RLC 的时域、s 域模型

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已于 2023-12-05 10:49:43 修改

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分类专栏：信号处理文章标签：算法

于 2022-11-15 21:44:05 首次发布

本文链接：https://blog.youkuaiyun.com/xzy3150787/article/details/127874585

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这篇博客详细介绍了RLC电路在时域和复频域的数学模型。时域中，电压u(t)与电流i(t)的关系通过欧姆定律和微分方程表示；复频域里，利用拉普拉斯变换将时域关系转换为U(s)和I(s)，并涉及初始条件。内容深入浅出地探讨了RLC元件的动态行为。

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                    RLC 的时域、s 域模型

元件时域复频域
Ru(t)=Ri(t)u(t)=Ri(t)u(t)=Ri(t)U=RI(s)U=RI(s)U=RI(s)
Lu(t)=Lddti(t)i(t)=1L∫−∞tu(τ)dτ\begin{aligned}u(t)&=L\frac{d}{dt}i(t)\\ i(t)&=\frac1L\int_{-\infty}^tu(\tau)d\tau\end{aligned}u(t)i(t)​=Ldtd​i(t)=L1​∫−∞t​u(τ)dτ​U(s)=sLI(s)−Li(0−)I(s)=U(s)sL+i(0−)s\begin{aligned} U(s)&=sLI(s)-Li(0^-) \\ I(s)&=\frac{U(s)}{sL}+\frac{i(0^-)}{s} \end{aligned}U(s)I(s)​=sLI(s)−Li(0−)=sLU(s)​+si(0−)​​
Cu(t)=1C∫−∞ti(τ)dτi(t)=Cddtu(t)\begin{aligned} u(t)&=\frac1C\int_{-\infty}^ti(\tau)d\tau \\ i(t)&=C\frac d{dt}u(t) \end{aligned} u(t)i(t)​=C1​∫−∞t​i(τ)dτ=Cdtd​u(t)​I(s)=sCU(s)−Cu(0−)U(s)=I(s)sC+u(0−)s\begin{aligned} I(s)&=sCU(s)-Cu(0^-)\\ U(s)&=\frac{I(s)}{sC}+\frac{u(0^-)}s \end{aligned}I(s)U(s)​=sCU(s)−Cu(0−)=sCI(s)​+su(0−)​​

元件	时域	复频域
R	$u (t) = R i (t)$	$U = R I (s)$
L	$u(t)=Lddti(t)i(t)=1L∫−∞tu(τ)dτ\begin{aligned}u(t)&=L\frac{d}{dt}i(t)\\ i(t)&=\frac1L\int_{-\infty}^tu(\tau)d\tau\end{aligned}$	$U(s)=sLI(s)−Li(0−)I(s)=U(s)sL+i(0−)s\begin{aligned} U(s)&=sLI(s)-Li(0^-) \\ I(s)&=\frac{U(s)}{sL}+\frac{i(0^-)}{s} \end{aligned}$
C	$u(t)=1C∫−∞ti(τ)dτi(t)=Cddtu(t)\begin{aligned} u(t)&=\frac1C\int_{-\infty}^ti(\tau)d\tau \\ i(t)&=C\frac d{dt}u(t) \end{aligned}$	$I(s)=sCU(s)−Cu(0−)U(s)=I(s)sC+u(0−)s\begin{aligned} I(s)&=sCU(s)-Cu(0^-)\\ U(s)&=\frac{I(s)}{sC}+\frac{u(0^-)}s \end{aligned}$