一阶RC低通滤波器（转）【滤波器学习笔记】一阶RC低通滤波

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我是靠谱客的博主阳光小天鹅，这篇文章主要介绍一阶RC低通滤波器（转）【滤波器学习笔记】一阶RC低通滤波，现在分享给大家，希望可以做个参考。

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【滤波器学习笔记】一阶RC低通滤波

2016年08月11日 20:38:40 吖勉阅读数：101584

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						<h1 id="一阶rc低通滤波"><a name="t0"></a>一阶RC低通滤波</h1>

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						<h1 id="一阶rc低通滤波"><a name="t0"></a>一阶RC低通滤波</h1>

从模拟到数字
本文整理自网络、《匠人手记》等书籍文章

模拟电路低通滤波时域、频域
软件低通滤波

典型电路

这里写图片描述
图1 典型RC电路
直流、交流、脉冲信号都可以用它

时域

电容电流：

I c = d q d t = d ( C ∙ U o ) d t = C d U o d t

假设电容初始电压值为0
R=1000Ω
C=4.7uF
Ui=1V
t=0.0001~0.1s
τ=RC
Vc(τ)=0.632
这里写图片描述

图2 一阶 RC系统的阶跃响应曲线

频域

u1=Ui；u2=Uo；
以电容电压作为输出，电路的网络函数为：

H (j ω) = U 2 U 1 = 1 j ω C R + 1 j ω C = 1 1 + j ω R C

ωc即为截止频率；

幅值和相角函数：

| H (j ω) | = 1 1 + ( ω ω c ) 2 --------\sqrt

θ (ω) = - a r c t a n ω ω c

各变量取值：
R=1000Ω
C=4.7uF

j = - 1 ---\sqrt

θ(fc)=-45

f=0.001、1、…….100000

幅频和相频特性图：
这里写图片描述
图3

图4
幅频特性图的对数表示：

图5

-当ω<ωc时，幅值是平行于坐标的直线，基本无衰减；

-当ω>>ωc时，是斜率与-20dB/十倍频成比例的一条直线；　

-当ω=ωc时，增益衰减至0.707，即-3dB，相位滞后45度，对应低通滤波器，该频率通常被称为截止频率。

缺点：
采用这种模拟滤波器抑制低频干扰时，要求滤波器有较大的时间常数和高精度的RC网络，增大时间常数要求增大R值，其漏电流也随之增大，从而降低了滤波效果；

软件上的一阶低通滤波

优点：

-采用数字滤波算法来实现动态的RC滤波，则能很好的克服模拟滤波器的缺点；
-在模拟常数要求较大的场合这种算法显得更为实用；
-其对于周期干扰有良好的抑制作用，
-比较节省RAM空间

缺点

-不足之处是带来了相位滞后，导致灵敏度低；
-同时它不能滤除频率高于采样频率的二分之一（称为奈奎斯特频率）的干扰（例如采样频率为100Hz，则它不能滤除50Hz以上的干扰信号）对于高于奈奎斯特频率的干扰信号，应该采用模拟滤波器。
-对没有乘、除法运算指令的单片机来说，程序运算工作量较大

基本滤波算法：

算法由来：
频率分析中一阶RC低通滤波在S域的传递函数：

V o u t V i n = 1 R C s + 1, (s = j ω)

带入S域传递函数中：

Y ( z ) X ( z ) = 1 R C 1 - z \land ( - 1 ) T + 1 = T R C ( 1 - z \land ( - 1 ) ) + T

通过Z变换把S域的传递函数转化成时域的差分方程，分析可得到

一阶RC数字滤波的基本算法

X为输入，Y为滤波后得输出值，则：

Y (n) = a * X (n) + (1 - a) * Y (n - 1)

a为与RC值有关的一个参数，称为滤波系数，其值决定新采样值在本次滤波结果中所占的权重，其值通常远小于1，当采样间隔t足够小的时候，

a = t R C

-滤波系数越小，滤波结果越平稳，但是灵敏度越低；

-滤波系数越大，灵敏度越高，但是滤波结果越不稳定

-本次输出值主要取决于上次滤波输出值，当前采样值对本次输出贡献比较小，起到修正作用；

-截止频率：

f l = a 2 π t

例如：t=0.5s (f=2Hz), a=1/32
则fl=(1/32)/(2*3.14*0.5)=0.01Hz;

基本程序：

按照一阶滤波的基本原理与公式写程序，如下：

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/*程序中整数运算比小数运算快，为加快程序的处理速度，为计算方便，a取一整数，1-a用256-a来代替，a则取0~255,代表新采样值在滤波结果中的权重（也可将1-a的基数改为100-a，计算结果做相应处理，这里不做说明）*/

#define a 128 

char value; //上次滤波值
char filter()
{
    char new_value;
    new_value=get_ad();//本次采样值
    return(256-a)*value/256+a*new_value/256；
}
 
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程序初步优化

减少乘、除的运算次数以提高运算速度。
具体优化办法：
先将新采样值与上次滤波结果进行比较，然后根据比较采用不同的公式计算，这样程序的运算效率提高了一倍；
化解基本公式可得:

当 X n < Y (n - 1) 时 ， Y n = Y (n - 1) - (Y (n - 1) - X n) \times a \div 256;

流程图：
这里写图片描述

程序：

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/*入口：NEW_DATA 新采样值
 OLD_DATA 上次滤波结果
 k 滤波系数(0~255)(代表在滤波结果中的权重)
 出口： 本次滤波结果
 */
 char filter_1(char NEW_DATA,char OLD_DATA,char k)
{
 int result;
 if(NEW_DATA<OLD_DATA)
 {
 result=OLD_DATA-NEW_DATA;
 result=result*k;
 result=result+128;//+128是为了四色五入
 result=result/256;
 result=OLD_DATA-result;<div id="article_content" class="article_content clearfix csdn-tracking-statistics" data-pid="blog" data-mod="popu_307" data-dsm="post">
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					版权声明：本文为博主原创文章，未经博主允许不得转载。					https://blog.csdn.net/qq_27334499/article/details/52186336				</div>
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							<h1 id="一阶rc低通滤波"><a name="t0"></a>一阶RC低通滤波</h1>

从模拟到数字 
本文整理自网络、<a href="https://www.baidu.com/s?wd=%E3%80%8A%E5%8C%A0%E4%BA%BA%E6%89%8B%E8%AE%B0%E3%80%8B&tn=24004469_oem_dg&rsv_dl=gh_pl_sl_csd" target="_blank">《匠人手记》</a>等书籍文章

<ul>
<li>模拟电路低通滤波时域、频域</li>
<li>软件低通滤波</li>
</ul>

<hr>

<img src="https://file2.kaopuke.com:8081/files_image/20230520/202305201048006935945.jpg" alt="这里写图片描述" title=""> 
图1 典型RC电路 
直流、交流、脉冲信号都可以用它

电容电流： 
<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mi>I</mi><mi>c</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><mi>q</mi><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>C</mi><mo>&#x2022;</mo><mi>U</mi><mi>o</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>=</mo><mi>C</mi><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><mi>U</mi><mi>o</mi><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle></math>" role="presentation">Ic=dqdt=d(C∙Uo)dt=CdUodt<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>I</mi><mi>c</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><mi>q</mi><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>C</mi><mo>•</mo><mi>U</mi><mi>o</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>=</mo><mi>C</mi><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><mi>U</mi><mi>o</mi><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle></math></div><script type="math/tex; mode=display" id="MathJax-Element-1">Ic = dfrac{dq {}}{dt} = dfrac{d(C•Uo) {}}{dt} = Cdfrac{dUo {}}{dt} </script> 
基尔霍夫电压定律得： 
<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mi>U</mi><mi>i</mi><mo>=</mo><mi>R</mi><mi>C</mi><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><mi>U</mi><mi>o</mi><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>+</mo><mi>U</mi><mi>o</mi></math>" role="presentation">Ui=RCdUodt+Uo<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>U</mi><mi>i</mi><mo>=</mo><mi>R</mi><mi>C</mi><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><mi>U</mi><mi>o</mi><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>+</mo><mi>U</mi><mi>o</mi></math></div><script type="math/tex; mode=display" id="MathJax-Element-2"> Ui=RC dfrac{dUo {}}{dt}+Uo </script> 
Ui的单位是伏特，RC的单位为秒，τ=RC； 
解得: 
<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mi>U</mi><mi>o</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>U</mi><mi>i</mi><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><msup><mi>e</mi><mo>&#x2227;</mo></msup><mo stretchy="false">(</mo><mo>&#x2212;</mo><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>R</mi><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math>" role="presentation">Uo(t)=Ui(1−e∧(−t/RC))<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>U</mi><mi>o</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>U</mi><mi>i</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mo>∧</mo></msup><mo stretchy="false">(</mo><mo>−</mo><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>R</mi><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></div><script type="math/tex; mode=display" id="MathJax-Element-3">Uo (t)= Ui(1-e ^∧ (-t/RC)) </script> 
假设电容初始电压值为0 
R=1000Ω 
C=4.7uF 
Ui=1V 
t=0.0001~0.1s 
τ=RC 
Vc(τ)=0.632 
<img src="https://file2.kaopuke.com:8081/files_image/20230520/202305201048004350772.png" alt="这里写图片描述" title=""> 
图2 一阶<a href="https://www.baidu.com/s?wd=RC%E7%B3%BB%E7%BB%9F&tn=24004469_oem_dg&rsv_dl=gh_pl_sl_csd" target="_blank">RC系统</a>的阶跃响应曲线

u1=Ui；u2=Uo； 
以电容电压作为输出，电路的网络函数为： 
<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mi>H</mi><mo stretchy="false">(</mo><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C9;</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>U</mi><mn>2</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>U</mi><mn>1</mn></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C9;</mo></mrow><mi>C</mi></mrow></mfrac></mstyle><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>R</mi><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C9;</mo></mrow><mi>C</mi></mrow></mfrac></mstyle></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mn>1</mn><mo>+</mo><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C9;</mo></mrow><mi>R</mi><mi>C</mi></mrow></mfrac></mstyle></math>" role="presentation">H(jω)=U2U1=1jωCR+1jωC=11+jωRC<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>H</mi><mo stretchy="false">(</mo><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>U</mi><mn>2</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>U</mi><mn>1</mn></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mi>C</mi></mrow></mfrac></mstyle><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>R</mi><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mi>C</mi></mrow></mfrac></mstyle></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mn>1</mn><mo>+</mo><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mi>R</mi><mi>C</mi></mrow></mfrac></mstyle></math></div><script type="math/tex; mode=display" id="MathJax-Element-4"> H(jω)= dfrac{U2 {}}{U1}=dfrac{dfrac{1 {}}{jωC} {}}{R+ dfrac{1 {}}{jωC}} = dfrac{1 {}}{1+ jωRC}</script> 
令ωc=<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>R</mi><mi>C</mi></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C4;</mo></mrow></mfrac></mstyle></math>" role="presentation">1RC=1τ<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>R</mi><mi>C</mi></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo>τ</mo></mrow></mfrac></mstyle></math></div><script type="math/tex; mode=display" id="MathJax-Element-5"> dfrac{1 {}}{RC}=dfrac{1 {}}{τ} </script> 
ωc即为截止频率；

幅值和相角函数： 
<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>H</mi><mo stretchy="false">(</mo><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C9;</mo></mrow><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C9;</mo></mrow><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C9;</mo></mrow><mi>c</mi></mrow></mfrac></mstyle><msup><mo stretchy="false">)</mo><mn>2</mn></msup></msqrt></mfrac></mstyle></math>" role="presentation">|H(jω)|=11+(ωωc)2−−−−−−−−√<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>H</mi><mo stretchy="false">(</mo><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mi>c</mi></mrow></mfrac></mstyle><msup><mo stretchy="false">)</mo><mn>2</mn></msup></msqrt></mfrac></mstyle></math></div><script type="math/tex; mode=display" id="MathJax-Element-6">	|H(jω)| = dfrac{1}{ sqrt{1 + ( dfrac{ω {}}{ωc}) ^2}} </script>

<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mrow class="MJX-TeXAtom-ORD"><mo>&#x3B8;</mo></mrow><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C9;</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><mo>&#x2212;</mo><mi>a</mi><mi>r</mi><mi>c</mi><mi>t</mi><mi>a</mi><mi>n</mi><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C9;</mo></mrow><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C9;</mo></mrow><mi>c</mi></mrow></mfrac></mstyle></math>" role="presentation">θ(ω)=−arctanωωc<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mo>θ</mo></mrow><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>a</mi><mi>r</mi><mi>c</mi><mi>t</mi><mi>a</mi><mi>n</mi><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mi>c</mi></mrow></mfrac></mstyle></math></div><script type="math/tex; mode=display" id="MathJax-Element-7"> θ(ω)=-arctan dfrac{ω {}}{ωc} </script>

各变量取值： 
R=1000Ω 
C=4.7uF 
<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mi>j</mi><mo>=</mo><msqrt><mo>&#x2212;</mo><mn>1</mn></msqrt></math>" role="presentation">j=−1−−−√<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>j</mi><mo>=</mo><msqrt><mo>−</mo><mn>1</mn></msqrt></math></div><script type="math/tex; mode=display" id="MathJax-Element-8">	j = sqrt{-1} </script> 
<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C9;</mo></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mi>R</mi><mi>C</mi></mrow></mfrac></mstyle></math>" role="presentation">ω=1RC<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mi>R</mi><mi>C</mi></mrow></mfrac></mstyle></math></div><script type="math/tex; mode=display" id="MathJax-Element-9">	ω = dfrac{1}{RC} </script> 
<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mi>f</mi><mi>c</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C0;</mo></mrow><mi>R</mi><mi>C</mi></mrow></mfrac></mstyle></math>" role="presentation">fc=12πRC<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mi>c</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>π</mo></mrow><mi>R</mi><mi>C</mi></mrow></mfrac></mstyle></math></div><script type="math/tex; mode=display" id="MathJax-Element-10">fc = dfrac{1}{2πRC} </script> 
<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mi>A</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>j</mi><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C0;</mo></mrow><mi>f</mi><mi>R</mi><mi>C</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></math>" role="presentation">A(f)=1j2πfRC+1<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>A</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>j</mi><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>π</mo></mrow><mi>f</mi><mi>R</mi><mi>C</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></math></div><script type="math/tex; mode=display" id="MathJax-Element-11"> A(f)=dfrac{1 {}}{j2πfRC+1} </script> 
|A(fc)|=0.707 
<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mrow class="MJX-TeXAtom-ORD"><mo>&#x3B8;</mo></mrow><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>180</mn><mi>a</mi><mi>r</mi><mi>g</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C0;</mo></mrow></mfrac></mstyle></math>" role="presentation">θ(f)=180arg(A(f))π<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mo>θ</mo></mrow><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>180</mn><mi>a</mi><mi>r</mi><mi>g</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>π</mo></mrow></mfrac></mstyle></math></div><script type="math/tex; mode=display" id="MathJax-Element-12">θ(f) = dfrac{180arg(A(f))}{π} </script> 
θ(fc)=-45

f=0.001、1、…….100000

幅频和相频特性图： 
<img src="https://file2.kaopuke.com:8081/files_image/20230520/202305201048004167361.png" alt="这里写图片描述" title=""> 
图3 
<img src="https://file2.kaopuke.com:8081/files_image/20230520/202305201048009933529.png" alt="这里写图片描述" title=""> 
图4 
幅频特性图的对数表示： 
<img src="https://file2.kaopuke.com:8081/files_image/20230520/202305201048003939517.png" alt="这里写图片描述" title=""> 
图5

-当ω<ωc时，幅值是平行于坐标的直线，基本无衰减；

-当ω>>ωc时，是斜率与-20dB/十倍频成比例的一条直线；

-当ω=ωc时，增益衰减至0.707，即-3dB，相位滞后45度，对应低通<a href="https://www.baidu.com/s?wd=%E6%BB%A4%E6%B3%A2%E5%99%A8&tn=24004469_oem_dg&rsv_dl=gh_pl_sl_csd" target="_blank">滤波器</a>，该频率通常被称为截止频率。

缺点： 
采用这种模拟滤波器抑制低频干扰时，要求滤波器有较大的时间常数和<a href="https://www.baidu.com/s?wd=%E9%AB%98%E7%B2%BE%E5%BA%A6&tn=24004469_oem_dg&rsv_dl=gh_pl_sl_csd" target="_blank">高精度</a>的RC网络，增大时间常数要求增大R值，其漏电流也随之增大，从而降低了滤波效果；

<h2 id="软件上的一阶低通滤波"><a name="t4"></a>软件上的一阶低通滤波</h2>

-采用数字滤波算法来实现动态的RC滤波，则能很好的克服模拟滤波器的缺点； 
-在模拟常数要求较大的场合这种算法显得更为实用； 
-其对于周期干扰有良好的抑制作用， 
-比较节省RAM空间

-不足之处是带来了相位滞后，导致灵敏度低； 
-同时它不能滤除频率高于采样频率的二分之一（称为奈奎斯特频率）的干扰（例如采样频率为100Hz，则它不能滤除50Hz以上的干扰信号）对于高于奈奎斯特频率的干扰信号，应该采用模拟滤波器。 
-对没有乘、除法运算指令的<a href="https://www.baidu.com/s?wd=%E5%8D%95%E7%89%87%E6%9C%BA&tn=24004469_oem_dg&rsv_dl=gh_pl_sl_csd" target="_blank">单片机</a>来说，程序运算工作量较大

<h3 id="基本滤波算法"><a name="t5"></a>基本滤波算法：</h3>

算法由来： 
频率分析中一阶RC低通滤波在S域的传递函数： 
<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>V</mi><mi>o</mi><mi>u</mi><mi>t</mi></mrow><mrow><mi>V</mi><mi>i</mi><mi>n</mi></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mi>R</mi><mi>C</mi><mi>s</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>,</mo><mo stretchy="false">(</mo><mi>s</mi><mo>=</mo><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x3C9;</mo></mrow><mo stretchy="false">)</mo></math>" role="presentation">VoutVin=1RCs+1,(s=jω)<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>V</mi><mi>o</mi><mi>u</mi><mi>t</mi></mrow><mrow><mi>V</mi><mi>i</mi><mi>n</mi></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mi>R</mi><mi>C</mi><mi>s</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>,</mo><mo stretchy="false">(</mo><mi>s</mi><mo>=</mo><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mo stretchy="false">)</mo></math></div><script type="math/tex; mode=display" id="MathJax-Element-13">	dfrac{Vout}{Vin}= dfrac{1}{RCs+1} ,(s=jω)</script> 
通过z变换（方法很多，如一阶前向差分、双线性变换等这里用一阶后向差分法） 
<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mi>s</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mo>&#x2212;</mo><msup><mi>z</mi><mo>&#x2227;</mo></msup><mo stretchy="false">(</mo><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>T</mi></mfrac></mstyle><mo>,</mo><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x8868;</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x793A;</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x91C7;</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x6837;</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x5468;</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x671F;</mo></mrow></math>" role="presentation">s=1−z∧(−1)T,T表示采样周期<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>s</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mo>−</mo><msup><mi>z</mi><mo>∧</mo></msup><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>T</mi></mfrac></mstyle><mo>,</mo><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mo>表</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>示</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>采</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>样</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>周</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>期</mo></mrow></math></div><script type="math/tex; mode=display" id="MathJax-Element-14">	s= dfrac{1-z^∧(-1)}{T} ,T表示采样周期</script>

带入S域传递函数中： 
<div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><div class="MathJax_Display" style="text-align: center;"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mi>R</mi><mi>C</mi><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mo>&#x2212;</mo><msup><mi>z</mi><mo>&#x2227;</mo></msup><mo stretchy="false">(</mo><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>T</mi></mfrac></mstyle><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>T</mi><mrow><mi>R</mi><mi>C</mi><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><msup><mi>z</mi><mo>&#x2227;</mo></msup><mo stretchy="false">(</mo><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>T</mi></mrow></mfrac></mstyle></math>" role="presentation">Y(z)X(z)=1RC1−z∧(−1)T+1=<span style="display: inline-block; position: relative; width: 9.203em; height: 0px; margi

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136
137
138
139
140
141
142
/*入口：NEW_DATA 新采样值
       OLD_DATA 上次滤波结果
       k        滤波系数(0~255)(代表在滤波结果中的权重)
  出口：         本次滤波结果
 */
 char filter_1(char NEW_DATA,char OLD_DATA,char k)
{
    int result;
    if(NEW_DATA<OLD_DATA)
    {
        result=OLD_DATA-NEW_DATA;
        result=result*k;
        result=result+128;//+128是为了四色五入
        result=result/256;
        result=OLD_DATA-result;<div id="article_content" class="article_content clearfix csdn-tracking-statistics" data-pid="blog" data-mod="popu_307" data-dsm="post">
								<div class="article-copyright">
                  					
					版权声明：本文为博主原创文章，未经博主允许不得转载。					https://blog.csdn.net/qq_27334499/article/details/52186336				</div>
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							<h1 id="一阶rc低通滤波"><a name="t0"></a>一阶RC低通滤波</h1>

<p>从模拟到数字 <br>
本文整理自网络、<a href="https://www.baidu.com/s?wd=%E3%80%8A%E5%8C%A0%E4%BA%BA%E6%89%8B%E8%AE%B0%E3%80%8B&amp;tn=24004469_oem_dg&amp;rsv_dl=gh_pl_sl_csd" target="_blank">《匠人手记》</a>等书籍文章</p>

<ul>
<li><strong>模拟电路低通滤波时域、频域</strong></li>
<li><strong>软件低通滤波</strong></li>
</ul>

<hr>

<h2 id="典型电路"><a name="t1"></a><strong>典型电路</strong></h2>

<p><img src="https://file2.kaopuke.com:8081/files_image/20230520/202305201048006935945.jpg" alt="这里写图片描述" title=""> <br>
图1 典型RC电路 <br>
直流、交流、脉冲信号都可以用它</p>



<h3 id="时域"><a name="t2"></a><strong>时域</strong></h3>

<p>电容电流： <br>
<span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;><mi>I</mi><mi>c</mi><mo>=</mo><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mrow><mi>d</mi><mi>q</mi><mrow class=&quot;MJX-TeXAtom-ORD&quot;></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mrow><mi>d</mi><mo stretchy=&quot;false&quot;>(</mo><mi>C</mi><mo>&amp;#x2022;</mo><mi>U</mi><mi>o</mi><mo stretchy=&quot;false&quot;>)</mo><mrow class=&quot;MJX-TeXAtom-ORD&quot;></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>=</mo><mi>C</mi><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mrow><mi>d</mi><mi>U</mi><mi>o</mi><mrow class=&quot;MJX-TeXAtom-ORD&quot;></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1" style="width: 17.089em; display: inline-block;"><span style="display: inline-block; position: relative; width: 13.66em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(0.631em 1013.66em 3.146em -999.997em); top: -2.283em; left: 0em;"><span class="mrow" id="MathJax-Span-2"><span class="mi" id="MathJax-Span-3" style="font-family: MathJax_Math-italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mi" id="MathJax-Span-4" style="font-family: MathJax_Math-italic;">c</span><span class="mo" id="MathJax-Span-5" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mstyle" id="MathJax-Span-6" style="padding-left: 0.289em;"><span class="mrow" id="MathJax-Span-7"><span class="mfrac" id="MathJax-Span-8"><span style="display: inline-block; position: relative; width: 1.089em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.146em 1000.97em 4.346em -999.997em); top: -4.683em; left: 50%; margin-left: -0.511em;"><span class="mrow" id="MathJax-Span-9"><span class="mi" id="MathJax-Span-10" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-11" style="font-family: MathJax_Math-italic;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="texatom" id="MathJax-Span-12"><span class="mrow" id="MathJax-Span-13"></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.146em 1000.86em 4.174em -999.997em); top: -3.311em; left: 50%; margin-left: -0.454em;"><span class="mrow" id="MathJax-Span-14"><span class="mi" id="MathJax-Span-15" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-16" style="font-family: MathJax_Math-italic;">t</span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1001.09em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.089em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-17" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mstyle" id="MathJax-Span-18" style="padding-left: 0.289em;"><span class="mrow" id="MathJax-Span-19"><span class="mfrac" id="MathJax-Span-20"><span style="display: inline-block; position: relative; width: 4.403em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.089em 1004.29em 4.403em -999.997em); top: -4.74em; left: 50%; margin-left: -2.111em;"><span class="mrow" id="MathJax-Span-21"><span class="mi" id="MathJax-Span-22" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-23" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-24" style="font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-25" style="font-family: MathJax_Main; padding-left: 0.231em;">∙</span><span class="mi" id="MathJax-Span-26" style="font-family: MathJax_Math-italic; padding-left: 0.231em;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mi" id="MathJax-Span-27" style="font-family: MathJax_Math-italic;">o</span><span class="mo" id="MathJax-Span-28" style="font-family: MathJax_Main;">)</span><span class="texatom" id="MathJax-Span-29"><span class="mrow" id="MathJax-Span-30"></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.146em 1000.86em 4.174em -999.997em); top: -3.311em; left: 50%; margin-left: -0.454em;"><span class="mrow" id="MathJax-Span-31"><span class="mi" id="MathJax-Span-32" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-33" style="font-family: MathJax_Math-italic;">t</span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1004.4em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.403em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-34" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mi" id="MathJax-Span-35" style="font-family: MathJax_Math-italic; padding-left: 0.289em;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mstyle" id="MathJax-Span-36"><span class="mrow" id="MathJax-Span-37"><span class="mfrac" id="MathJax-Span-38"><span style="display: inline-block; position: relative; width: 1.889em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.146em 1001.77em 4.174em -999.997em); top: -4.683em; left: 50%; margin-left: -0.911em;"><span class="mrow" id="MathJax-Span-39"><span class="mi" id="MathJax-Span-40" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-41" style="font-family: MathJax_Math-italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mi" id="MathJax-Span-42" style="font-family: MathJax_Math-italic;">o</span><span class="texatom" id="MathJax-Span-43"><span class="mrow" id="MathJax-Span-44"></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.146em 1000.86em 4.174em -999.997em); top: -3.311em; left: 50%; margin-left: -0.454em;"><span class="mrow" id="MathJax-Span-45"><span class="mi" id="MathJax-Span-46" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-47" style="font-family: MathJax_Math-italic;">t</span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1001.89em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.889em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.289em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.925em; border-left: 0px solid; width: 0px; height: 2.861em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>I</mi><mi>c</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><mi>q</mi><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>C</mi><mo>•</mo><mi>U</mi><mi>o</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>=</mo><mi>C</mi><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><mi>U</mi><mi>o</mi><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-1">Ic = dfrac{dq  {}}{dt} = dfrac{d(C•Uo)  {}}{dt} = Cdfrac{dUo  {}}{dt} </script> <br>
基尔霍夫电压定律得： <br>
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;><mi>U</mi><mi>i</mi><mo>=</mo><mi>R</mi><mi>C</mi><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mrow><mi>d</mi><mi>U</mi><mi>o</mi><mrow class=&quot;MJX-TeXAtom-ORD&quot;></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>+</mo><mi>U</mi><mi>o</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-48" style="width: 10.574em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.46em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(0.746em 1008.46em 3.146em -999.997em); top: -2.283em; left: 0em;"><span class="mrow" id="MathJax-Span-49"><span class="mi" id="MathJax-Span-50" style="font-family: MathJax_Math-italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mi" id="MathJax-Span-51" style="font-family: MathJax_Math-italic;">i</span><span class="mo" id="MathJax-Span-52" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mi" id="MathJax-Span-53" style="font-family: MathJax_Math-italic; padding-left: 0.289em;">R</span><span class="mi" id="MathJax-Span-54" style="font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mstyle" id="MathJax-Span-55"><span class="mrow" id="MathJax-Span-56"><span class="mfrac" id="MathJax-Span-57"><span style="display: inline-block; position: relative; width: 1.889em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.146em 1001.77em 4.174em -999.997em); top: -4.683em; left: 50%; margin-left: -0.911em;"><span class="mrow" id="MathJax-Span-58"><span class="mi" id="MathJax-Span-59" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-60" style="font-family: MathJax_Math-italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mi" id="MathJax-Span-61" style="font-family: MathJax_Math-italic;">o</span><span class="texatom" id="MathJax-Span-62"><span class="mrow" id="MathJax-Span-63"></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.146em 1000.86em 4.174em -999.997em); top: -3.311em; left: 50%; margin-left: -0.454em;"><span class="mrow" id="MathJax-Span-64"><span class="mi" id="MathJax-Span-65" style="font-family: MathJax_Math-italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-66" style="font-family: MathJax_Math-italic;">t</span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1001.89em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.889em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-67" style="font-family: MathJax_Main; padding-left: 0.231em;">+</span><span class="mi" id="MathJax-Span-68" style="font-family: MathJax_Math-italic; padding-left: 0.231em;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mi" id="MathJax-Span-69" style="font-family: MathJax_Math-italic;">o</span></span><span style="display: inline-block; width: 0px; height: 2.289em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.925em; border-left: 0px solid; width: 0px; height: 2.718em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>U</mi><mi>i</mi><mo>=</mo><mi>R</mi><mi>C</mi><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><mi>U</mi><mi>o</mi><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mstyle><mo>+</mo><mi>U</mi><mi>o</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-2"> Ui=RC dfrac{dUo  {}}{dt}+Uo  </script> <br>
Ui的单位是伏特，RC的单位为秒，τ=RC； <br>
解得: <br>
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;><mi>U</mi><mi>o</mi><mo stretchy=&quot;false&quot;>(</mo><mi>t</mi><mo stretchy=&quot;false&quot;>)</mo><mo>=</mo><mi>U</mi><mi>i</mi><mo stretchy=&quot;false&quot;>(</mo><mn>1</mn><mo>&amp;#x2212;</mo><msup><mi>e</mi><mo>&amp;#x2227;</mo></msup><mo stretchy=&quot;false&quot;>(</mo><mo>&amp;#x2212;</mo><mi>t</mi><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>/</mo></mrow><mi>R</mi><mi>C</mi><mo stretchy=&quot;false&quot;>)</mo><mo stretchy=&quot;false&quot;>)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-70" style="width: 15.317em; display: inline-block;"><span style="display: inline-block; position: relative; width: 12.231em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.26em 1012.12em 2.689em -999.997em); top: -2.283em; left: 0em;"><span class="mrow" id="MathJax-Span-71"><span class="mi" id="MathJax-Span-72" style="font-family: MathJax_Math-italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mi" id="MathJax-Span-73" style="font-family: MathJax_Math-italic;">o</span><span class="mo" id="MathJax-Span-74" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-75" style="font-family: MathJax_Math-italic;">t</span><span class="mo" id="MathJax-Span-76" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-77" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mi" id="MathJax-Span-78" style="font-family: MathJax_Math-italic; padding-left: 0.289em;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mi" id="MathJax-Span-79" style="font-family: MathJax_Math-italic;">i</span><span class="mo" id="MathJax-Span-80" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-81" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-82" style="font-family: MathJax_Main; padding-left: 0.231em;">−</span><span class="msubsup" id="MathJax-Span-83" style="padding-left: 0.231em;"><span style="display: inline-block; position: relative; width: 1.031em; height: 0px;"><span style="position: absolute; clip: rect(3.374em 1000.46em 4.174em -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-84" style="font-family: MathJax_Math-italic;">e</span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; top: -4.397em; left: 0.46em;"><span class="mo" id="MathJax-Span-85" style="font-size: 70.7%; font-family: MathJax_Main;">∧</span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span></span></span><span class="mo" id="MathJax-Span-86" style="font-family: MathJax_Main;">(</span><span class="mo" id="MathJax-Span-87" style="font-family: MathJax_Main;">−</span><span class="mi" id="MathJax-Span-88" style="font-family: MathJax_Math-italic;">t</span><span class="texatom" id="MathJax-Span-89"><span class="mrow" id="MathJax-Span-90"><span class="mo" id="MathJax-Span-91" style="font-family: MathJax_Main;">/</span></span></span><span class="mi" id="MathJax-Span-92" style="font-family: MathJax_Math-italic;">R</span><span class="mi" id="MathJax-Span-93" style="font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-94" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-95" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.289em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left: 0px solid; width: 0px; height: 1.504em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>U</mi><mi>o</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>U</mi><mi>i</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mo>∧</mo></msup><mo stretchy="false">(</mo><mo>−</mo><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>R</mi><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-3">Uo (t)= Ui(1-e ^∧ (-t/RC)) </script> <br>
假设电容初始电压值为0 <br>
R=1000Ω <br>
C=4.7uF <br>
Ui=1V <br>
t=0.0001~0.1s <br>
τ=RC <br>
Vc(τ)=0.632 <br>
<img src="https://file2.kaopuke.com:8081/files_image/20230520/202305201048004350772.png" alt="这里写图片描述" title=""> <br>
图2 一阶<a href="https://www.baidu.com/s?wd=RC%E7%B3%BB%E7%BB%9F&amp;tn=24004469_oem_dg&amp;rsv_dl=gh_pl_sl_csd" target="_blank">RC系统</a>的阶跃响应曲线<p></p>



<h3 id="频域"><a name="t3"></a><strong>频域</strong></h3>

<p>u1=Ui；u2=Uo； <br>
以电容电压作为输出，电路的网络函数为： <br>
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mode=display" id="MathJax-Element-4"> H(jω)= dfrac{U2  {}}{U1}=dfrac{dfrac{1  {}}{jωC}  {}}{R+ dfrac{1  {}}{jωC}}  = dfrac{1  {}}{1+ jωRC}</script> <br>
令ωc=<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mrow><mn>1</mn><mrow class=&quot;MJX-TeXAtom-ORD&quot;></mrow></mrow><mrow><mi>R</mi><mi>C</mi></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mrow><mn>1</mn><mrow class=&quot;MJX-TeXAtom-ORD&quot;></mrow></mrow><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x3C4;</mo></mrow></mfrac></mstyle></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-170" style="width: 5.089em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.06em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(0.803em 1004.06em 3.146em -999.997em); top: -2.283em; left: 0em;"><span class="mrow" id="MathJax-Span-171"><span class="mstyle" id="MathJax-Span-172"><span class="mrow" id="MathJax-Span-173"><span class="mfrac" id="MathJax-Span-174"><span style="display: inline-block; position: relative; width: 1.66em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.146em 1000.52em 4.174em -999.997em); top: -4.683em; left: 50%; margin-left: -0.226em;"><span class="mrow" id="MathJax-Span-175"><span class="mn" id="MathJax-Span-176" style="font-family: MathJax_Main;">1</span><span class="texatom" id="MathJax-Span-177"><span class="mrow" id="MathJax-Span-178"></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.146em 1001.55em 4.174em -999.997em); top: -3.311em; left: 50%; margin-left: -0.74em;"><span class="mrow" id="MathJax-Span-179"><span class="mi" id="MathJax-Span-180" style="font-family: MathJax_Math-italic;">R</span><span class="mi" id="MathJax-Span-181" style="font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1001.66em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.66em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-182" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mstyle" id="MathJax-Span-183" style="padding-left: 0.289em;"><span class="mrow" id="MathJax-Span-184"><span class="mfrac" id="MathJax-Span-185"><span style="display: inline-block; position: relative; width: 0.631em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.146em 1000.52em 4.174em -999.997em); top: -4.683em; left: 50%; margin-left: -0.226em;"><span class="mrow" id="MathJax-Span-186"><span class="mn" id="MathJax-Span-187" style="font-family: MathJax_Main;">1</span><span class="texatom" id="MathJax-Span-188"><span class="mrow" id="MathJax-Span-189"></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.374em 1000.52em 4.174em -999.997em); top: -3.311em; left: 50%; margin-left: -0.226em;"><span class="texatom" id="MathJax-Span-190"><span class="mrow" id="MathJax-Span-191"><span class="mo" id="MathJax-Span-192" style="font-family: MathJax_Math-italic;">τ</span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1000.63em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.631em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.289em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.925em; border-left: 0px solid; width: 0px; height: 2.718em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>R</mi><mi>C</mi></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mo>τ</mo></mrow></mfrac></mstyle></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-5">  dfrac{1  {}}{RC}=dfrac{1  {}}{τ}  </script> <br>
ωc即为截止频率；<p></p>

<p><strong>幅值和相角函数：</strong> <br>
<span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo stretchy=&quot;false&quot;>|</mo></mrow><mi>H</mi><mo stretchy=&quot;false&quot;>(</mo><mi>j</mi><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x3C9;</mo></mrow><mo stretchy=&quot;false&quot;>)</mo><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo stretchy=&quot;false&quot;>|</mo></mrow><mo>=</mo><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mo stretchy=&quot;false&quot;>(</mo><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mrow><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x3C9;</mo></mrow><mrow class=&quot;MJX-TeXAtom-ORD&quot;></mrow></mrow><mrow><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x3C9;</mo></mrow><mi>c</mi></mrow></mfrac></mstyle><msup><mo stretchy=&quot;false&quot;>)</mo><mn>2</mn></msup></msqrt></mfrac></mstyle></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-193" style="width: 12.86em; display: inline-block;"><span style="display: inline-block; position: relative; width: 10.289em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(0.803em 1010.29em 4.86em -999.997em); top: -2.283em; left: 0em;"><span class="mrow" id="MathJax-Span-194"><span class="texatom" id="MathJax-Span-195"><span class="mrow" id="MathJax-Span-196"><span class="mo" id="MathJax-Span-197" style="font-family: MathJax_Main;">|</span></span></span><span class="mi" id="MathJax-Span-198" style="font-family: MathJax_Math-italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-199" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-200" style="font-family: MathJax_Math-italic;">j</span><span class="texatom" id="MathJax-Span-201"><span class="mrow" id="MathJax-Span-202"><span class="mo" id="MathJax-Span-203" style="font-family: MathJax_Math-italic;">ω</span></span></span><span class="mo" id="MathJax-Span-204" style="font-family: MathJax_Main;">)</span><span class="texatom" id="MathJax-Span-205"><span class="mrow" id="MathJax-Span-206"><span class="mo" id="MathJax-Span-207" style="font-family: MathJax_Main;">|</span></span></span><span class="mo" id="MathJax-Span-208" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mstyle" id="MathJax-Span-209" style="padding-left: 0.289em;"><span class="mrow" id="MathJax-Span-210"><span class="mfrac" id="MathJax-Span-211"><span style="display: inline-block; position: relative; width: 5.489em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.146em 1000.4em 4.174em -999.997em); top: -4.683em; left: 50%; margin-left: -0.226em;"><span class="mn" id="MathJax-Span-212" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(2.403em 1005.37em 5.146em -999.997em); top: -2.569em; left: 50%; margin-left: -2.683em;"><span class="msqrt" id="MathJax-Span-213"><span style="display: inline-block; position: relative; width: 5.374em; height: 0px;"><span style="position: absolute; clip: rect(2.689em 1004.35em 4.86em -999.997em); top: -3.997em; left: 1.031em;"><span class="mrow" id="MathJax-Span-214"><span class="mn" id="MathJax-Span-215" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-216" style="font-family: MathJax_Main; padding-left: 0.231em;">+</span><span class="mo" id="MathJax-Span-217" style="font-family: MathJax_Main; padding-left: 0.231em;">(</span><span class="mstyle" id="MathJax-Span-218"><span class="mrow" id="MathJax-Span-219"><span class="mfrac" id="MathJax-Span-220"><span style="display: inline-block; position: relative; width: 1.203em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.374em 1000.63em 4.174em -999.997em); top: -4.683em; left: 50%; margin-left: -0.283em;"><span class="mrow" id="MathJax-Span-221"><span class="texatom" id="MathJax-Span-222"><span class="mrow" id="MathJax-Span-223"><span class="mo" id="MathJax-Span-224" style="font-family: MathJax_Math-italic;">ω</span></span></span><span class="texatom" id="MathJax-Span-225"><span class="mrow" id="MathJax-Span-226"></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.374em 1001.03em 4.174em -999.997em); top: -3.311em; left: 50%; margin-left: -0.511em;"><span class="mrow" id="MathJax-Span-227"><span class="texatom" id="MathJax-Span-228"><span class="mrow" id="MathJax-Span-229"><span class="mo" id="MathJax-Span-230" style="font-family: MathJax_Math-italic;">ω</span></span></span><span class="mi" id="MathJax-Span-231" style="font-family: MathJax_Math-italic;">c</span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1001.2em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.203em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span><span class="msubsup" id="MathJax-Span-232"><span style="display: inline-block; position: relative; width: 0.803em; height: 0px;"><span style="position: absolute; clip: rect(3.089em 1000.29em 4.403em -999.997em); top: -3.997em; left: 0em;"><span class="mo" id="MathJax-Span-233" style="font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; top: -4.283em; left: 0.403em;"><span class="mn" id="MathJax-Span-234" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.546em 1004.4em 3.946em -999.997em); top: -5.14em; left: 1.031em;"><span style="display: inline-block; position: relative; width: 4.403em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.997em; left: -0.054em;">−<span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -3.997em; left: 3.717em;">−<span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.997em; left: 0.46em;">−<span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.997em; left: 0.974em;">−<span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.997em; left: 1.546em;">−<span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.997em; left: 2.06em;">−<span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.997em; left: 2.631em;">−<span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.997em; left: 3.146em;">−<span style="display: inline-block; width: 0px; height: 4.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(2.403em 1001.03em 5.146em -999.997em); top: -3.94em; left: 0em;"><span style="font-family: MathJax_Size3;">√</span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1005.49em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 5.489em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.289em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -3.068em; border-left: 0px solid; width: 0px; height: 4.789em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>H</mi><mo stretchy="false">(</mo><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mi>c</mi></mrow></mfrac></mstyle><msup><mo stretchy="false">)</mo><mn>2</mn></msup></msqrt></mfrac></mstyle></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-6">	|H(jω)| = dfrac{1}{ sqrt{1 + ( dfrac{ω  {}}{ωc}) ^2}} </script><p></p>



<p><span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-7-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x3B8;</mo></mrow><mo stretchy=&quot;false&quot;>(</mo><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x3C9;</mo></mrow><mo stretchy=&quot;false&quot;>)</mo><mo>=</mo><mo>&amp;#x2212;</mo><mi>a</mi><mi>r</mi><mi>c</mi><mi>t</mi><mi>a</mi><mi>n</mi><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mrow><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x3C9;</mo></mrow><mrow class=&quot;MJX-TeXAtom-ORD&quot;></mrow></mrow><mrow><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x3C9;</mo></mrow><mi>c</mi></mrow></mfrac></mstyle></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-235" style="width: 10.403em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.289em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(0.974em 1008.29em 3.146em -999.997em); top: -2.283em; left: 0em;"><span class="mrow" id="MathJax-Span-236"><span class="texatom" id="MathJax-Span-237"><span class="mrow" id="MathJax-Span-238"><span class="mo" id="MathJax-Span-239" style="font-family: MathJax_Math-italic;">θ</span></span></span><span class="mo" id="MathJax-Span-240" style="font-family: MathJax_Main;">(</span><span class="texatom" id="MathJax-Span-241"><span class="mrow" id="MathJax-Span-242"><span class="mo" id="MathJax-Span-243" style="font-family: MathJax_Math-italic;">ω</span></span></span><span class="mo" id="MathJax-Span-244" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-245" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mo" id="MathJax-Span-246" style="font-family: MathJax_Main; padding-left: 0.289em;">−</span><span class="mi" id="MathJax-Span-247" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-248" style="font-family: MathJax_Math-italic;">r</span><span class="mi" id="MathJax-Span-249" style="font-family: MathJax_Math-italic;">c</span><span class="mi" id="MathJax-Span-250" style="font-family: MathJax_Math-italic;">t</span><span class="mi" id="MathJax-Span-251" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-252" style="font-family: MathJax_Math-italic;">n</span><span class="mstyle" id="MathJax-Span-253"><span class="mrow" id="MathJax-Span-254"><span class="mfrac" id="MathJax-Span-255"><span style="display: inline-block; position: relative; width: 1.203em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.374em 1000.63em 4.174em -999.997em); top: -4.683em; left: 50%; margin-left: -0.283em;"><span class="mrow" id="MathJax-Span-256"><span class="texatom" id="MathJax-Span-257"><span class="mrow" id="MathJax-Span-258"><span class="mo" id="MathJax-Span-259" style="font-family: MathJax_Math-italic;">ω</span></span></span><span class="texatom" id="MathJax-Span-260"><span class="mrow" id="MathJax-Span-261"></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.374em 1001.03em 4.174em -999.997em); top: -3.311em; left: 50%; margin-left: -0.511em;"><span class="mrow" id="MathJax-Span-262"><span class="texatom" id="MathJax-Span-263"><span class="mrow" id="MathJax-Span-264"><span class="mo" id="MathJax-Span-265" style="font-family: MathJax_Math-italic;">ω</span></span></span><span class="mi" id="MathJax-Span-266" style="font-family: MathJax_Math-italic;">c</span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1001.2em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.203em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.289em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.925em; border-left: 0px solid; width: 0px; height: 2.432em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mo>θ</mo></mrow><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>a</mi><mi>r</mi><mi>c</mi><mi>t</mi><mi>a</mi><mi>n</mi><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mi>c</mi></mrow></mfrac></mstyle></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-7"> θ(ω)=-arctan dfrac{ω  {}}{ωc}  </script><p></p>

<p>各变量取值： <br>
R=1000Ω <br>
C=4.7uF <br>
<span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-8-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;><mi>j</mi><mo>=</mo><msqrt><mo>&amp;#x2212;</mo><mn>1</mn></msqrt></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-267" style="width: 4.803em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.831em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.374em 1003.83em 2.689em -999.997em); top: -2.283em; left: 0em;"><span class="mrow" id="MathJax-Span-268"><span class="mi" id="MathJax-Span-269" style="font-family: MathJax_Math-italic;">j</span><span class="mo" id="MathJax-Span-270" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="msqrt" id="MathJax-Span-271" style="padding-left: 0.289em;"><span style="display: inline-block; position: relative; width: 2.117em; height: 0px;"><span style="position: absolute; clip: rect(3.146em 1001.2em 4.231em -999.997em); top: -3.997em; left: 0.86em;"><span class="mrow" id="MathJax-Span-272"><span class="mo" id="MathJax-Span-273" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-274" style="font-family: MathJax_Main;">1</span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.546em 1001.32em 3.946em -999.997em); top: -4.454em; left: 0.86em;"><span style="display: inline-block; position: relative; width: 1.317em; height: 0px;"><span style="position: absolute; font-family: MathJax_Main; top: -3.997em; left: -0.054em;">−<span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; font-family: MathJax_Main; top: -3.997em; left: 0.631em;">−<span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="font-family: MathJax_Main; position: absolute; top: -3.997em; left: 0.231em;">−<span style="display: inline-block; width: 0px; height: 4.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.031em 1000.86em 4.403em -999.997em); top: -3.94em; left: 0em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.289em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.354em; border-left: 0px solid; width: 0px; height: 1.432em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>j</mi><mo>=</mo><msqrt><mo>−</mo><mn>1</mn></msqrt></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-8">	j = sqrt{-1} </script> <br>
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x3C9;</mo></mrow><mo>=</mo><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mn>1</mn><mrow><mi>R</mi><mi>C</mi></mrow></mfrac></mstyle></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-275" style="width: 4.746em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.774em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(0.803em 1003.77em 3.146em -999.997em); top: -2.283em; left: 0em;"><span class="mrow" id="MathJax-Span-276"><span class="texatom" id="MathJax-Span-277"><span class="mrow" id="MathJax-Span-278"><span class="mo" id="MathJax-Span-279" style="font-family: MathJax_Math-italic;">ω</span></span></span><span class="mo" id="MathJax-Span-280" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mstyle" id="MathJax-Span-281" style="padding-left: 0.289em;"><span class="mrow" id="MathJax-Span-282"><span class="mfrac" id="MathJax-Span-283"><span style="display: inline-block; position: relative; width: 1.66em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.146em 1000.4em 4.174em -999.997em); top: -4.683em; left: 50%; margin-left: -0.226em;"><span class="mn" id="MathJax-Span-284" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.146em 1001.55em 4.174em -999.997em); top: -3.311em; left: 50%; margin-left: -0.74em;"><span class="mrow" id="MathJax-Span-285"><span class="mi" id="MathJax-Span-286" style="font-family: MathJax_Math-italic;">R</span><span class="mi" id="MathJax-Span-287" style="font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1001.66em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.66em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.289em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.925em; border-left: 0px solid; width: 0px; height: 2.718em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mi>R</mi><mi>C</mi></mrow></mfrac></mstyle></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-9">	ω = dfrac{1}{RC} </script> <br>
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-10-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;><mi>f</mi><mi>c</mi><mo>=</mo><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mn>1</mn><mrow><mn>2</mn><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x3C0;</mo></mrow><mi>R</mi><mi>C</mi></mrow></mfrac></mstyle></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-288" style="width: 6.517em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.203em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(0.803em 1005.2em 3.146em -999.997em); top: -2.283em; left: 0em;"><span class="mrow" id="MathJax-Span-289"><span class="mi" id="MathJax-Span-290" style="font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mi" id="MathJax-Span-291" style="font-family: MathJax_Math-italic;">c</span><span class="mo" id="MathJax-Span-292" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mstyle" id="MathJax-Span-293" style="padding-left: 0.289em;"><span class="mrow" id="MathJax-Span-294"><span class="mfrac" id="MathJax-Span-295"><span style="display: inline-block; position: relative; width: 2.689em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.146em 1000.4em 4.174em -999.997em); top: -4.683em; left: 50%; margin-left: -0.226em;"><span class="mn" id="MathJax-Span-296" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.146em 1002.57em 4.174em -999.997em); top: -3.311em; left: 50%; margin-left: -1.311em;"><span class="mrow" id="MathJax-Span-297"><span class="mn" id="MathJax-Span-298" style="font-family: MathJax_Main;">2</span><span class="texatom" id="MathJax-Span-299"><span class="mrow" id="MathJax-Span-300"><span class="mo" id="MathJax-Span-301" style="font-family: MathJax_Math-italic;">π</span></span></span><span class="mi" id="MathJax-Span-302" style="font-family: MathJax_Math-italic;">R</span><span class="mi" id="MathJax-Span-303" style="font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1002.69em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.689em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.289em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.925em; border-left: 0px solid; width: 0px; height: 2.718em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mi>c</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>π</mo></mrow><mi>R</mi><mi>C</mi></mrow></mfrac></mstyle></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-10">fc = dfrac{1}{2πRC} </script> <br>
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-11-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;><mi>A</mi><mo stretchy=&quot;false&quot;>(</mo><mi>f</mi><mo stretchy=&quot;false&quot;>)</mo><mo>=</mo><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mrow><mn>1</mn><mrow class=&quot;MJX-TeXAtom-ORD&quot;></mrow></mrow><mrow><mi>j</mi><mn>2</mn><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x3C0;</mo></mrow><mi>f</mi><mi>R</mi><mi>C</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-304" style="width: 11.317em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.031em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(0.803em 1009.03em 3.374em -999.997em); top: -2.283em; left: 0em;"><span class="mrow" id="MathJax-Span-305"><span class="mi" id="MathJax-Span-306" style="font-family: MathJax_Math-italic;">A</span><span class="mo" id="MathJax-Span-307" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-308" style="font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-309" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-310" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mstyle" id="MathJax-Span-311" style="padding-left: 0.289em;"><span class="mrow" id="MathJax-Span-312"><span class="mfrac" id="MathJax-Span-313"><span style="display: inline-block; position: relative; width: 5.431em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.146em 1000.52em 4.174em -999.997em); top: -4.683em; left: 50%; margin-left: -0.226em;"><span class="mrow" id="MathJax-Span-314"><span class="mn" id="MathJax-Span-315" style="font-family: MathJax_Main;">1</span><span class="texatom" id="MathJax-Span-316"><span class="mrow" id="MathJax-Span-317"></span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.146em 1005.2em 4.403em -999.997em); top: -3.311em; left: 50%; margin-left: -2.626em;"><span class="mrow" id="MathJax-Span-318"><span class="mi" id="MathJax-Span-319" style="font-family: MathJax_Math-italic;">j</span><span class="mn" id="MathJax-Span-320" style="font-family: MathJax_Main;">2</span><span class="texatom" id="MathJax-Span-321"><span class="mrow" id="MathJax-Span-322"><span class="mo" id="MathJax-Span-323" style="font-family: MathJax_Math-italic;">π</span></span></span><span class="mi" id="MathJax-Span-324" style="font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mi" id="MathJax-Span-325" style="font-family: MathJax_Math-italic;">R</span><span class="mi" id="MathJax-Span-326" style="font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-327" style="font-family: MathJax_Main; padding-left: 0.231em;">+</span><span class="mn" id="MathJax-Span-328" style="font-family: MathJax_Main; padding-left: 0.231em;">1</span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1005.43em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 5.431em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.289em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.211em; border-left: 0px solid; width: 0px; height: 2.932em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>A</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow><mi>j</mi><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>π</mo></mrow><mi>f</mi><mi>R</mi><mi>C</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-11"> A(f)=dfrac{1  {}}{j2πfRC+1}  </script> <br>
|A(fc)|=0.707 <br>
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-12-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x3B8;</mo></mrow><mo stretchy=&quot;false&quot;>(</mo><mi>f</mi><mo stretchy=&quot;false&quot;>)</mo><mo>=</mo><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mrow><mn>180</mn><mi>a</mi><mi>r</mi><mi>g</mi><mo stretchy=&quot;false&quot;>(</mo><mi>A</mi><mo stretchy=&quot;false&quot;>(</mo><mi>f</mi><mo stretchy=&quot;false&quot;>)</mo><mo stretchy=&quot;false&quot;>)</mo></mrow><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x3C0;</mo></mrow></mfrac></mstyle></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-329" style="width: 11.603em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.26em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(0.631em 1009.26em 3.146em -999.997em); top: -2.283em; left: 0em;"><span class="mrow" id="MathJax-Span-330"><span class="texatom" id="MathJax-Span-331"><span class="mrow" id="MathJax-Span-332"><span class="mo" id="MathJax-Span-333" style="font-family: MathJax_Math-italic;">θ</span></span></span><span class="mo" id="MathJax-Span-334" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-335" style="font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-336" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-337" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mstyle" id="MathJax-Span-338" style="padding-left: 0.289em;"><span class="mrow" id="MathJax-Span-339"><span class="mfrac" id="MathJax-Span-340"><span style="display: inline-block; position: relative; width: 5.946em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.089em 1005.72em 4.403em -999.997em); top: -4.74em; left: 50%; margin-left: -2.911em;"><span class="mrow" id="MathJax-Span-341"><span class="mn" id="MathJax-Span-342" style="font-family: MathJax_Main;">180</span><span class="mi" id="MathJax-Span-343" style="font-family: MathJax_Math-italic;">a</span><span class="mi" id="MathJax-Span-344" style="font-family: MathJax_Math-italic;">r</span><span class="mi" id="MathJax-Span-345" style="font-family: MathJax_Math-italic;">g<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-346" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-347" style="font-family: MathJax_Math-italic;">A</span><span class="mo" id="MathJax-Span-348" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-349" style="font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mo" id="MathJax-Span-350" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-351" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.374em 1000.57em 4.174em -999.997em); top: -3.311em; left: 50%; margin-left: -0.283em;"><span class="texatom" id="MathJax-Span-352"><span class="mrow" id="MathJax-Span-353"><span class="mo" id="MathJax-Span-354" style="font-family: MathJax_Math-italic;">π</span></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1005.95em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 5.946em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.289em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.925em; border-left: 0px solid; width: 0px; height: 2.861em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mo>θ</mo></mrow><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>180</mn><mi>a</mi><mi>r</mi><mi>g</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>π</mo></mrow></mfrac></mstyle></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-12">θ(f) = dfrac{180arg(A(f))}{π} </script> <br>
θ(fc)=-45<p></p>

<p>f=0.001、1、…….100000</p>

<p>幅频和相频特性图： <br>
<img src="https://file2.kaopuke.com:8081/files_image/20230520/202305201048004167361.png" alt="这里写图片描述" title=""> <br>
图3 <br>
<img src="https://file2.kaopuke.com:8081/files_image/20230520/202305201048009933529.png" alt="这里写图片描述" title=""> <br>
图4 <br>
幅频特性图的对数表示： <br>
<img src="https://file2.kaopuke.com:8081/files_image/20230520/202305201048003939517.png" alt="这里写图片描述" title=""> <br>
图5</p>

<p><strong>-当ω&lt;ωc时，幅值是平行于坐标的直线，基本无衰减；</strong>　</p>

<p><strong>-当ω&gt;&gt;ωc时，是斜率与-20dB/十倍频成比例的一条直线；</strong>　</p>

<p><strong>-当ω=ωc时，增益衰减至0.707，即-3dB，相位滞后45度，对应低通<a href="https://www.baidu.com/s?wd=%E6%BB%A4%E6%B3%A2%E5%99%A8&amp;tn=24004469_oem_dg&amp;rsv_dl=gh_pl_sl_csd" target="_blank">滤波器</a>，该频率通常被称为截止频率。</strong></p>

<p><strong>缺点：</strong> <br>
采用这种模拟滤波器抑制低频干扰时，要求滤波器有较大的时间常数和<a href="https://www.baidu.com/s?wd=%E9%AB%98%E7%B2%BE%E5%BA%A6&amp;tn=24004469_oem_dg&amp;rsv_dl=gh_pl_sl_csd" target="_blank">高精度</a>的RC网络，增大时间常数要求增大R值，其漏电流也随之增大，从而降低了滤波效果；</p>



<h2 id="软件上的一阶低通滤波"><a name="t4"></a><strong>软件上的一阶低通滤波</strong></h2>



<h4 id="优点"><strong>优点：</strong></h4>

<p>-采用数字滤波算法来实现动态的RC滤波，则能很好的克服模拟滤波器的缺点； <br>
-在<strong>模拟常数要求较大的场合这种算法显得更为实用；</strong> <br>
-<strong>其对于周期干扰有良好的抑制作用，</strong> <br>
-<strong>比较节省RAM空间</strong></p>



<h4 id="缺点"><strong>缺点</strong></h4>

<p>-<strong>不足之处</strong>是带来了<strong>相位滞后</strong>，导致<strong>灵敏度低</strong>； <br>
-同时它<strong>不能滤除频率高于采样频率的二分之一（称为奈奎斯特频率）的干扰</strong>（例如采样频率为100Hz，则它不能滤除50Hz以上的干扰信号）对于高于奈奎斯特频率的干扰信号，应该采用模拟滤波器。 <br>
-对没有乘、除法运算指令的<a href="https://www.baidu.com/s?wd=%E5%8D%95%E7%89%87%E6%9C%BA&amp;tn=24004469_oem_dg&amp;rsv_dl=gh_pl_sl_csd" target="_blank">单片机</a>来说，程序运算工作量较大</p>



<h3 id="基本滤波算法"><a name="t5"></a><strong>基本滤波算法：</strong></h3>

<p><strong>算法由来：</strong> <br>
频率分析中一阶RC低通滤波在S域的传递函数： <br>
<span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-13-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mrow><mi>V</mi><mi>o</mi><mi>u</mi><mi>t</mi></mrow><mrow><mi>V</mi><mi>i</mi><mi>n</mi></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mn>1</mn><mrow><mi>R</mi><mi>C</mi><mi>s</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>,</mo><mo stretchy=&quot;false&quot;>(</mo><mi>s</mi><mo>=</mo><mi>j</mi><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x3C9;</mo></mrow><mo stretchy=&quot;false&quot;>)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-355" style="width: 14.86em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.889em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(0.746em 1011.77em 3.203em -999.997em); top: -2.283em; left: 0em;"><span class="mrow" id="MathJax-Span-356"><span class="mstyle" id="MathJax-Span-357"><span class="mrow" id="MathJax-Span-358"><span class="mfrac" id="MathJax-Span-359"><span style="display: inline-block; position: relative; width: 2.289em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.146em 1002.17em 4.174em -999.997em); top: -4.683em; left: 50%; margin-left: -1.083em;"><span class="mrow" id="MathJax-Span-360"><span class="mi" id="MathJax-Span-361" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.174em;"></span></span><span class="mi" id="MathJax-Span-362" style="font-family: MathJax_Math-italic;">o</span><span class="mi" id="MathJax-Span-363" style="font-family: MathJax_Math-italic;">u</span><span class="mi" id="MathJax-Span-364" style="font-family: MathJax_Math-italic;">t</span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.146em 1001.72em 4.174em -999.997em); top: -3.311em; left: 50%; margin-left: -0.854em;"><span class="mrow" id="MathJax-Span-365"><span class="mi" id="MathJax-Span-366" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.174em;"></span></span><span class="mi" id="MathJax-Span-367" style="font-family: MathJax_Math-italic;">i</span><span class="mi" id="MathJax-Span-368" style="font-family: MathJax_Math-italic;">n</span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1002.29em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.289em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-369" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mstyle" id="MathJax-Span-370" style="padding-left: 0.289em;"><span class="mrow" id="MathJax-Span-371"><span class="mfrac" id="MathJax-Span-372"><span style="display: inline-block; position: relative; width: 3.831em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.146em 1000.4em 4.174em -999.997em); top: -4.683em; left: 50%; margin-left: -0.226em;"><span class="mn" id="MathJax-Span-373" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.146em 1003.66em 4.231em -999.997em); top: -3.311em; left: 50%; margin-left: -1.883em;"><span class="mrow" id="MathJax-Span-374"><span class="mi" id="MathJax-Span-375" style="font-family: MathJax_Math-italic;">R</span><span class="mi" id="MathJax-Span-376" style="font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span class="mi" id="MathJax-Span-377" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-378" style="font-family: MathJax_Main; padding-left: 0.231em;">+</span><span class="mn" id="MathJax-Span-379" style="font-family: MathJax_Main; padding-left: 0.231em;">1</span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1003.83em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.831em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-380" style="font-family: MathJax_Main;">,</span><span class="mo" id="MathJax-Span-381" style="font-family: MathJax_Main; padding-left: 0.174em;">(</span><span class="mi" id="MathJax-Span-382" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-383" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mi" id="MathJax-Span-384" style="font-family: MathJax_Math-italic; padding-left: 0.289em;">j</span><span class="texatom" id="MathJax-Span-385"><span class="mrow" id="MathJax-Span-386"><span class="mo" id="MathJax-Span-387" style="font-family: MathJax_Math-italic;">ω</span></span></span><span class="mo" id="MathJax-Span-388" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.289em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.996em; border-left: 0px solid; width: 0px; height: 2.789em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>V</mi><mi>o</mi><mi>u</mi><mi>t</mi></mrow><mrow><mi>V</mi><mi>i</mi><mi>n</mi></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mi>R</mi><mi>C</mi><mi>s</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>,</mo><mo stretchy="false">(</mo><mi>s</mi><mo>=</mo><mi>j</mi><mrow class="MJX-TeXAtom-ORD"><mo>ω</mo></mrow><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-13">	dfrac{Vout}{Vin}= dfrac{1}{RCs+1} ,(s=jω)</script> <br>
通过z变换（方法很多，如一阶前向差分、双线性变换等这里用一阶后向差分法） <br>
<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-14-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;><mi>s</mi><mo>=</mo><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mrow><mn>1</mn><mo>&amp;#x2212;</mo><msup><mi>z</mi><mo>&amp;#x2227;</mo></msup><mo stretchy=&quot;false&quot;>(</mo><mo>&amp;#x2212;</mo><mn>1</mn><mo stretchy=&quot;false&quot;>)</mo></mrow><mi>T</mi></mfrac></mstyle><mo>,</mo><mi>T</mi><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x8868;</mo></mrow><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x793A;</mo></mrow><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x91C7;</mo></mrow><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x6837;</mo></mrow><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x5468;</mo></mrow><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>&amp;#x671F;</mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-389" style="width: 16.117em; display: inline-block;"><span style="display: inline-block; position: relative; width: 12.86em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(0.574em 1012.86em 3.146em -999.997em); top: -2.283em; left: 0em;"><span class="mrow" id="MathJax-Span-390"><span class="mi" id="MathJax-Span-391" style="font-family: MathJax_Math-italic;">s</span><span class="mo" id="MathJax-Span-392" style="font-family: MathJax_Main; padding-left: 0.289em;">=</span><span class="mstyle" id="MathJax-Span-393" style="padding-left: 0.289em;"><span class="mrow" id="MathJax-Span-394"><span class="mfrac" id="MathJax-Span-395"><span style="display: inline-block; position: relative; width: 4.974em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.031em 1004.75em 4.403em -999.997em); top: -4.74em; left: 50%; margin-left: -2.454em;"><span class="mrow" id="MathJax-Span-396"><span class="mn" id="MathJax-Span-397" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-398" style="font-family: MathJax_Main; padding-left: 0.231em;">−</span><span class="msubsup" id="MathJax-Span-399" style="padding-left: 0.231em;"><span style="display: inline-block; position: relative; width: 1.089em; height: 0px;"><span style="position: absolute; clip: rect(3.374em 1000.46em 4.174em -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-400" style="font-family: MathJax_Math-italic;">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; top: -4.34em; left: 0.517em;"><span class="mo" id="MathJax-Span-401" style="font-size: 70.7%; font-family: MathJax_Main;">∧</span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span></span></span><span class="mo" id="MathJax-Span-402" style="font-family: MathJax_Main;">(</span><span class="mo" id="MathJax-Span-403" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-404" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-405" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.146em 1000.69em 4.174em -999.997em); top: -3.311em; left: 50%; margin-left: -0.34em;"><span class="mi" id="MathJax-Span-406" style="font-family: MathJax_Math-italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.117em;"></span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(0.86em 1004.97em 1.26em -999.997em); top: -1.311em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.974em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.089em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-407" style="font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-408" style="font-family: MathJax_Math-italic; padding-left: 0.174em;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.117em;"></span></span><span class="texatom" id="MathJax-Span-409"><span class="mrow" id="MathJax-Span-410"><span class="mo" id="MathJax-Span-411"><span style="font-family: STIXGeneral, &quot;Arial Unicode MS&quot;, serif; font-size: 80%; font-style: normal; font-weight: normal;">表</span></span></span></span><span class="texatom" id="MathJax-Span-412"><span class="mrow" id="MathJax-Span-413"><span class="mo" id="MathJax-Span-414"><span style="font-family: STIXGeneral, &quot;Arial Unicode MS&quot;, serif; font-size: 80%; font-style: normal; font-weight: normal;">示</span></span></span></span><span class="texatom" id="MathJax-Span-415"><span class="mrow" id="MathJax-Span-416"><span class="mo" id="MathJax-Span-417"><span style="font-family: STIXGeneral, &quot;Arial Unicode MS&quot;, serif; font-size: 80%; font-style: normal; font-weight: normal;">采</span></span></span></span><span class="texatom" id="MathJax-Span-418"><span class="mrow" id="MathJax-Span-419"><span class="mo" id="MathJax-Span-420"><span style="font-family: STIXGeneral, &quot;Arial Unicode MS&quot;, serif; font-size: 80%; font-style: normal; font-weight: normal;">样</span></span></span></span><span class="texatom" id="MathJax-Span-421"><span class="mrow" id="MathJax-Span-422"><span class="mo" id="MathJax-Span-423"><span style="font-family: STIXGeneral, &quot;Arial Unicode MS&quot;, serif; font-size: 80%; font-style: normal; font-weight: normal;">周</span></span></span></span><span class="texatom" id="MathJax-Span-424"><span class="mrow" id="MathJax-Span-425"><span class="mo" id="MathJax-Span-426"><span style="font-family: STIXGeneral, &quot;Arial Unicode MS&quot;, serif; font-size: 80%; font-style: normal; font-weight: normal;">期</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.289em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.925em; border-left: 0px solid; width: 0px; height: 2.932em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>s</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>1</mn><mo>−</mo><msup><mi>z</mi><mo>∧</mo></msup><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>T</mi></mfrac></mstyle><mo>,</mo><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mo>表</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>示</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>采</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>样</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>周</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>期</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-14">	s= dfrac{1-z^∧(-1)}{T} ,T表示采样周期</script><p></p>

<p>带入S域传递函数中： <br>
<span class="MathJax_Preview"></span></p><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"></div><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-15-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mrow><mi>Y</mi><mo stretchy=&quot;false&quot;>(</mo><mi>z</mi><mo stretchy=&quot;false&quot;>)</mo></mrow><mrow><mi>X</mi><mo stretchy=&quot;false&quot;>(</mo><mi>z</mi><mo stretchy=&quot;false&quot;>)</mo></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mn>1</mn><mrow><mi>R</mi><mi>C</mi><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mrow><mn>1</mn><mo>&amp;#x2212;</mo><msup><mi>z</mi><mo>&amp;#x2227;</mo></msup><mo stretchy=&quot;false&quot;>(</mo><mo>&amp;#x2212;</mo><mn>1</mn><mo stretchy=&quot;false&quot;>)</mo></mrow><mi>T</mi></mfrac></mstyle><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle=&quot;true&quot; scriptlevel=&quot;0&quot;><mfrac><mi>T</mi><mrow><mi>R</mi><mi>C</mi><mo stretchy=&quot;false&quot;>(</mo><mn>1</mn><mo>&amp;#x2212;</mo><msup><mi>z</mi><mo>&amp;#x2227;</mo></msup><mo stretchy=&quot;false&quot;>(</mo><mo>&amp;#x2212;</mo><mn>1</mn><mo stretchy=&quot;false&quot;>)</mo><mo stretchy=&quot;false&quot;>)</mo><mo>+</mo><mi>T</mi></mrow></mfrac></mstyle></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-427" style="width: 29.26em; display: inline-block;"><span style="display: inline-block; position: relative; width: 23.374em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(0.631em 1023.37em 4.631em -999.997em); top: -2.283em; left: 0em;"><span class="mrow" id="MathJax-Span-428"><span class="mstyle" id="MathJax-Span-429"><span class="mrow" id="MathJax-Span-430"><span class="mfrac" id="MathJax-Span-431"><span style="display: inline-block; position: relative; width: 2.231em; height: 0px; margin-right: 0.117em; margin-left: 0.117em;"><span style="position: absolute; clip: rect(3.089em 1001.89em 4.403em -999.997em); top: -4.74em; left: 50%; margin-left: -1.026em;"><span class="mrow" id="MathJax-Span-432"><span class="mi" id="MathJax-Span-433" style="font-family: MathJax_Math-italic;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.174em;"></span></span><span class="mo" id="MathJax-Span-434" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-435" style="font-family: MathJax_Math-italic;">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-436" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 4.003em;"></span></span><span style="position: absolute; clip: rect(3.089em 1002em 4.403em -999.997em); top: -3.254em; left: 50%; margin-left: -1.026em;"><span class="mrow" id="MathJax-Span-437"><span class="mi" id="MathJax-Span-438" style="font-family: MathJax_Math-italic;">X<span style="display: inline-block; 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本文分类：RC滤波器
浏览次数：106 次浏览
发布日期：2023-05-22 10:10:01
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一阶RC低通滤波器（转）【滤波器学习笔记】一阶RC低通滤波

【滤波器学习笔记】一阶RC低通滤波

典型电路

时域

频域

软件上的一阶低通滤波

优点：

缺点

基本滤波算法：

一阶RC数字滤波的基本算法

-滤波系数越小，滤波结果越平稳，但是灵敏度越低；

-滤波系数越大，灵敏度越高，但是滤波结果越不稳定

-截止频率：

基本程序：

程序初步优化

最后

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一阶RC低通滤波器（转）【滤波器学习笔记】一阶RC低通滤波

【滤波器学习笔记】一阶RC低通滤波

典型电路

时域

频域

软件上的一阶低通滤波

优点：

缺点

基本滤波算法：

一阶RC数字滤波的基本算法

-滤波系数越小，滤波结果越平稳，但是灵敏度越低；

-滤波系数越大，灵敏度越高，但是滤波结果越不稳定

-截止频率：

基本程序：

程序初步优化

最后

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