From 106eea13c4e6b21242c6bdce4db368d768f0e83f Mon Sep 17 00:00:00 2001 From: Kenryu Shibata Date: Mon, 10 Aug 2026 11:11:13 +0900 Subject: [PATCH] finished a-3 --- assets/a-3/exp2-1-bode.eps | 68 +++++++-------- assets/a-3/exp2-1-bode.tex | 4 +- assets/a-3/exp2-1.gnuplot | 8 +- assets/a-3/exp2-2-bode.eps | 36 ++++---- assets/a-3/exp2-2-bode.tex | 4 +- assets/a-3/exp2-2.gnuplot | 8 +- bibs/a-3.bib | 22 +++++ out/report_a-3.pdf | Bin 394098 -> 481688 bytes out/report_a-3.synctex.gz | Bin 22484 -> 76897 bytes report_a-3.tex | 10 ++- sections/a-3/exp-detail.tex | 16 ++-- sections/a-3/exp-result.tex | 38 ++++++--- sections/a-3/reflection.tex | 145 ++++++++++++++++++++++++++++++++ sections/a-3/theory.tex | 162 +++++++++++++++++++++++++++++++++++- 14 files changed, 430 insertions(+), 91 deletions(-) diff --git a/assets/a-3/exp2-1-bode.eps b/assets/a-3/exp2-1-bode.eps index 7263b7f..831be79 100644 --- a/assets/a-3/exp2-1-bode.eps +++ b/assets/a-3/exp2-1-bode.eps @@ -1,7 +1,7 @@ %!PS-Adobe-2.0 EPSF-2.0 %%Title: exp2-1-bode.tex %%Creator: gnuplot 6.0 patchlevel 4 -%%CreationDate: Fri Jul 17 14:19:04 2026 +%%CreationDate: Mon Jul 20 22:49:34 2026 %%DocumentFonts: %%BoundingBox: 50 50 390 276 %%EndComments @@ -441,7 +441,7 @@ SDict begin [ /Creator (gnuplot 6.0 patchlevel 4) % /Producer (gnuplot) % /Keywords () - /CreationDate (Fri Jul 17 14:19:04 2026) + /CreationDate (Mon Jul 20 22:49:34 2026) /DOCINFO pdfmark end } ifelse @@ -898,14 +898,14 @@ LTb 1.000 UP 1.000 UL LTb -2066 3945 Pls -2664 3628 Pls -2921 3492 Pls -3519 3170 Pls -3777 3034 Pls -4375 2717 Pls -4632 2590 Pls -6241 3510 Pls +2066 3945 Crs +2664 3628 Crs +2921 3492 Crs +3519 3170 Crs +3777 3034 Crs +4375 2717 Crs +4632 2590 Crs +6241 3510 Crs LTw % End plot #1 % Begin plot #2 @@ -917,14 +917,14 @@ LT1 1.000 UL LTb LT1 -2066 3736 Pls -2664 3736 Pls -2921 3725 Pls -3519 3722 Pls -3777 3720 Pls -4375 3718 Pls -4632 3719 Pls -6241 3290 Pls +2066 3736 Box +2664 3736 Box +2921 3725 Box +3519 3722 Box +3777 3720 Box +4375 3718 Box +4632 3719 Box +6241 3290 Box LTw % End plot #2 2.000 UL @@ -1279,14 +1279,14 @@ LTb 1.000 UP 1.000 UL LTb -2066 1778 Pls -2664 1868 Pls -2921 1778 Pls -3519 1760 Pls -3777 1760 Pls -4375 1689 Pls -4632 1599 Pls -6241 1485 Pls +2066 1778 Crs +2664 1868 Crs +2921 1778 Crs +3519 1760 Crs +3777 1760 Crs +4375 1689 Crs +4632 1599 Crs +6241 1485 Crs LTw % End plot #1 % Begin plot #2 @@ -1298,14 +1298,14 @@ LT1 1.000 UL LTb LT1 -2066 973 Pls -2664 973 Pls -2921 973 Pls -3519 973 Pls -3777 973 Pls -4375 964 Pls -4632 937 Pls -6241 1265 Pls +2066 973 Box +2664 973 Box +2921 973 Box +3519 973 Box +3777 973 Box +4375 964 Box +4632 937 Box +6241 1265 Box LTw % End plot #2 2.000 UL diff --git a/assets/a-3/exp2-1-bode.tex b/assets/a-3/exp2-1-bode.tex index 594a43e..b2500d3 100644 --- a/assets/a-3/exp2-1-bode.tex +++ b/assets/a-3/exp2-1-bode.tex @@ -100,7 +100,7 @@ \csname LTb\endcsname%% \put(5814,3510){\makebox(0,0)[r]{\strut{}1nF}}% \put(5814,3290){\makebox(0,0)[r]{\strut{}100uF}}% - \put(209,3400){\rotatebox{-270.00}{\makebox(0,0){\strut{}Impedance (Ohm)}}}% + \put(209,3400){\rotatebox{-270.00}{\makebox(0,0){\strut{}Impedance ($\Omega$)}}}% }% \gplgaddtomacro\gplbacktext{% \csname LTb\endcsname%% @@ -120,7 +120,7 @@ \csname LTb\endcsname%% \put(5814,1485){\makebox(0,0)[r]{\strut{}1nF}}% \put(5814,1265){\makebox(0,0)[r]{\strut{}100uF}}% - \put(209,1375){\rotatebox{-270.00}{\makebox(0,0){\strut{}Phase (deg)}}}% + \put(209,1375){\rotatebox{-270.00}{\makebox(0,0){\strut{}Phase ($\ \textdegree$)}}}% \put(3050,154){\makebox(0,0){\strut{}Frequency (Hz)}}% }% \gplbacktext diff --git a/assets/a-3/exp2-1.gnuplot b/assets/a-3/exp2-1.gnuplot index 9490b94..edec230 100644 --- a/assets/a-3/exp2-1.gnuplot +++ b/assets/a-3/exp2-1.gnuplot @@ -12,21 +12,21 @@ set xrange [1e2:2e6] set multiplot layout 2,1 rowsfirst set xlabel -set ylabel "Impedance (Ohm)" +set ylabel "Impedance ($\\Omega$)" set format x "" set format y "%.1g" -plot "exp2-1_1nF-Z.txt" u 1:2 title "1nF" with points, "exp2-1_100uF-Z.txt" u 1:2 title "100uF" with points +plot "exp2-1_1nF-Z.txt" u 1:2 title "1nF" with points pt 2, "exp2-1_100uF-Z.txt" u 1:2 title "100uF" with points pt 4 unset log y set xtics add ("1k" 1e3, "10k" 1e4, "100k" 1e5, "1M" 1e6) set yrange [-30:120] set ytics 30 set xlabel "Frequency (Hz)" -set ylabel "Phase (deg)" +set ylabel "Phase ($\\ \\textdegree$)" set format y " %.0f" -plot "exp2-1_1nF.txt" u 1:4 title "1nF" with points, "exp2-1_100uF.txt" u 1:4 title "100uF" with points +plot "exp2-1_1nF.txt" u 1:4 title "1nF" with points pt 2, "exp2-1_100uF.txt" u 1:4 title "100uF" with points pt 4 unset multiplot diff --git a/assets/a-3/exp2-2-bode.eps b/assets/a-3/exp2-2-bode.eps index c471949..4526964 100644 --- a/assets/a-3/exp2-2-bode.eps +++ b/assets/a-3/exp2-2-bode.eps @@ -1,7 +1,7 @@ %!PS-Adobe-2.0 EPSF-2.0 %%Title: exp2-2-bode.tex %%Creator: gnuplot 6.0 patchlevel 4 -%%CreationDate: Fri Jul 17 14:19:08 2026 +%%CreationDate: Mon Jul 20 22:49:34 2026 %%DocumentFonts: %%BoundingBox: 50 50 390 276 %%EndComments @@ -441,7 +441,7 @@ SDict begin [ /Creator (gnuplot 6.0 patchlevel 4) % /Producer (gnuplot) % /Keywords () - /CreationDate (Fri Jul 17 14:19:08 2026) + /CreationDate (Mon Jul 20 22:49:34 2026) /DOCINFO pdfmark end } ifelse @@ -855,14 +855,14 @@ LTb 1.000 UP 1.000 UL LTb -2418 2944 Pls -3262 3356 Pls -3626 3526 Pls -4470 3767 Pls -4834 4267 Pls -5678 4004 Pls -6041 3801 Pls -2693 4030 Pls +2418 2944 BoxF +3262 3356 BoxF +3626 3526 BoxF +4470 3767 BoxF +4834 4267 BoxF +5678 4004 BoxF +6041 3801 BoxF +2693 4030 BoxF LTw % End plot #1 2.000 UL @@ -1213,14 +1213,14 @@ LTb 1.000 UP 1.000 UL LTb -2418 872 Pls -3262 791 Pls -3626 785 Pls -4470 785 Pls -4834 771 Pls -5678 1980 Pls -6041 1946 Pls -2693 1764 Pls +2418 872 BoxF +3262 791 BoxF +3626 785 BoxF +4470 785 BoxF +4834 771 BoxF +5678 1980 BoxF +6041 1946 BoxF +2693 1764 BoxF LTw % End plot #1 2.000 UL diff --git a/assets/a-3/exp2-2-bode.tex b/assets/a-3/exp2-2-bode.tex index b523117..af38a32 100644 --- a/assets/a-3/exp2-2-bode.tex +++ b/assets/a-3/exp2-2-bode.tex @@ -98,7 +98,7 @@ \gplgaddtomacro\gplfronttext{% \csname LTb\endcsname%% \put(2266,4030){\makebox(0,0)[r]{\strut{}100mH}}% - \put(209,3400){\rotatebox{-270.00}{\makebox(0,0){\strut{}Impedance (Ohm)}}}% + \put(209,3400){\rotatebox{-270.00}{\makebox(0,0){\strut{}Impedance ($\Omega$)}}}% }% \gplgaddtomacro\gplbacktext{% \csname LTb\endcsname%% @@ -116,7 +116,7 @@ \gplgaddtomacro\gplfronttext{% \csname LTb\endcsname%% \put(2266,1764){\makebox(0,0)[r]{\strut{}100mH}}% - \put(209,1375){\rotatebox{-270.00}{\makebox(0,0){\strut{}Phase (deg)}}}% + \put(209,1375){\rotatebox{-270.00}{\makebox(0,0){\strut{}Phase ($\ \textdegree$)}}}% \put(3807,154){\makebox(0,0){\strut{}Frequency (Hz)}}% }% \gplbacktext diff --git a/assets/a-3/exp2-2.gnuplot b/assets/a-3/exp2-2.gnuplot index 62e445b..1c9953a 100644 --- a/assets/a-3/exp2-2.gnuplot +++ b/assets/a-3/exp2-2.gnuplot @@ -12,20 +12,20 @@ set xrange [1e2:2e6] set multiplot layout 2,1 rowsfirst set xlabel -set ylabel "Impedance (Ohm)" +set ylabel "Impedance ($\\Omega$)" set format x "" set format y "%.1g" -plot "exp2-2-Z.txt" u 1:2 title "100mH" with points +plot "exp2-2-Z.txt" u 1:2 title "100mH" with points pt 5 unset log y set xtics add ("1k" 1e3, "10k" 1e4, "100k" 1e5, "1M" 1e6) set yrange [-100:100] set ytics 40 set xlabel "Frequency (Hz)" -set ylabel "Phase (deg)" +set ylabel "Phase ($\\ \\textdegree$)" set format y " %.0f" -plot "exp2-2.txt" u 1:4 title "100mH" with points +plot "exp2-2.txt" u 1:4 title "100mH" with points pt 5 unset multiplot diff --git a/bibs/a-3.bib b/bibs/a-3.bib index e69de29..b7f7100 100644 --- a/bibs/a-3.bib +++ b/bibs/a-3.bib @@ -0,0 +1,22 @@ +@book{brit-time_constant, + title={ブリタニカ国際大百科事典 小項目電子辞書版}, + author={Ltd., Britannica Japan Co. and Inc., Encyclop\ae{}dia Britannica}, + publisher={Britannica Japan Co., Ltd.}, + year={2016} +} +@inbook{ac-theory:impedance, + title={基礎からの交流理論}, + author={小郷 寛 and 小亀 英己 and 石亀 篤司}, + publisher={電気学会 and オーム社}, + year={2023}, + month={04}, + chapter={3} +} +@inbook{ac-theory:rlc-parallel, + title={基礎からの交流理論}, + author={小郷 寛 and 小亀 英己 and 石亀 篤司}, + publisher={電気学会 and オーム社}, + year={2023}, + month={04}, + pages={77-80} +} diff --git a/out/report_a-3.pdf b/out/report_a-3.pdf index a08848a8604e04318be56a7ace6367d867847440..853d9bd7ff44940705252920d13cb4d249c68c88 100644 GIT binary patch delta 184935 zcmZshQ*b5>(4}MBwryi#+qP}zon&I$wylY6zp-uGv)@+j#lN+C(N%rdH+`zkbGjeG za59<+6Sx8FLPBtE?yeRl4shNZ%bF8$7_10AkC^X4aGnRlks&9)A*8n?pAcOtQsLb) z(EH}9d#bv8JE2j~@FX(Z2tMC^=pXIZP=3}b*zuYb)|Zq}{0H9yU%!9McLL3mYt67{ zD%taLj?WZGcT_!3miz;xI6YFtWIV``cNO#RhYtZVafiug4kXBKf0$hQ?}540-?i}A zHF72^^_$Md$?F#@mQMA24I?{cDVBnme=X%S^%gg~Z0zEmht4xvjJ0dh&%(-?bP&tN zg9S<$t@9iW2nJ)L6QP@OJ+ipvl?lJZ22yZ@1EeY`=t@avC~Y)UD###dsrCF|3~One z1_b~Qw1rS>qXg8)^vyDHswmT`d$?bFMIA)2uo#cH?z(4Ew{JPqFbL_zRHmA&1G3 z{@0T83Nj7`LKtzymzPA1D?vGJM^i}m`%8epZu-0V>`v?pwhu3d4QK 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zmOH8toPpb^rjwpiL5R~SXXF3&%{1j3@yVc_?69-T`Ey^=?F6tsf2O$S{HZ~0U;QJN zE97za74oZFKt69iT%A8|9^T$>zTDkE{@3}L{o(v8ULn`7{;#`-$MetcZ#M5<{p;P= l+sAj{0{d5g0^@|pzdT&rUwwJ}`uW|f{~zd^Z_}5E0ssVfKcD~r diff --git a/report_a-3.tex b/report_a-3.tex index 359c06f..203c227 100644 --- a/report_a-3.tex +++ b/report_a-3.tex @@ -36,7 +36,7 @@ \reportauthor{柴田健琉} \reporttitle{抵抗・コンデンサ・インダクタの特性} \reportdate{2026年}{07月}{17日} -\turnindate{2026年}{07月}{21日} +\turnindate{2026年}{08月}{10日} \schoolyear{2026} \grade{3} \department{電子制御工学科} @@ -44,7 +44,7 @@ \reportid{A-3} \expgroup{4班} \seatingnum{15} -\addResearcher{後藤 } +\addResearcher{後藤 昊大} \addResearcher{佐藤 暖斗} \addResearcher{高橋 健太} \addExperimentDate{2026年 06月 30日} @@ -67,6 +67,8 @@ \input{sections/a-3/exp-detail} \resetrefcounter + \newpage + \input{sections/a-3/exp-result} \resetrefcounter @@ -79,7 +81,9 @@ 今回の実験より以下の事が分かった: \begin{itemize} - \item {} + \item {回路素子の特性は周波数ごとに様々な様子を見せる} + \item {インダクタやコンデンサの定数は有効な周波数での特性図から読み取れる} + \item {フィルタの設計にはQ値や半値幅の設定が重要となる} \end{itemize} \printbibliography[title={参考文献}]{} diff --git a/sections/a-3/exp-detail.tex b/sections/a-3/exp-detail.tex index e918ab9..b78aadb 100644 --- a/sections/a-3/exp-detail.tex +++ b/sections/a-3/exp-detail.tex @@ -49,16 +49,14 @@ ADALM2000を使用してRL直列回路に矩形波を入力し, 応答波形を \subsection{実験2-2} -\Cref{fig:exp22-cd}のRL直列回路を作成し, ファンクションジェネレータを使用して正弦波を回路に入力し, オシロスコープとマルチメータで1 kHzから1 MHzまでの回路のインピーダンス特性を測定する. -オシロスコープで入力信号と抵抗器の電圧との位相差を測定し, マルチメータで抵抗器の電圧を測定する. 電流は抵抗器の抵抗値と電圧から求める. -なお, 入力周波数が10 kHzを超えるとマルチメータの読みが不正確になるのでオシロスコープで10 kHz以上の信号を測定した. +\Cref{fig:exp22-cd}のRL直列回路を作成し, 実験2-1同様の手順でインダクタのインピーダンス特性を測定する. インダクタは100 mHのものを使用した. \begin{figure}[H] \centering \begin{circuitikz} \ctikzset{bipoles/oscope/waveform=sin} - \draw (0,0) node[ground]{} to [vsourcesin, l={Fn}] ++(0,4) -- ++(2,0) to [L, l={$100 \ \text{mH}$}] ++(0,-2) coordinate (vr) to [R, l={$47 \ \Omega$}] ++(0,-2) node[ground]{}; + \draw (0,0) node[ground]{} to [vsourcesin, l={Fn}] ++(0,4) -- ++(2,0) to [american inductor, l={$100 \ \text{mH}$}] ++(0,-2) coordinate (vr) to [R, l={$47 \ \Omega$}] ++(0,-2) node[ground]{}; \draw (vr) to [short, *-] ++(2,0) to [short, -*] ++(0,0.5) node[oscopeshape,anchor=in 1](os){$V_R$} (os.in 2) to [short, *-] ++(0,-0.5) node[ground]{}; \end{circuitikz} \caption{Circuit Diagram for Experiment 2-2} @@ -67,19 +65,21 @@ ADALM2000を使用してRL直列回路に矩形波を入力し, 応答波形を \subsection{実験2-3} -\Cref{fig:exp23-cd}のRLC並列回路を作成し, ADALM2000のネットワークアナライザ機能を使用してコンデンサでの応答特性を測定する. - -抵抗器には$R_i$は$10 \ \text{k}\Omega$を, $R$は$100 \ \text{k}\Omega$を, コンデンサ$C$には1 nFを, そしてインダクタ$L$には100 mHのものを使用した. +\Cref{fig:exp23-cd}のRLC並列回路を作成し, ADALM2000のネットワークアナライザ機能を使用して回路の応答特性を測定する. +なお, 回路中に使用した素子は$R_i$には$10 \ \text{k}\Omega$を, $R$には$100 \ \text{k}\Omega$を, $C$には1 nFを, そして$L$には100 mHのものを使用した. \begin{figure}[H] \centering \begin{circuitikz} \draw (0,0) node[ground]{} to [vsourcesin, l={ADALM2000}] ++(0,4) to [R, l={$R_i$}] ++(4,0) coordinate (a) ++(0,-4) coordinate (b); \draw (a) |- ++(-2, -1) to [R, l={$R$}] ++(0,-2) -| (b); - \draw (a) |- ++(0,-1) to [L, l={$L$}] ++(0,-2) -| (b); + \draw (a) |- ++(0,-1) to [american inductor, l={$L$}] ++(0,-2) -| (b); \draw (a) |- ++(2,-1) to [C, l={$C$}] ++(0,-2) -| (b); \draw (a) ++(0,-1) node[circ]{} (b) ++(0,1) node[circ]{}; \draw (b) node[ground]{}; + \draw (0,0) ++(0,4) ++(1,0) to [short, *-o] ++(0,0.5) node[above]{$1+$}; + \draw (0,0) ++(0,4) ++(3,0) to [short, *-o] ++(0,0.5) node[above]{$2+$}; + \draw (b) ++(0,0.25) to[short, *-o] ++(0.5,0) node[right]{$1-,\ 2-$}; \end{circuitikz} \caption{Circuit Diagram for Experiment 2-3} \label{fig:exp23-cd} diff --git a/sections/a-3/exp-result.tex b/sections/a-3/exp-result.tex index b129c16..1bc4c94 100644 --- a/sections/a-3/exp-result.tex +++ b/sections/a-3/exp-result.tex @@ -7,20 +7,23 @@ \begin{figure}[H] \centering - \begin{minipage}[h]{0.33\textwidth} + \begin{minipage}[h]{0.45\textwidth} \centering - \includegraphics[width=4cm]{./assets/a-3/RC-10kOhm-1nF-1t.png} + \includegraphics[width=6cm]{./assets/a-3/RC-10kOhm-1nF-1t.png} \subcaption{$T_{\text{in}} = 2 \times \tau$} + \label{fig:exp11-res-a} \end{minipage} - \begin{minipage}[h]{0.33\textwidth} + \begin{minipage}[h]{0.45\textwidth} \centering - \includegraphics[width=4cm]{./assets/a-3/RC-10kOhm-1nF-5t.png} + \includegraphics[width=6cm]{./assets/a-3/RC-10kOhm-1nF-5t.png} \subcaption{$T_{\text{in}} = 2 \times 5\tau$} + \label{fig:exp11-res-b} \end{minipage} - \begin{minipage}[h]{0.33\textwidth} + \begin{minipage}[h]{0.90\textwidth} \centering - \includegraphics[width=4cm]{./assets/a-3/RC-10kOhm-1nF-15t.png} + \includegraphics[width=6cm]{./assets/a-3/RC-10kOhm-1nF-15t.png} \subcaption{$T_{\text{in}} = 2 \times 15\tau$} + \label{fig:exp11-res-c} \end{minipage} \caption{Response of RC Circuit, Used $10 \ \text{k}\Omega$ Resistor and $1 \ \text{nF}$ Capacitor} \label{fig:exp11-res} @@ -33,20 +36,23 @@ \begin{figure}[H] \centering - \begin{minipage}[h]{0.33\textwidth} + \begin{minipage}[h]{0.45\textwidth} \centering - \includegraphics[width=4cm]{./assets/a-3/RL-10kOhm-100mH-0.5t.png} + \includegraphics[width=6cm]{./assets/a-3/RL-10kOhm-100mH-0.5t.png} \subcaption{$T_{\text{in}} = 2 \times 0.5\tau$} + \label{fig:exp12-res-a} \end{minipage} - \begin{minipage}[h]{0.33\textwidth} + \begin{minipage}[h]{0.45\textwidth} \centering - \includegraphics[width=4cm]{./assets/a-3/RL-10kOhm-100mH.png} + \includegraphics[width=6cm]{./assets/a-3/RL-10kOhm-100mH.png} \subcaption{$T_{\text{in}} = 2 \times 5\tau$} + \label{fig:exp12-res-b} \end{minipage} - \begin{minipage}[h]{0.33\textwidth} + \begin{minipage}[h]{0.90\textwidth} \centering - \includegraphics[width=4cm]{./assets/a-3/RL-10kOhm-100mH-25t.png} + \includegraphics[width=6cm]{./assets/a-3/RL-10kOhm-100mH-25t.png} \subcaption{$T_{\text{in}} = 2 \times 25\tau$} + \label{fig:exp12-res-c} \end{minipage} \caption{Response of RL Circuit, Used $10 \ \text{k}\Omega$ Resistor and $100 \ \text{mH}$ Inductor} \label{fig:exp12-res} @@ -54,6 +60,8 @@ \subsection{実験2-1} +測定範囲を通じて, インピーダンス・位相ともに一定の傾向を確認した. + \begin{figure}[H] \centering \input{./assets/a-3/exp2-1-bode} @@ -63,6 +71,8 @@ \subsection{実験2-2} +100 kHzから500 kHzの範囲を境にインピーダンスは減少, 位相は$180 \textdegree$変化した. + \begin{figure}[H] \centering \input{./assets/a-3/exp2-2-bode} @@ -72,9 +82,11 @@ \subsection{実験2-3} +ゲインは緩やかな曲線を描き, 15 kHz付近で最大となった. + \begin{figure}[H] \centering - \includegraphics[width=9cm]{./assets/a-3/100kOhm-1nF-100mH.png} + \includegraphics[width=10cm]{./assets/a-3/100kOhm-1nF-100mH.png} \caption{Bode Plot of RLC Parallel Circuit Response} \label{fig:exp23-res} \end{figure} diff --git a/sections/a-3/reflection.tex b/sections/a-3/reflection.tex index 20a2d41..d0c79c5 100644 --- a/sections/a-3/reflection.tex +++ b/sections/a-3/reflection.tex @@ -1,7 +1,152 @@ \section{考察} \subsection{実験1-1} + +時定数とはある変動を加えた系の状態が平衡に達するまでの時間の尺度である. +この時間が短ければ信号は直ぐに平衡状態になり, 長ければ信号の変化はゆるやかとなり, 平衡状態に達するまでの時間が長くなる\supercite{brit-time_constant}. + +\Cref{equ:rc-v}の指数関数項の指数が-1になる時間が時定数$\tau$である, よって時定数は\cref{equ:tc-c}となる. +\begin{equation} + \label{equ:tc-c} + \begin{split} + v_C(\tau) &= v_i \left(1 - e^{-\frac{\tau}{RC}}\right) = v_i \left(1 - e^{-1}\right) \\ + \tau &= RC + \end{split} +\end{equation} + +\Cref{equ:tc-c}から時定数は抵抗値とキャパシタンスに比例することが分かる. + +今回の実験では, 時定数は$10 \ \text{k}\Omega \times 1 \ \text{nF} = 10 \ \mu\text{s}$となる. +実際に\cref{fig:exp11-res-b}では$10 \ \mu\text{s}$で$v_i$の$(1 - e^{-1})\text{倍} \approx 60\%$まで充電されている. + \subsection{実験1-2} + +\Cref{equ:rl-v}の指数関数項の指数が-1になる時間が時定数$\tau$なので, 時定数は\cref{equ:tc-l}となる. +\begin{equation} + \label{equ:tc-l} + \begin{split} + v_L(\tau) &= v_i e^{-\frac{R}{L}\tau} = v_i e^{-1} \\ + \tau &= \frac{L}{R} + \end{split} +\end{equation} + +\Cref{equ:tc-l}から時定数はインダクタンスに比例し, 抵抗値に反比例することが分かる. + +今回の実験では, 時定数は$\frac{100 \ \text{mH}}{10 \ \text{k}\Omega} = 10 \ \mu\text{s}$となる. +実際に\cref{fig:exp12-res-b}では$10 \ \mu\text{s}$で$v_i$の$(1 - e^{-1})\text{倍} \approx 60\%$まで充電されている. + \subsection{実験2-1} + +\Cref{fig:exp21-cd}の回路内のシャント抵抗$R$を無視するには, $R \ll \frac{1}{2\pi{}fC}$を満たす周波数領域内である必要がある. +コンデンサのインピーダンスがシャント抵抗の$\frac{1}{100}$倍になる場合を十分小さいとみなし, 1 nFと100 $\mu$Fのコンデンサでは\cref{tab:cap-f-range}で示した範囲となった. + +\begin{table}[H] + \centering + \caption{Frequency Range for a Shunt Resistor being Insignificant in the Circuit} + \label{tab:cap-f-range} + \begin{tabular}{r|l} + \hline + Capacitor & Range \\ + \hline + 1 nF & $f < 33.9 \ \text{kHz}$ \\ + 100 $\mu$F & $f < 0.338 \ \text{Hz}$ \\ + \hline + \end{tabular} +\end{table} + +\cref{tab:cap-f-range}から分かるように, 100 $\mu$Fでは計測範囲の下限1 kHzより更に低い周波数でシャント抵抗の抵抗成分が主となるため, 実験結果\cref{fig:exp21-res}の100 $\mu$Fのインピーダンスは一定となり, 電流と電圧は同相を示す理由に説明がつく. なので, 結果からこのコンデンサの静電容量の推定は不可能となる. + +一方, 1 nFのコンデンサは計測範囲内に納まっているので結果から静電容量の推定が可能である. +実験結果\cref{fig:exp21-res}から, 目測できる範囲で計算すると\cref{equ:calc-capacitance-1nf}となる. +\begin{equation} + \label{equ:calc-capacitance-1nf} + \begin{split} + f &= 5 \ \text{kHz} \\ + Z &= \frac{1}{2\pi{}fC} \approx 3 \times 10^4 \\ + C &= \frac{1}{2\pi{}fZ} \approx 1.061 \ \text{nF} + \end{split} +\end{equation} +これは定格容量の$+6.1\%$の誤差となる. + \subsection{実験2-2} + +実験結果\cref{fig:exp22-res}より, 目測できる範囲でインダクタンスを計算すると\cref{equ:calc-inductance-100mh}となる. +\begin{equation} + \label{equ:calc-inductance-100mh} + \begin{split} + f &= 5 \ \text{kHz} \\ + Z &= 2\pi{}fL \approx 3 \times 10^3 \\ + L &= \frac{Z}{2\pi{}f} \approx 95.49 \ \text{mH} + \end{split} +\end{equation} +これは定格インダクタンスの$-4.51\%$の誤差となる. + +また実験結果より, 100 kHzから500 kHzの間でインピーダンスが減少する傾向に転じた. また, 位相では180度増加した. +この現象は現実のインダクターはRLC並列回路の一種であるとすれば説明できる. +銅線は値が低くとも抵抗値を持ち, 巻かれているので使用する長さも長くなり抵抗値が増える. +また, 巻線間のピッチが短くなるので銅線どうしでコンデンサのように振舞うことがある. +この仮想的なコンデンサが持つ静電容量を寄生キャパシタンスという. + +マルチメータで測定したインダクターの端子間抵抗値は138 $\Omega$であったので等価回路は\cref{equ:l-equivalent-cd}となる. +\begin{figure}[tbh] + \centering + \begin{circuitikz} + \draw (0,0) to[R={$138 \ \Omega$}, o-*] ++(2,0) coordinate (p); + \draw (p) -- ++(0,1) to[american inductor={$100 \ \text{mH}$}] ++(2,0) -- ++(0,-1); + \draw (p) -- ++(0,-1) to[C={$C$}] ++(2,0) -- ++(0,1); + \draw (p) ++(2,0) to[short, *-o] ++(1,0); + \end{circuitikz} + \caption{Equivalent Circuit of Inductor} + \label{equ:l-equivalent-cd} +\end{figure} + +この回路のインピーダンスは\cref{equ:r-lc-z}となり, 共振周波数は\cref{equ:resfreq-r-lc}となる. +\begin{equation} + \label{equ:r-lc-z} + \dot{Z} = R + \frac{1}{j\omega{}C + \frac{1}{j\omega{}L}} = R - j\frac{\omega{}L}{\omega{}^{2}CL - 1} +\end{equation} +\begin{equation} + \label{equ:resfreq-r-lc} + \begin{split} + Z &= |\dot{Z}| = \sqrt{R^2 + \left(\frac{\omega{}L}{\omega{}^{2}CL - 1}\right)^{2}} \\ + Z_\text{max} &= Z(f_a) = \infty \\ + 0 &= \omega{}^{2}LC - 1 \\ + \omega{}^{2} &= \frac{1}{LC} \\ + \omega{} &= \frac{1}{\sqrt{LC}} = 2\pi{}f_a \\ + f_a &= \frac{1}{2\pi{}\sqrt{LC}} + \end{split} +\end{equation} + +$100 \ \text{kHz} < f_a < 500 \ \text{kHz}$を満たす寄生キャパシタンスは$1.061 \ \text{pF} < C < 26.53 \ \text{pF}$となる. +また, $f_a$を\\中間の300 kHzとすると寄生キャパシタンスは2.947 pFとなる. + +この共振周波数を境に回路が誘導性を示すか容量性を示すかが決まってくる. 今回の実験では共振周波数より低い周波数では回路は誘導性を示し, 電流は電圧より90\textdegree 遅れ, 共振周波数より高い周波数では回路は容量性を示し, 電流は電圧より90\textdegree 進む. + \subsection{実験2-3} + +今回の実験のパラメータから, 抵抗成分は信号抵抗と主抵抗の合成並列抵抗であることに注意して計算すると, 共振周波数は\cref{equ:res-f-exp23}となり, Q値は\cref{equ:qf-exp23}となった. +\begin{equation} + \label{equ:res-f-exp23} + f_r = \frac{1}{2\pi\sqrt{100 \ \text{mH} \times 1 \ \text{nF}}} = 15.915 \ \text{kHz} +\end{equation} +\begin{equation} + \label{equ:qf-exp23} + \begin{split} + r &= \frac{1}{R^{-1} + {R_i}^{-1}} \approx 9090.91 \ \Omega \\ + Q &= r \times \sqrt{\frac{1 \ \text{nF}}{100 \ \text{mH}}} = 0.9091 + \end{split} +\end{equation} +そして, これらの値から半値幅の理論値を計算すると\cref{equ:bw-exp23-theory}となる. +\begin{equation} + \label{equ:bw-exp23-theory} + \text{BW}_{\text{theory}} = \frac{f_r}{Q} \approx 17.51 \ \text{kHz} +\end{equation} + +実験結果\Cref{fig:exp23-res}から目測で周波数を読み, 半値幅を概算すると\cref{equ:bw-exp23-measured}となる. +\begin{equation} + \label{equ:bw-exp23-measured} + \text{BW} = f_2 - f_1 = 26 \ \text{kHz} - 9 \ \text{kHz} = 17 \ \text{kHz} +\end{equation} + +詳細な周波数を取得していなかったので500 Hz以上の誤差が出てしまったが, 共振周波数・半値幅ともに理論値とおおかた一致した. +これら共振周波数や半値幅はフィルタ回路の設計で, どの周波数の信号をどれだけの振幅で通過させるかを決定するのに重要な指標である. diff --git a/sections/a-3/theory.tex b/sections/a-3/theory.tex index 7753e33..a6cc8ea 100644 --- a/sections/a-3/theory.tex +++ b/sections/a-3/theory.tex @@ -2,10 +2,166 @@ \subsection{コンデンサ} -\subsubsection{時定数} +コンデンサとは電圧を加えると電荷を保持する素子である. +コンデンサは電圧$v(t)$に比例して電流$i(t)$の時間微分である電荷$q(t)$を保持するので\cref{equ:cap-i}が成り立つ. +\begin{equation} + \label{equ:cap-i} + \frac{dq}{dt} = i(t) = C \frac{dv}{dt} +\end{equation} +\Cref{equ:cap-i}を電圧$v(t)$で示すと\cref{equ:cap-v}が得られる. +\begin{equation} + \label{equ:cap-v} + v(t) = \frac{1}{C} \int^{t}_{0} i(\delta)d\delta +\end{equation} +ここで, $C$はコンデンサの容量を表す\supercite{ac-theory:impedance}. + +抵抗器とコンデンサの直列回路をRC回路という. +この回路の時刻$t$における電圧を考える. +抵抗値$R$を持つ抵抗器と容量$C$を持つコンデンサの直列回路に電流$i(t)$を流した際の回路全体の電圧$v(t)$は\cref{equ:rc-v-1}となる. +\begin{equation} + \label{equ:rc-v-1} + v(t) = Ri(t) + \frac{1}{C}\int^{t}_{0}i(\delta)d\delta +\end{equation} + +ここで, \cref{equ:cap-i}を用いて\cref{equ:rc-v-1}を書き換えると一階線形微分方程式\cref{equ:rc-v-2}を得る. +\begin{equation} + \label{equ:rc-v-2} + v_i = RC\frac{d v_{C}}{dt} + v_{C}(t) +\end{equation} +\Cref{equ:rc-v-2}の入力電圧$v_i$は直流電圧で任意の定数とする. + +\Cref{equ:rc-v-2}の微分方程式を変数分離形として一般解を求めると\cref{equ:rc-v-solv}となる. +\begin{equation} + \label{equ:rc-v-solv} + \begin{aligned} + RC\frac{d v_{C}}{dt} &= v_i - v_{C} \\ + \frac{d v_{C}}{v_i - v_{C}} &= \frac{1}{RC} dt \\ + \int \frac{d v_{C}}{v_i - v_{C}} &= \int \frac{1}{RC} dt \\ + \log{(v_i - v_C)} + C_v &= -\frac{t}{RC} + C_t \\ + e^{C_v}(v_i - v_C) &= e^{C_t} e^{-\frac{t}{RC}} \\ + v_C &= v_i - \left(e^{C_t - C_v}\right) e^{-\frac{t}{RC}} + \end{aligned} +\end{equation} + +ここで$e^{C_t - C_v}$を$C_0$と置き, $v_{C}(0) = 0$という条件を満たす$C_0$を与えると\cref{equ:rc-v}となる. +\begin{align} + \label{equ:rc-v} + v_C(t) = v_i - C_{0}e^{-\frac{t}{RC}} = v_i\left(1 - e^{-\frac{t}{RC}}\right) && \text{where, } \ C_0 = v_i +\end{align} + \subsection{インダクタ} -\subsubsection{時定数} +インダクタは電流を磁束としてエネルギーを保持する素子である. +磁束が電流$i(t)$に比例する場合のインダクタにかかる電圧は\cref{equ:l-v}で示される. +\begin{equation} + \label{equ:l-v} + v(t) = L\frac{di}{dt} +\end{equation} +ここで, $L$はインダクタンスを表す\supercite{ac-theory:impedance}. -\subsection{共振・半幅値} +抵抗器とインダクタの直列回路をRL回路という. この回路の時刻$t$における電圧を考える. +抵抗値$R$を持つ抵抗器とインダクタンス$L$を持つインダクタの直列回路に電流$i(t)$を流した際の回路全体の電圧$v(t)$は\cref{equ:rl-v-1}となる. +\begin{equation} + \label{equ:rl-v-1} + v(t) = Ri(t) + L\frac{di}{dt} +\end{equation} + +\Cref{equ:rl-v-1}の$v(t)$を直流電圧$v_i$の任意の定数関数とすると\cref{equ:rl-v-2}の一階微分方程式が得られる. +\begin{equation} + \label{equ:rl-v-2} + v_i = Ri + L\frac{di}{dt} +\end{equation} + +\Cref{equ:rl-v-2}の微分方程式を変数分離形として一般解を求めると\cref{equ:rl-v-solv}となる. +\begin{equation} + \label{equ:rl-v-solv} + \begin{aligned} + L\frac{di}{dt} &= v_i - Ri = v_L \\ + \frac{di}{v_i - Ri} &= \frac{dt}{L} \\ + \int \frac{di}{v_i - Ri} &= \int \frac{dt}{L} \\ + -\frac{1}{R}\log{(v_i - Ri)} + C_i &= \frac{1}{L}t + C_t \\ + \log{(v_i - Ri)} + C_i &= -\frac{R}{L}t + C_t \\ + e^{C_i}(v_i - Ri) &= e^{C_t}e^{-\frac{R}{L}t} \\ + v_i - Ri &= e^{C_t - C_i}e^{-\frac{R}{L}t} = v_L \\ + i &= \frac{v_i}{R} - \frac{1}{R}e^{C_t - C_i}e^{-\frac{R}{L}t} + \end{aligned} +\end{equation} +ここで, $e^{C_t - C_i}$を$C_0$とし, $i(0) = 0$という条件を満たす$C_0$を与えると\cref{equ:rl-v}となる. +\begin{align} + \label{equ:rl-i} i(t) &= \frac{v_i}{R} - \frac{C_0}{R}e^{-\frac{R}{L}t} = \frac{v_i}{R}\left(1 - e^{-\frac{R}{L}t}\right) && \text{where, } \ C_0 = v_i \\ + \label{equ:rl-v} v_{L}(t) &= C_0 e^{-\frac{R}{L}t} = v_i e^{-\frac{R}{L}t} && \text{where, } \ C_0 = v_i +\end{align} + +\subsection{共振周波数・Q値・半値幅} + +RLCの交流回路網でインピーダンスが純抵抗性・純コンダクタンスを示す時の周波数を共振周波数という. + +直列回路の場合, インピーダンスは\cref{equ:rlc-s-z}となる. +\begin{equation} + \label{equ:rlc-s-z} + Z = \sqrt{R^2 + \left(\omega{}L - \frac{1}{\omega{}C}\right)^2} +\end{equation} +このインピーダンスが純抵抗性を示すのは括弧内の値が0になれば良いので共振周波数$f_r$は\cref{equ:rlc-s-fr}となる\supercite{ac-theory:impedance}. +\begin{equation} + \label{equ:rlc-s-fr} + f_r = \frac{\omega{}_r}{2\pi} = \frac{1}{2\pi{}\sqrt{LC}} +\end{equation} + +並列回路の場合, アドミタンスは\cref{equ:rlc-p-z}となる. +\begin{equation} + \label{equ:rlc-p-z} + Y = \sqrt{\frac{1}{R^2} + \left(\omega{}C - \frac{1}{\omega{}L}\right)^2} +\end{equation} +このアドミタンスが純コンダクタンス性を示すのは括弧内の値が0になれば良いので共振周波数$f_a$は\cref{equ:rlc-p-fa}となる\supercite{ac-theory:rlc-parallel}. +\begin{equation} + \label{equ:rlc-p-fa} + f_a = \frac{\omega{}_a}{2\pi} = \frac{1}{2\pi\sqrt{LC}} +\end{equation} + +Q値とはQuality Factorのことで, 共振時の共振のよさを示す指標である. +直列回路ではインダクタのインピーダンスと抵抗器の抵抗値との比がコンデンサのインピーダンスと抵抗器の抵抗値との比が等しい時の値で, +並列回路ではインダクタのアドミタンスと抵抗器のコンダクタンスとの比がコンデンサのアドミタンスと抵抗器のコンダクタンスとの比が等しい時の値である\supercite{ac-theory:impedance}\supercite{ac-theory:rlc-parallel}. +\begin{equation} + \label{equ:q-factor-s} + Q_{\text{Series}} = \frac{\omega{}_r L}{R} = \frac{1}{\omega{}_r CR} = \frac{1}{R} \frac{L}{\sqrt{LC}} = \frac{1}{R} \frac{\sqrt{LC}}{C} = \frac{1}{R}\sqrt{\frac{L}{C}} +\end{equation} +\begin{equation} + \label{equ:q-factor-p} + Q_{\text{Parallel}} = \omega{}_a CR = \frac{R}{\omega{}_a L} = R \frac{C}{\sqrt{LC}} = R \frac{\sqrt{LC}}{L} = R \sqrt{\frac{C}{L}} +\end{equation} + +半値幅とは共振周波数を中心に電力が半分, または電流・電圧のゲインが$\frac{1}{\sqrt{2}}$倍になる2つの周波数の差を表す値で, 共振が鋭ければ鋭いほどこの値は小さくなる. +直列回路の場合, 半値幅BWは\cref{equ:bw-s}となる\supercite{ac-theory:impedance}. +\begin{equation} + \label{equ:bw-s} + \begin{split} + Z_r &= R \\ + \sqrt{2}Z_r &= \sqrt{R^2 + \left(\omega{}L - \frac{1}{\omega{}C}\right)^2} \\ + 2{Z_r}^2 &= 2R^2 = R^2 + \left(\omega{}L - \frac{1}{\omega{}C}\right)^2 \\ + \pm{}R &= \left(\omega{}L - \frac{1}{\omega{}C}\right) = \frac{\omega{}^{2}LC - 1}{\omega{}C} \\ + 0 &= \omega{}^{2}LC \pm \omega{}RC - 1, (\omega{} > 0) \\ + \omega{} &= \frac{\pm{}RC + \sqrt{D}}{2LC} \\ + \omega{}_1 &= \frac{-RC + \sqrt{D}}{2LC} \\ + \omega{}_2 &= \frac{RC + \sqrt{D}}{2LC} \\ + \text{BW} &= f_2 - f_1 = \frac{\omega{}_2}{2\pi} - \frac{\omega{}_1}{2\pi} = \frac{R}{2\pi{}L} = \frac{f_r}{Q_{\text{Series}}} + \end{split} +\end{equation} + +\newpage + +並列回路の場合, 半値幅BWは\cref{equ:bw-p}となる\supercite{ac-theory:rlc-parallel}. +\begin{equation} + \label{equ:bw-p} + \begin{split} + Y_r &= \frac{1}{R} = G \\ + \sqrt{2}Y_r &= \sqrt{G^2 + \left(\omega{}C - \frac{1}{\omega{}L}\right)^2} \\ + 2{Y_r}^2 &= 2G^2 = G^2 + \left(\omega{}C - \frac{1}{\omega{}L}\right)^2 \\ + \pm{}G &= \frac{\omega{}^{2}LC - 1}{\omega{}L} \\ + 0 &= \omega{}^{2}LC \pm \omega{}LG - 1, (\omega{} > 0) \\ + \omega{} &= \frac{\pm{}LG + \sqrt{D}}{2LC} \\ + \omega{}_1 &= \frac{-LG + \sqrt{D}}{2LC} \\ + \omega{}_2 &= \frac{LG + \sqrt{D}}{2LC} \\ + \text{BW} &= f_2 - f_1 = \frac{\omega{}_2}{2\pi} - \frac{\omega{}_1}{2\pi} = \frac{G}{2\pi{}C} = \frac{1}{2\pi{}RC} = \frac{f_a}{Q_{\text{Parallel}}} + \end{split} +\end{equation}