CIRCUIT ANALYSIS FOR COMPLETE IDIOTS
电路分析完全傻瓜书

All Rights Reserved.
版权所有。
No part of this publication may be reproduced in any form or by any means, including scanning, photocopying, or otherwise without prior written permission of the copyright holder.
本出版物的任何部分不得以任何形式或任何手段复制，包括扫描、复印或其他方式，未经版权持有者事先书面许可。
Copyright © 2019
版权 © 2019

Other books in the Series:
系列中的其他书籍：

Digital Signal Processing for Complete Idiots
数字信号处理入门
Control Systems for Complete Idiots Basic Electronics for Complete Idiots
完全傻瓜的控制系统 完全傻瓜的基础电子学
Digital Electronics for Complete Idiots Electromagnetic Theory for Complete Idiots
完全傻瓜的数字电子 完全傻瓜的电磁理论

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In today’s world, there’s an electronic gadget for everything and inside these gadgets are circuits, little components wired together to perform some meaningful function.
在当今世界，几乎每样东西都有电子设备，这些设备内部有电路，小组件相互连接以执行某些有意义的功能。
Have you wondered how a led display sign works or how a calculator works or toy cars work?
你有没有想过 LED 显示标志是如何工作的，计算器是如何工作的，玩具车是如何工作的？
How is it possible?? Answer, all because of electrical circuits.
这怎么可能？？答案，全都因为电路。
These tiny components when arranged in certain manner can do wonders. Fascinating isn’t it?
这些微小的组件以某种方式排列时可以创造奇迹。很迷人，不是吗？
Our fascination with gadgets and reliance on machinery is only growing day by day and hence from an engineering perspective, it is absolutely crucial to be familiar with the analysis and designing of such Circuits, at least identify components.
我们对小工具的迷恋和对机器的依赖与日俱增，因此从工程的角度来看，熟悉此类电路的分析和设计，至少识别组件是绝对重要的。
Circuit analysis is one of basic subjects in engineering and particularly important for Electrical and Electronics students.
电路分析是工程学中的基本学科之一，对于电气和电子专业的学生尤其重要。
So circuit analysis is a good starting point for anyone wanting to get into the field.
因此，电路分析是任何想要进入该领域的人的一个良好起点。
It is a very easy subject to learn and understand, but messing up these ideas or misunderstanding them, will lead to a lot of headache in other subjects.
这是一个非常容易学习和理解的科目，但搞混这些概念或误解它们，会在其他科目中带来很多头痛。
In this book we provide a concise introduction into basic Circuit analysis.
在本书中，我们提供了对基本电路分析的简明介绍。
A basic knowledge of Calculus and some Physics are the only prerequisites required to follow the topics discussed in the book.
学习本书讨论的主题只需要基本的微积分知识和一些物理知识。
We’ve tried to explain the various fundamental concepts of Circuit theory in the simplest manner without an over reliance on math.
我们尝试以最简单的方式解释电路理论的各种基本概念，而不依赖过多的数学。
Also, we have tried to connect the various topics with real life situations wherever possible.
此外，我们尽可能将各种主题与现实生活中的情况联系起来。
This way even first timers can learn the basics of Circuit theory with minimum effort.
这样，即使是初学者也能以最小的努力学习电路理论的基础知识。
Hopefully the students will enjoy this different approach to Circuit Analysis.
希望学生们会喜欢这种不同的电路分析方法。
The various concepts of the
各种概念的
subject are arranged logically and explained in a simple reader-friendly language with illustrative figures.
主题以逻辑方式安排，并用简单易懂的语言和插图进行解释。
This book is not meant to be a replacement for those standard Circuit theory textbooks, rather this book should be viewed as an introductory text for beginners to come in grips with advanced level topics covered in those books.
这本书并不是为了替代那些标准的电路理论教科书，而是应该被视为一本入门书籍，帮助初学者掌握那些书中涵盖的高级主题。
This book will hopefully serve as inspiration to learn Circuit theory in greater depths.
这本书希望能激励人们更深入地学习电路理论。
Readers are welcome to give constructive suggestions for the improvement of the book and please do leave a review.
欢迎读者提出建设性的建议以改进本书，并请留下评论。

Have you ever wondered what Electricity is and where it comes from?
你有没有想过电是什么以及它来自哪里？
To answer these questions, we have to start with the atom.
要回答这些问题，我们必须从原子开始。
Although we are more interested in the properties of electricity than the phenomenon itself, it wouldn’t hurt us to quickly discuss the basics.
虽然我们对电的性质比现象本身更感兴趣，但快速讨论一下基础知识也无妨。
Everything in the universe is made of atoms and every atom consists of 3 types of particles, neutrons, protons and electrons.
宇宙中的一切都是由原子构成的，每个原子由三种粒子组成：中子、质子和电子。
Neutrons and protons are packed together in the nucleus and make up the center of an atom, whereas the electrons move around the nucleus in a constant motion.
中子和质子聚集在原子核中，构成原子的中心，而电子则以恒定的运动围绕原子核移动。
For this discussion, we are only concerned about protons and electrons or more specifically, a property these two particles possess called the Electric Charge.
在本次讨论中，我们只关注质子和电子，或者更具体地说，这两个粒子所具有的一个属性，称为电荷。
Although it is very unlikely you’ll ever come across a proper definition for charge, the best we can come up with is, that charge is a form of electrical energy.
虽然你很可能永远不会遇到电荷的正式定义，但我们能想到的最好解释是，电荷是一种电能形式。
Protons have a positive charge and Electrons have a negative charge.
质子带有正电荷，电子带有负电荷。
In a normal atom, the number of protons is equal to the number of electron and thus the atom as a whole is electrically neutral.
在一个正常的原子中，质子的数量等于电子的数量，因此整个原子是电中性的。
Neutral objects aren’t of much interest to us, we are more interested in charged bodies.
中性物体对我们并不太感兴趣，我们更关注带电物体。
Electric Charge is denoted by the letter Q.
电荷用字母 Q 表示。

The SI unit of electric charge is Coulomb © and it is the charge possessed by $6.24\times {10}^{18}$$6.24\times {10}^{18}$6.24 xx10^(18)6.24 \times 10^{18} electrons.
电荷的国际单位是库仑（C），它是 $6.24\times {10}^{18}$$6.24\times {10}^{18}$6.24 xx10^(18)6.24 \times 10^{18} 个电子所拥有的电荷。

Previously we mentioned that free electrons are responsible for the flow of Electric Current.
之前我们提到，自由电子负责电流的流动。
The concept behind this phenomenon is very simple, whenever a charged particle moves, it produces an Electric Current.
这个现象背后的概念非常简单，任何带电粒子移动时，它都会产生电流。
Obviously the protons can’t move, because they are inside the nucleus.
显然，质子不能移动，因为它们在原子核内。
And the electrons close to the nucleus are held tightly by the force of attraction, so they can’t move either.
而靠近原子核的电子被吸引力紧紧束缚，因此它们也无法移动。
So the only way an Electric current is produced is through movement of outer electrons, called the free electrons (it’s a little different in electronics though).
因此，电流产生的唯一方式是通过外层电子的运动，称为自由电子（不过在电子学中稍有不同）。

To understand this better, consider the inside section of a Conductor as shown below.
要更好地理解这一点，请考虑下面所示的导体内部部分。

Conductors have tons of free electrons and they keep moving in random direction (due to thermal energy), and each of these small movements contribute to an Electric current.
导体中有大量自由电子，它们在随机方向上不断移动（由于热能），这些微小的运动共同产生电流。
You might be thinking, if an electric current is produced this easily in a conductor, why do we need batteries and generators and power plants and stuff.
你可能在想，如果在导体中如此容易产生电流，为什么我们还需要电池、发电机和电厂等东西。
Can’t we just hook up a small piece of copper wire to a bulb and be done with it.
我们不能只把一小段铜线连接到灯泡上就完成了吗？
Unfortunately, that won’t work.
不幸的是，那行不通。
That’s because the currents produced by each free electron are in random direction (in accordance with the direction of their motion) and when we consider the conductor as a whole, these currents cancel each other out and net current is zero.
这是因为每个自由电子产生的电流方向是随机的（与它们的运动方向一致），当我们将导体视为一个整体时，这些电流相互抵消，净电流为零。
The way out of this problem is to make all the free electrons drift in one direction and thus the net Electric Current adds up to a non-zero value.
解决这个问题的方法是使所有自由电子朝一个方向漂移，从而净电流累加到一个非零值。
To do this all we need is a little effort, a force of sorts, called the EMF or the Electromotive Force.
要做到这一点，我们只需要一点努力，一种被称为电动势（EMF）的力量。
We will discuss more about the EMF in the next section.
我们将在下一节中讨论更多关于 EMF 的内容。

So Electric Current can be defined as the flow of charge (electrons) when subjected to an EMF.
电流可以定义为在电动势作用下电荷（电子）的流动。
Or the more accurate definition would be, Current is the rate of flow of charge.
或者更准确的定义是，电流是电荷流动的速率。
Mathematically, Current I is equal to,
在数学上，电流 I 等于，

The unit of current is Ampere, named after French mathematician and physicist André-Marie Ampère.
电流的单位是安培，以法国数学家和物理学家安德烈-玛丽·安培的名字命名。
One ampere of current represents one coulomb of electrical charge moving past a specific point in one second.
一安培的电流表示在一秒钟内有一个库仑的电荷经过一个特定点。

EMF stands for Electromotive force.
EMF 代表电动势。
The name may give you the impression that electromotive force is a type of force.
这个名称可能会给你留下电动势是一种力的印象。
Actually, it is not.
实际上，情况并非如此。
As mentioned in the previous section, EMF or the Electromagnetic force is an energy that can cause current to flow in an electrical circuit or device.
如前一节所述，电磁场或电磁力是一种能在电路或设备中引起电流流动的能量。
This means that a current can flow in a circuit or a device, only if an EMF is provided.
这意味着电流只能在电路或设备中流动，如果提供了电动势。
Sources of EMF can be batteries, solar cells, generators etc. EMF is denoted by the symbol E and is measured in unit Volt (V).
电磁场的来源可以是电池、太阳能电池、发电机等。电磁场用符号 E 表示，单位为伏特（V）。

Both EMF and Potential Difference are closely related and are often used interchangeably in many places, but they aren’t the same quantities.
电动势和电位差密切相关，许多地方常常可以互换使用，但它们并不是相同的量。
When a current flows through a material, the electrons are accelerated due to the applied EMF.
当电流通过材料时，电子由于施加的电动势而加速。
But these electrons don’t gain much velocity, because they keep colliding with ions in the material and due to this, the kinetic energy of the electrons is converted to heat.
但是这些电子并没有获得太大的速度，因为它们不断与材料中的离子碰撞，因此电子的动能转化为热能。
What this means is that, the electrons at one of the material has more energy than the electrons at the other end, which leads to a potential difference.
这意味着，材料一端的电子比另一端的电子具有更高的能量，从而导致了电位差。
This obviously is a rough explanation, the actual physics behind phenomenon is more complex and beyond the scope of this book.
这显然是一个粗略的解释，现象背后的实际物理学更复杂，超出了本书的范围。
It is important to note that, Potential difference is always measured between 2 points and never at a single point.
重要的是要注意，电位差总是测量两个点之间的，而不是在单个点上。

To sum up, the EMF is the driving force that keeps electrons in motion and Potential difference is the difference in energy of the electrons as a current is passed through a material.
总之，电动势是使电子运动的驱动力，而电位差是电流通过材料时电子能量的差异。
Both EMF and Potential difference have the common unit Volt (V).
电动势和电位差的共同单位是伏特（V）。
The term Voltage can be used in place of Potential difference or EMF.
电压一词可以用来代替电位差或电动势。

From the previous sections itself, it must be pretty clear that the Voltage and the Current are two closely related quantities.
从前面的部分来看，电压和电流是两个密切相关的量。
They have a cause effect relation as given by this general equation:
它们之间存在因果关系，如下所示的通用方程：

Where the Voltage is the cause and the Current is the effect.
电压是原因，电流是结果。
Now the question is, what could possibly be the opposition to current?
现在的问题是，可能对电流的反对是什么？
This is where we introduce a quantity called Resistance.
这是我们引入一个叫做电阻的量的地方。
The concept of Resistance is analogous to friction in mechanics.
阻力的概念类似于力学中的摩擦。
Every material has a tendency to oppose current, but some more than the others.
每种材料都有抵抗电流的倾向，但有些材料的抵抗力更强。
Materials with large no.
材料数量大。
of free electrons like metals have low resistance or a low tendency to oppose current.
自由电子像金属一样具有低电阻或低反对电流的倾向。
Such materials are called Conductors. Whereas materials with small no.
这种材料被称为导体。而具有少量的材料。
of free electrons like plastic have high resistance. Such materials are called Insulators.
像塑料这样的自由电子具有高电阻。这种材料被称为绝缘体。
And some materials fall in between, they offer some resistance, but not very high either.
有些材料介于两者之间，它们提供了一定的阻力，但也不是很高。
They are called Semi-conductors.
它们被称为半导体。

Now let’s substitute the terms we introduced so far into our general equation from earlier.
现在让我们将到目前为止介绍的术语代入我们之前的通用方程中。

The result is this beautiful equation called the Ohm’s Law, after the German physicist and mathematician Georg_Simon Ohm (weird name right??).
结果是这个美丽的方程，称为欧姆定律，以德国物理学家和数学家乔治·西蒙·欧姆的名字命名（名字很奇怪，对吧？？）。
It’s one of the most fundamental things there is in electrical engineering. Get used to it,
这是电气工程中最基本的事情之一。习惯它吧，
because it will remain with you as long as you do anything electrical related.
因为只要你做任何与电相关的事情，它就会伴随你。

The Ohm’s law essentially implies that, the current flowing through a material/circuit is directly proportional to the Voltage applied across it, provided that the resistance of the material remain fixed.
欧姆定律基本上意味着，流经材料/电路的电流与施加在其上的电压成正比，前提是材料的电阻保持不变。
So if we were to apply twice the voltage across a bulb, twice the amount of current would flow through it or if we apply one third the voltage, then one third the current would flow.
所以如果我们在灯泡两端施加两倍的电压，流过它的电流将是两倍；如果我们施加三分之一的电压，那么流过的电流将是三分之一。
Graphically the Ohm’s law would look like,
欧姆定律在图形上看起来是这样的，

The Unit of Resistance is Ohm and is denoted by the Greek letter $\mathrm{\Omega }$$\mathrm{\Omega }$Omega\Omega.
电阻的单位是欧姆，用希腊字母 $\mathrm{\Omega }$$\mathrm{\Omega }$Omega\Omega 表示。

While we are at it, let’s define one more new quantity called Conductance.
在此期间，让我们定义一个新的量，称为电导。
Conductance is the inverse of Resistance.
导电性是电阻的倒数。
It’s a measure of how well a material allows current to flow
它是衡量材料允许电流流动的能力
through it. The Unit of Conductance is Siemens and is denoted by ${\mathrm{\Omega }}^{-1}$${\mathrm{\Omega }}^{-1}$Omega^(-1)\Omega^{-1}.
通过它。电导的单位是西门子，用 ${\mathrm{\Omega }}^{-1}$${\mathrm{\Omega }}^{-1}$Omega^(-1)\Omega^{-1} 表示。

Have you seen one of these tiny components in an electronic circuit before??
你见过电子电路中的这些微小组件吗？

Those are resistors. A Resistor is a device that provide resistance in an electrical circuit. WHAT??
那些是电阻器。电阻器是一个在电路中提供电阻的设备。什么？？
But isn’t resistance a bad thing?
但抵抗不是一件坏事吗？
Yes, resistance does oppose current and it does cause energy loss.
是的，电阻确实会对电流产生阻碍，并导致能量损失。
But when used the right way it isn’t always a bad thing.
但当以正确的方式使用时，这并不总是一件坏事。
Do you know that resistance is the reason we have bulbs and heaters?
你知道电阻是我们有灯泡和加热器的原因吗？
Resistors are electrical components that help control the flow of current in a circuit.
电阻器是帮助控制电路中电流流动的电气元件。
A high resistance means there is less current available for a given voltage.
高电阻意味着在给定电压下可用的电流较少。
It is widely used in heating applications, for biasing, voltage dividers and tons of other applications.
它广泛用于加热应用、偏置、电压分压器以及许多其他应用。
The symbol for resistor is:
电阻器的符号是：

Electrical power is defined as the rate at which electrical energy is transferred from an energy source to a circuit.
电力被定义为电能从能源源转移到电路的速率。
When current is passed through a resistor, energy is dissipated as heat.
当电流通过电阻时，能量以热的形式散失。
It is easy to calculate Electrical power, it is simply the product of the current (I) flowing through a component and the voltage ( V ) across the component.
计算电功率很简单，它只是流过一个元件的电流（I）与该元件两端电压（V）的乘积。

Applying the Ohm’s law, 2 other forms of equation can be obtained,
应用欧姆定律，可以得到另外两种形式的方程

Unit of electrical power is Watts.
电功率的单位是瓦特。

A series circuit is a circuit in which any number of components are connected one after the other, such that there is a single path for the flow of current.
串联电路是指多个元件一个接一个连接在一起的电路，从而形成一个电流流动的单一路径。
For example, in the circuit shown in the figure below, the Resistors $R_1$$R_1$R_(1)R_{1} and $R_2$$R_2$R_(2)R_{2} are in series, because they are connected at a common point b. Similarly, Resistor ${\mathrm{R}}_2$${\mathrm{R}}_2$R_(2)\mathrm{R}_{2} and the Voltage source are also in series, with the common point c.
例如，在下图所示的电路中，电阻器 $R_1$$R_1$R_(1)R_{1} 和 $R_2$$R_2$R_(2)R_{2} 是串联的，因为它们连接在一个公共点 b。类似地，电阻器 ${\mathrm{R}}_2$${\mathrm{R}}_2$R_(2)\mathrm{R}_{2} 和电压源也是串联的，公共点为 c。

$C$$C$CC

If there were any other components (that carry current) connected at any of these nodes ( $a,b$$a,b$a,ba, b or $c$$c$cc ), then this circuit wouldn’t be a series circuit anymore. For instance, if there
如果在这些节点（ $a,b$$a,b$a,ba, b 或 $c$$c$cc ）的任何一个连接了其他任何带电流的组件，那么这个电路就不再是串联电路了。例如，如果有
had been a third resistor ${\mathrm{R}}_3$${\mathrm{R}}_3$R_(3)\mathrm{R}_{3} connected between nodes a and b , as shown in the figure below, this is no longer a series circuit.
在下图中，节点 a 和 b 之间连接了一个第三个电阻 ${\mathrm{R}}_3$${\mathrm{R}}_3$R_(3)\mathrm{R}_{3} ，这不再是一个串联电路。
Clearly there are 2 paths for the current to flow, through $R_1\mathrm{\&}{R}_3$$R_1\mathrm{\&}{R}_3$R_(1)&R_(3)R_{1} \& R_{3}.
显然，电流有两条路径可以流动，通过 $R_1\mathrm{\&}{R}_3$$R_1\mathrm{\&}{R}_3$R_(1)&R_(3)R_{1} \& R_{3} 。

$C$$C$CC

Kirchhoff’s Law’s…Wait!! "Laws " you say?? You mean there’s more than one law??
基尔霍夫定律……等等！！“定律”你说？？你是说不止一条定律？？
Yes, there are 2 Kirchhoff’s Law’s: Kirchhoff’s Voltage law & the Kirchhoff’s Current Law.
是的，有两个基尔霍夫定律：基尔霍夫电压定律和基尔霍夫电流定律。
Kirchhoff’s laws are the most fundamental laws, next to the Ohm’s law, in Electrical engineering.
基尔霍夫定律是电气工程中最基本的定律，仅次于欧姆定律。
But fortunately, just like the Ohm’s law, these are 2 really simple laws. Even
但幸运的是，就像欧姆定律一样，这两个法则真的很简单。即使
simpler than the Ohm’s Law I would say, because there is no formula, just a simple statement.
比欧姆定律简单，我会说，因为没有公式，只有一个简单的陈述。
The entire basis of Circuit analysis are these 2 laws and the Ohm’s law.
电路分析的整个基础是这两个定律和欧姆定律。
They are basically spin offs to the energy and charge conservation laws.
它们基本上是能量和电荷守恒定律的衍生物。
We’ll get to the Kirchhoff’s Current Law in later section.
我们将在后面的章节中讨论基尔霍夫电流定律。
For now, we’ll focus on the Kirchhoff’s Voltage Law or the KVL.
目前，我们将重点关注基尔霍夫电压定律或 KVL。
Kirchhoff’s voltage law (KVL) states that “the algebraic sum of the potential rises and drops around a closed loop (or path) is zero”.
基尔霍夫电压定律（KVL）指出：“一个闭合回路（或路径）中电势升高和降低的代数和为零。”
Symbolically,
象征性地，

Closed Path
闭合路径

In layman’s terms Kirchhoff’s voltage law essentially means: “Voltage supplied = Voltage used up, around a closed loop”.
用通俗的话来说，基尔霍夫电压定律基本上意味着：“供电电压 = 使用电压，围绕一个闭合回路。”
Forming a KVL equation is really easy, start at a certain point of the circuit and note down all the potential changes (either rises or drops) in one particular direction, till the starting point is reached once again.
形成一个 KVL 方程真的很简单，从电路的某个点开始，记录在一个特定方向上的所有电位变化（无论是上升还是下降），直到再次到达起始点。
Then equate the resulting expression to zero. That’s it.
然后将结果表达式等于零。就这样。

For the above Circuit, KVL equation is $\mathbf{E}-{\mathbf{V}}_{\mathbf{1}}-{\mathbf{V}}_{\mathbf{2}}\mathbf{=}\mathbf{0}$$\mathbf{E}-{\mathbf{V}}_{\mathbf{1}}-{\mathbf{V}}_{\mathbf{2}}\mathbf{=}\mathbf{0}$E-V_(1)-V_(2)=0\mathbf{E}-\mathbf{V}_{\mathbf{1}}-\mathbf{V}_{\mathbf{2}} \mathbf{= 0} or $\mathbf{E}$$\mathbf{E}$E\mathbf{E} (Voltage supplied) $={\mathbf{V}}_{\mathbf{1}}+{\mathbf{V}}_{\mathbf{2}}$$={\mathbf{V}}_{\mathbf{1}}+{\mathbf{V}}_{\mathbf{2}}$=V_(1)+V_(2)=\mathbf{V}_{\mathbf{1}}+\mathbf{V}_{\mathbf{2}} (Voltage Used up).
对于上述电路，KVL 方程是 $\mathbf{E}-{\mathbf{V}}_{\mathbf{1}}-{\mathbf{V}}_{\mathbf{2}}\mathbf{=}\mathbf{0}$$\mathbf{E}-{\mathbf{V}}_{\mathbf{1}}-{\mathbf{V}}_{\mathbf{2}}\mathbf{=}\mathbf{0}$E-V_(1)-V_(2)=0\mathbf{E}-\mathbf{V}_{\mathbf{1}}-\mathbf{V}_{\mathbf{2}} \mathbf{= 0} 或 $\mathbf{E}$$\mathbf{E}$E\mathbf{E} （供电电压） $={\mathbf{V}}_{\mathbf{1}}+{\mathbf{V}}_{\mathbf{2}}$$={\mathbf{V}}_{\mathbf{1}}+{\mathbf{V}}_{\mathbf{2}}$=V_(1)+V_(2)=\mathbf{V}_{\mathbf{1}}+\mathbf{V}_{\mathbf{2}} （消耗电压）。
Do note that KVL is applicable to all loops or closed paths, however complex the circuit maybe.
请注意，基尔霍夫电压定律适用于所有回路或闭合路径，无论电路多么复杂。

When dealing with a circuit containing large no of components, it’s a smart thing to simplify the circuit.
在处理包含大量元件的电路时，简化电路是一个明智的选择。
This applies to resistors as well.
这同样适用于电阻。
A combination of resistors, be it series or parallel or otherwise can be replaced by a single resistance, called the equivalent or the effective resistance of the circuit.
一组电阻，无论是串联、并联还是其他方式，都可以用一个单一的电阻来替代，称为电路的等效电阻或有效电阻。
For a series combination of resistors, the equivalent resistance is found by simply adding the individual resistance values.
对于电阻的串联组合，等效电阻通过简单地将各个电阻值相加来找到。
Mathematically,
数学上，

The proof for this is pretty straight forward.
这个证明相当简单。
Consider our example (first one) from section 2.1. Let ${\mathrm{V}}_1\mathrm{\&}{\mathrm{V}}_2$${\mathrm{V}}_1\mathrm{\&}{\mathrm{V}}_2$V_(1)&V_(2)\mathrm{V}_{1} \& \mathrm{~V}_{2} are the voltages across the resistors $R_1$$R_1$R_(1)R_{1} and $R_2$$R_2$R_(2)R_{2} respectively. Using KVL , we know $\mathbf{V}={\mathbf{V}}_{\mathbf{1}}+{\mathbf{V}}_{\mathbf{2}}$$\mathbf{V}={\mathbf{V}}_{\mathbf{1}}+{\mathbf{V}}_{\mathbf{2}}$V=V_(1)+V_(2)\mathbf{V}=\mathbf{V}_{\mathbf{1}}+\mathbf{V}_{\mathbf{2}}. Therefore,
考虑我们在 2.1 节中的例子（第一个）。设 ${\mathrm{V}}_1\mathrm{\&}{\mathrm{V}}_2$${\mathrm{V}}_1\mathrm{\&}{\mathrm{V}}_2$V_(1)&V_(2)\mathrm{V}_{1} \& \mathrm{~V}_{2} 是电阻 $R_1$$R_1$R_(1)R_{1} 和 $R_2$$R_2$R_(2)R_{2} 上的电压。根据基尔霍夫电压定律，我们知道 $\mathbf{V}={\mathbf{V}}_{\mathbf{1}}+{\mathbf{V}}_{\mathbf{2}}$$\mathbf{V}={\mathbf{V}}_{\mathbf{1}}+{\mathbf{V}}_{\mathbf{2}}$V=V_(1)+V_(2)\mathbf{V}=\mathbf{V}_{\mathbf{1}}+\mathbf{V}_{\mathbf{2}} 。因此，

In the last section, we saw that in a series connection, the resistors share a common current, but have different voltage drops across them.
在最后一节中，我们看到在串联连接中，电阻器共享一个共同的电流，但它们之间的电压降却不同。
Now we will try to find out the exact magnitude of the voltage drops.
现在我们将尝试找出电压降的确切大小。
For that we use the Voltage Divider Rule.
为此，我们使用电压分压规则。

From Ohm’s law,
根据欧姆定律，

Then the Voltage drops across the resistors are:
然后电阻上的电压降为：

To sum up, the Voltage drop across a Resistor in series connection is given,
总之，串联电阻上的电压降是给定的，

A parallel circuit is a circuit in which any number of components are connected across 2 common terminals, such that they share a common voltage.
并联电路是指任何数量的元件连接在两个公共端子之间的电路，使它们共享一个共同的电压。
For example, in the circuit shown in the figure below, the Resistors $R_1$$R_1$R_(1)R_{1} and $R_2$$R_2$R_(2)R_{2} are in parallel, because they are connected between the same terminals a and b .
例如，在下图所示的电路中，电阻器 $R_1$$R_1$R_(1)R_{1} 和 $R_2$$R_2$R_(2)R_{2} 是并联的，因为它们连接在相同的端子 a 和 b 之间。
The current will be divided amongst the resistors, according as their resistance values.
电流将根据电阻值在电阻器之间分配。

According to the Kirchhoff’s Current Law, the algebraic sum of the currents entering and leaving a node or a junction of a circuit is zero.
根据基尔霍夫电流定律，进入和离开电路节点或连接点的电流代数和为零。
It’s easily evident that this law is derived from the Law of conservation of charge.
显然，这条法律源于电荷守恒定律。
The idea is really simple, once a current is generated in a circuit, it is distributed throughout the circuit.
这个想法非常简单，一旦在电路中产生电流，它就会在整个电路中分布。
It cannot just accumulate in a wire or vanish into thin air.
它不能仅仅积累在一根电线上或消失在空气中。

Symbolically,
象征性地，

Consider the example shown below and let’s formulate the KCL equation for node a.
考虑下面的示例，让我们为节点 a 公式化 KCL 方程。
At node a, there are 3 currents, one entering and 2 leaving. Hence the KCL equation is, $\mathrm{I}={\mathrm{I}}_1$$\mathrm{I}={\mathrm{I}}_1$I=I_(1)\mathrm{I}=\mathrm{I}_{1} $+{\mathrm{I}}_2$$+{\mathrm{I}}_2$+I_(2)+\mathrm{I}_{2}.
在节点 a，有 3 个电流，一个进入，两个离开。因此，KCL 方程是， $\mathrm{I}={\mathrm{I}}_1$$\mathrm{I}={\mathrm{I}}_1$I=I_(1)\mathrm{I}=\mathrm{I}_{1} $+{\mathrm{I}}_2$$+{\mathrm{I}}_2$+I_(2)+\mathrm{I}_{2} 。

For a parallel combination of resistors, the reciprocal of the equivalent resistance is the sum of the reciprocals of the
对于电阻的并联组合，等效电阻的倒数是各个电阻倒数的总和
individual resistances. Mathematically,
个体电阻。数学上，

Consider our example from section 2.1. Let ${\mathrm{I}}_1\mathrm{\&}{\mathrm{I}}_2$${\mathrm{I}}_1\mathrm{\&}{\mathrm{I}}_2$I_(1)&I_(2)\mathrm{I}_{1} \& \mathrm{I}_{2} be the currents flowing through the resistors R1 and R2 respectively. Using KCL, we know $\mathbf{I}={\mathbf{I}}_{\mathbf{1}}+{\mathbf{I}}_{\mathbf{2}}$$\mathbf{I}={\mathbf{I}}_{\mathbf{1}}+{\mathbf{I}}_{\mathbf{2}}$I=I_(1)+I_(2)\mathbf{I}=\mathbf{I}_{\mathbf{1}}+\mathbf{I}_{\mathbf{2}}.
考虑我们在第 2.1 节中的例子。设 ${\mathrm{I}}_1\mathrm{\&}{\mathrm{I}}_2$${\mathrm{I}}_1\mathrm{\&}{\mathrm{I}}_2$I_(1)&I_(2)\mathrm{I}_{1} \& \mathrm{I}_{2} 为流过电阻 R1 和 R2 的电流。根据基尔霍夫电流定律，我们知道 $\mathbf{I}={\mathbf{I}}_{\mathbf{1}}+{\mathbf{I}}_{\mathbf{2}}$$\mathbf{I}={\mathbf{I}}_{\mathbf{1}}+{\mathbf{I}}_{\mathbf{2}}$I=I_(1)+I_(2)\mathbf{I}=\mathbf{I}_{\mathbf{1}}+\mathbf{I}_{\mathbf{2}} 。

The Current Divider Rule is used to determine the magnitude of current entering each branch of a parallel connection.
电流分配规则用于确定进入并联连接每个支路的电流大小。

From Ohm’s law,
根据欧姆定律，

Then the Currents flowing through resistors are:
然后通过电阻的电流是：

To sum up, the Current flowing through a Resistor in parallel connection is given by,
总之，流过并联连接中的电阻器的电流由以下公式给出，

Short Circuit is a condition where two points in a circuit are directly connected to each other through a path of zero resistance.
短路是指电路中的两个点通过零电阻的路径直接连接在一起的状态。
The voltage across the 2 points will be always zero in case of a short circuit.
在短路的情况下，两个点之间的电压将始终为零。

Short Circuit
短路

Open Circuit
开路

Open Circuit is exactly the opposite condition as short circuit.
开路正好与短路相反。
In case of an open circuit, there is no connection between two points in a circuit and hence no current flows between the 2 points.
在开路的情况下，电路中的两个点之间没有连接，因此在这两个点之间没有电流流动。

There are basically 2 types of energy sources: Voltage source & Current source.
基本上有两种类型的能源来源：电压源和电流源。
Again they can be classified as ideal & practical sources.
它们可以再次被分类为理想和实用的来源。
First we’ll discuss ideal sources then consider practical sources.
首先我们将讨论理想的来源，然后考虑实际的来源。

An ideal Voltage source is an Energy source which gives constant Voltage across its terminals irrespective of the current drawn by the load connected to its terminals.
理想电压源是一个能量源，它在其端子之间提供恒定电压，而不管连接到其端子的负载所抽取的电流。
At any instant of time, the voltage across the terminals remain the same.
在任何时刻，端子之间的电压保持不变。
Thus the V-I Characteristics of an ideal voltage source is a straight line as shown.
因此，理想电压源的电压-电流特性是一条直线，如图所示。

But it is not possible to make such Voltage sources in practice.
但在实践中不可能制造这样的电压源。
Practically, all Voltage sources have a small internal resistance.
实际上，所有电压源都有一个小的内阻。
For analysis purposes, we assume that this internal resistance is in series with the voltage source and is represented by ${\mathrm{R}}_{\mathrm{se}}$${\mathrm{R}}_{\mathrm{se}}$R_(se)\mathrm{R}_{\mathrm{se}}. Because of $R_{se}$$R_{se}$R_(se)R_{s e}, the voltage across the terminals decreases slightly with the increase in the current.
为了分析的目的，我们假设这个内部电阻与电压源串联，并用 ${\mathrm{R}}_{\mathrm{se}}$${\mathrm{R}}_{\mathrm{se}}$R_(se)\mathrm{R}_{\mathrm{se}} 表示。由于 $R_{se}$$R_{se}$R_(se)R_{s e} ，端子之间的电压随着电流的增加略微下降。

V-I Characteristics
V-I 特性

Usually, Voltage sources are manufactured keeping the internal resistance to the minimum, such that it acts more or less like an ideal voltage source (till a max load current limit).
通常，电压源的制造是将内部电阻保持在最低限度，使其在最大负载电流限制内表现得或多或少像一个理想电压源。
Batteries are an example of Voltage source.
电池是电压源的一个例子。

No prizes for guessing what a current source is, an ideal current source is a power source that gives constant current, irrespective of the voltage appearing across its terminals
没有奖品可以猜测什么是电流源，理想电流源是一种提供恒定电流的电源，无论其端子之间出现的电压如何

But a practical Current source hardly ever functions this way.
但实际的电流源几乎从不以这种方式工作。
In a practical Current source, the current decreases slightly as the Voltage across the load terminals increase.
在实际的电流源中，随着负载端子上的电压增加，电流略微减少。
This behavior can be analyzed by considering a high internal resistance, represented by ${\mathrm{R}}_{\mathrm{sh}}$${\mathrm{R}}_{\mathrm{sh}}$R_(sh)\mathrm{R}_{\mathrm{sh}} in parallel with the source.
这种行为可以通过考虑一个高内阻来分析，表示为 ${\mathrm{R}}_{\mathrm{sh}}$${\mathrm{R}}_{\mathrm{sh}}$R_(sh)\mathrm{R}_{\mathrm{sh}}