RLC电路的谐振条件

nyy

1. Clear["Global`*"];
   
2. Z1=R1+X1*I (*电感与电阻串联*)
   
3. Z2=R2+X2*I (*电容与电阻串联*)
   
4. Z=Z1*Z2/(Z1+Z2)(*分别串联之后,再求并联阻抗*)
   
5. aa=FullSimplify@ComplexExpand@ReIm[Z](*求出阻抗的实部与虚部*)
   
6. bb=FullSimplify@ComplexExpand@ReIm[1/Z](*求出导纳的实部与虚部*)
   

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计算结果
阻抗的虚部与实部
\[\left\{\frac{\text{R2} \left(\text{R1} (\text{R1}+\text{R2})+\text{X1}^2\right)+\text{R1} \text{X2}^2}{(\text{R1}+\text{R2})^2+(\text{X1}+\text{X2})^2},\frac{\text{X2} \left(\text{R1}^2+\text{X1} (\text{X1}+\text{X2})\right)+\text{R2}^2 \text{X1}}{(\text{R1}+\text{R2})^2+(\text{X1}+\text{X2})^2}\right\}\]

导纳的虚部与实部
\[\left\{\frac{\text{R1}}{\text{R1}^2+\text{X1}^2}+\frac{\text{R2}}{\text{R2}^2+\text{X2}^2},-\frac{\text{X1}}{\text{R1}^2+\text{X1}^2}-\frac{\text{X2}}{\text{R2}^2+\text{X2}^2}\right\}\]

1. Clear["Global`*"];
   
2. Z1=R1*(X1*I)/(R1+X1*I)(*电感与电阻并联*)
   
3. Z2=R2*(X2*I)/(R2+X2*I)(*电容与电阻并联*)
   
4. Z=Z1+Z2(*两者分别并联之后,再串联后的阻抗*)
   
5. aa=FullSimplify@ComplexExpand@ReIm[Z](*求出阻抗的实部与虚部*)
   

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计算结果，实数部分与虚数部分
\[\left\{\frac{\text{R1} \text{X1}^2}{\text{R1}^2+\text{X1}^2}+\frac{\text{R2} \text{X2}^2}{\text{R2}^2+\text{X2}^2},\frac{\text{R1}^2 \text{X1}}{\text{R1}^2+\text{X1}^2}+\frac{\text{R2}^2 \text{X2}}{\text{R2}^2+\text{X2}^2}\right\}\]

nyy

第一个谐振的条件是
R2^2 X1 + R1^2 X2 + X1 X2 (X1 + X2)=0

第二个谐振的条件是
X1 X2 (R2^2 X1 + R1^2 X2) + R1^2 R2^2 (X1 + X2)=0

nyy

1. Clear["Global`*"];
   
2. Z1=R1+X1*I (*电感与电阻串联*)
   
3. Z2=R2+X2*I (*电容与电阻串联*)
   
4. Z=Z1*Z2/(Z1+Z2)(*分别串联之后,再求并联阻抗*)
   
5. aa=FullSimplify@ComplexExpand@ReIm[Z](*求出阻抗的实部与虚部*)
   
6. bb=FullSimplify@ComplexExpand@ReIm[1/Z](*求出导纳的实部与虚部*)
   
7. ans=Solve[(bb[[2]]/.{X1->w*L,X2->-1/(w*C)})==0,{w}]//FullSimplify(*求出谐振频率*)
   

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与电阻串联的
并联谐振的频率。
\[\left\{w\to \frac{\sqrt{L-C \text{R1}^2}}{\sqrt{C L \left(L-C \text{R2}^2\right)}}\right\}\]
由于电阻可以无穷大，所以改写一下。
\[\left\{w\to \frac{\sqrt{C \text{R1}^2-L}}{\sqrt{C L \left(C \text{R2}^2-L\right)}}\right\}\]

1. Clear["Global`*"];
   
2. Z1=R1*(X1*I)/(R1+X1*I)(*电感与电阻并联*)
   
3. Z2=R2*(X2*I)/(R2+X2*I)(*电容与电阻并联*)
   
4. Z=Z1+Z2(*两者分别并联之后,再串联后的阻抗*)
   
5. aa=FullSimplify@ComplexExpand@ReIm[Z](*求出阻抗的实部与虚部*)
   
6. ans=Solve[(aa[[2]]/.{X1->w*L,X2->-1/(w*C)})==0,{w}]//FullSimplify(*求出谐振频率*)
   

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与电阻并联，再串联谐振的频率
\[\left\{w\to \frac{\text{R1}}{\text{R2} \sqrt{\frac{C L \left(L-C \text{R1}^2\right)}{L-C \text{R2}^2}}}\right\}\]

改写一下
\[\left\{w\to \frac{\text{R1}}{\text{R2} \sqrt{\frac{C L \left(C \text{R1}^2-L\right)}{C \text{R2}^2-L}}}\right\}\]