电路原理一阶电路_一阶电路和二阶电路怎么区分

7.1 Introduction

Two types of simple circuits:a circuit comprising a resistor and capactior and a circuit comprising a resistor and an inductor.
These are called RC and RL circuits,respectively.
Resistive circuits


注释:损变

RC circuits(动态电路及其阶数)

动态电路:含有动态元件(电容或电感)的电路
特征:当电路的结构或参数改变时,电路可能从一种工作状态转变到另一种工作状态
换路:电路的结构或参数改变

Before switching k,circuit has reached steady state:
$$i=0,u_c=0$$

Since k has been closed for a long time,the capacitor voltage has reached steady state again:
$$i=0,u_c=U_s$$
过渡过程:电路由一个稳态过渡到另一个稳态的过程
过渡状态(瞬态,暂态,动态)
过渡过程产生原因:电路内部含有储能元件 ,电路在换时能量 电路在换时能量 电路在换时能量 发生变化，而能量的储存和释放都需要时间

研究过渡过程具有实际意义
利用:产生各种波形
提防:暂态过程瞬间可能出现高电压,大电流,使仪器设备损坏

动态电路的阶数
描述动态电路的方程是微分方程
方程阶数=电路阶数
一阶电路:描述电路的方程是一阶微分方程,一阶电路中一般只有一个动态元件
二阶电路:描述电路的方程是二阶微分方程,一般有二个动态元件

;
$$\begin{gathered} 应用KVL:Ri+u_c=u_s \\ 电容VCR:i=C\frac{d{u}_c}{dt} \\ RC\frac{d{u}_c}{dt}+u_c=u_s \\ 一阶微分方程也即一阶电路 \\ 方程的阶数通常等于电路中动态元件的个数 \end{gathered}$$

$R(C_1+C_2)\frac{d{u}_c}{dt}+u_c=u_s$
一阶电路

动态电路分析方法:

1. 时域分析法
   经典法:直接求解常微分方程
   状态变量法
   数值法
2. 变换法
   傅里叶变换法:频域分析法
   拉普拉斯变换法:复频域分析法

RL circuits

Before switching k,circuit has reached steady state:
$$i=0,u_L=0$$

Since k has been closed for a long time,the capacitor voltage has reached steady state again
$$u_L=0,i=\frac{U_s}{R}$$


Applying Kirchhoff’s laws to purely resistive circuits results in algebraic equations

While applying Kirchhoff’s laws to RC and RL circuits produces differential equations(微分方程)

The differential equations resulting from analyzing RC and RL circuits are of the first order.Hence.the circuits are collectively known as first-order circuits.

A first-order circuit is characterized by a first-order differential equation.

RC circuits

Applying KVL:
$$Ri+u_c=u_s(t)$$
By definition(定义)(VCR):
$$\begin{gathered} i=C\frac{d{u}_c}{dt} \\ \to RC\frac{d{u}_c}{dt}+u_c=u_s(t) \\ \to R\frac{di}{dt}+\frac{i}{C}=\frac{d{u}_s(t)}{dt} \end{gathered}$$

RL circuits

Applying KVL:
$$Ri+u_L=u_s(t)$$
By definition(定义)(VCR):
$$\begin{gathered} u_L=L\frac{di}{dt} \\ \to Ri+L\frac{di}{dt}=u_s(t) \end{gathered}$$