ãã®èšäºã§ã¯RCçŽååè·¯ã®ãã©ãã©ã¹å€æã«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
- é»æµ\(i(t)\)ãæµããŠãã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ãå¢ããã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ãäžæããã
- ããçšåºŠæéãçµéãããšãé»æµ\(i(t)\)ãæµããªããªã(ã€ãŸããäžå®å€\(0{\mathrm{[A]}}\)ã«ãªã)ããŸãããã®æãã³ã³ãã³ãµ\(C\)ãéæŸããããããªç¶æ ã§ãããã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ãé»æºé»å§ã®é»å§\(E\)ãšçãããªãã
ãã®æãé»æµ\(i(t)\)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«æµããé»æµ\(i(t)\)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¯äžåŒãã©ãã©ã¹å€æãçšããŠè§£ããŠãããŸãããªããäžåŒã¯åŸ®åæ¹çšåŒã解ãæãåºæ¬çãªãã¿ãŒã³ã®å€æ°åé¢åœ¢ã®åŸ®åæ¹çšåŒã«ããŠãçŽæ¥è§£ãããšãå¯èœã§ãã
å€æ°åé¢åœ¢ã®åŸ®åæ¹çšåŒã«ããŠãçŽæ¥è§£ãæ¹æ³ã«ã€ããŠã¯ä»¥äžã®èšäºã«è©³ãã説æããŠããŸãã®ã§ãåèã«ããŠãã ããã
ãRCçŽååè·¯ã®åŸ®åæ¹çšåŒããéæž¡çŸè±¡ãã®è§£ãæ¹ïŒ
ç¶ããèŠã
ãRCçŽååè·¯ããã©ãã©ã¹å€æãã«ãã解ãæ¹
ã©ãã©ã¹å€æãçšããŠRCçŽååè·¯ã®éæž¡çŸè±¡ã解ãå Žåã以äžã®â ïœâ£ã®æé ã§è¡ããŸãã
ã©ãã©ã¹å€æã®æé
- åè·¯æ¹çšåŒãããŠã
- ã©ãã©ã¹å€æããsé åã®æ¹çšåŒã«ãã
- sé åã®æ¹çšåŒã解ã
- ã©ãã©ã¹éå€æãã
â察象ãšãªãåè·¯(ä»åã¯RCçŽååè·¯)ã®té¢æ°ã«ããåè·¯æ¹çšåŒãããŠãŸãã
ââ ã§æ±ããåè·¯æ¹çšåŒãã©ãã©ã¹å€æããŠãsé åã®æ¹çšåŒã«ããŸãããã®éãåææ¡ä»¶ãèæ ®ããå¿ èŠããããŸãã
âæ±ãããsé¢æ°ã®åŒã«ããŸããä»åã¯ã\(I(s)={\cdots}\)ãã®åŒã«ããŸãã
ââ¢ã§æ±ããåŒãã©ãã©ã¹éå€æããŠãté åã®æ¹çšåŒã«ããŸãã
ã§ã¯ãããããåæé ã«ã€ããŠé çªã«èª¬æããŠãããŸãã
ãRCçŽååè·¯ãåè·¯æ¹çšåŒãããŠã
RCçŽååè·¯ãäžå³ã«ç€ºããŸãã
äžå³ã®RCçŽååè·¯ã«ãã«ããããã®é»å§å(ãã«ããããã®ç¬¬äºæ³å)ãçšãããšæ¬¡åŒãæãç«ã¡ãŸãã
\begin{eqnarray}
E=v_{R}(t)+v_{C}(t)\tag{1}
\end{eqnarray}
(1)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)\tag{2}\\
v_{C}(t)&=&\frac{1}{C}{\displaystyle\int}i(t)dt\tag{3}
\end{eqnarray}
(2)åŒãš(3)åŒã(1)åŒã«ä»£å ¥ãããšã次åŒãåŸãããŸãã
\begin{eqnarray}
E&=&v_{R}(t)+v_{C}(t)\\
&=&Ri(t)+\frac{1}{C}{\displaystyle\int}i(t)dt\tag{4}
\end{eqnarray}
äžåŒããRCçŽååè·¯ã®té åã®åè·¯æ¹çšåŒãšãªããŸãã
ãRCçŽååè·¯ãã©ãã©ã¹å€æããsé åã®æ¹çšåŒã«ãã
(4)åŒã®å·ŠèŸºã¯ã\(E=E{\;}{\cdot}{\;}1\)ãã§ããããã(4)åŒãã©ãã©ã¹å€æãããšã次åŒãšãªããŸãã
\begin{eqnarray}
E{\;}{\cdot}{\;}\frac{1}{s}=RI(s)+\frac{1}{C}\left(\frac{I(s)}{s}+\frac{q(0)}{s}\right)\tag{5}
\end{eqnarray}
(5)åŒã«ãããŠãã\(q(0)\)ãã¯ã\(t=0\)ãã®æã«ãããã³ã³ãã³ãµ\(C\)ã«èããããé»è·ã§ããã¹ã€ãã\(SW\)ããªã³ããåã¯ãé»è·ã¯èããããŠããªãããã次åŒãæãç«ã¡ãŸãã
\begin{eqnarray}
q(0)=0\tag{6}
\end{eqnarray}
(6)åŒã(5)åŒã«ä»£å ¥ãããšã次åŒãšãªããŸãã
\begin{eqnarray}
E{\;}{\cdot}{\;}\frac{1}{s}&=&RI(s)+\frac{1}{C}\left(\frac{I(s)}{s}+\frac{0}{s}\right)\\
{\Leftrightarrow}\frac{E}{s}&=&RI(s)+\frac{1}{sC}I(s)\tag{7}
\end{eqnarray}
äžåŒããRCçŽååè·¯ã®sé åã®æ¹çšåŒãšãªããŸãã
ãRCçŽååè·¯ãsé åã®æ¹çšåŒã解ã
(7)åŒãã\(I(s)={\cdots}\)ãã®åŒã«å€æããŸãã
(7)åŒã\(I(s)\)ã§æŽçãããšã次åŒãšãªããŸãã
\begin{eqnarray}
\left(R+\frac{1}{sC}\right)I(s)=\frac{E}{s}\tag{8}
\end{eqnarray}
(8)åŒã®äž¡èŸºã\(R+\displaystyle\frac{1}{sC}\)ã§å²ããšæ¬¡åŒãšãªããŸãã
\begin{eqnarray}
I(s)=\frac{E}{s\left(R+\displaystyle\frac{1}{sC}\right)}\tag{9}
\end{eqnarray}
(9)åŒã®å³èŸºã®ååãšåæ¯ã\(R\)ã§å²ããšæ¬¡åŒãšãªããŸãã
\begin{eqnarray}
I(s)&=&\frac{\displaystyle\frac{E}{R}}{s\left(1+\displaystyle\frac{1}{sCR}\right)}\\
&=&\frac{E}{R}\frac{1}{s+\displaystyle\frac{1}{CR}}\tag{10}
\end{eqnarray}
ãã®ããã«ãå€åœ¢ããããšã§ãã\(I(s)={\cdots}\)ãã«ããããšãã§ããŸãããªãã(10)åŒã¯ãã®åŸã«èª¬æããã©ãã©ã¹éå€æãããããããã«å€åœ¢ããŠããŸãã
ãRCçŽååè·¯ãã©ãã©ã¹éå€æãã
(10)åŒãã©ãã©ã¹éå€æãããšã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{1}{CR}}\right]\\
&=&\frac{E}{R}{\mathcal{L}}^{-1}\left[\frac{1}{s+\displaystyle\frac{1}{CR}}\right]\\
&=&\frac{E}{R}e^{-\frac{1}{CR}t}\tag{11}
\end{eqnarray}
以äžãããRCçŽååè·¯ã«æµããé»æµ\(i(t)\)ã®åŒãå°åºããããšãã§ããŸããã
RCçŽååè·¯ã«æµããé»æµ\(i(t)\)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{1}{CR}t}\tag{12}
\end{eqnarray}
ãŸãã(11)åŒã(3)åŒã«ä»£å ¥ãããšãã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
v_{C}(t)&=&\frac{1}{C}{\displaystyle\int}i(t)dt\\
&=&\frac{1}{C}{\displaystyle\int}\frac{E}{R}e^{-\frac{1}{CR}t}dt\\
&=&\frac{E}{CR}{\displaystyle\int}e^{-\frac{1}{CR}t}dt\\
&=&\frac{E}{CR}\left(\frac{1}{-\displaystyle\frac{1}{CR}}e^{-\frac{1}{CR}t}\right)+D\\
&=&\frac{E}{CR}\left(-CRe^{-\frac{1}{CR}t}\right)+D\\
&=&-Ee^{-\frac{1}{CR}t}+D\tag{13}
\end{eqnarray}
(13)åŒã«ãããŠã\(D\)ã¯ç©åå®æ°ãšãªã£ãŠããŸãã
次ã«ã(13)åŒã®ç©åå®æ°\(D\)ãæ±ããå¿ èŠããããŸãã
ç©åå®æ°\(D\)ã¯ä»¥äžã®ããã«æ±ããŸãã
ç©åå®æ°Dã®æ±ãæ¹
ç©åå®æ°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®é»å§\(v_{C}(t)\)ã¯ãŒããšãªããŸãã
ãã®ãããåææ¡ä»¶ã¯ã\(t=0ãv_{C}(0)=0\)ããšãªããŸãã
(13)åŒã«åææ¡ä»¶ãä»£å ¥ãããšã
\begin{eqnarray}
v_{C}(0)&=&-Ee^{-\frac{1}{CR}Ã0}+D\\
{\Leftrightarrow}0&=&-Ee^0+D\\
0&=&-EÃ1+D\\
0&=&-E+D\tag{14}
\end{eqnarray}
ãšãªããŸãã
ã€ãŸããã\(t=0\)ãã®æãã\(D=E\)ããšãªããŸãã
(14)åŒã(13)åŒã«ä»£å ¥ãããšã次åŒãšãªããã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ãå°åºããããšãã§ããŸãã
\begin{eqnarray}
v_{C}(t)&=&-Ee^{-\frac{1}{CR}t}+E\\
&=&E\left(1-e^{-\frac{1}{CR}t}\right)\tag{15}
\end{eqnarray}
ãã®èšäºã§ã¯ãã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ã¯RCçŽååè·¯ã«æµããé»æµ\(i(t)\)ãç©åããããšã§æ±ããŸãããããã«ããããã®é»å§åã§ãæ±ããããšãã§ããŸãã(1)åŒãš(12)åŒãçšãããš(15)åŒãšåãçµæã«ãªããŸããã
ã\(Q=CV\)ããããã³ã³ãã³ãµ\(C\)ã®é»å§\(v_{C}(t)\)ãåãããšãã³ã³ãã³ãµ\(C\)ã«èããããé»è·\(q(t)\)ã次åŒã®ããã«ç°¡åã«æ±ããããšãã§ããŸãã
\begin{eqnarray}
q(t)&=&Cv_{C}(t)\\
&=&CE\left(1-e^{-\frac{1}{CR}t}\right)\tag{16}
\end{eqnarray}
ãªããåã ã®åŒã®ã°ã©ãã«ã€ããŠã¯ä»¥äžã®èšäºã«è©³ãã説æããŠããŸãã®ã§ãåèã«ããŠãã ããã
-
ãRCçŽååè·¯ã®åŸ®åæ¹çšåŒããéæž¡çŸè±¡ãã®è§£ãæ¹ïŒ
ç¶ããèŠã
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·¯ããã©ãã©ã¹å€æãã®è§£ãæ¹
ãèªã¿é ãããããšãããããŸããã
åœãµã€ãã§ã¯é»æ°ã«é¢ããæ§ã
ãªæ
å ±ãèšèŒããŠããŸããåœãµã€ãã®å
šèšäºäžèŠ§ã«ã¯ä»¥äžã®ãã¿ã³ãã移åããããšãã§ããŸãã