ãã®èšäºã§ã¯ãCRãã€ãã¹ãã£ã«ã¿ãã«ã€ããŠ
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- CRãã€ãã¹ãã£ã«ã¿ã®ãäŒé颿°ã,ãã²ã€ã³ã,ãã«ãããªãåšæ³¢æ°ã,ãäœçžã
- CRãã€ãã¹ãã£ã«ã¿ã®ãåšæ³¢æ°ç¹æ§ã
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CRãã€ãã¹ãã£ã«ã¿ãšã¯
äžå³ã«CRãã€ãã¹ãã£ã«ã¿ã®åè·¯æ§æã瀺ããŠããŸãã
CRãã€ãã¹ãã£ã«ã¿ã¯ãã³ã³ãã³ãµ\(C\)ãšæµæ\(R\)ã®ã¿ã§æ§æããããã€ãã¹ãã£ã«ã¿ã§ããå ¥åé»å§\(V_{IN}\)ã®é«åšæ³¢æåãééãããäœåšæ³¢æåã鮿ããŸãã
åŸã»ã©å°åºæ¹æ³ãªã©è©³çްã«èª¬æããŸãããCRãã€ãã¹ãã£ã«ã¿ã®ãäŒé颿°ã,ãã²ã€ã³ã,ãã«ãããªãåšæ³¢æ°ã,ãäœçžãã®åŒãšãåšæ³¢æ°ç¹æ§ãããŸãšãããšãäžèšã®ããã«ãªããŸãã
CRãã€ãã¹ãã£ã«ã¿ã®ãŸãšã
- äŒé颿°\(G(j{\omega})\)
- ã²ã€ã³\(|G(j{\omega})|\)
- ã«ãããªãåšæ³¢æ°\(f_C\)
- äœçž\({\theta}\)
âå
¥åé»å§\(V_{IN}\)ãšåºåé»å§\(V_{OUT}\)ã®æ¯ãäŒé颿°\(G(j{\omega})\)ã§ãããæ¬¡åŒãšãªãã
\begin{eqnarray}
G(j{\omega})=\frac{V_{OUT}}{V_{IN}}=\frac{j{\omega}CR}{1+j{\omega}CR}=\frac{j2{\pi}fCR}{1+j2{\pi}fCR}\tag{1-1}
\end{eqnarray}
âäŒé颿°\(G(j{\omega})\)ã®çµ¶å¯Ÿå€ãã²ã€ã³\(|G(j{\omega})|\)ã§ãããæ¬¡åŒãšãªãã
\begin{eqnarray}
|G(j{\omega})|=\frac{{\omega}CR}{\sqrt{1+({\omega}CR)^2}}=\frac{2{\pi}fCR}{\sqrt{1+(2{\pi}fCR)^2}}\tag{1-2}
\end{eqnarray}
âã²ã€ã³\(|G(j{\omega})|\)ã\(\displaystyle\frac{1}{\sqrt{2}}({\;}{\approx}{\;}0.707)\)ã«ãªãåšæ³¢æ°ãã«ãããªãåšæ³¢æ°\(f_C\)ã§ãããæ¬¡åŒãšãªãã
\begin{eqnarray}
f_C=\frac{1}{2{\pi}CR}\tag{1-3}
\end{eqnarray}
âå
¥åé»å§\(V_{IN}\)ã«å¯Ÿããåºåé»å§\(V_{OUT}\)ã®äœçžã§ãããæ¬¡åŒãšãªãã
\begin{eqnarray}
{\theta}={\tan}^{-1}\left(\frac{1}{{\omega}CR}\right)={\tan}^{-1}\left(\frac{1}{2{\pi}fCR}\right)\tag{1-4}
\end{eqnarray}
å ¥åé»å§\(V_{IN}\)ã®åšæ³¢æ°ãäœãå Žåãã³ã³ãã³ãµ\(C\)ã®ã€ã³ããŒãã³ã¹ã倧ããã®ã§ãåºåé»å§\(V_{OUT}\)ãäœããªã(ããªãã¡ãäœåšæ³¢æåã鮿ãã)ãšããããšã¯èŠåœãã€ããšæããŸãã
è£è¶³
- ãã€ãã¹ãã£ã«ã¿ã¯ãé«åééãã£ã«ã¿ãããããŒã«ãããã£ã«ã¿ããšãåŒã°ããŠããŸãã
CRãã€ãã¹ãã£ã«ã¿ã®ãäŒé颿°ããšãã²ã€ã³ã
CRãã€ãã¹ãã£ã«ã¿ã®ãäŒé颿°ããšãã²ã€ã³ãã®å°åºæ¹æ³ã«ã€ããŠèª¬æããŸãã
ãã³ã³ãã³ãµ\(C\)ã®ã€ã³ããŒãã³ã¹\({\dot{Z}_C}\)ããšãæµæ\(R\)ã®ã€ã³ããŒãã³ã¹\({\dot{Z}_R}\)ãã¯æ¬¡åŒã§è¡šãããŸãã
\begin{eqnarray}
{\dot{Z}_C}&=&\frac{1}{j{\omega}C}\tag{2-1}\\
\\
{\dot{Z}_R}&=&R\tag{2-2}
\end{eqnarray}
ãããã£ãŠãåºåé»å§\(V_{OUT}\)ã¯å ¥åé»å§\(V_{IN}\)ã\({\dot{Z}_C}\)ãš\({\dot{Z}_R}\)ã§åå§ããŠããã®ã§ã次åŒã§è¡šãããŸãã
\begin{eqnarray}
V_{OUT}=\displaystyle\frac{{\dot{Z}_R}}{{\dot{Z}_C}+{\dot{Z}_R}}V_{IN}=\frac{R}{\displaystyle\frac{1}{j{\omega}C}+R}V_{IN}=\frac{j{\omega}CR}{1+j{\omega}CR}V_{IN}\tag{2-3}
\end{eqnarray}
CRãã€ãã¹ãã£ã«ã¿ã®äŒé颿°\(G(j{\omega})\)ã¯å ¥åé»å§\(V_{IN}\)ãšåºåé»å§\(V_{OUT}\)ã®æ¯ã§ãããã®ããã(2-3)åŒãå€åœ¢ãããšãäŒé颿°\(G(j{\omega})\)ã¯æ¬¡åŒã§è¡šãããšãã§ããŸãã
\begin{eqnarray}
G(j{\omega})=\frac{V_{OUT}}{V_{IN}}=\frac{j{\omega}CR}{1+j{\omega}CR}\tag{2-4}
\end{eqnarray}
(2-4)åŒã®åæ¯ã«ã¯èæ°åäœ\(j\)ããããŸããããã§ååã®ã¿ã«èæ°åäœ\(j\)ãããããã«ããããã«ã忝ãšååã«ã\(1-j{\omega}CR\)ããæããŸãããããš(2-4)åŒã¯æ¬¡åŒã«å€åœ¢ããããšãã§ããŸãã
\begin{eqnarray}
G(j{\omega})&=&\frac{j{\omega}CR}{1+j{\omega}CR}Ã\frac{1-j{\omega}CR}{1-j{\omega}CR}\\
\\
&=&\frac{j{\omega}CR+({\omega}CR)^2}{1+({\omega}CR)^2}\\
\\
&=&\frac{({\omega}CR)^2}{1+({\omega}CR)^2}+j\frac{{\omega}CR}{1+({\omega}CR)^2}\tag{2-5}
\end{eqnarray}
äŒé颿°\(G(j{\omega})\)ã®çµ¶å¯Ÿå€ãCRãã€ãã¹ãã£ã«ã¿ã®ã²ã€ã³\(|G(j{\omega})|\)ãšãªããŸããããå°ã詳ãã説æãããšãCRãã€ãã¹ãã£ã«ã¿ã®ã²ã€ã³\(|G(j{\omega})|\)ã¯(2-5)åŒã«ãããŠããå®éš\(\left\{\displaystyle\frac{({\omega}CR)^2}{1+({\omega}CR)^2}\right\}\)ã®2ä¹ããšãèéš\(\left\{\displaystyle\frac{{\omega}CR}{1+({\omega}CR)^2}\right\}\)ã®2ä¹ããè¶³ããŠãå¹³æ¹æ ¹ãåãããšã§æ±ããããšãã§ããŸãããã®ãããã²ã€ã³\(|G(j{\omega})|\)ã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
|G(j{\omega})|&=&\sqrt{\left\{\frac{({\omega}CR)^2}{1+({\omega}CR)^2}\right\}^2+\left\{\frac{{\omega}CR}{1+({\omega}CR)^2}\right\}^2}\\
\\
&=&\sqrt{\frac{({\omega}CR)^2\left\{1+({\omega}CR)^2\right\}}{\left\{1+({\omega}CR)^2\right\}^2}}\\
\\
&=&\sqrt{\frac{({\omega}CR)^2}{1+({\omega}CR)^2}}\\
\\
&=&\frac{{\omega}CR}{\sqrt{1+({\omega}CR)^2}}\tag{2-6}
\end{eqnarray}
ããã§ãè§åšæ³¢æ°\({\omega}\)ã¯\({\omega}=2{\pi}f\)ã®é¢ä¿ãããã®ã§ã(2-6)åŒã®\({\omega}\)ã\(2{\pi}f\)ã«æžãæãããšã次åŒãšãªããŸãã
\begin{eqnarray}
|G(j{\omega})|=\frac{2{\pi}fCR}{\sqrt{1+(2{\pi}fCR)^2}}\tag{2-7}
\end{eqnarray}
ãªããCRãã€ãã¹ãã£ã«ã¿ã®ã²ã€ã³\(|G(j{\omega})|\)ããã·ãã«è¡šç€ºã«ãããã®ã\(G_{dB}(j{\omega})\)ãšãããšã\(G_{dB}(j{\omega})\)ã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
G_{dB}(j{\omega})&=&20{\log}_{10}|G(j{\omega})|\\
\\
&=&20{\log}_{10}\frac{{\omega}CR}{\sqrt{1+({\omega}CR)^2}}{\mathrm{[dB]}}\\
\\
&=&20{\log}_{10}\frac{2{\pi}fCR}{\sqrt{1+(2{\pi}fCR)^2}}{\mathrm{[dB]}}\tag{2-8}
\end{eqnarray}
ããã§ãCRãã€ãã¹ãã£ã«ã¿ã®ãäŒé颿°ããšãã²ã€ã³ãã®å°åºã¯çµããã§ãã
ãã©ãã©ã¹æŒç®å\(s=j{\omega}\)ããšãCRãã€ãã¹ãã£ã«ã¿ã®æå®æ°\({\tau}=RC\)ããçšãããšãäŒé颿°\(G(j{\omega})\)ã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
G(j{\omega})=\frac{j{\omega}CR}{1+j{\omega}CR}=\frac{s{\tau}}{1+s{\tau}}
\end{eqnarray}
äŒé颿°\(G(j{\omega})\)ãäžåŒã§è¡šããŠããè³æãããèŠãããŸãã
ããããŠèªã¿ãã
(2-8)åŒã«ç€ºãããã«ãã²ã€ã³\(|G(j{\omega})|\)ããã·ãã«ã§è¡šãå Žåã«ã¯ãã²ã€ã³ã®åžžçšå¯Ÿæ°(\({\log}_{10}\))ã20åããŸãããã·ãã«ã«ã€ããŠè©³ããã¯äžèšã®èšäºã§èª¬æããŠããŸãã®ã§ããåèã«ãªãã°å¹žãã§ãã
-
é»å§ãé»åã®ããã·ãã«(dB)ããšã¯ïŒèšç®æ¹æ³ãå€ææ¹æ³ã«ã€ããŠ
ç¶ããèŠã
CRãã€ãã¹ãã£ã«ã¿ã®ãã«ãããªãåšæ³¢æ°ã
ã«ãããªãåšæ³¢æ°\(f_C\)ã¯ãCRãã€ãã¹ãã£ã«ã¿ã®ã²ã€ã³\(|G(j{\omega})|\)ã\(\displaystyle\frac{1}{\sqrt{2}}({\;}{\approx}{\;}0.707)\)ã«ãªãåšæ³¢æ°ã§ãããæ¬¡åŒãšãªããŸãã
\begin{eqnarray}
\frac{1}{\sqrt{2}}&=&|G(j{\omega})|\\
\\
&=&\frac{2{\pi}f_CCR}{\sqrt{1+(2{\pi}f_CCR)^2}}\\
\\
{\Leftrightarrow}\sqrt{1+(2{\pi}f_CCR)^2}&=&\sqrt{2}Ã2{\pi}f_CCR\\
\\
1+(2{\pi}f_CCR)^2&=&2Ã(2{\pi}f_CCR)^2\\
\\
(2{\pi}f_CCR)^2&=&1\\
\\
2{\pi}f_CCR&=&1\\
\\
f_C&=&\frac{1}{2{\pi}CR}\tag{3-1}
\end{eqnarray}
å ¥åé»å§\(V_{IN}\)ã¯CRãã€ãã¹ãã£ã«ã¿ã«ãã£ãŠãã«ãããªãåšæ³¢æ°\(f_C\)ããé«ãæåã®åšæ³¢æ°ã¯ã»ãšãã©ééããã«ãããªãåšæ³¢æ°\(f_C\)ããäœãæåã®åšæ³¢æ°ã¯æžè¡°ããŸãã
ããããŠèªã¿ãã
ãã«ãããªãåšæ³¢æ°ã£ãŠäœïŒããäœã§ã²ã€ã³\(|G(j{\omega})|\)ã\(\displaystyle\frac{1}{\sqrt{2}}({\;}{\approx}{\;}0.707)\)ã«ãªãåšæ³¢æ°ãã«ãããªãåšæ³¢æ°ãªã®ïŒããšããæ¹ã¯äžèšã®èšäºã圹ã«ç«ã€ãšæããŸãã®ã§ããåèã«ããŠãã ããã
-
ãã«ãããªãåšæ³¢æ°(é®æåšæ³¢æ°)ããšã¯ïŒããã£ã«ã¿åè·¯ã
ç¶ããèŠã
CRãã€ãã¹ãã£ã«ã¿ã®æå®æ°\({\tau}\)ã¯ã\({\tau}=RC\)ããªã®ã§ã(3-1)åŒã®\(RC\)ã\({\tau}\)ã«çœ®ãæãããšæ¬¡åŒãšãªããŸãã
\begin{eqnarray}
f_C=\frac{1}{2{\pi}CR}=\frac{1}{2{\pi}{\tau}}\tag{3-2}
\end{eqnarray}
ã€ãŸããã«ãããªãåšæ³¢æ°\(f_C\)ã¯æå®æ°\({\tau}\)ã«åæ¯äŸããŸããæµæã®æµæå€\(R\)ã倧ããã»ã©ãã³ã³ãã³ãµã®éé»å®¹é\(C\)ã倧ããã»ã©ãã«ãããªãåšæ³¢æ°\(f_C\)ã¯äœããªããŸãã
è£è¶³
- CRãã€ãã¹ãã£ã«ã¿ã®ã²ã€ã³\(|G(j{\omega})|\)ã\(\displaystyle\frac{1}{\sqrt{2}}({\;}{\approx}{\;}0.707)\)ã«ãªãæã¯ããã·ãã«åäœã§è¡šããšæ¬¡åŒã«ç€ºãããã«çŽïŒ3dBãšãªããŸãã
\begin{eqnarray}
G_{dB}(j{\omega})&=&20{\log}_{10}\frac{1}{\sqrt{2}}\\
\\
&=&-3.01029{\cdots}{\mathrm{[dB]}}\\
\\
&{\approx}&-3{\mathrm{[dB]}}
\end{eqnarray} - ã«ãããªãåšæ³¢æ°ã¯ãé®æåšæ³¢æ°ããšãåŒã°ããŠããŸãã
ãã«ãããªãè§åšæ³¢æ°ãã«ã€ããŠ
ã«ãããªãè§åšæ³¢æ°\({\omega}_C\)ã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
{\omega}_C=\frac{1}{CR}
\end{eqnarray}
äžåŒãçšãããšãäŒé颿°\(G(j{\omega})\)ãšã²ã€ã³\(|G(j{\omega})|\)ã¯æ¬¡åŒãšãªããŸãã
\begin{eqnarray}
G(j{\omega})&=&\frac{j{\omega}CR}{1+j{\omega}CR}=\frac{j\displaystyle\frac{{\omega}}{{\omega}_C}}{1+j\displaystyle\frac{{\omega}}{{\omega}_C}}\\
\\
|G(j{\omega})|&=&\frac{{\omega}CR}{\sqrt{1+({\omega}CR)^2}}=\frac{\displaystyle\frac{{\omega}}{{\omega}_C}}{\sqrt{1+\left(\displaystyle\frac{{\omega}}{{\omega}_C}\right)^2}}
\end{eqnarray}
äŒé颿°\(G(j{\omega})\)ãšã²ã€ã³\(|G(j{\omega})|\)ãäžåŒã§è¡šããŠããè³æãããèŠãããŸãã
CRãã€ãã¹ãã£ã«ã¿ã®ãäœçžã
ç¹°ãè¿ãã«ãªããŸãããCRãã€ãã¹ãã£ã«ã¿ã®äŒé颿°\(G(j{\omega})\)ã¯æ¬¡åŒã§è¡šãããŸãã
\begin{eqnarray}
G(j{\omega})=\frac{({\omega}CR)^2}{1+({\omega}CR)^2}+j\frac{{\omega}CR}{1+({\omega}CR)^2}\tag{4-1}
\end{eqnarray}
è€çŽ å¹³é¢(暪軞ã¯å®æ°ã®ç®çã瞊軞ã¯èæ°ã®ç®çã§ãããã¬ãŠã¹å¹³é¢ãšãåŒã°ããŠãã)äžã«(4-1)åŒã®ãã¯ãã«ãæããšäžå³ã®ããã«ãªããŸãããã®ãã¯ãã«å³ããCRãã€ãã¹ãã£ã«ã¿ã®äœçž\({\theta}\)ãæ±ããããšãã§ããæ¬¡åŒã§è¡šãããŸãã
\begin{eqnarray}
{\tan}{\theta}&=&\frac{\displaystyle\frac{{\omega}CR}{1+({\omega}CR)^2}}{\displaystyle\frac{({\omega}CR)^2}{1+({\omega}CR)^2}}\\
\\
&=&\frac{1}{{\omega}CR}\\
\\
{\Leftrightarrow}{\theta}&=&{\tan}^{-1}\left(\frac{1}{{\omega}CR}\right)\\
\\
&=&{\tan}^{-1}\left(\frac{1}{2{\pi}fCR}\right)\tag{4-2}
\end{eqnarray}
ãªããã«ãããªãåšæ³¢æ°\(f_C\)ã«ãããäœçž\({\theta}_C\)ã¯ä»¥äžã®å€ãšãªããŸãã
\begin{eqnarray}
{\theta}_C&=&{\tan}^{-1}\left(\frac{1}{2{\pi}f_CCR}\right)\\
\\
&=&{\tan}^{-1}\left(\frac{1}{2{\pi}Ã\displaystyle\frac{1}{2{\pi}CR}ÃCR}\right)\\
\\
&=&{\tan}^{-1}(1)\\
\\
&=&0.7853{\cdots}{\mathrm{[rad]}}\tag{4-3}
\end{eqnarray}
äžåŒã®åäœã¯[rad](ã©ãžã¢ã³)ãªã®ã§ã[rad]ã[°(床)]ã«å€æãããšã次åŒã«ç€ºãããã«45°(床)ãšãªããŸãã
\begin{eqnarray}
{\theta}_C&=&{\tan}^{-1}(1)Ã\frac{180}{{\pi}}\\
\\
&=&45{\mathrm{°}}\tag{4-4}
\end{eqnarray}
[rad]ã[°(床)]ã«å€æããããã«ã¯ã\(\displaystyle\frac{180}{{\pi}}\)ãæããŸãã
CRãã€ãã¹ãã£ã«ã¿ã®ãåšæ³¢æ°ç¹æ§ã
äžäŸãšããŠãã³ã³ãã³ãµ\(C=1{\mathrm{[ÎŒF]}}\)ãæµæ\(R=1{\mathrm{[kΩ]}}\)ã®CRãã€ãã¹ãã£ã«ã¿ã«ãããŠãã²ã€ã³\(|G(j{\omega})|\)ãšäœçž\({\theta}\)ã®åšæ³¢æ°ç¹æ§ãäžå³ã«ç€ºããŠããŸãã
CRãã€ãã¹ãã£ã«ã¿ã®ã«ãããªãåšæ³¢æ°\(f_C\)ã¯ä»¥äžã®å€ãšãªããŸãã
\begin{eqnarray}
f_C&=&\frac{1}{2{\pi}CR}\\
\\
&=&\frac{1}{2{\pi}Ã1Ã10^{-6}Ã1Ã10^{3}}\\
\\
&=&159.154{\cdots}\\
\\
&{\approx}&159{\mathrm{[Hz]}}\tag{5-1}
\end{eqnarray}
äžå³ãèŠããšãã«ãããªãåšæ³¢æ°\(f_C{\;}{\approx}{\;}159{\mathrm{[Hz]}}\)ã§ã²ã€ã³\(|G(j{\omega})|\)ãçŽïŒ3dBãäœçž\({\theta}\)ã45°ã«ãªã£ãŠããããšã確èªã§ããŸãã
ãŸããåšæ³¢æ°\(f\)ãäœããŠã\(1{\;}{\ll}{\;}\displaystyle\frac{1}{(2{\pi}fCR)^2}\)ããšã¿ãªããå Žåãã1ããç¡èŠãããšãã²ã€ã³\(|G(j{\omega})|\)ã¯æ¬¡åŒã§è¡šãããšãã§ããŸãã
\begin{eqnarray}
|G(j{\omega})|&=&\frac{2{\pi}fCR}{\sqrt{1+(2{\pi}fCR)^2}}\\
\\
&=&\frac{1}{\sqrt{\displaystyle\frac{1}{(2{\pi}fCR)^2}+1}}\\
\\
&{\approx}&\frac{1}{\sqrt{\displaystyle\frac{1}{(2{\pi}fCR)^2}}}\\
\\
&{\approx}&2{\pi}fCR\tag{5-2}\\
\end{eqnarray}
äžåŒãããåšæ³¢æ°\(f\)ã10åã«ãªããšãã²ã€ã³\(|G(j{\omega})|\)ã10åã«ãªããŸã(ãã·ãã«è¡šèšã§ã¯ãã\(G_{dB}(j{\omega})=20{\log}_{10}10=20{\mathrm{[dB]}}\)ããšãªããŸã)ãã€ãŸããåšæ³¢æ°ãäœãé åã§ã¯ã20[dB/dec]ã®åŸãã§ã²ã€ã³\(|G(j{\omega})|\)ãå¢å ããŠããŸãã
åæ§ã«ãåšæ³¢æ°fã2åã«ãªããšãã²ã€ã³\(|G(j{\omega})|\)ã2åã«ãªããŸã(ãã·ãã«è¡šèšã§ã¯ãã\(G_{dB}(j{\omega})=20{\log}_{10}2=6{\mathrm{[dB]}}\)ããšãªããŸã)ãã€ãŸããåšæ³¢æ°ãäœãé åã§ã¯ã6[dB/oct]ã®åŸãã§ã²ã€ã³\(|G(j{\omega})|\)ãå¢å ããŠãããšãèšããŸãã
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