数字逻辑chapter3-1

3.1.2 Deriving Switching Equations

Each input variable group that produces a logic 1 in a truth table output column can form a term in a Boolean switching equation. (每一个在真值表中输入
C2 C3 C1
A NASA system consists of three computers, if one computer experiences a problem it is take off-line and another computer is brought on line. Self-checking diagnostics determine each computer fails it must be switched off-line. No more than two computers are to be on-line at any given time. Design the control logic to connect or disconnect the computer. In the event that two computers are unavailable, generate a warning and allow the third computer to come online. If all three computers are unavailable, generate a second warning signal that invoke emergency procedures.
3.1.1 Problem Statements to Truth Tables

E.g. A NASA system(Ex. 3.2)
C2 C1
C3
3.1.1 Problem Statements to Truth Tables

E.g. Conveyor system( ex. 3.4)
A conveyor system brings raw material in form three different sources. The three sources converge into a single output conveyor. Sensors mounted adjacent to each S3 sourcem3 conveyor indicate the present of raw material.
S3 m3
S2
m2
m4
S1
m1
3.1.1 Problem Statements to Truth Tables

In summary, the process of converting a verbal problem statement into a truth table involves the following steps(总之，将一个书面问题描述转换成真值表的 过程包含以下几步):
CHAPTER 3: PRINCIPLES OF COMBINATIONAL LOGIC
(Sections 3.1 – 3.5)
3.1 DEFINITION OF COMBINATIONAL LOGIC
Logic circuits without feedback from output to the input, constructed from a functionally complete gate set, are said to be combinational.
Determine the input variables and ouFra Baidu bibliotekput variables that are
involved (确定所包含的输入、输出变量) Assign mnemonic or letter or number symbols to each variable(为每个变量分配助记符或字母或标记) determine the size of the truth table(确定真值表的大小); how many input combinations exist: 2x=y where x=number of input variables and y=number of combinations. Construct a truth table containing all of the input variable combinations(构造一个包含所有输入变量组合的真值表) By careful reading of the problem statement determine the combinations of inputs that cause a given output to be true(仔细研究问题描述，确定使给定输出为真的输入组合)
Inputs
· ·
Combinational Logic Functions
·
· · ·
Outputs
3.1.1 Problem Statements to Truth Tables

Before any combinational logic can be designed it must be defined. Proper statement of a problem is the most important part of any digital design task. Figure 3.2 illustrates the sequence of design tasks in a general way
3.1.2 Deriving Switching Equations

Definitions Canonical sum of products(标准积之和): A canonical sum of products is a complete set of minterms that defines when an output variable is a logical 1. the SOP for the output M in table 3.1 is M=a’bms+ab’ms+abms Canonical product of sums(标准和之积): A canonical product of sums is a complete set of maxterms that defines when an output is logical 0. the POS for the output O1 in table 3.2 is: O1’=(C1+C2+C3)(C’1+C2+C3)(C’1+C2+C’3)(C’1+C’2+ C3)(C’1+C’2+C’3)
Boolean equations can be derived directly from a truth table or from the logic diagrams A truth table or logic diagram can be constructed from the Boolean equations.
3.1.1 Problem Statements to Truth Tables

E.g : Conveyor system (Ex. 3.1)
M=a’bms+ab’ms+abms
3.1.1 Problem Statements to Truth Tables

E.g. A NASA system (Ex. 3.2)
M1=S3‘S2’S1 + S3‘S2S1 + S3S2’S1 + S3S2S1
3.1.2 Deriving Switching Equations

Definitions Literal(字母): A literal is a Boolean variable or its complement Product term(乘积项): literals or the logical product(AND) of multiple literals. connected by • Sum term: literals connected by + Sum of products: A sum of product (SOP) is the logic OR of multiple product Product of sums: A product of sum (POS) is the logic AND of multiple OR term Minterm: A minterm is a product term in which all the variables appear exactly once Maxterm: A maxterm is a sum term in which all the variables appear exactly once, either complemented or uncomplemented
The first task is to identify the input and output variables and to assign names to them. Solution: a: operator 1 is in position b: operator 2 is in position M is the signal to turn the motor on and off s means the interlock switch is closed m means material is present a,b,s and m are input variable; M is an output variable
1 0 0 1 1
1 0 1 0 1
1 0 1 1 1
3.1.2 Deriving Switching Equations
导出开关方程

Logic can be described in several ways Truth table Logic diagrams Boolean equation(s)
C1 is computer1; C2 is computer2; C3 is computer3; When C1=1, for example, then that computer has failed；let o1 ,o2, and o3 be the computer disconnected control signal outputs. If Ox=1, for example, then that computer is connected. Let W1 and W2 be the two warning output signals. If Wx=1, then the warning is activated
列产生逻辑1的输入变量组可形成布尔转换方程的一项)
Example 3.4, Table 3.4 output variable M1 is a 1 in four case: {S3′，S2 ′ ，S1} {S3′，S2，S1} {S3，S2′，S1} {S3，S2，S1} For M1 the Boolean equation:
问题陈述 构建真值表 写出开关方程
构建逻辑电路
画出逻辑图
方程化简
3.1.1 Problem Statements to Truth Tables

E.g : Conveyor system ( Ex. 3.1)
一个由电动马达带动的输送 原料的传输装置，如果有原 料要传送且保护联合开关没 有打开，两个操作员之一在 位时可被启动。
3.1.1 PROBLEM STATEMENTS TO TRUTH TABLES

Exe: Design a truth table to indicate a majority of three inputs is true
I3
0 0 0
I2
0 0 1
I1
0 1 0
O1
0 0 0
0 1 1 1 1
S2 m2 m4
一个传送系统从三个不同来源运输原材料。三个源汇集为一个 S1 单输出装置。与每个源传输装置相连的监测器显示原材料的出 m1 现。四个传输装置有分离的马达，可各自分开控制。
3.1.1 Problem Statements to Truth Tables

E.g. Conveyor system( ex. 3.4)