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ããã§ãæå®æ°ã®ãã€ã³ãã«ã€ããŠãŸãšããŸã(åŸã§å³ãçšããŠåããããã説æããŸã)ã
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- æå®æ°ãšã¯ãéæž¡çŸè±¡ãã©ã®ãããç¶ãã®ãã衚ãç®å®ã衚ããŠãããåäœã¯[s]ãšãªããŸãã
- æå®æ°ã¯ãã®ãªã·ã£æåã®\({\tau}\)(ã¿ãŠ)ã§è¡šãããŸãã
- RLåè·¯ã®æå®æ°\({\tau}\)ã¯ãã€ã³ãã¯ã¿\(L\)ãæµæ\(R\)ã§å²ã£ãå€ãšãªãã\({\tau}=\displaystyle\frac{L}{R}\)ããšãªããŸãã
- æé\(t\)ããæå®æ°\({\tau}\left(=\displaystyle\frac{L}{R}\right)\)ããšãªã£ãæãRLåè·¯ã«æµãã黿µ\(i(t)\)ã¯ã\(0.632\displaystyle\frac{E}{R}\)ããšãªããŸãã
- æå®æ°\({\tau}\)ã倧ãããšéæž¡çŸè±¡ãé·ãç¶ããå°ãããšéæž¡çŸè±¡ãæ©ãçµãããŸã(æ©ãå®åžžç¶æ ã«ãªããŸã)ã
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ããå°ã詳ããïŒ
ç¹°ãè¿ãã«ãªããŸãããäžå³ã®RLåè·¯ã«ãããŠãã\(t=0{\mathrm{[s]}}\)ãã§ã¹ã€ãã\(SW\)ãONã«ãããšãRLåè·¯ã«æµãã黿µ\(i(t)\)ã\(0{\mathrm{[A]}}\)ããåŸã ã«äžæããããçšåºŠæéãçµéãããšãRLåè·¯ã«æµãã黿µ\(i(t)\)ã®å€åããªããªããäžå®å€\(\displaystyle\frac{E}{R}{\mathrm{[A]}}\)ãšãªããŸãã
ãã®æãRLåè·¯ã«æµãã黿µ\(i(t)\)ãå€åããäžå®å€\(\left(\displaystyle\frac{E}{R}{\mathrm{[A]}}\right)\)ãšãªã£ãç¶æ ããå®åžžç¶æ ãããå®åžžç¶æ ãã«è³ããŸã§ã®ç¶æ ããéæž¡ç¶æ ãããã®éçšã§èŠãããçŸç¶ããéæž¡çŸè±¡ããšãããŸãã
ãŸãããã®RLåè·¯ã«æµãã黿µ\(i(t)\)ãåŒã§è¡šããšæ¬¡åŒã§è¡šãããŸãã
\begin{eqnarray}
i(t)=\frac{E}{R}\left(1-e^{-\frac{R}{L}t}\right)\tag{1}
\end{eqnarray}
ãªãã(1)åŒã®å°åºã«ã€ããŠã¯ã以äžã®èšäºã§èª¬æããŠããŸããå°åºæ¹æ³ã«ã€ããŠç¥ãããæ¹ã¯ä»¥äžã®èšäºãåèã«ããŠãã ããã
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ãRLçŽååè·¯ã®åŸ®åæ¹çšåŒããéæž¡çŸè±¡ãã®è§£ãæ¹ïŒ
ç¶ããèŠã
(1)åŒãããRLåè·¯ã«æµãã黿µ\(i(t)\)ã¯æé\(t\)ããæå®æ°\({\tau}\left(=\displaystyle\frac{L}{R}\right)\)ããšãªã£ãæã次åŒãšãªããŸãã
\begin{eqnarray}
i({\tau})&=&\frac{E}{R}\left(1-e^{-\frac{R}{L}Ã\frac{L}{R}}\right)\\
&=&\frac{E}{R}\left(1-e^{-1}\right)\\
&=&\frac{E}{R}\left(1-\frac{1}{e}\right)\tag{2}
\end{eqnarray}
ããã§ã(2)åŒã«åºãŠãã\(e\)ã¯èªç¶å®æ°\(\log_{e}\)ã®åºã§ããããã€ãã¢æ°ãšåŒã°ãããã®ã§ãããã€ãã¢æ°\(e\)ã®å€ã¯æ¬¡åŒã§è¡šãããŸãã
\begin{eqnarray}
e=2.71828{\;}18284{\;}59045{\;}23536{\;}{\cdots}\tag{3}
\end{eqnarray}
ãã®ãã€ãã¢æ°\(e\)ã(2)åŒã«ä»£å ¥ãããšã次åŒãšãªããŸãã
\begin{eqnarray}
i({\tau})&=&\frac{E}{R}\left(1-\frac{1}{e}\right)\\
&{\approx}&\frac{E}{R}\left(1-\frac{1}{2.71828{\;}{\cdots}}\right)\\
&{\approx}&\frac{E}{R}\left(1-0.368\right)\\
&{\approx}&0.632\frac{E}{R}\tag{4}
\end{eqnarray}
ã€ãŸããRLåè·¯ã«æµãã黿µ\(i(t)\)ã¯æé\(t\)ããæå®æ°\({\tau}\left(=\displaystyle\frac{L}{R}\right)\)ããšãªã£ãæãå®åžžç¶æ ã«ããã黿µå€\(\displaystyle\frac{E}{R}\)ã®\(63.2{\%}\)ãšãªããŸãã
èšãæãããšãå®åžžç¶æ ã«ããã黿µå€\(\displaystyle\frac{E}{R}\)ã®\(63.2{\%}\)ã«éãããŸã§ã®æéãæå®æ°\({\tau}\)ãšããããšã«ãªããŸãã
ãŸãã(4)åŒãããæå®æ°\({\tau}\left(=\displaystyle\frac{L}{R}\right)\)ãã®å€§ããã«ãã£ãŠä»¥äžã®ããšãåãããŸãã
- ãæå®æ°\({\tau}\left(=\displaystyle\frac{L}{R}\right)\)ãã倧ããæ
- ãæå®æ°\({\tau}\left(=\displaystyle\frac{L}{R}\right)\)ããå°ããæ
RLåè·¯ã«æµãã黿µ\(i(t)\)ã\(\displaystyle0.632\frac{E}{R}\)ã«ãªãã®ã«æéãããããã€ãŸããéæž¡çŸè±¡ãé·ãç¶ãã
RLåè·¯ã«æµãã黿µ\(i(t)\)ãæ©ã\(\displaystyle0.632\frac{E}{R}\)ã«ãªããã€ãŸããéæž¡çŸè±¡ãæ©ãçµãã(æ©ãå®åžžç¶æ ãšãªã)ã
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- æå®æ°ã¯è±èªã§ã¯ãTime ConstantããšæžããŸãã
- æå®æ°ã¯äžè¬çã«ã¯ãããŠãããããšèªã¿ãŸããããããJISã§ã¯æå®æ°ã®æ¥æ¬èªã®èªã¿æ¹ã¯ããšãããããããã§ãããšå®ããããŠãŸãããŸãããTime Constantãã®éŠèš³èªãšããŠã¯ããšããŠãããããªã®ã§ããšããŠãããããšèªã人ãããŸãã
RLåè·¯ã®æå®æ°ã®æ±ãæ¹
RLåè·¯ã®æå®æ°\({\tau}\)ãã\({\tau}=\displaystyle\frac{L}{R}\)ããšãªãã®ã¯ãªãã§ããããïŒ
ããã§ã¯ãã®çç±ã説æããŸãã
å ã«çµè«ããèšããšã»ã»ã»
RLåè·¯ã«æµãã黿µ\(i(t)\)ã®ã\(t=0\)ãã«ãããæ¥ç·ãšå®åžžç¶æ ã«ããã黿µå€\(\displaystyle\frac{E}{R}\)ã®äº€ããæéããæå®æ°\({\tau}\left(=\displaystyle\frac{L}{R}\right)\)ããšãªãã®ã§ãã
ã§ã¯å®éã«å°åºããŠã¿ãŸãããã
ç¹°ãè¿ãã«ãªããŸãããäžå³ã®RLåè·¯ã«ãããŠãã\(t=0{\mathrm{[s]}}\)ãã§ã¹ã€ãã\(SW\)ãONã«ãããšãRLåè·¯ã«æµãã黿µ\(i(t)\)ã\(0{\mathrm{[A]}}\)ããåŸã ã«äžæããããçšåºŠæéãçµéãããšãRLåè·¯ã«æµãã黿µ\(i(t)\)ã®å€åããªããªããäžå®å€\(\displaystyle\frac{E}{R}{\mathrm{[A]}}\)ãšãªããŸãã
(1)åŒã\(t\)ã§åŸ®åããŠãã\(t=0\)ããä»£å ¥ãããšãRLåè·¯ã«æµãã黿µ\(i(t)\)ã®ã\(t=0\)ãã«ãããæ¥ç·ã®åŸããæ±ããããšãã§ããŸãã
(1)åŒã\(t\)ã§åŸ®åãããšæ¬¡åŒãšãªããŸãã
\begin{eqnarray}
\frac{di(t)}{dt}&=&\frac{1}{dt}\left[\frac{E}{R}\left(1-e^{-\frac{R}{L}t}\right)\right]\\
&=&\frac{1}{dt}\left(\frac{E}{R}-\frac{E}{R}e^{-\frac{R}{L}t}\right)\\
&=&\frac{E}{R}\frac{1}{dt}(1)-\frac{E}{R}\frac{1}{dt}\left(e^{-\frac{R}{L}t}\right)\\
&=&\frac{E}{R}t-\frac{E}{R}\left(-\frac{R}{L}\right)e^{-\frac{R}{L}t}\\
&=&\frac{E}{R}t+\frac{E}{L}e^{-\frac{R}{L}t}\tag{5}
\end{eqnarray}
(5)åŒã«ã\(t=0\)ããä»£å ¥ãããšã次åŒãšãªããŸãã
\begin{eqnarray}
\frac{di(0)}{dt}&=&\frac{E}{R}Ã0+\frac{E}{L}e^{-\frac{R}{L}Ã0}\\
&=&0+\frac{E}{L}e^{0}\\
&=&\frac{E}{L}Ã1\\
&=&\frac{E}{L}\tag{6}
\end{eqnarray}
ã€ãŸããRLåè·¯ã«æµãã黿µ\(i(t)\)ã®ã\(t=0\)ãã«ãããæ¥ç·ã®åŸãã¯ã\(\displaystyle\frac{E}{L}\)ããšãªããŸãã
RLåè·¯ã«æµãã黿µ\(i(t)\)ã®ã\(t=0\)ãã«ãããæ¥ç·ã¯ã\((0,0)\)ããéãã®ã§ã次åŒãšãªããŸãã
\begin{eqnarray}
I=\frac{E}{L}t\tag{7}
\end{eqnarray}
(7)åŒã®RLåè·¯ã«æµãã黿µ\(i(t)\)ã®ã\(t=0\)ãã«ãããæ¥ç·ãšå®åžžç¶æ ã«ããã黿µå€\(\displaystyle\frac{E}{R}\)ã亀ããæéãæ±ããŸãã
(7)åŒã®\(I\)ã«\(\displaystyle\frac{E}{R}\)ãä»£å ¥ãããšæ¬¡åŒãšãªããŸãã
\begin{eqnarray}
\frac{E}{R}&=&\frac{E}{L}t\\
{\Leftrightarrow}t&=&\frac{L}{R}\tag{8}
\end{eqnarray}
ãã®æéãæå®æ°ãšãªããããRLåè·¯ã®æå®æ°\({\tau}\)ã¯ã\({\tau}=\displaystyle\frac{L}{R}\)ããšãªããŸãã
(8)åŒããåããããã«ãRLåè·¯ã§ã¯ã€ã³ãã¯ã¿\(L\)ã®ã€ã³ãã¯ã¿ã³ã¹ã倧ãããªããšãæå®æ°\({\tau}\)ã倧ãããªããæµæ\(R\)ã®æµæå€ã倧ãããªããšãæå®æ°\({\tau}\)ãå°ãããªããšããããšãåãããŸãã(RLåè·¯ã®æå®æ°\({\tau}\)ã¯ã€ã³ãã¯ã¿\(L\)ã®ã€ã³ãã¯ã¿ã³ã¹ã«æ¯äŸããæµæ\(R\)ã®æµæå€ã«åæ¯äŸãããšããããšã§ã)ã
æå®æ°ã®åäœã[s]ã®çç±
æå®æ°\({\tau}\)ã®åäœã\({\mathrm{[s]}}\)ãšãªãã®ã¯ãªãã§ããããïŒ
äŸãã°ãRLåè·¯ã®å Žåãã€ã³ãã¯ã¿\(L\)ã®ã€ã³ãã¯ã¿ã³ã¹ã®åäœã¯\({\mathrm{[H]}}\)ãæµæ\(R\)ã®æµæå€ã®åäœã¯\({\mathrm{[Ω]}}\)ãªã®ã«ããªãæå®æ°\({\tau}=\displaystyle\frac{L}{R}\)ã®åäœã¯\({\mathrm{[s]}}\)ãšãªãã®ã§ããããïŒ
ããã§ã¯ãã®çç±ã説æããŸãã
ã€ã³ãã¯ã¿\(L\)ã®ã€ã³ãã¯ã¿ã³ã¹ã®åäœã𿵿\(R\)ã®æµæå€ã®åäœã«ã€ããŠå¥ã ã«è©³ããèŠãŠãããŸãã
ã€ã³ãã¯ã¿Lã®ã€ã³ãã¯ã¿ã³ã¹ã®åäœã«ã€ããŠ
ã€ã³ãã¯ã¿ã«ãããé»å§\(v{\mathrm{[V]}}\)ãšã€ã³ãã¯ã¿ã«æµãã黿µ\(i{\mathrm{[A]}}\)ãšã€ã³ãã¯ã¿ã®ã€ã³ãã¯ã¿ã³ã¹\(L{\mathrm{[H]}}\)ã®é¢ä¿ã¯æ¬¡åŒã§è¡šãããŸãã
\begin{eqnarray}
v&=&L\frac{di}{dt}\\
{\Leftrightarrow}L&=&vÃ\displaystyle\frac{dt}{di}\tag{9}
\end{eqnarray}
(9)åŒãåäœã§è¡šããšã次åŒãšãªããŸãã
\begin{eqnarray}
{\mathrm{[H]}}&=&\frac{{\mathrm{[V]}}{\mathrm{[s]}}}{{\mathrm{[A]}}}\tag{10}
\end{eqnarray}
æµæRã®æµæå€ã®åäœã«ã€ããŠ
ãªãŒã ã®æ³åãããæµæ\(R\)ã®æµæå€\(R{\mathrm{[Ω]}}\)ãšæµæ\(R\)ã«ãããé»å§\(v{\mathrm{[V]}}\)ãšæµæ\(R\)ã«æµãã黿µ\(i{\mathrm{[A]}}\)ã®é¢ä¿ã¯æ¬¡åŒã§è¡šãããŸãã
\begin{eqnarray}
R=\frac{v}{i}\tag{11}
\end{eqnarray}
(11)åŒãåäœã§è¡šããšã次åŒãšãªããŸãã
\begin{eqnarray}
{\mathrm{[Ω]}}=\frac{{\mathrm{[V]}}}{{\mathrm{[A]}}}\tag{12}
\end{eqnarray}
ã€ãŸããæå®æ°\({\tau}=\displaystyle\frac{L}{R}\)ãåäœã¯(10)åŒãš(12)åŒãçšãããšæ¬¡åŒãšãªããŸãã
\begin{eqnarray}
æå®æ°{\tau}ã®åäœ&=&\frac{{\mathrm{[H]}}}{{\mathrm{[Ω]}}}\\
&=&\frac{\displaystyle\frac{{\mathrm{[V]}}{\mathrm{[s]}}}{{\mathrm{[A]}}}}{\displaystyle\frac{{\mathrm{[V]}}}{{\mathrm{[A]}}}}\\
&=&{{\mathrm{[s]}}}\tag{13}
\end{eqnarray}
以äžãããã€ã³ãã¯ã¿\(L\)ã®ã€ã³ãã¯ã¿ã³ã¹ã®åäœã𿵿\(R\)ã®æµæå€ã®åäœãåè§£ãããšãæå®æ°\({\tau}\)ã®åäœã\({\mathrm{[s]}}\)ã«ãªãããšãåãããŸãã
RCåè·¯ã®æå®æ°ã2åã3åã»ã»ã»ãšããæã®å€
RLåè·¯ã«æµãã黿µ\(i(t)\)ã¯æé\(t\)ããæå®æ°\({\tau}(=\displaystyle\frac{L}{R})\)ããšãªã£ãæãå®åžžç¶æ ã«ããã黿µå€\(\displaystyle\frac{E}{R}\)ã®\(63.2{\%}\)ã«ãªããŸãã
ã§ã¯ãæé\(t\)ãæå®æ°\({\tau}\)ã2åã®æã3åã®æã¯RLåè·¯ã«æµãã黿µ\(i(t)\)ã¯ã©ããããã«ãªãã®ã§ããããã
æå®æ°Ïã1åã®æ
æå®æ°\({\tau}\)ã1åã®æ(ã€ãŸããã\(t={\tau}ïŒ\displaystyle\frac{L}{R}\)ãã®æ)ãRLåè·¯ã«æµãã黿µ\(i(t)\)ã¯ä»¥äžã®å€ãšãªããŸãã
\begin{eqnarray}
i({\tau})&=&\frac{E}{R}\left(1-e^{-\frac{R}{L}Ã\frac{L}{R}}\right)\\
&=&\frac{E}{R}\left(1-e^{-1}\right)\\
&=&\frac{E}{R}\left(1-\frac{1}{e}\right)\\
&{\approx}&\frac{E}{R}\left(1-\frac{1}{2.71828{\;}{\cdots}}\right)\\
&{\approx}&0.632\frac{E}{R}\tag{14}
\end{eqnarray}
ã€ãŸããRLåè·¯ã«æµãã黿µ\(i(t)\)ã¯æé\(t\)ãæå®æ°\({\tau}\)ã®1åã®æãå®åžžç¶æ ã«ããã黿µå€\(\displaystyle\frac{E}{R}\)ã®\(63.2{\%}\)ãšãªããŸãã
æå®æ°Ïã2åã®æ
æå®æ°\({\tau}\)ã2åã®æ(ã€ãŸããã\(t={\tau}ïŒ2\displaystyle\frac{L}{R}\)ãã®æ)ãRLåè·¯ã«æµãã黿µ\(i(t)\)ã¯ä»¥äžã®å€ãšãªããŸãã
\begin{eqnarray}
i(2{\tau})&=&\frac{E}{R}\left(1-e^{-\frac{R}{L}Ã2\frac{L}{R}}\right)\\
&=&\frac{E}{R}\left(1-e^{-2}\right)\\
&=&\frac{E}{R}\left(1-\frac{1}{e^2}\right)\\
&{\approx}&\frac{E}{R}\left(1-\frac{1}{2.71828{\;}{\cdots}^2}\right)\\
&{\approx}&0.865\frac{E}{R}\tag{15}
\end{eqnarray}
ã€ãŸããRLåè·¯ã«æµãã黿µ\(i(t)\)ã¯æé\(t\)ãæå®æ°\({\tau}\)ã®2åã®æãå®åžžç¶æ ã«ããã黿µå€\(\displaystyle\frac{E}{R}\)ã®\(86.5{\%}\)ãšãªããŸãã
æå®æ°Ïã3åã®æ
æå®æ°\({\tau}\)ã3åã®æ(ã€ãŸããã\(t={\tau}ïŒ3\displaystyle\frac{L}{R}\)ãã®æ)ãRLåè·¯ã«æµãã黿µ\(i(t)\)ã¯ä»¥äžã®å€ãšãªããŸãã
\begin{eqnarray}
i(3{\tau})&=&\frac{E}{R}\left(1-e^{-\frac{R}{L}Ã3\frac{L}{R}}\right)\\
&=&\frac{E}{R}\left(1-e^{-3}\right)\\
&=&\frac{E}{R}\left(1-\frac{1}{e^3}\right)\\
&{\approx}&\frac{E}{R}\left(1-\frac{1}{2.71828{\;}{\cdots}^3}\right)\\
&{\approx}&0.950\frac{E}{R}\tag{16}
\end{eqnarray}
ã€ãŸããRLåè·¯ã«æµãã黿µ\(i(t)\)ã¯æé\(t\)ãæå®æ°\({\tau}\)ã®3åã®æãå®åžžç¶æ ã«ããã黿µå€\(\displaystyle\frac{E}{R}\)ã®\(95.0{\%}\)ãšãªããŸãã
æå®æ°Ïã4åã®æ
æå®æ°\({\tau}\)ã4åã®æ(ã€ãŸããã\(t={\tau}ïŒ4\displaystyle\frac{L}{R}\)ãã®æ)ãRLåè·¯ã«æµãã黿µ\(i(t)\)ã¯ä»¥äžã®å€ãšãªããŸãã
\begin{eqnarray}
i(4{\tau})&=&\frac{E}{R}\left(1-e^{-\frac{R}{L}Ã4\frac{L}{R}}\right)\\
&=&\frac{E}{R}\left(1-e^{-4}\right)\\
&=&\frac{E}{R}\left(1-\frac{1}{e^4}\right)\\
&{\approx}&\frac{E}{R}\left(1-\frac{1}{2.71828{\;}{\cdots}^4}\right)\\
&{\approx}&0.982\frac{E}{R}\tag{17}
\end{eqnarray}
ã€ãŸããRLåè·¯ã«æµãã黿µ\(i(t)\)ã¯æé\(t\)ãæå®æ°\({\tau}\)ã®4åã®æãå®åžžç¶æ ã«ããã黿µå€\(\displaystyle\frac{E}{R}\)ã®\(98.2{\%}\)ãšãªããŸãã
æé\(t\)ãæå®æ°\({\tau}\)ã®4åãšãªããšãRLåè·¯ã«æµãã黿µ\(i(t)\)ã¯å®åžžç¶æ ã«ããã黿µå€\(\displaystyle\frac{E}{R}\)ã®çŽ\(98.2{\%}\)ãšãªãã®ã§ãã»ãŒå®åžžç¶æ ãšãããŸãã
æå®æ°Ïã5åã®æ
æå®æ°\({\tau}\)ã5åã®æ(ã€ãŸããã\(t={\tau}ïŒ5\displaystyle\frac{L}{R}\)ãã®æ)ãRLåè·¯ã«æµãã黿µ\(i(t)\)ã¯ä»¥äžã®å€ãšãªããŸãã
\begin{eqnarray}
i(5{\tau})&=&\frac{E}{R}\left(1-e^{-\frac{R}{L}Ã5\frac{L}{R}}\right)\\
&=&\frac{E}{R}\left(1-e^{-5}\right)\\
&=&\frac{E}{R}\left(1-\frac{1}{e^5}\right)\\
&{\approx}&\frac{E}{R}\left(1-\frac{1}{2.71828{\;}{\cdots}^5}\right)\\
&{\approx}&0.993\frac{E}{R}\tag{18}
\end{eqnarray}
ã€ãŸããRLåè·¯ã«æµãã黿µ\(i(t)\)ã¯æé\(t\)ãæå®æ°\({\tau}\)ã®5åã®æãå®åžžç¶æ ã«ããã黿µå€\(\displaystyle\frac{E}{R}\)ã®\(99.3{\%}\)ãšãªããŸãã
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