuit and explains the operation of the logic circuit. The following is the truth table of the full-adder circuit −

Inputs	Outputs
A	B
Cin	S (Sum)
Cout (Carry)	0
0	0
0	0
0	0
1	1
0	0
1	0
1	0
0	1
1	0
1	1
0	0
1	0
1	0
1	0
1	1
1	0
0	1
1	1
1	1

因此，从真值表可以清楚地看到，当只有一个输入等于 1 或所有输入都等于 1 时，全加器的和输出等于 1。而当两个或三个输入等于 1 时，进位输出带有进位 1。

Hence, from the truth table, it is clear that the sum output of the full adder is equal to 1 when only 1 input is equal to 1 or when all the inputs are equal to 1. While the carry output has a carry of 1 if two or three inputs are equal to 1.

K-Map for Full Adder

K-Map（卡诺图）是简化二进制复杂布尔代数表达式的工具。全加器的 K-Map 如图 2 所示。

K-Map (Karnaugh Map) is a tool for simplifying binary complex Boolean algebraic expressions. The K-Map for full adder is shown in Figure-2.

Characteristic Equations of Full Adder

全加器的特性方程，即和（S）和进位输出（Cout）的方程，是根据二进制加法的规则得到的。这些方程如下 −

The characteristic equations of the full adder, i.e. equations of sum (S) and carry output (Cout) are obtained according to the rules of binary addition. These equations are given below −

全加器的和（S）是 A、B 和 Cin 的异或。因此，

The sum (S) of the full-adder is the XOR of A, B, and Cin. Therefore,

\mathrm{Sum, \: S \: = \: A \: \oplus \: B \: \oplus \: C_{in} \: = \: A’B’C_{in} \: + \: A’BC' {in} \: + \: AB’C' {in} \: + \: ABC_{in} }

\mathrm{Sum, \: S \: = \: A \: \oplus \: B \: \oplus \: C_{in} \: = \: A’B’C_{in} \: + \: A’BC'{in} \: + \: AB’C'{in} \: + \: ABC_{in} }

半加法器的进位（C）是 A 和 B 的与。因此，

\mathrm{Carry, \: C \: = \: AB \: + \: AC_{in} \: + \: BC_{in}}

Advantages of Full Adder

以下是全加器相对于半加器的优势 −

The following are the important advantages of full adder over half adder −

Full adder provides facility to add the carry from the previous stage.
The power consumed by the full adder is relatively less as compared to half adder.
Full adder can be easily converted into a half subtractor just by adding a NOT gate in the circuit.
Full adder produces higher output that half adder.
Full adder is one of the essential part of critic digital circuits like multiplexers.
Full adder performs operation at higher speed.

Applications of Full Adder

以下是全加器的重要应用 −

The following are the important applications of full adder −

Full adders are used in ALUs (arithmetic logic units) of CPUs of computers.
Full adders are used in calculators.
Full adders also help in carrying out multiplication of binary numbers.
Full adders are also used to realize critic digital circuits like multiplexers.
Full adders are used to generate memory addresses.
Full adders are also used in generation of program counterpoints.
Full adders are also used in GPU (Graphical Processing Unit).

Conclusion

在本教程中，我们讨论了与数字电子学中的全加器相关的所有关键概念。全加器在许多数字电子电路中发挥着重要作用，因为全加器可用于实现其他一些关键数字电路。

In this tutorial, we discussed all the key concepts related to full adders in digital electronics. Full adders play an important role in many digital electronic circuits because a full adder can be used realize several other critical digital circuits.