ãã®èšäºã§ã¯RCçŽååè·¯ã®ãåŸ®åæ¹çšåŒã«ããéæž¡çŸè±¡ã®è§£ãæ¹ãã«ã€ããŠèª¬æããŠããŸãã
åããããã説æããããã«ãå³ãå€ãçšããŠãããåŒã®å°åºéçšã现ããæžãããã«æèããŠããŸãã
ãRCçŽååè·¯ããéæž¡çŸè±¡ãã®åŒãšã°ã©ã
äžå³ã¯æµæ\(R{\mathrm{[Ω]}}\)ãã³ã³ãã³ãµ\(C{\mathrm{[F]}}\)ãçŽæµé»æº\(E{\mathrm{[V]}}\)ãã¹ã€ãã\(SW\)ãããªãRCçŽååè·¯ã§ãã
ãã®èšäºã§ã¯ã以äžã®æ¡ä»¶ã«ããããéæž¡çŸè±¡ãã®åŒãå°åºããŸãã
æ¡ä»¶
- ã¹ã€ãã\(SW\)ããªã³ããæã®æé\(t\)ã\(t=0{\mathrm{[s]}}\)ãšããã
- ã¹ã€ãã\(SW\)ããªã³ããåã«ã¯ãã³ã³ãã³ãµ\(C\)ã«èããããŠããé»è·\(q(t)\)ã¯ãŒããšããã
âã€ãŸããã\(q(0)=0\)ããšããããšã
ã¹ã€ãã\(SW\)ããªã³ãããšã以äžã®éæž¡çŸè±¡ãçããŸãã
- 黿µ\(i(t)\)ãæµããŠãã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ãå¢ããã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ãäžæããã
- ããçšåºŠæéãçµéãããšã黿µ\(i(t)\)ãæµããªããªã(ã€ãŸããäžå®å€\(0{\mathrm{[A]}}\)ã«ãªã)ããŸãããã®æãã³ã³ãã³ãµ\(C\)ãéæŸããããããªç¶æ ã§ãããã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ã黿ºé»å§ã®é»å§\(E\)ãšçãããªãã
ãã®æã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
ãŸããRCçŽååè·¯ã«æµãã黿µ\(i(t)\)ãã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ãæµæ\(R\)ã®é»å§\(v_{R}(t)\)ãã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ã®åŒãšã°ã©ãã¯äžèšãšãªããŸãã
\begin{eqnarray}
i(t)&=&\frac{E}{R}e^{-\frac{1}{CR}t}\\
q(t)&=&CE\left(1-e^{-\frac{1}{CR}t}\right)\\
v_{R}(t)&=&Ee^{-\frac{1}{CR}t}\\
v_{C}(t)&=&E\left(1-e^{-\frac{1}{CR}t}\right)
\end{eqnarray}
ãã®èšäºã§ã¯äžåŒãåŸ®åæ¹çšåŒãè§£ãæãåºæ¬çã®å€æ°åé¢åœ¢ã®åŸ®åæ¹çšåŒã§è§£ããŠãããŸãããªããäžåŒã¯ã©ãã©ã¹å€æã§ãè§£ãããšãã§ããŸãã
ã©ãã©ã¹å€æã§è§£ãæ¹æ³ã«ã€ããŠã¯ä»¥äžã®èšäºã«è©³ãã説æããŠããŸãã®ã§ãåèã«ããŠãã ããã
-
ãRCçŽååè·¯ã®ã©ãã©ã¹å€æããéæž¡çŸè±¡ãã®è§£ãæ¹ïŒ
ç¶ããèŠã
ãRCçŽååè·¯ããåŸ®åæ¹çšåŒãã®è§£ãæ¹
ãRCçŽååè·¯ã黿µi(t)ã®æ±ãæ¹
RCçŽååè·¯ã«æµãã黿µ\(i(t)\)ãšã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã®é¢ä¿ã¯æ¬¡åŒã§è¡šãããŸãã
\begin{eqnarray}
i(t)=\frac{dq(t)}{dt}\tag{1}
\end{eqnarray}
ãŸããRCçŽååè·¯ã«ãã«ããããã®é»å§å(ãã«ããããã®ç¬¬äºæ³å)ãçšãããšæ¬¡åŒãæãç«ã¡ãŸãã
\begin{eqnarray}
E=v_{R}(t)+v_{C}(t)\tag{2}
\end{eqnarray}
(2)åŒã«ãããŠãæµæ\(R\)ã®é»å§\(v_{R}(t)\)ãšã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ã¯æ¬¡åŒã§è¡šãããŸãã
\begin{eqnarray}
v_{R}(t)&=&Ri(t)=R\frac{dq(t)}{dt}\tag{3}\\
v_{C}(t)&=&\frac{1}{C}{\displaystyle\int}i(t)dt=\frac{1}{C}{\displaystyle\int}\left(\frac{dq(t)}{dt}\right)dt=\frac{q(t)}{C}\tag{4}
\end{eqnarray}
(3)åŒãš(4)åŒã(2)åŒã«ä»£å ¥ãããšã次åŒãåŸãããŸãã
\begin{eqnarray}
E&=&v_{R}(t)+v_{C}(t)\\
&=&R\frac{dq(t)}{dt}+\frac{q(t)}{C}\tag{5}
\end{eqnarray}
(5)åŒã¯ã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã«é¢ãããåŸ®åæ¹çšåŒãã§ãã
ãã®ãåŸ®åæ¹çšåŒããè§£ããšãã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ãå°åºããããšãã§ããæ¬¡åŒã®ææ°é¢æ°ãšãªããŸã(次åŒã®å°åºæ¹æ³ã«ã€ããŠã¯ãå°åºéçšãããªãé·ããªãããããã®èšäºã®åŸåã«è©³ãã説æããŠããŸã)ã
\begin{eqnarray}
q(t)=CE\left(1-e^{-\frac{1}{CR}t}\right)\tag{6}
\end{eqnarray}
ããã§ã(6)åŒã(1)åŒã«ä»£å ¥ãããšãRCçŽååè·¯ã«æµãã黿µ\(i(t)\)ãæ±ããããšãã§ããæ¬¡åŒã®ææ°é¢æ°ãšãªããŸã(次åŒã®å°åºæ¹æ³ã«ã€ããŠã¯ãå°åºéçšãããªãé·ããªãããããã®èšäºã®åŸåã«è©³ãã説æããŠããŸã)ã
\begin{eqnarray}
i(t)&=&\frac{E}{R}e^{-\frac{1}{CR}t}\tag{7}
\end{eqnarray}
ãRCçŽååè·¯ã黿µi(t)ã®ã°ã©ã
RCçŽååè·¯ã«æµãã黿µ\(i(t)\)ã®ã°ã©ãã¯äžå³ã®ããã«ãªããŸãããã®ã°ã©ãã«ã€ããŠèª¬æããŸãã
ç¹°ãè¿ãã«ãªããŸãããRCçŽååè·¯ã«æµãã黿µ\(i(t)\)ã®åŒã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
i(t)&=&\frac{E}{R}e^{-\frac{1}{CR}t}\tag{7}
\end{eqnarray}
(7)åŒã®\(t\)ã«ã\(t=0\)ããšã\(t=â\)ããä»£å ¥ããæãèããŠã¿ãŸãããã
ãt=0ããä»£å ¥ããæ
ã\(t=0\)ããä»£å ¥ãããšã(7)åŒã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
i(0)&=&\frac{E}{R}e^{-\frac{1}{CR}Ã0}\\
&=&\frac{E}{R}e^{0}\\
&=&\frac{E}{R}Ã1\\
&=&\frac{E}{R}\tag{8}
\end{eqnarray}
ã€ãŸãã\(t=0\)ãã®æãRCçŽååè·¯ã«æµãã黿µ\(i(t)\)ã¯ã\(i(0)=\displaystyle\frac{E}{R}\)ããšãªããŸãã
ããã¯ãã¹ã€ãã\(SW\)ããªã³ããç¬éãã³ã³ãã³ãµ\(C\)ã¯ç絡ããããããªç¶æ ã§ãããæµæ\(R\)ã«ãã£ãŠRCçŽååè·¯ã«æµãã黿µãå¶éãããŠãããšããããšã衚ããŠããŸãã
ãt=âããä»£å ¥ããæ(å®åžžç¶æ ã®æ)
ã\(t=â\)ããä»£å ¥ãããšã(7)åŒã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
i(â)&=&\frac{E}{R}e^{-\frac{1}{CR}Ãâ}\\
&=&\frac{E}{R}e^{-â}\\
&=&\frac{E}{R}\frac{1}{e^{â}}\\
&=&\frac{E}{R}\frac{1}{â}\\
&=&\frac{E}{R}Ã0\\
&=&0\tag{9}
\end{eqnarray}
ã€ãŸããã\(t=â\)ãã®æãRCçŽååè·¯ã«æµãã黿µ\(i(t)\)ã¯ã\(i(â)=0\)ããšãªããŸãã
ããã¯ãã¹ã€ãã\(SW\)ããªã³ãããã°ããæéãçµéãããšãã³ã³ãã³ãµ\(C\)ã«é»è·\(q(t)\)ãèãããããã®ã³ã³ãã³ãµ\(C\)ã®é»è·\(q(t)\)ãæºãŸããããã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ãçŽæµé»æºã®é»å§\(E\)ãšçãããªããšãRCçŽååè·¯ã«æµãã黿µããŒãã«ãªãããšã衚ããŠããŸãã
ãããã£ãŠãRCçŽååè·¯ã«æµãã黿µ\(i(t)\)ã®ã°ã©ãã¯ã\(i(0)=\displaystyle\frac{E}{R}\)ãããã\(i(â)=0\)ãã«ãªãããã«æžå°ããŠããã®ã§ããããã®æžå°å ·åã¯ãã³ã³ãã³ãµ\(C\)ãšæµæ\(R\)ã®ç©\(CR\)ãã«ãã£ãŠå€ãããŸãã
ãã®ã\(CR\)ã¯äžè¬çã«æå®æ°Ï(ã¿ãŠ)ãšåŒã°ããŠããŸãã
æå®æ°Ï(ã¿ãŠ)ã¯éæž¡çŸè±¡ãã©ã®ãããç¶ãã®ãã衚ãç®å®ã衚ããŠãããåäœã¯[s]ãšãªããŸãã
ä»åã®RCçŽååè·¯ã«æµãã黿µ\(i(t)\)ã®ã°ã©ãã®å Žåãã\({\tau}=CR\)ãã®å€§ããã«ãã£ãŠä»¥äžã®ããã«å€ãããŸãã
- ã\({\tau}=CR\)ãã倧ããæ
- ã\({\tau}=CR\)ããå°ããæ
éæž¡çŸè±¡ãé·ãç¶ããŸãã
éæž¡çŸè±¡ãæ©ãçµãããŸããããªãã¡ãæ©ãå®åžžç¶æ ãšãªããŸãã
ãŸããæétãæå®æ°Ï(ã¿ãŠ)ãšçãããªãæãRCçŽååè·¯ã«æµãã黿µ\(i(t)\)ã¯ä»¥äžã®å€ãšãªããŸãã
- ã\(t={\tau}=CR\)ãã®æ
- ã\(t=4{\tau}=4CR\)ãã®æ
ã\(\displaystyle\frac{E}{R}\)ãã®çŽ37ïŒ
ã\(\displaystyle\frac{E}{R}\)ãã®çŽ2ïŒ
ãRCçŽååè·¯ãã³ã³ãã³ãµCã«èããããé»è·q(t)ã®ã°ã©ã
ã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã®ã°ã©ãã¯äžå³ã®ããã«ãªããŸãããã®ã°ã©ãã«ã€ããŠèª¬æããŸãã
ç¹°ãè¿ãã«ãªããŸãããã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã®åŒã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
q(t)=CE\left(1-e^{-\frac{1}{CR}t}\right)\tag{6}
\end{eqnarray}
(6)åŒã®\(t\)ã«ã\(t=0\)ããšã\(t=â\)ããä»£å ¥ããæãèããŠã¿ãŸãããã
ãt=0ããä»£å ¥ããæ
ã\(t=0\)ããä»£å ¥ãããšã(6)åŒã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
q(0)&=&CE\left(1-e^{-\frac{1}{CR}Ã0}\right)\\
&=&CE\left(1-e^{0}\right)\\
&=&CE\left(1-1\right)\\
&=&0\tag{10}
\end{eqnarray}
ã€ãŸãã\(t=0\)ãã®æãã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã¯ã\(q(0)=0\)ããšãªããŸãã
ããã¯ãã¹ã€ãã\(SW\)ããªã³ããç¬éãã³ã³ãã³ãµ\(C\)ã«èããããŠããé»è·ã¯ãŒãã§ãããšããããšã衚ããŠããŸãã
ãt=âããä»£å ¥ããæ(å®åžžç¶æ ã®æ)
ã\(t=â\)ããä»£å ¥ãããšã(6)åŒã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
q(â)&=&CE\left(1-e^{-\frac{1}{CR}Ãâ}\right)\\
&=&CE\left(1-e^{-â}\right)\\
&=&CE\left(1-\frac{1}{e^{â}}\right)\\
&=&CE\left(1-0\right)\\
&=&CE\tag{11}
\end{eqnarray}
ã€ãŸããã\(t=â\)ãã®æãã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã¯ã\(q(â)=CE\)ããšãªããŸãã
ããã¯ãã¹ã€ãã\(SW\)ããªã³ãããã°ããæéãçµéãããšãã³ã³ãã³ãµ\(C\)ã«é»è·\(q(t)\)ãèãããããã®ã³ã³ãã³ãµ\(C\)ã®é»è·\(q(t)\)ãæºãŸããããã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ãçŽæµé»æºã®é»å§\(E\)ãšçãããªããšãã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã\(CE\)ã«ãªãããšã衚ããŠããŸãã
ãããã£ãŠãã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã®ã°ã©ãã¯ã\(q(0)=0\)ãããã\(q(â)=CE\)ãã«ãªãããã«å¢å ããŠããã®ã§ããããã®å¢å å ·åã¯æå®æ°Ï(ã¿ãŠ)ã«ãã£ãŠå€ãããŸãã
ãŸããæétãæå®æ°Ï(ã¿ãŠ)ãšçãããªãæãã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã¯ä»¥äžã®å€ãšãªããŸãã
- ã\(t={\tau}=CR\)ãã®æ
- ã\(t=4{\tau}=4CR\)ãã®æ
ã\(CE\)ãã®çŽ63ïŒ
ã\(CE\)ãã®çŽ98ïŒ
ãŸããå®åžžç¶æ ã«ãããŠãã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ãå€åããªããšããããšã¯ã(7)åŒãããRCçŽååè·¯ã«æµãã黿µ\(i(t)\)ããŒããšããããšã«ãªããŸããåã ã®ã°ã©ããèŠããšãã®ããã«ãªã£ãŠããããšã確èªã§ãããšæããŸãã
ãRCçŽååè·¯ãæµæRã®é»å§VR(t)ã®æ±ãæ¹
RCçŽååè·¯ã«æµãã黿µ\(i(t)\)ãåãããšãæµæ\(R\)ã®é»å§\(v_{R}(t)\)ãç°¡åã«æ±ããããšãã§ããŸãã
(7)åŒã(3)åŒã«ä»£å ¥ãããšãæµæ\(R\)ã®é»å§\(v_{R}(t)\)ã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
v_{R}(t)&=&Ri(t)\\
&=&E{\;}e^{-\frac{1}{CR}t}\tag{12}
\end{eqnarray}
ãRCçŽååè·¯ãæµæRã®é»å§VR(t)ã®ã°ã©ã
æµæ\(R\)ã®é»å§\(v_{R}(t)\)ã®ã°ã©ãã¯äžå³ã®ããã«ãªããŸãããã®ã°ã©ãã«ã€ããŠèª¬æããŸãã
ãã®ã°ã©ãã§ãããæµæ\(R\)ã®é»å§\(v_{R}(t)\)ã¯RCçŽååè·¯ã«æµãã黿µ\(i(t)\)ã«\(R\)ãæããã ãã§ãã
ããªãã¡ãRCçŽååè·¯ã«æµãã黿µ\(i(t)\)ã®ã°ã©ããšåããããªã°ã©ããšãªããŸãã
ã¹ã€ãã\(SW\)ããªã³ããç¬éãã³ã³ãã³ãµ\(C\)ã¯ç絡ããããããªç¶æ ã§ãããããæµæ\(R\)ã«çŽæµé»æºã®é»å§\(E\)ãããããŸãã
ã¹ã€ãã\(SW\)ããªã³ãããã°ããæéãçµéãããš(å®åžžç¶æ ã®æ)ãRCçŽååè·¯ã«æµãã黿µ\(i(t)\)ããŒããªã®ã§ãæµæ\(R\)ã®é»å§\(v_{R}(t)\)ããŒããšãªããŸãã
ãRCçŽååè·¯ãã³ã³ãã³ãµCã®é»å§VC(t)ã®æ±ãæ¹
ã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ã¯ãã«ããããã®é»å§å(ãã«ããããã®ç¬¬äºæ³å)ãçšãããšç°¡åã«æ±ããããšãã§ããŸãã
(2)åŒãå€åœ¢ãããšã次åŒãšãªããŸãã
\begin{eqnarray}
&&E=v_{R}(t)+v_{C}(t)\\
{\Leftrightarrow}&&v_{C}(t)=E-v_{R}(t)\tag{13}
\end{eqnarray}
(13)åŒã«(12)åŒãä»£å ¥ãããšãã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
v_{C}(t)&=&E-v_{R}(t)\\
&=&E-E{\;}e^{-\frac{1}{CR}t}\\
&=&E\left(1-e^{-\frac{1}{CR}t}\right)\tag{14}
\end{eqnarray}
ãRCçŽååè·¯ãã³ã³ãã³ãµCã®é»å§VC(t)ã®ã°ã©ã
ã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ã®ã°ã©ãã¯äžå³ã®ããã«ãªããŸãããã®ã°ã©ãã«ã€ããŠèª¬æããŸãã
ãã®ã°ã©ãã§ãããã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ã¯ã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã«\(\displaystyle\frac{1}{C}\)ãæããã ãã§ãã
ããªãã¡ãã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã®ã°ã©ããšåããããªã°ã©ããšãªããŸãã
ã¹ã€ãã\(SW\)ããªã³ããç¬éãã³ã³ãã³ãµ\(C\)ã¯ç絡ããããããªç¶æ ã§ãããããã³ã³ãã³ãµ\(C\)ã®é»å§ã¯ãŒããšãªããŸãã
ã¹ã€ãã\(SW\)ããªã³ãããã°ããæéãçµéãããš(å®åžžç¶æ ã®æ)ãã³ã³ãã³ãµ\(C\)ã«é»è·\(q(t)\)ãèãããããã®ã³ã³ãã³ãµ\(C\)ã®é»è·\(q(t)\)ãæºãŸããããã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ãçŽæµé»æºã®é»å§\(E\)ãšçãããªããŸãã
ãåŸ®åæ¹çšåŒãã®è§£ãæ¹
ç¹°ãè¿ãã«ãªããŸããã(5)åŒãš(6)åŒãããäžåºŠç€ºããŸãã
(5)åŒã¯ã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã«é¢ãããåŸ®åæ¹çšåŒããšãªã次åŒãšãªããŸãã
\begin{eqnarray}
E=R\frac{dq(t)}{dt}+\frac{q(t)}{C}\tag{5}
\end{eqnarray}
ãã®ãåŸ®åæ¹çšåŒããè§£ããšã(6)åŒã®ã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ãå°åºããããšãã§ããŸãã
\begin{eqnarray}
q(t)=CE\left(1-e^{-\frac{1}{CR}t}\right)\tag{6}
\end{eqnarray}
ãã®å°åºæ¹æ³ã«ã€ããŠèª¬æããŸãã
å°åºæ¹æ³
ããã§ã¯ãåŸ®åæ¹çšåŒãè§£ãæãåºæ¬çãªãã¿ãŒã³ã®äžã€ã§ããã倿°åé¢åœ¢ã®åŸ®åæ¹çšåŒãã§è§£ããŠãããŸãã
ã倿°åé¢åœ¢ã®åŸ®åæ¹çšåŒããšã¯ãã®åã®éãã倿°ã巊蟺ãšå³èŸºã«åé¢ããåŸ®åæ¹çšåŒã®ããšã§ãã
(5)åŒã®å Žåãé»è·\(q(t)\)ãšæé\(t\)ã倿°ãªã®ã§ãé»è·\(q(t)\)ã«é¢ãããã®ã巊蟺ã«ãæé\(t\)ã«é¢ãããã®ãå³èŸºã«ãªãããã«åé¢ããŸãã
ããªãã¡ã(5)åŒã次åŒã®åœ¢ã«ãªãããã«å€åœ¢ããŸãã
\begin{eqnarray}
â¡dq(t)=â¡dt\tag{15}
\end{eqnarray}
å€åœ¢ã¯ä»¥äžã®ããã«è¡ããŸãã
倿°ã巊蟺ãšå³èŸºã«åé¢ããæ¹æ³
ãŸãã(5)åŒã®\(R\displaystyle\frac{dq(t)}{dt}\)ã巊蟺ã«ã\(E\)ãå³èŸºã«ç§»åããŠã䞡蟺ã«ãã€ãã¹ãæãããšã次åŒãšãªããŸãã
\begin{eqnarray}
R\frac{dq(t)}{dt}=\frac{CE-q(t)}{C}\tag{16}
\end{eqnarray}
(16)åŒã®äž¡èŸºã\(R\)ã§å²ããšã次åŒãšãªããŸãã
\begin{eqnarray}
\frac{dq(t)}{dt}=\frac{CE-q(t)}{CR}\tag{17}
\end{eqnarray}
(17)åŒã®äž¡èŸºã\(CE-q(t)\)ã§å²ããšã次åŒãšãªããŸãã
\begin{eqnarray}
\frac{1}{CE-q(t)}\frac{dq(t)}{dt}=\frac{1}{CR}\tag{18}
\end{eqnarray}
(18)åŒã®äž¡èŸºã«\(dt\)ãæãããšã次åŒãšãªããŸãã
\begin{eqnarray}
\frac{1}{CE-q(t)}dq(t)=\frac{1}{CR}dt\tag{19}
\end{eqnarray}
以äžãããé»è·\(q(t)\)ã«é¢ãããã®ã巊蟺ã«ãæé\(t\)ã«é¢ãããã®ãå³èŸºã«ãªãããã«åé¢ã§ããŸããã
ããªãã¡ãã\(â¡dq(t)=â¡dt\)ãã®åœ¢ã«ãªãããã«å€åœ¢ããããšãã§ããŸããããªããã³ã³ãã³ãµ\(C\)ãšæµæ\(R\)ãšçŽæµé»æºã®é»å§\(E\)ã¯å®æ°ãªã®ã§ã巊蟺ã«ãã£ãŠãå³èŸºã«ãã£ãŠãã©ã£ã¡ã§ãè¯ãã§ãã
(19)åŒã®äž¡èŸºãç©åãããšã次åŒãšãªããŸãã
\begin{eqnarray}
{\displaystyle\int}\frac{1}{CE-q(t)}dq(t)={\displaystyle\int}\frac{1}{CR}dt\tag{20}
\end{eqnarray}
(20)åŒã®å·ŠèŸºãšå³èŸºã¯å¥ã ã«è§£ããŠãããŸãã
巊蟺ã®è§£ãæ¹
ã\(k=CE-q(t)\)ããšçœ®ããšã\(q(t)\)ã¯ã³ã³ãã³ãµ\(C\)ã«èããããŠããé»è·ã§ããã\(CE\)ã¯å®åžžç¶æ ã«ãããŠã³ã³ãã³ãµ\(C\)ã«èããããŠããé»è·ãªã®ã§ãã\(k=CE-q(t){≥}0\)ããšãªããŸãã
ãŸããã\(dk=-dq(t)\)ããšãªãã®ã§ã(20)åŒã®å·ŠèŸºã¯æ¬¡åŒã®ããã«å€åœ¢ã§ããŸãã
\begin{eqnarray}
(20)åŒã®å·ŠèŸº&=&{\displaystyle\int}\frac{1}{CE-q(t)}dq(t)\\
&=&-{\displaystyle\int}\frac{1}{k}dk\\
&=&-\log_{e}k+A\\
&=&-\log_{e}\left(CE-q(t)\right)+A\tag{21}
\end{eqnarray}
(21)åŒã«ãããŠã\(A\)ã¯ç©å宿°ãšãªã£ãŠããŸãã
å³èŸºã®è§£ãæ¹
\(\displaystyle\frac{1}{CR}\)ã¯å®æ°ãªã®ã§ãç©åã®å€ã«åºãããšãã§ããã®ã§ã(20)åŒã®å³èŸºã¯æ¬¡åŒã®ããã«å€åœ¢ã§ããŸãã
\begin{eqnarray}
(20)åŒã®å³èŸº&=&{\displaystyle\int}\frac{1}{CR}dt\\
&=&\frac{1}{CR}{\displaystyle\int}dt\\
&=&\frac{1}{CR}t+B\tag{22}
\end{eqnarray}
(22)åŒã«ãããŠã\(B\)ã¯ç©å宿°ãšãªã£ãŠããŸãã
(21)åŒãš(22)åŒã(20)åŒã«æ»ããšã次åŒãšãªããŸãã
\begin{eqnarray}
-\log_{e}\left(CE-q(t)\right)+A=\frac{1}{CR}t+B\tag{23}
\end{eqnarray}
(23)åŒã§ã¯\(A\)ãš\(B\)ã®2ã€ã®ç©å宿°ããããŸããããã§ã\(A-B=D\)ãšçœ®ããšã(23)åŒã¯æ¬¡åŒã®ããã«å€åœ¢ã§ããŸãã
\begin{eqnarray}
\log_{e}\left(CE-q(t)\right)=-\frac{1}{CR}t+D\tag{24}
\end{eqnarray}
(24)åŒãå€åœ¢ãããšã次åŒãšãªããŸãã
\begin{eqnarray}
CE-q(t)&=&e^{-\frac{1}{CR}t+D}\\
&=&e^{-\frac{1}{CR}t}Ãe^{D}\tag{25}
\end{eqnarray}
(25)åŒã«ãããŠã\(CE\)ãå³èŸºã«ç§»åããŠã䞡蟺ã«ãã€ãã¹ãæãããšã次åŒãšãªããŸãã
\begin{eqnarray}
q(t)=CE-e^{-\frac{1}{CR}t}Ãe^{D}\tag{26}
\end{eqnarray}
次ã«ã(26)åŒã®ç©å宿°\(D\)ãæ±ããå¿ èŠããããŸãã
ç©å宿°\(D\)ã¯ä»¥äžã®ããã«æ±ããŸãã
ç©å宿°Dã®æ±ãæ¹
ç©å宿°ã¯åè·¯ã®åææ¡ä»¶ãçšããããšã§æ±ããããšãã§ããŸãã
ãã®åè·¯ã®å Žåãã\(t=0\)ãã®æãããªãã¡ãã¹ã€ãã\(SW\)ããªã³ããç¬éã¯ãã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã¯ãŒããšãªããŸãã
ãã®ãããåææ¡ä»¶ã¯ã\(t=0ãq(0)=0\)ããšãªããŸãã
(26)åŒã«åææ¡ä»¶ãä»£å ¥ãããšã
\begin{eqnarray}
q(0)&=&CE-e^{-\frac{1}{CR}Ã0}Ãe^{D}\\
{\Leftrightarrow}0&=&CE-e^0Ãe^{D}\\
0&=&CE-1Ãe^{D}\\
0&=&CE-e^{D}\tag{27}
\end{eqnarray}
ãšãªããŸãã
ã€ãŸããã\(t=0\)ãã®æãã\(e^{D}=CE\)ããšãªããŸãããªããç©å宿°\(D\)ãæ±ããŠãè¯ãã§ããã(26)åŒã«ãããŠãã\(e^{D}\)ãããã®ãŸãŸä»£å ¥ã§ããŸãã
ãã®ãããä»åã¯ã\(e^{D}=CE\)ããŸã§ã®å°åºã§å€§äžå€«ã§ãã
(27)åŒã(26)åŒã«ä»£å ¥ãããšã次åŒãšãªããã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ãå°åºããããšãã§ããŸããã
\begin{eqnarray}
q(t)&=&CE-e^{-\frac{1}{CR}t}Ãe^{D}\\
&=&CE-e^{-\frac{1}{CR}t}ÃCE\\
&=&CE\left(1-e^{-\frac{1}{CR}t}\right)\\
&=&CE-CEe^{-\frac{1}{CR}t}\tag{28}
\end{eqnarray}
ãªãã(28)åŒã®å³èŸºã®ç¬¬1é ã¯å®åžžè§£ã第2é ã¯é枡解ãšåŒã°ããŠããŸãã
RCçŽååè·¯ã«æµãã黿µi(t)ã®å°åºæ¹æ³
ç¹°ãè¿ãã«ãªããŸããã(1)åŒãš(6)åŒãš(7)åŒãããäžåºŠç€ºããŸãã
(1)åŒã¯RCçŽååè·¯ã«æµãã黿µ\(i(t)\)ãšã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã®é¢ä¿ã§ãããæ¬¡åŒã§è¡šãããŸãã
\begin{eqnarray}
i(t)=\frac{dq(t)}{dt}\tag{1}
\end{eqnarray}
(6)åŒã¯ã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã§ãããæ¬¡åŒãšãªããŸãã
\begin{eqnarray}
q(t)=CE\left(1-e^{-\frac{1}{CR}t}\right)\tag{6}
\end{eqnarray}
ããã§ã(6)åŒã(1)åŒã«ä»£å ¥ãããšãRCçŽååè·¯ã«æµãã黿µ\(i(t)\)ãæ±ããããšãã§ããæ¬¡åŒã®ææ°é¢æ°ãšãªããŸã
\begin{eqnarray}
i(t)&=&\frac{E}{R}e^{-\frac{1}{CR}t}\tag{7}
\end{eqnarray}
ãã®å°åºæ¹æ³ã«ã€ããŠèª¬æããŸãã
å°åºæ¹æ³
(6)åŒã(1)åŒã«ä»£å ¥ãããšã次åŒãšãªããŸãã
\begin{eqnarray}
i(t)=\frac{d}{dt}CE\left(1-e^{-\frac{1}{CR}t}\right)\tag{29}
\end{eqnarray}
ããã§ã\(CE\)ã¯å®æ°ãªã®ã§ã埮åã®å€ã«åºããšæ¬¡åŒãšãªããŸãã
\begin{eqnarray}
i(t)&=&CE\frac{d}{dt}\left(1-e^{-\frac{1}{CR}t}\right)\\
&=&CE\left(\frac{d}{dt}1-\frac{d}{dt}e^{-\frac{1}{CR}t}\right)\\
&=&CE\left\{0-\left(-\frac{1}{CR}e^{-\frac{1}{CR}t}\right)\right\}\\
&=&CE\left(\frac{1}{CR}e^{-\frac{1}{CR}t}\right)\\
&=&\frac{E}{R}e^{-\frac{1}{CR}t}\tag{30}
\end{eqnarray}
以äžãããRCçŽååè·¯ã«æµãã黿µ\(i(t)\)ãæ±ããããšãã§ããŸããã
è£è¶³
- æå®æ°\({\tau}=CR\)ã®åäœããªã\({\mathrm{[s]}}\)ãªã®ã
ã³ã³ãã³ãµ\(C\)ã®åäœã¯
\begin{eqnarray}
\frac{{\mathrm{[C]}}}{{\mathrm{[V]}}}=\frac{{\mathrm{[A]}}{\mathrm{[s]}}}{{\mathrm{[V]}}}
\end{eqnarray}
ãšãªããŸãã
æµæ\(R\)ã®åäœã¯
\begin{eqnarray}
\frac{{\mathrm{[V]}}}{{\mathrm{[A]}}}
\end{eqnarray}
ãšãªããŸãã
ãããã£ãŠãæå®æ°\({\tau}=CR\)ã®åäœã¯
\begin{eqnarray}
\frac{{\mathrm{[A]}}{\mathrm{[s]}}}{{\mathrm{[V]}}}Ã\frac{{\mathrm{[V]}}}{{\mathrm{[A]}}}={\mathrm{[s]}}
\end{eqnarray}
ãšãªããŸãã
ãŸãšã
ãã®èšäºã§ã¯RCçŽååè·¯ã«ã€ããŠã以äžã®å 容ã説æããŸããã
åœèšäºã®ãŸãšã
- ãRCçŽååè·¯ããéæž¡çŸè±¡ãã®åŒãšã°ã©ã
- ãRCçŽååè·¯ããåŸ®åæ¹çšåŒãã®è§£ãæ¹
ãèªã¿é ãããããšãããããŸããã
åœãµã€ãã§ã¯é»æ°ã«é¢ããæ§ã
ãªæ
å ±ãèšèŒããŠããŸããåœãµã€ãã®å
šèšäºäžèЧã«ã¯ä»¥äžã®ãã¿ã³ããç§»åããããšãã§ããŸãã