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运算法求解动态电路
source link: https://z-rui.github.io/post/2016/04/dynamic-circuit/
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运算法求解动态电路
Fri Apr 8, 2016
如图所示的电路，在t=0时刻，有iLr(0)=I0, uCr(0)=0。试求t>0时的iLr(t)和uCr(t)。
这电路含有动态元件，可以使用运算法来求解。
如图，作出运算电路，在电感和电容之间的节点处，由基尔霍夫电流定律得 Ui/s−uCr(s)sLr+iLr(0)s−uCr(s)1/(sCr)−I0s=0.
解得uCr(s)=Uis11+LrCrs2=Uiωr2s(s2+ωr2), 其中ωr=1LrCr。
电感电流是运算电路中的电感电流加上附带的电流源的电流，即 iLr(s)=Ui/s−uCr(s)sLr+iLr(0)s=UiCr1+LrCrs2+iLr(0)s=UiZωrs2+ωr2+iLr(0)s, 其中Z=LrCr。
根据拉普拉斯变换表， L{1}=1s,L{sinωt}=ωs2+ω2,L{1−cosωt}=ω2s(s2+ω2).
可知时域的解为 iLr(t)=UiZsinωrt+I0,uCr(t)=Ui(1−cosωrt).
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