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- ãã®åŸãã¹ã€ãã\(SW\)ãã\(b\)ãã«åãæ¿ããããã®æã®æé\(t\)ã\(t=0{\mathrm{[s]}}\)ãšããã
âã€ãŸããRLæŸé»åè·¯ã«æµãã黿µ\(i(t)\)ãã\(i(t)=\displaystyle\frac{E}{R}{\mathrm{[A]}}\)ããšãªãã
ã¹ã€ãã\(SW\)ãã\(b\)ãã«åãæ¿ãããšã以äžã®éæž¡çŸè±¡ãçããŸãã
- 黿µ\(i(t)\)ã\(\displaystyle\frac{E}{R}{\mathrm{[A]}}\)ããæžå°ããã
- ããçšåºŠæéãçµéãããšã黿µ\(i(t)\)ãæµããªããªã(ã€ãŸããäžå®å€\(0{\mathrm{[A]}}\)ãšãªã)ããŸãããã®æãæµæ\(R\)ã®é»å§\(v_{R}(t)\)ãã€ã³ãã¯ã¿\(L\)ã®é»å§\(v_{L}(t)\)ã\(0{\mathrm{[V]}}\)ã«ãªãã
ãªãã黿µ\(i(t)\)ãäžå®å€\(0{\mathrm{[A]}}\)ãšãªã£ãç¶æ ããå®åžžç¶æ ãããå®åžžç¶æ ãã«è³ããŸã§ã®ç¶æ ããéæž¡ç¶æ ãããã®éçšã§èŠãããçŸç¶ããéæž¡çŸè±¡ããšãããŸãã
ãŸããRLæŸé»åè·¯ã«æµãã黿µ\(i(t)\)ãæµæ\(R\)ã®é»å§\(v_{R}(t)\)ãã€ã³ãã¯ã¿\(L\)ã®é»å§\(v_{L}(t)\)ã®åŒãšã°ã©ãã¯äžèšãšãªããŸãã
\begin{eqnarray}
i(t)&=&\frac{E}{R}e^{-\frac{R}{L}t}\\
v_{R}(t)&=&Ee^{-\frac{R}{L}t}\\
v_{L}(t)&=&-Ee^{-\frac{R}{L}t}
\end{eqnarray}
ãã®èšäºã§ã¯äžåŒãã©ãã©ã¹å€æãçšããŠè§£ããŠãããŸãããªããäžåŒã¯åŸ®åæ¹çšåŒãè§£ãæãåºæ¬çãªãã¿ãŒã³ã®å€æ°åé¢åœ¢ã®åŸ®åæ¹çšåŒã«ããŠãçŽæ¥è§£ãããšãå¯èœã§ãã
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- sé åã®æ¹çšåŒãè§£ã
- ã©ãã©ã¹é倿ãã
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ââ ã§æ±ããåè·¯æ¹çšåŒãã©ãã©ã¹å€æããŠãsé åã®æ¹çšåŒã«ããŸãããã®éãåææ¡ä»¶ãèæ ®ããå¿ èŠããããŸãã
âæ±ãããs颿°ã®åŒã«ããŸããä»åã¯ã\(I(s)={\cdots}\)ãã®åŒã«ããŸãã
ââ¢ã§æ±ããåŒãã©ãã©ã¹é倿ããŠãté åã®æ¹çšåŒã«ããŸãã
ã§ã¯ãããããåæé ã«ã€ããŠé çªã«èª¬æããŠãããŸãã
ãRLæŸé»åè·¯ãåè·¯æ¹çšåŒãããŠã
RLæŸé»åè·¯ãäžå³ã«ç€ºããŸãã
äžå³ã®RLæŸé»åè·¯ã«ãã«ããããã®é»å§å(ãã«ããããã®ç¬¬äºæ³å)ãçšãããšæ¬¡åŒãæãç«ã¡ãŸãã
\begin{eqnarray}
0=v_{R}(t)+v_{L}(t)\tag{1}
\end{eqnarray}
(1)åŒã«ãããŠãæµæ\(R\)ã®é»å§\(v_{R}(t)\)ãšã€ã³ãã¯ã¿\(L\)ã®é»å§\(v_{L}(t)\)ã¯æ¬¡åŒã§è¡šãããŸãã
\begin{eqnarray}
v_{R}(t)&=&Ri(t)\tag{2}\\
v_{L}(t)&=&L\frac{di(t)}{dt}\tag{3}
\end{eqnarray}
(2)åŒãš(3)åŒã(1)åŒã«ä»£å ¥ãããšã次åŒãåŸãããŸãã
\begin{eqnarray}
0&=&v_{R}(t)+v_{L}(t)\\
&=&Ri(t)+L\frac{di(t)}{dt}\tag{4}
\end{eqnarray}
äžåŒããRLæŸé»åè·¯ã®té åã®åè·¯æ¹çšåŒãšãªããŸãã
ãRLæŸé»åè·¯ãã©ãã©ã¹å€æããsé åã®æ¹çšåŒã«ãã
(4)åŒãã©ãã©ã¹å€æãããšã次åŒãšãªããŸãã
\begin{eqnarray}
0=RI(s)+L\left(sI(s)-i(0)\right)\tag{5}
\end{eqnarray}
(5)åŒã«ãããŠãã\(i(0)\)ãã¯ã\(t=0\)ãã®æã«ãããRLæŸé»åè·¯ã«æµãã黿µã§ããã¹ã€ãã\(SW\)ãã\(b\)ãã«åãæ¿ããåã¯ãRLæŸé»åè·¯ã«æµãã黿µ\(i(t)\)ãã\(\displaystyle\frac{E}{R}\)ãã§ãããããæ¬¡åŒãæãç«ã¡ãŸãã
\begin{eqnarray}
i(0)=\frac{E}{R}\tag{6}
\end{eqnarray}
(6)åŒã(5)åŒã«ä»£å ¥ãããšã次åŒãšãªããŸãã
\begin{eqnarray}
0&=&RI(s)+L\left(sI(s)-\frac{E}{R}\right)\\
&=&RI(s)+sLI(s)-\frac{E}{R}L\tag{7}
\end{eqnarray}
äžåŒããRLæŸé»åè·¯ã®sé åã®æ¹çšåŒãšãªããŸãã
ãRLæŸé»åè·¯ãsé åã®æ¹çšåŒãè§£ã
(7)åŒãã\(I(s)={\cdots}\)ãã®åŒã«å€æããŸãã
(7)åŒã\(I(s)\)ã§æŽçãããšã次åŒãšãªããŸãã
\begin{eqnarray}
\left(R+sL\right)I(s)=\frac{E}{R}L\tag{8}
\end{eqnarray}
(8)åŒã®äž¡èŸºã\(R+sL\)ã§å²ããšæ¬¡åŒãšãªããŸãã
\begin{eqnarray}
I(s)=\frac{\displaystyle\frac{E}{R}L}{R+sL}\tag{9}
\end{eqnarray}
(9)åŒã®å³èŸºã®ååãšåæ¯ã\(L\)ã§å²ããšæ¬¡åŒãšãªããŸãã
\begin{eqnarray}
I(s)=\frac{E}{R}\frac{1}{s+\displaystyle\frac{R}{L}}\tag{10}
\end{eqnarray}
ãã®ããã«ãå€åœ¢ããããšã§ãã\(I(s)={\cdots}\)ãã«ããããšãã§ããŸãããªãã(10)åŒã¯ãã®åŸã«èª¬æããã©ãã©ã¹é倿ãããããããã«å€åœ¢ããŠããŸãã
ãRLæŸé»åè·¯ãã©ãã©ã¹é倿ãã
(10)åŒãã©ãã©ã¹é倿ãããšã次åŒãšãªããŸãã
\begin{eqnarray}
i(t)&=&{\mathcal{L}}^{-1}\left[I(s)\right]\\
&=&{\mathcal{L}}^{-1}\left[\frac{E}{R}\frac{1}{s+\displaystyle\frac{R}{L}}\right]\\
&=&\frac{E}{R}{\mathcal{L}}^{-1}\left[\frac{1}{s+\displaystyle\frac{R}{L}}\right]\\
&=&\frac{E}{R}e^{-\frac{R}{L}t}\tag{11}
\end{eqnarray}
以äžãããRLæŸé»åè·¯ã«æµãã黿µ\(i(t)\)ã®åŒãå°åºããããšãã§ããŸããã
RLæŸé»åè·¯ã«æµãã黿µ\(i(t)\)ãåãããšãæµæ\(R\)ã®é»å§\(v_{R}(t)\)ãç°¡åã«æ±ããããšãã§ããŸãã
(11)åŒã(2)åŒã«ä»£å ¥ãããšãæµæ\(R\)ã®é»å§\(v_{R}(t)\)ã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
v_{R}(t)&=&Ri(t)\\
&=&Ee^{-\frac{R}{L}t}\tag{12}
\end{eqnarray}
ãŸãã(11)åŒã(3)åŒã«ä»£å ¥ãããšãã€ã³ãã¯ã¿\(L\)ã®é»å§\(v_{L}(t)\)ã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
v_{L}(t)&=&L\frac{di(t)}{dt}\\
&=&L\frac{d}{dt}\left[\frac{E}{R}e^{-\frac{R}{L}t}\right]\\
&=&L\frac{E}{R}\frac{d}{dt}\left(e^{-\frac{R}{L}t}\right)\\
&=&L\frac{E}{R}\left(-\frac{R}{L}\right)e^{-\frac{R}{L}t}\\
&=&-Ee^{-\frac{R}{L}t}\tag{13}
\end{eqnarray}
ãã®èšäºã§ã¯ãã€ã³ãã¯ã¿\(L\)ã®é»å§\(v_{L}(t)\)ã¯RLæŸé»åè·¯ã«æµãã黿µ\(i(t)\)ã埮åããããšã§æ±ããŸãããããã«ããããã®é»å§åã§ãæ±ããããšãã§ããŸãã(1)åŒãš(12)åŒãçšãããš(13)åŒãšåãçµæã«ãªããŸããã
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