A-level 计算机科学学习记录：逻辑电路基础（logic gate）

Introduction

logic gates are fundamental while essential to computer hardware. A logic gate can be featured multiple inputs but only leads to a single output.

Be careful! Boolean circuit wise, when we say "give a value", it means the logic gate produces a TRUE value. Logic True and logic False are physically represented by voltage - no or low voltage denotes False (0), and high voltage indicates True (1). Hence, give a value means "high voltage".

In the article, the terminologies 1s / 0s, high / low voltage, and give a value / not give a value will be used interchangeably. Additionally, the website we use to show logic gate is: Logic.lyhttps://logic.ly/demo/Feel free to have a play with it.

basic gates

AND gate

 what is logic AND? It gives a value only when both inputs hold true, meaning that only for inputs (1, 1) does it produce 1. For example, you say: I own a extravagent mansion AND I am paid 100,000 pounds annually. So only when both conditions are satisfied, your statement is true.

 

OR gate

waht is logic OR? the logic gate produces 1 when at least one of all inputs is true, which indicates that input combinations (0,1), (1,0), and (1,1) can lead to a true value. For example, you tell a fairy: I want a high salary or a big house. She only needs to realise at least one of your wishes.

 

NOT gate

Not operation is unary and the simplest to learn. Given any value inputed, the gate returns a reversed output. For example you input 0 it outputs 1.

 

XOR gate

XOR stands for "excluded OR". It returns a value when either of the inputs is true. It can be represented by a combination of logic gates.

The combinational logic gate shown above simulates the functionality of a XOR gate. Some people may find it fairly complicated. Later on we will devise how these circuits are devised.

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Truth table

Truth table is simply a table that enumerates all possible input combinations and the according output under the law of a basic/combinational logic gate.

truth table of AND gate

 

truth table of OR gate

 

truth table for NOT gate

 

the truth table for XOR gate

truth table are usually used to build logic gate. How?

Inputs and intended output are entered for all possible values. We are to find the rows of inputs that make the output true. For example, in the truth table for XOR gate, we can see that the output will be true when input A is 0, B is 1, or when input A is 1, B is 0.

In the two valid pairs of inputs, we need to identify the input that is 0 and turn it to 1 by not gate, and use AND gate to connect the two inputs. For example, in the second row of the truth table of XOR gate, the input A is 0 while the input B is 1, then we do: !A*B; similarly, for the last row, we have A*!B.

Two possible combinations of inputs both return an output of 1, so we utilise OR gate to connect the two -- then we have the expression for XOR gate by OR, AND, and NOT gate: !A*B+A*!B.

However, usually, this is not the last step. Some more complicated boolean expressions can be simplified via certain techniques. It makes sure that we can use as few gates as possible to achieve the same function for cost-effective purposes.

 

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Boolean Algebra

A AND B is represented as A*B, or A∧B,  A&B which is logic multiplication, which makes sense in a set operation...

A OR B is A+B, A∨B, A|B which is logic addition

NOT A is usually represented as ¬A, Ā, 'A, !A

A XOR B is: A⊕B, A⊻B, A^B

similar to number operation, we do have some rules of arithmetics in boolean operation

general rules

A*0=0, A*1=1, A*A=A, A*~A=0

A+1=1, A+0=A, A+A=A, A+~A=1

A*B=B*A, A+B=B+A (commutative rules)

A+B+C=A+(B+C), A*B*C=A*(B*C) (associative rules)

A*(B+C)=A*B+A+C, A+(B*C)=(A+B)*(A+C) (distributive rules)

A+(A*B)=A, A*(A+B)=A (absorption rules) (proof: try (A*1)+(A*B) , the second equation can be transformed to the first)

~(A*B)=~A+~B, ~(A+B)=~A*~B (De Morgan) (can be proved via Venn diagram)

A=~~A (double negative)

 Use the powerful internet if you want to prove some of them.

 

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NAND gate

why do we sometimes use a combination of logic gates to represent another basic logic gate?

Mass production of a circuit costs less than producing all in a small amount. Thus, it's wise to use one circuit to represent the others. However, in some scenarios, representing a circuit by a combination of logic circuit is inefficient and would become too unnecessarily complicated.

nand gate is the AND gate followed by a NOT gate. It is said that NAND gate can represent all the basic circuits.

Use NAND gate to represent all the basic gates. How do we figure out why they make sense? use boolean algebra to prove it. Dont worry, it's quite easy!

XOR

This is purely obtained from the boolean expression, by the truth table.

 

Notice that this one achieves the function with one less gate! this can be proved to function using boolean algebra. We firstly look at the gate above and use all possible combinations of inputs and correspond outputs to obtain a truth table, from which we can get a boolean expression. Then, it is simplified and turns out to be !A*B+A*!B.

NOT

AND

can you draw a truth table for it?

OR

to prove this, we use De Morgan's law: !A*!B=!(A+B)

then: !(!A*!B)=!(!(A+B))

then: !(!A*!B)=A+B (double negative)

 

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Useful circuits

half adder and full adder

The full adder is the combinational logic gate that enables the computer to calculate binary numbers physically.

half adder

Half adder is a simplified version of a full adder. This is a machine that add one digit of one binary with another. How does it do addtion?

Assume that all of you have learnt about binary arithmetics. 

binary addtion, one digit

Here is a truth table that takes all possible combinations of inputs and produces intended outputs.

now we make a boolean expression that relates the digit of A and the digit of B to the output: A*!B + !A*B->Sum, A*B->Carry out. The first expression can be simplified to A⊕B. So, we make logic gates base on these expressions!

 

Full adder

What if we want to calculate several digits? for example 10+11 (bin)? Then we need to consider the carry in.

The truth table above shows all useful combinations of three inputs and according outputs. We make boolean expressions out of which and simplified it as usual. Note 'carry in' as C. For Sum, we have: !C*!A*B+!C*A*!B+C*!A*!B+C*A*B. For carry out, we have: !C*A*B+C*!A*B+C*A*!B+C*A*B

 This is wayyyyy too complicated -- any factories who adopt this expression will bankrupt immdiately. If not, consumers suffer - opening a web page may need minutes, and the game design will be impractical attributed to its low computational performance. 

so we need to simplify the boolean expressoin above!

that's a one digit full adder. Usually, we use much bigger full adders with several bytes. Here is a 8-digit full adder (one byte)

 

D-latch

J-latch

The purpose of this D-latch is to store data when there is no current (input)

If you enter an input, it gives an ouput, which will hold on even if the input is turned off

 

D-latch A-level version

Registers hold data temporarily and release them at regular time intervals according the clock speed. So we need to add a clock into this model so its change in output responses to the input and the clock.

 it is said to be a simplified version of a D-latch.