电子技术基础-电路

只看该作者 · 2025-7-22 22:42

本帖最后由 ttianyer 于 2025-7-22 23:53 编辑

第一章集总参数电路

集总电路

一个抽象的模型(仅含电源电阻电感电容)
电流电压功率

可以设置参考方向(实际方向与之无关) 实际方向与参考方向相同则为正

电流:设置流动方向(正电荷流动的方向是实际电流方向)

电压:设置正负极

关联参考方向:电流方向从+到-即为一致方向

电源提供能量电阻吸收能量

先给功率假设一个正方向一致则功率为正电流和电压关联且功率参考方向向电路内时若计算的功率为正则表示与功率方向一致吸收能量
基尔霍夫定律

KCL:节点电流入=出(电荷守恒)

KVL:回路电压绕圈为0(电荷守恒和能量守恒)

双下标记法:前后顺序表示电压降的方向<span class="MathJax_SVG" id="MathJax-Element-2-Frame" tabindex="0" data-mathml="uab" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uab表示从a到b的电压降
特勒根定理

电路各元件吸收功率的代数和为0(功率守恒)
电阻

VCR:电压电流关系

电导:<span class="MathJax_SVG" id="MathJax-Element-1-Frame" tabindex="0" data-mathml="G=1/R" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">G=1/R

任何一个二端元件只要u(t)和i(t)之间存在代数关系都是无**的

开路:R=∞

短路:R=0
电压源

端电压与电流无关是它本身确定的
电流源

电流与端电压无关是它本身确定的
受控源

受控源是一种双口电路控制支路(开路或短路) 受控支路(电压源或电流源)

类似电阻的伏安特性受控源有转移特性

功率:<span class="MathJax_SVG" id="MathJax-Element-3-Frame" tabindex="0" data-mathml="p(t)=u2(t)i2(t)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">p(t)=u2(t)i2(t)(就是受控支路的功率)

含受控源电路仍需满足两类约束在列KVL、KCL方程时，要把受控源暂时看作独立源
分压分流公式

串联同流分压(按电阻分) 并联同压分流(按电阻的倒数分)
拓扑约束和元件约束

KCL、KVL(拓扑约束)和元件VCR(元件约束)是对电路中各电压变量、电流变量施加的全部约束

拓扑约束:只取决于互连形式的约束

元件约束:只取决于元件特性的约束

电路分析的典型问题:给定电路的结构、元件的特性以及各独立电源的电压或电流，求出电路中所有的支路电压和支路电流，或某些指定的支路电压、支路电流

11. 支路电流法支路电压法

以电阻支路电流(或支路电压)和电压源支路电流为未知量列方程

第二章独立电路电压变量的分析方法

网孔分析法

对于具有b条支路和n个节点的平面连通电路来说它的(b-n+1)个网孔电流就是一组独立电流变量用网孔电流作变量建立的电路方程，称为网孔方程

网孔电流:在网孔内闭合流动的电流

它是一组能确定全部支路电流的独立电流变量

网孔方程一般表达式:

<span class="MathJax_SVG" id="MathJax-Element-12-Frame" tabindex="0" data-mathml="R11iM1+R12iM2+R13iM3=us11" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R11iM1+R12iM2+R13iM3=us11

<span class="MathJax_SVG" id="MathJax-Element-9-Frame" tabindex="0" data-mathml="R12iM1+R22iM2+R23iM3=us22" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R12iM1+R22iM2+R23iM3=us22

<span class="MathJax_SVG" id="MathJax-Element-7-Frame" tabindex="0" data-mathml="R13iM1+R32iM2+R33iM3=us11" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R13iM1+R32iM2+R33iM3=us11

2. <span class="MathJax_SVG" id="MathJax-Element-8-Frame" tabindex="0" data-mathml="R11" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R11 <span class="MathJax_SVG" id="MathJax-Element-6-Frame" tabindex="0" data-mathml="R22" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R22 <span class="MathJax_SVG" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="R33" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R33为网孔1 2 3的自电阻

<span class="MathJax_SVG" id="MathJax-Element-11-Frame" tabindex="0" data-mathml="R13" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R13表示网孔1 2的互电阻(回路电流方向一样则为正值方向相反为负值)

<span class="MathJax_SVG" id="MathJax-Element-13-Frame" tabindex="0" data-mathml="us11" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">us11 <span class="MathJax_SVG" id="MathJax-Element-10-Frame" tabindex="0" data-mathml="us22" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">us22 <span class="MathJax_SVG" id="MathJax-Element-5-Frame" tabindex="0" data-mathml="us33" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">us33 为网孔1 2 3的回路电压升

基本步骤:

1.电路图上标明网孔电流及其参考方向(若全部为一个方向则互电阻全部为负值)

2.观察电路图给出网孔方程
(a)对角线元为自电阻

(b)非对角线元为互电阻(注意取负值)

(c)方程右端元为电压升

3.解方程得到网孔电流

4.通过网孔电流和支路电流的代数关系得到支路电流

5.通过VCR解得支路电压

节点分析法

在电路中任选一个节点为参考点，其余的每一节点到参考点的电压降，就称为这个节点的节点电压

节点电压为一组完备的独立电压变量

<span class="MathJax_SVG" id="MathJax-Element-18-Frame" tabindex="0" data-mathml="G11uN1+G12uN2+G13uN3=is11" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">G11uN1+G12uN2+G13uN3=is11

<span class="MathJax_SVG" id="MathJax-Element-23-Frame" tabindex="0" data-mathml="G12uN1+G22uN2+G23uN3=is22" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">G12uN1+G22uN2+G23uN3=is22

<span class="MathJax_SVG" id="MathJax-Element-19-Frame" tabindex="0" data-mathml="G13uN1+G32uN2+G33uN3=is11" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">G13uN1+G32uN2+G33uN3=is11

<span class="MathJax_SVG" id="MathJax-Element-14-Frame" tabindex="0" data-mathml="G11" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">G11<span class="MathJax_SVG" id="MathJax-Element-20-Frame" tabindex="0" data-mathml="G22" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">G22<span class="MathJax_SVG" id="MathJax-Element-22-Frame" tabindex="0" data-mathml="G33" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">G33 为节点1 2 3的自电导(为该节点上所有电导的总和)

<span class="MathJax_SVG" id="MathJax-Element-16-Frame" tabindex="0" data-mathml="G13" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">G13 表示节点1 3的互电导(互电导为两节点公有电导的负值)

<span class="MathJax_SVG" id="MathJax-Element-17-Frame" tabindex="0" data-mathml="is11" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">is11<span class="MathJax_SVG" id="MathJax-Element-21-Frame" tabindex="0" data-mathml="is22" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">is22<span class="MathJax_SVG" id="MathJax-Element-15-Frame" tabindex="0" data-mathml="is33" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">is33 为电流源输送给节点1 2 3的电流的代数和(电压源也不要忽略)

网孔分析法只适用于平面网络节点法无限制
含运算放大器的电阻电路

模电2 8章
电路的对偶性

第三章叠加方法与网络函数

叠加原理

由线性电阻、线性受控源及独立源组成的电路中每一元件的电流或电压可以看成是电路中每一个独立源单独作用于电路时在该元件上产生的电流和电压的代数和

某一独立源单独作用时其他独立源应为零值即独立电压源用短路代替独立电流源用开路代替

功率不等于各电源单独作用的功率之和

第四章分解方法及单双口网络

一个元件的电压/电流关系是由元件本身所确定的与外接的电路无关

分解的基本步骤

把网络划分为两个单口网络<span class="MathJax_SVG" id="MathJax-Element-26-Frame" tabindex="0" data-mathml="N1" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">N1<span class="MathJax_SVG" id="MathJax-Element-27-Frame" tabindex="0" data-mathml="N2" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">N2

分别求出<span class="MathJax_SVG" id="MathJax-Element-25-Frame" tabindex="0" data-mathml="N1" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">N1<span class="MathJax_SVG" id="MathJax-Element-24-Frame" tabindex="0" data-mathml="N2" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">N2的VCR(计算或者测量)

联立两者的VCR式或通过他们的伏安曲线交点来求<span class="MathJax_SVG" id="MathJax-Element-29-Frame" tabindex="0" data-mathml="N1" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">N1<span class="MathJax_SVG" id="MathJax-Element-28-Frame" tabindex="0" data-mathml="N2" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">N2的端电压和电流

分别求他们的内部电压和电流
单口网络的电压电流关系

单口网络的三种表现形式

1.电路模型

2.端口电压和电流的约束关系

3.等效电路

外施电流源求电压法和外施电压源求电流法式常用的方法
置换原理

定义：若网络N由两个单口网络<span class="MathJax_SVG" id="MathJax-Element-32-Frame" tabindex="0" data-mathml="N1" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">N1和<span class="MathJax_SVG" id="MathJax-Element-31-Frame" tabindex="0" data-mathml="N2" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">N2连接组成且各支路电压、电流均有惟一解设已知端口电压和电流值分别为α和β 则<span class="MathJax_SVG" id="MathJax-Element-35-Frame" tabindex="0" data-mathml="N2" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">N2(或<span class="MathJax_SVG" id="MathJax-Element-34-Frame" tabindex="0" data-mathml="N1" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">N1)可以用一个电压为α的电压源或一个电流为α的电流源置换不影响<span class="MathJax_SVG" id="MathJax-Element-33-Frame" tabindex="0" data-mathml="N1" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">N1(或<span class="MathJax_SVG" id="MathJax-Element-30-Frame" tabindex="0" data-mathml="N2" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">N2)内各支路电压、电流原有数值只要在置换后网络仍有惟一解
单口网络的等效电路

伏安曲线重叠则等效
等效规律和公式

电压源电流源电阻三种元件取两个串联或者并联共12种

含受控源的单口网络一般也要外施电源求VCR的方法处理

电阻串联电压源电阻并联电流源可以互相等效
戴维南定理

单口网络都可以等效为一个电压源串联电阻电压源电压<span class="MathJax_SVG" id="MathJax-Element-36-Frame" tabindex="0" data-mathml="uoc" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uoc是开路电压电阻<span class="MathJax_SVG" id="MathJax-Element-37-Frame" tabindex="0" data-mathml="Ro" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Ro是独立源为0时的等效电阻电流<span class="MathJax_SVG" id="MathJax-Element-38-Frame" tabindex="0" data-mathml="isc" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">isc是短路电流

<span class="MathJax_SVG" id="MathJax-Element-39-Frame" tabindex="0" data-mathml="Ro=uocisc" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Ro=uocisc
用戴维南定理求含受控源电路时求等效电阻<span class="MathJax_SVG" id="MathJax-Element-40-Frame" tabindex="0" data-mathml="Ro" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Ro时受控源要保留计算
诺顿定理
单口网络等效为一个电流源并联一个电阻
最大功率传递定理

0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">RL=Ro>0时 <span class="MathJax_SVG" id="MathJax-Element-44-Frame" tabindex="0" data-mathml="Ro" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Ro吸收功率与<span class="MathJax_SVG" id="MathJax-Element-41-Frame" tabindex="0" data-mathml="RL" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">RL吸收功率相等对电压源而言功率传输效率为<span class="MathJax_SVG" id="MathJax-Element-43-Frame" tabindex="0" data-mathml="η=50" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">η=50 对单口网络中的独立源而言效率可能更低

电力系统要求尽可能提高效率测量电子信息中着眼于从微弱信号中获得最大功率而不在乎效率

三角叉电阻变换

▲->Y:

<span class="MathJax_SVG" id="MathJax-Element-45-Frame" tabindex="0" data-mathml="R1=R12R13R12+R13+R23" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R1=R12R13R12+R13+R23

<span class="MathJax_SVG" id="MathJax-Element-46-Frame" tabindex="0" data-mathml="R2=R21R23R12+R13+R23" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R2=R21R23R12+R13+R23

<span class="MathJax_SVG" id="MathJax-Element-47-Frame" tabindex="0" data-mathml="R3=R31R32R12+R13+R23" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R3=R31R32R12+R13+R23

Y->▲:

<span class="MathJax_SVG" id="MathJax-Element-48-Frame" tabindex="0" data-mathml="R12=R1R2+R1R3+R2R3R3" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R12=R1R2+R1R3+R2R3R3

<span class="MathJax_SVG" id="MathJax-Element-49-Frame" tabindex="0" data-mathml="R13=R1R2+R1R3+R2R3R2" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R13=R1R2+R1R3+R2R3R2

<span class="MathJax_SVG" id="MathJax-Element-50-Frame" tabindex="0" data-mathml="R23=R1R2+R1R3+R2R3R1" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R23=R1R2+R1R3+R2R3R1

9. 双口网络

有两个端口的网络

第五章电容和电感

动态元件:电压电流关系涉及对电流电压的微分或积分的
基尔霍夫定律施加于电路的约束关系只取决于电路的连接方式而与构成电路的元件性质无关

电容元件

理想介质不导电

定义:一个二端元件如果在任一时刻t 它的电荷q(t)同它的端电压u(t)之间的关系可以用u-q平面上的一条曲线来确定则此二端元件称为电容元件

<span class="MathJax_SVG" id="MathJax-Element-51-Frame" tabindex="0" data-mathml="q(t)=Cu(t)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">q(t)=Cu(t)

q(t)与u(t)是关联方向
电流指向电容正极板是关联方向(也可以通过电流和电压来看)

电容的VCR

<span class="MathJax_SVG" id="MathJax-Element-52-Frame" tabindex="0" data-mathml="i(t)=Cdudt" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">i(t)=Cdudt

<span class="MathJax_SVG" id="MathJax-Element-53-Frame" tabindex="0" data-mathml="u(t)=u(t0)+1C∫t0ti(ξ)dξ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">u(t)=u(t0)+1C∫t0ti(ξ)dξ

2. 电容电压的连续和**性质

电容电压不能跃变

利用初始电压

<span class="MathJax_SVG" id="MathJax-Element-57-Frame" tabindex="0" data-mathml="uc(t0)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uc(t0)对<span class="ztext-math" data-eeimg="1" data-tex="t<span class="MathJax_SVG" id="MathJax-Element-56-Frame" tabindex="0" data-mathml="t<t0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t<t0时电流的**作用不需考虑<span class="ztext-math" data-eeimg="1" data-tex="t<span class="MathJax_SVG" id="MathJax-Element-55-Frame" tabindex="0" data-mathml="t<t0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t<t0时电流的具体情况下即能解决t_0">t0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t>t0时的电容电压<span class="MathJax_SVG" id="MathJax-Element-58-Frame" tabindex="0" data-mathml="uC(t)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC(t)的问题

电容的初始电压是一个必须具备的条件

对已被充电的电容若已知<span class="MathJax_SVG" id="MathJax-Element-60-Frame" tabindex="0" data-mathml="uC(t0)=U0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC(t0)=U0，则在t_0">t0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t>t0时可等效为一个未充电的电容与电压源相串联的电路电压源的电压值即为<span class="MathJax_SVG" id="MathJax-Element-64-Frame" tabindex="0" data-mathml="t0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t0时电容两端的电压<span class="MathJax_SVG" id="MathJax-Element-61-Frame" tabindex="0" data-mathml="U0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">U0 电压<span class="MathJax_SVG" id="MathJax-Element-62-Frame" tabindex="0" data-mathml="U0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">U0称为电容电压<span class="MathJax_SVG" id="MathJax-Element-63-Frame" tabindex="0" data-mathml="uC" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC的初始状态

电容的储能
电容的储能只与该时刻的电压有关

<span class="MathJax_SVG" id="MathJax-Element-65-Frame" tabindex="0" data-mathml="wC=12Cu2(t)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">wC=12Cu2(t)

电感元件

定义:一个二端元件在任一时刻<span class="MathJax_SVG" id="MathJax-Element-69-Frame" tabindex="0" data-mathml="t" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t 它的电流<span class="MathJax_SVG" id="MathJax-Element-68-Frame" tabindex="0" data-mathml="i(t)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">i(t)同它的磁链<span class="MathJax_SVG" id="MathJax-Element-67-Frame" tabindex="0" data-mathml="Ψ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Ψ(t)之间的关系可以用<span class="MathJax_SVG" id="MathJax-Element-66-Frame" tabindex="0" data-mathml="i(t)−Ψ(t)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">i(t)−Ψ(t)平面上的一条曲线来确定

<span class="MathJax_SVG" id="MathJax-Element-70-Frame" tabindex="0" data-mathml="Ψ(t)=Li(t)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Ψ(t)=Li(t)

3. 电感的VCR

电压与磁链的参考方向符合右手螺旋法则时

<span class="MathJax_SVG" id="MathJax-Element-71-Frame" tabindex="0" data-mathml="u=dΨdt=Ldidt" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">u=dΨdt=Ldidt

<span class="MathJax_SVG" id="MathJax-Element-72-Frame" tabindex="0" data-mathml="i(t)=i(t0)+1L∫t0tu(ξ)dξ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">i(t)=i(t0)+1L∫t0tu(ξ)dξ

在直流电路中电感相当于一个短路

电磁学中的感应电动势<span class="MathJax_SVG" id="MathJax-Element-73-Frame" tabindex="0" data-mathml="e" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">e与<span class="MathJax_SVG" id="MathJax-Element-74-Frame" tabindex="0" data-mathml="u" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">u的参考方向规定相反(e是电压升)

电感电流的连续和**性质

电感的电流不能跃变

在t_0">t0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t>t0时可等效为一个初始电流为零的电感与电流源的并联电路电流源的电流值即为<span class="MathJax_SVG" id="MathJax-Element-76-Frame" tabindex="0" data-mathml="t0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t0时电感的电流<span class="MathJax_SVG" id="MathJax-Element-77-Frame" tabindex="0" data-mathml="I0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">I0

4. 电感的储能

<span class="MathJax_SVG" id="MathJax-Element-78-Frame" tabindex="0" data-mathml="wL=12Li2()t" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">wL=12Li2()t

在动态电路的各个电压、电流变量中电容电压<span class="MathJax_SVG" id="MathJax-Element-79-Frame" tabindex="0" data-mathml="uC(t)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC(t)和电感电流<span class="MathJax_SVG" id="MathJax-Element-80-Frame" tabindex="0" data-mathml="iL(t)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">iL(t)占有特殊重要的地位称为电路的状态变量

状态变量是指一组最少的变量若已知它们在<span class="MathJax_SVG" id="MathJax-Element-81-Frame" tabindex="0" data-mathml="t0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t0时的数值则连同所有在<span class="MathJax_SVG" id="MathJax-Element-82-Frame" tabindex="0" data-mathml="t≥t0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t≥t0时的输入就能确定在<span class="MathJax_SVG" id="MathJax-Element-83-Frame" tabindex="0" data-mathml="t≥t0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t≥t0时电路中的任何电路变量

4. 电容电感的对偶性

第六章一阶电路

只含一个动态元件的线性时不变电路用线性常系数一阶常微分方程来描述用一阶微分方程来描述的电路称为一阶电路

分解方法在动态电路分析中的运用

一阶电路可以看成两个单口网络其一含有所有电源及电阻元件另一个含一个动态元件

状态方程本身体现了电路状态演变的情况即状态的变化率是当前状态和当前输入的函数仅由动态元件初始条件引起的响应称为零输入响应
仅由独立电源引起的响应称为零状态响应

动态电路分析的基本方法是建立微分方程然后用数学方法求解微分方程得到电压电流响应的表达式
零输入响应

零输入响应:电容或电感的放电过程

<span class="MathJax_SVG" id="MathJax-Element-84-Frame" tabindex="0" data-mathml="u" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">u和<span class="MathJax_SVG" id="MathJax-Element-85-Frame" tabindex="0" data-mathml="i" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">i均为一个随时间衰变的指数函数电流在<span class="MathJax_SVG" id="MathJax-Element-86-Frame" tabindex="0" data-mathml="t=0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t=0发生跃变

函数表达式中<span class="MathJax_SVG" id="MathJax-Element-89-Frame" tabindex="0" data-mathml="e−tRC" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">e−tRC是无量纲的故<span class="MathJax_SVG" id="MathJax-Element-88-Frame" tabindex="0" data-mathml="RC" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">RC是时间量纲通常用<span class="MathJax_SVG" id="MathJax-Element-87-Frame" tabindex="0" data-mathml="τ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">τ表示称为时间常数<span class="MathJax_SVG" id="MathJax-Element-90-Frame" tabindex="0" data-mathml="τ=RC" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">τ=RC
<span class="MathJax_SVG" id="MathJax-Element-91-Frame" tabindex="0" data-mathml="τ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">τ越大衰减越慢

<span class="MathJax_SVG" id="MathJax-Element-92-Frame" tabindex="0" data-mathml="法拉·欧姆=秒" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">法拉欧姆秒法拉·欧姆=秒

<span class="MathJax_SVG" id="MathJax-Element-93-Frame" tabindex="0" data-mathml="τ=LR" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">τ=LR

在求解时，首先求电容电压/电感电流的初始值

<span class="MathJax_SVG" id="MathJax-Element-94-Frame" tabindex="0" data-mathml="RCduCdt+uC=0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">RCduCdt+uC=0

<span class="MathJax_SVG" id="MathJax-Element-96-Frame" tabindex="0" data-mathml="−1RC" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">−1RC为特征方程<span class="MathJax_SVG" id="MathJax-Element-95-Frame" tabindex="0" data-mathml="RCS+1=0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">RCS+1=0的根

<span class="MathJax_SVG" id="MathJax-Element-97-Frame" tabindex="0" data-mathml="LdiLdt+RiL=0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">LdiLdt+RiL=0

<span class="MathJax_SVG" id="MathJax-Element-99-Frame" tabindex="0" data-mathml="−RL" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">−RL为特征方程<span class="MathJax_SVG" id="MathJax-Element-98-Frame" tabindex="0" data-mathml="Ls+R=0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Ls+R=0的根

特征根具有时间倒数或频率的量纲故称为固有频率

在电路理论中固有频率用来表明网络的固有性质

零状态响应

电容

<span class="MathJax_SVG" id="MathJax-Element-100-Frame" tabindex="0" data-mathml="CduC(t)dt+1RuC=IS" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">CduC(t)dt+1RuC=IS

通解为:

<span class="MathJax_SVG" id="MathJax-Element-101-Frame" tabindex="0" data-mathml="uC(t)=Ke−tRC+RIS" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC(t)=Ke−tRC+RIS

零状态解

<span class="MathJax_SVG" id="MathJax-Element-102-Frame" tabindex="0" data-mathml="uC(t)=RIS(1−e−tRC)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC(t)=RIS(1−e−tRC)

<span class="MathJax_SVG" id="MathJax-Element-103-Frame" tabindex="0" data-mathml="τ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">τ越小电容电压达到稳定值越快

电感

<span class="MathJax_SVG" id="MathJax-Element-104-Frame" tabindex="0" data-mathml="iL(t)=USR(1−e−RtL)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">iL(t)=USR(1−e−RtL)

线性动态电路的叠加原理

多个独立电源作用于线性动态电路零状态响应为各个独立电源单独作用时所产生的零状态响应的代数和

完全响应:初始状态和输入共同作用下的响应

<span class="MathJax_SVG" id="MathJax-Element-105-Frame" tabindex="0" data-mathml="完全响应=零输入响应+零状态响应" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">完全响应零输入响应零状态响应完全响应=零输入响应+零状态响应

此结论来源于线性电路的叠加性

称为线性动态电路的叠加原理

RC电路完全响应

<span class="MathJax_SVG" id="MathJax-Element-106-Frame" tabindex="0" data-mathml="uC(t)=U0e−tRC(零输入响应)+RIS(1−e−tRC)(零状态响应)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">零输入响应零状态响应uC(t)=U0e−tRC(零输入响应)+RIS(1−e−tRC)(零状态响应)

<span class="MathJax_SVG" id="MathJax-Element-107-Frame" tabindex="0" data-mathml="uC(t)=RIS(强制响应/稳态响应)+(U0−RIS)e(−tRC)(固有响应/暂态响应)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">强制响应稳态响应固有响应暂态响应uC(t)=RIS(强制响应/稳态响应)+(U0−RIS)e(−tRC)(固有响应/暂态响应)

3. 三要素法

直流电源驱动的只含一个动态元件的一阶电路全响应的一般表达式

此方法适用于直流输入情况

<span class="MathJax_SVG" id="MathJax-Element-110-Frame" tabindex="0" data-mathml="uC(t)−uC(∞)=(uC(t0)−uC(∞))e−tτ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC(t)−uC(∞)=(uC(t0)−uC(∞))e−tτ

<span class="MathJax_SVG" id="MathJax-Element-111-Frame" tabindex="0" data-mathml="uC(t)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC(t)是由<span class="MathJax_SVG" id="MathJax-Element-109-Frame" tabindex="0" data-mathml="uC(0)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC(0)、<span class="MathJax_SVG" id="MathJax-Element-108-Frame" tabindex="0" data-mathml="uC(∞)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC(∞)和<span class="MathJax_SVG" id="MathJax-Element-112-Frame" tabindex="0" data-mathml="τ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">τ这三个参量确定的只要算得这三个参量就可把解答直接写出不必求解微分方程

任意支路电流电压的通式:

<span class="MathJax_SVG" id="MathJax-Element-113-Frame" tabindex="0" data-mathml="f(t)−f(∞)=(f(0)−f(∞))e−tτ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">f(t)−f(∞)=(f(0)−f(∞))e−tτ

解题步骤:
1.用电压为<span class="MathJax_SVG" id="MathJax-Element-116-Frame" tabindex="0" data-mathml="uC(0)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC(0)的直流电压源置换电容或用电流为<span class="MathJax_SVG" id="MathJax-Element-117-Frame" tabindex="0" data-mathml="iL(0)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">iL(0)的直流电流源置换电感所得电路为<span class="MathJax_SVG" id="MathJax-Element-118-Frame" tabindex="0" data-mathml="t=0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t=0时的等效电路由此电路得到任意电压或电流的初始值<span class="MathJax_SVG" id="MathJax-Element-115-Frame" tabindex="0" data-mathml="f(0)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">f(0)

2.用开路代替电容或短路代替电感得到<span class="MathJax_SVG" id="MathJax-Element-114-Frame" tabindex="0" data-mathml="t=∞" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t=∞时的等效电路求得任意电压或电流的稳态值<span class="MathJax_SVG" id="MathJax-Element-119-Frame" tabindex="0" data-mathml="f(∞)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">f(∞)

3.求<span class="MathJax_SVG" id="MathJax-Element-121-Frame" tabindex="0" data-mathml="N1" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">N1的等效电路来计算电路的时间常数

(1)求从动态元件两端看进去的戴维南等效电阻

1.独立源为0 用电阻串并联化简

2.独立源为0 外加电压u 求端口流i 等效电阻:<span class="MathJax_SVG" id="MathJax-Element-120-Frame" tabindex="0" data-mathml="Ro=ui" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Ro=ui

3.开路电压比短路电流(独立源要保留)

(2)含受控源的电路只能用2 3方法

4.<span class="MathJax_SVG" id="MathJax-Element-122-Frame" tabindex="0" data-mathml="f(t)−f(∞)=(f(0)−f(∞))e−tτ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">f(t)−f(∞)=(f(0)−f(∞))e−tτ

4. 阶跃响应和冲击响应

单位阶跃函数定义:

0,t<0 1,t>0 " role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ϵ(t)={ 0,t<0 1,t>0

<span class="MathJax_SVG" id="MathJax-Element-125-Frame" tabindex="0" data-mathml="ϵ(t)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ϵ(t)是奇异函数 <span class="MathJax_SVG" id="MathJax-Element-124-Frame" tabindex="0" data-mathml="t=0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t=0时无定义可取0或1

延迟单位阶跃:

t0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ϵ(t−t0)={0,t<t01,t>t0

分段常量信号可以表示为一系列阶跃信号之和

<span class="MathJax_SVG" id="MathJax-Element-127-Frame" tabindex="0" data-mathml="f(t)=ϵ(t)−ϵ(t−t0)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">f(t)=ϵ(t)−ϵ(t−t0)

<span class="MathJax_SVG" id="MathJax-Element-128-Frame" tabindex="0" data-mathml="f(t)=Aϵ(t)−Aϵ(t−t0)+Aϵ(t−2t0)−Aϵ(t−3t0)+..." role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">f(t)=Aϵ(t)−Aϵ(t−t0)+Aϵ(t−2t0)−Aϵ(t−3t0)+...

<span class="MathJax_SVG" id="MathJax-Element-129-Frame" tabindex="0" data-mathml="f(t)=Aϵ(t)−2ϵ(t−t0)+ϵ(t−2t0)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">f(t)=Aϵ(t)−2ϵ(t−t0)+ϵ(t−2t0)

<span class="MathJax_SVG" id="MathJax-Element-130-Frame" tabindex="0" data-mathml="f(t)=A1ϵ(t−t0)+(A2−A1)ϵ(t−t1)−(A1+A2)ϵ(t−t2)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">f(t)=A1ϵ(t−t0)+(A2−A1)ϵ(t−t1)−(A1+A2)ϵ(t−t2)

零状态电路对单位阶跃信号的响应称为(单位)阶跃响应并用s(t)表示

单位冲激函数(狄拉克函数)定义:

δ(t)=0,∀t≠0 ∫−∞∞δ(t)=1 " role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">{ δ(t)=0,∀t≠0 ∫−∞∞δ(t)=1

冲激函数的面积称为冲激函数的强度

对冲激电流强度量纲为库仑

延时同理

δ(t−t0)=0,∀t≠t0 ∫−∞∞δ(t−t0)=1 " role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">{ δ(t−t0)=0,∀t≠t0 ∫−∞∞δ(t−t0)=1

在<span class="MathJax_SVG" id="MathJax-Element-135-Frame" tabindex="0" data-mathml="t0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t0处强度为A的冲激函数记为<span class="MathJax_SVG" id="MathJax-Element-134-Frame" tabindex="0" data-mathml="Aδ(t−t0)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Aδ(t−t0)

电流:<span class="MathJax_SVG" id="MathJax-Element-136-Frame" tabindex="0" data-mathml="Qδ(t−t0)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Qδ(t−t0)
电压:<span class="MathJax_SVG" id="MathJax-Element-133-Frame" tabindex="0" data-mathml="Ψδ(t−t0)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Ψδ(t−t0)

其他形状也可以作为单位脉冲的近似

冲激函数是阶跃函数的导数

<span class="MathJax_SVG" id="MathJax-Element-137-Frame" tabindex="0" data-mathml="∫−∞∞f(t)δ(t−t0)dt=f(t0)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">∫−∞∞f(t)δ(t−t0)dt=f(t0)

零状态电路对单位冲激信号的响应称为(单位)冲激响应用h(t)表示

计算冲激响应<span class="MathJax_SVG" id="MathJax-Element-138-Frame" tabindex="0" data-mathml="h(t)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">h(t)时先计算由<span class="MathJax_SVG" id="MathJax-Element-139-Frame" tabindex="0" data-mathml="δ(t)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">δ(t)产生的在<span class="MathJax_SVG" id="MathJax-Element-140-Frame" tabindex="0" data-mathml="t=0+" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t=0+时的初始状态然后求解由这一初始状态所产生的零输入响应

线性非时变电路有一个性质:
若激励x产生响应y

则激励<span class="MathJax_SVG" id="MathJax-Element-143-Frame" tabindex="0" data-mathml="dxdt" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">dxdt产生响应<span class="MathJax_SVG" id="MathJax-Element-141-Frame" tabindex="0" data-mathml="dydt" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">dydt

则激励<span class="MathJax_SVG" id="MathJax-Element-145-Frame" tabindex="0" data-mathml="∫xdt" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">∫xdt产生响应<span class="MathJax_SVG" id="MathJax-Element-142-Frame" tabindex="0" data-mathml="∫ydt+k" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">∫ydt+k <span class="MathJax_SVG" id="MathJax-Element-144-Frame" tabindex="0" data-mathml="k" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">k为积分常数

线性非时变电路的冲激响应是它的阶跃响应的导数

<span class="MathJax_SVG" id="MathJax-Element-146-Frame" tabindex="0" data-mathml="h(t)=ds(t)dt" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">h(t)=ds(t)dt

第八章交流动态电路相量法

正弦激励的过渡过程和稳态

电路的微分方程:
<span class="MathJax_SVG" id="MathJax-Element-147-Frame" tabindex="0" data-mathml="CduCdt+1RuC=Ismcos(ωt+θ)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">CduCdt+1RuC=Ismcos(ωt+θ)

完全解:

<span class="MathJax_SVG" id="MathJax-Element-148-Frame" tabindex="0" data-mathml="uC(t)=uC(0)e−tRC−UCmcosθu·e−tRC+UCmcos(ωt+θu)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC(t)=uC(0)e−tRC−UCmcosθu·e−tRC+UCmcos(ωt+θu)

<span class="MathJax_SVG" id="MathJax-Element-150-Frame" tabindex="0" data-mathml="uC(0)e−tRC" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC(0)e−tRC 0输入响应

<span class="MathJax_SVG" id="MathJax-Element-149-Frame" tabindex="0" data-mathml="−UCmcosθu·e−tRC+UCmcos(ωt+θu)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">−UCmcosθu·e−tRC+UCmcos(ωt+θu) 0状态响应

<span class="MathJax_SVG" id="MathJax-Element-151-Frame" tabindex="0" data-mathml="uC(0)e−tRC−UCmcosθu·e−tRC" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC(0)e−tRC−UCmcosθu·e−tRC 暂态响应

<span class="MathJax_SVG" id="MathJax-Element-152-Frame" tabindex="0" data-mathml="UCmcos(ωt+θu)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">UCmcos(ωt+θu) 稳态响应

由于特征根是负的因此解的第一项是衰减的当t趋于无限大该项也趋于零特征根的性质是与电路的激励方式无关的第一项代表暂态响应分量

解的第二项即微分方程的特解为稳态响应分量这一分量与外施激励形式相同差别仅在具体的振幅和相位值不同当t趋于无限大时电路的响应只由该项确定

暂态响应分量的出现是为了使电路的响应满足初始条件保证换路瞬间电容电压不能跃变

暂态响应分量初始值K应为电容初始电压<span class="MathJax_SVG" id="MathJax-Element-153-Frame" tabindex="0" data-mathml="uC(0)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">uC(0)与稳态响应分量初始值之差
变换方法的概念

原来的问题->变换域中较简单的问题->变换域中问题的解答->原问题的解答

振幅相量

相量法是建立在用复数来表示正弦波的基础上

<span class="MathJax_SVG" id="MathJax-Element-154-Frame" tabindex="0" data-mathml="eiθ=cosθ+isinθ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">eiθ=cosθ+isinθ

正弦电压表示:

<span class="MathJax_SVG" id="MathJax-Element-155-Frame" tabindex="0" data-mathml="cosωt=Re(ejωt)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">cosωt=Re(ejωt)

<span class="MathJax_SVG" id="MathJax-Element-156-Frame" tabindex="0" data-mathml="sinωt=Im(ejωt)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">sinωt=Im(ejωt)

<span class="MathJax_SVG" id="MathJax-Element-157-Frame" tabindex="0" data-mathml="u(t)=Umcos(ωt+θ)=Re(Umejωt)=Re(Um∠ωt)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">u(t)=Umcos(ωt+θ)=Re(Umejωt)=Re(Um∠ωt)

电压振幅相量
<span class="MathJax_SVG" id="MathJax-Element-158-Frame" tabindex="0" data-mathml="Umejθ=Um∠θ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Umejθ=Um∠θ

模为电压的振幅辐角为初相

相量在复平面上的图示是相量图

相量与<span class="MathJax_SVG" id="MathJax-Element-160-Frame" tabindex="0" data-mathml="ejωt" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ejωt的乘积则是时间<span class="MathJax_SVG" id="MathJax-Element-159-Frame" tabindex="0" data-mathml="t" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">t的复值函数在复平面上可用以恒定角速度<span class="MathJax_SVG" id="MathJax-Element-161-Frame" tabindex="0" data-mathml="ω" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ω逆时针方向旋转的相量表示

相量只能表征或代表正弦波并不等于正弦波

<span class="MathJax_SVG" id="MathJax-Element-162-Frame" tabindex="0" data-mathml="u(t)=Re(Um∠θ∠ωt)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">u(t)=Re(Um∠θ∠ωt)

3. 有效值有效值相量

周期电流、电压的瞬时值是随时间而变化的有时并不需要知道它们每一瞬间的大小在这种情况下就需要为它们规定一个表征大小的特定值

平均值是0不合适

最大值也不合适(仅表征某一瞬间的大小)

从周期电流(电压)和直流电流(电压)施加于电阻时电阻消耗电能考虑可以为周期波规定一个表征其大小的特定值即有效值

<span class="MathJax_SVG" id="MathJax-Element-163-Frame" tabindex="0" data-mathml="R∫0Ti2dt=RI2T" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">R∫0Ti2dt=RI2T

<span class="MathJax_SVG" id="MathJax-Element-164-Frame" tabindex="0" data-mathml="I=1T∫0Tu2dt" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">I=1T∫0Tu2dt

用于正弦电流 <span class="MathJax_SVG" id="MathJax-Element-165-Frame" tabindex="0" data-mathml="I=12Im=0.707Im" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">I=12Im=0.707Im

交流电表测读值都是有效值

生活中的220V是有效值

除非特别声明相量均系指有效值相量

在正弦激励的动态电路中若各电压、电流均为与激励同频率的正弦波称为正弦稳态电路

利用相量可以使微分方程的求解问题简化为复数方程的求解问题

引入阻抗和导纳的概念不仅能把电阻电路的分析方法推广应用于分析正弦稳态电路还能表征元件和电路在正弦稳态时的性能

4. 基尔霍夫定律的相量形式

在正弦稳态电路中基尔霍夫定律可直接用电流相量和电压相量写出也可直接用电流振幅相量和电压振幅相量写

三种基本元件的VCR的相量形式

电阻:两端的正弦电压和流过的正弦电流是同相的

电容:<span class="MathJax_SVG" id="MathJax-Element-166-Frame" tabindex="0" data-mathml="I.=jωCU." role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">I.=jωCU.

<span class="MathJax_SVG" id="MathJax-Element-168-Frame" tabindex="0" data-mathml="I=ωCU" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">I=ωCU <span class="MathJax_SVG" id="MathJax-Element-169-Frame" tabindex="0" data-mathml="θu=θi+90o" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">θu=θi+90o

当C值一定时对一定的U来说 <span class="MathJax_SVG" id="MathJax-Element-167-Frame" tabindex="0" data-mathml="ω" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ω越高则<span class="MathJax_SVG" id="MathJax-Element-172-Frame" tabindex="0" data-mathml="I" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">I越大也就是说电流越容易通过当<span class="MathJax_SVG" id="MathJax-Element-170-Frame" tabindex="0" data-mathml="ω=0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ω=0(相当于直流激励)时 <span class="MathJax_SVG" id="MathJax-Element-171-Frame" tabindex="0" data-mathml="I=0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">I=0 电容相当于开路这正是直流稳态时电容的表现

电流超前电压<span class="MathJax_SVG" id="MathJax-Element-173-Frame" tabindex="0" data-mathml="90o" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">90o

<span class="MathJax_SVG" id="MathJax-Element-174-Frame" tabindex="0" data-mathml="u(t)=2Ucos(ωt+θu)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">u(t)=2Ucos(ωt+θu)

<span class="MathJax_SVG" id="MathJax-Element-176-Frame" tabindex="0" data-mathml="i(t)=2UωCcos(ωt+θu+90o)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">i(t)=2UωCcos(ωt+θu+90o)

电感:<span class="MathJax_SVG" id="MathJax-Element-177-Frame" tabindex="0" data-mathml="U.=jωLI." role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">U.=jωLI.

<span class="MathJax_SVG" id="MathJax-Element-175-Frame" tabindex="0" data-mathml="U=ωLI" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">U=ωLI <span class="MathJax_SVG" id="MathJax-Element-178-Frame" tabindex="0" data-mathml="θu=θu+90o" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">θu=θu+90o

当L值一定时对一定的I来说 <span class="MathJax_SVG" id="MathJax-Element-182-Frame" tabindex="0" data-mathml="ω" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ω越高则<span class="MathJax_SVG" id="MathJax-Element-181-Frame" tabindex="0" data-mathml="U" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">U越大当<span class="MathJax_SVG" id="MathJax-Element-180-Frame" tabindex="0" data-mathml="ω=0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ω=0(相当于直流激励)时 <span class="MathJax_SVG" id="MathJax-Element-179-Frame" tabindex="0" data-mathml="U=0" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">U=0 电感相当于短路

电流滞后电压<span class="MathJax_SVG" id="MathJax-Element-183-Frame" tabindex="0" data-mathml="90o" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">90o

<span class="MathJax_SVG" id="MathJax-Element-184-Frame" tabindex="0" data-mathml="u(t)=2Icos(ωt+θi)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">u(t)=2Icos(ωt+θi)

<span class="MathJax_SVG" id="MathJax-Element-185-Frame" tabindex="0" data-mathml="i(t)=2IωLcos(ωt+θi+90o)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">i(t)=2IωLcos(ωt+θi+90o)

5. VCR相量形式的统一阻抗和导纳的引入

在关联参考方向时:

U.=1jωCI. U.=jωLI." role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">U.=RI. U.=1jωCI. U.=jωLI.

如果我们把元件在正弦稳态时电压相量与电流相量之比定义为该元件的阻抗记为<span class="MathJax_SVG" id="MathJax-Element-187-Frame" tabindex="0" data-mathml="Z" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Z

ZC=1jωC=−j1ωC ZL=jωL" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ZR=R ZC=1jωC=−j1ωC ZL=jωL

元件的阻抗也可定义为电压振幅相量与电流振幅相量之比

阻抗(Z)的倒数称为导纳(Y)

6. 正弦稳态混联电路分析

<span class="MathJax_SVG" id="MathJax-Element-190-Frame" tabindex="0" data-mathml="Z=Σk=1nZk" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Z=Σk=1nZk串联阻抗

<span class="MathJax_SVG" id="MathJax-Element-189-Frame" tabindex="0" data-mathml="Y=Σk=1nYk" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Y=Σk=1nYk并联阻抗

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