Intro. to Circuits

Chapter 1: Basic Concepts

Charge and Currents

Charge:

• an electrical property of the atomic particles of which matter consists
• Coulomb = Ampere $\times$ Second
• $1$ electron = $1.602\times 10^{-19}$ C
• $1$ C = $6.24\times 10 ^{18}$ electrons
• Law of conservation of charge 电荷守恒定律

电荷和电流的关系：

• $i=\frac{dq}{dt}$
• $Q=\int_{t_0}^{t}dt$

direct current 直流电

• the unidirectional flow of electric charge 单向电流
• remains constant with time 常电流 (textbook)

alternating current 交流电

• the movement of electric charge periodically reverses direction 周期变向电流
• a current that varies sinusoidally with time 正弦电流 (textbook)

例题：

$$ i=\begin{cases} 2\text{A}, 0 < t < 1 \text{s} \\ 2t^2\text{A}, t>1\text{s} \end{cases} $$

求 0-2s charge entering the element.

Solution:

$Q=\int_0^2i \, dt=\int_0^1 i\,dt+\int_1^2 i\,dt=2t\big|_0^1+2\frac{t^3}{3}\Big|_1^2=2+\frac{14}{3}\approx 6.667$ (C)

直接利用公式代入求解即可

Voltage

Voltage 电压，即电势差/电动势 (potential difference):

• the energy (or work) required to move a unit charge through an element, measured in volts (V)
• $V_{ab}=\frac{dw}{dq}$
• "+""-" is to define the reference direction/polarity of the voltage

Signal

Current and voltage are the two basic variables in electric circuits. The common term signal is used for an electric quantity such as a current or a voltage (or even electromagnetic wave) when it is used for conveying information.

信号 是电学物理量 (电压/电流/电磁波) 被用于 传递信息 时的常规代称

Power and Energy

Power:

• the time rate of expending or absorbing energy, measured in watts (W).
• the instantaneous power absorbed by an element:
  • $p=\frac{dw}{dt}=\frac{dw}{dq}\frac{dq}{dt}=vi$

电路功率标记方法：

• Passive sign convention (PSC)

  • When the current enters through the positive terminal of an element and $p = +vi$.
  • 正进负出用电器 (passive), $p>0$ for absorbing power.
  • If the current enters through the negative terminal, $p= −vi$.
  • 负进正出给电器 (active), $p<0$ for supplying power.

• Active sign convention (ASC): Opposite to PSC.

According to the conservation of energy, $\sum p = 0$.

Energy:

• the capacity to do work, measured in joules (J)
• Energy absorbed: $w = \int_{t0}^t p\,dt = \int{t_0}^t vi\,dt$

例题：

Find the power delivered to an element at $t=5$ ms if the current entering its positive terminal is $i=5\cos60\pi t$ A and the voltage is:
(a) $v=2i$ V.
(b) $v=\left( 10+5\int_0^t i\,dt \right)$ V.

Solution:

(a) The power delivered to the element is 17.27W:
$p=vi=2i^2=2\times(5\cos 60 \pi t)^2\ =50\cos^2 60 \pi t = 50\cos^2 (60\pi\times 5\times 10^{-3} ) \ \approx 17.27\, (\text{W})$

(b)

$$ \begin{align} v & =\left( 10+5\int_0^t i \, dt \right)=10+5\int_0^t 5\cos 60 \pi t \\ & =10+\frac{5}{12}\sin 60 \pi t\Big|_0^t=10+\frac{5}{12\pi}\sin 60 \pi t \\ & = 10+\frac{5}{12\pi}\sin (60\pi\times0.005)\approx 10.1073 \,(\text{V}) \\ i & =5 \cos (60\pi t)=5\cos(60\pi\times0.005)\\ &\approx 2.9389 \, (\text{A}) \\ p & =vi=10.1073\times 2.9389\approx 29.70 \, (\text{W}) \end{align} $$

Circuit Elements 电子元件

Two type:

• Passive elements model physical devices
  • cannot generate electric energy
  • resistors 电阻, capacitors 电容, inductors 电感, ...
• Active elements model devices
  • capable of generating electric energy
  • generators 发电机, batteries 电池, operational amplifiers 运算放大器, ...

Sources (active elements, 形状决定独立, 符号决定电学量):

• Independent sources 独立源

  • ideal independent voltage source (理想恒压电源, 圆交/直, 线直, $+-$ )

  • ideal independent current source (理想恒流电源, 圆箭头, $\uparrow$ )

• Dependent (or controlled) sources 受控源

  • the source quantity is controlled by another voltage or current. (菱形受控，符号源类，公式显示受控类型及相关量)
    eg. op-amp 运算放大器，transformers 变压器，transistors 晶体管
  • voltage-controlled voltage source (VCVS, 压控电压源)
  • C C V S (流控电压源)
  • V C C S (压控电流源)
  • C C C S (流控电流源)
  • symbol:
  • example: a CCVS: 菱形表示受控， $+-$ 表示电压源，10 $i$ 表示其两端电压数值大小(V)为流过中路元件的电流 ($i$) 数值的10倍(A)

Chapter 2: Basic Laws

Ohm's Law 欧姆定律：