{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# No.25 Twin-T型発振回路\n", "大人の科学 Vol.32の付録、電子ブロックを使ってTwin-T型発信回路に挑戦してみました。\n", "\n", "矢印の抵抗とコンデサーが繋がっているところが、Tの字が２つ並んでみえることからTwin-T型と呼ばれています。\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 動かしてオシロスコープでみてみよう\n", "それでは、回路を電子ブロックで組んで、動かしてみましょう。ポーとやさしい音が聞こえてきます。\n", "\n", "\n", "\n", "九州工大のArduinoオシロスコープで波形とスペクトルをみてみましょう。ちょっと歪んでいますが、正弦波に近いに波形を示しています。FFTの結果から周波数は1kHz近くであることが分かります。\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## LTSpiceで計算してみよう\n", "上記の回路をLTSpiceで計算して波形をみてみましょう。\n", "ちょっと分かりづらいですが、ベース電圧V(n004)の青い線とコレクター電圧V(n002)の緑の線で波形が反転しているのが分かります。\n", "\n", "- [No.25.asc](data/No.25.asc)\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Twin-T CRフィルタの計算\n", "抵抗とコンデンサーで作られるTwin-T CRフィルタの計算をしてくれる便利なサイトがありました。\n", "\n", "- [Twin-T CR・フィルタ計算ツール](http://sim.okawa-denshi.jp/TwinTCRtool.php)\n", "\n", "このサイトで計算した伝達関数は、以下のようになります。\n", "\n", "$$\n", "G(s) = \\frac{s^3+2000s^2+12000000s+200000000000}{s^3+114000s^2+412000000s+200000000000\n", "}\n", "$$\n", "\n", "伝達関数のグラフは、バンドパスフィルタがY軸で反転した形を持ち、\n", "振幅が最も小さいところで、位相-180度（反転）となっています。\n", "\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Sageでノッチフィルタの伝達関数をプロット\n", "せっかくSageを使っているので、ノッチフィルタの伝達関数のグラフを求めてみましょう。\n", "\n", "s, R1, R2, R3, C1, C2, C3の変数を定義して、伝達関数Hを定義します。\n", "伝達関数は、\n", "[ツインＴノッチフィルタの解析](https://note.chiebukuro.yahoo.co.jp/detail/n327729?fr=pc_fb_share_n)\n", "を参考にしました。" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "## ノッチフィルタ\n", "s, R1, R2, R3, C1, C2, C3 = var('s R1 R2 R3 C1 C2 C3')\n", "\n", "H =( s^3*R1*R2*R3*C1*C2*C3 + s^2*(R2*R3*C2*C3 + R1*R3*C2*C3) + s*(C2*R3 + C3*R3) + 1)/(s^3*(C1*C2*C3*R1*R2*R3) + s^2*(C1*C3*R1*R2 + C2*C3*R1*R3 + C2*C3*R2*R3 + C1*C2*R1*R3 + C1*C3*R1*R3) + s*(C3*R1 + C3*R2 + C2*R3 + C1*R1 + C3*R3) + 1)" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "(C1*C2*C3*R1*R2*R3*s^3 + (C2*C3*R1*R3 + C2*C3*R2*R3)*s^2 + (C2*R3 + C3*R3)*s + 1)/(C1*C2*C3*R1*R2*R3*s^3 + (C1*C3*R1*R2 + C1*C2*R1*R3 + C1*C3*R1*R3 + C2*C3*R1*R3 + C2*C3*R2*R3)*s^2 + (C1*R1 + C3*R1 + C3*R2 + C2*R3 + C3*R3)*s + 1)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(H)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "抵抗とコンデンサーの値をHに代入し、$s=2 \\pi i f$として、伝達関数の振幅と位相をグラフに表示してみましょう。振幅はtoDb関数デジベルに変換、位相はtoDeg関数で度に変換して方対数グラフでプロットします。\n" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "f = var('f')\n", "vals = {s: 2*i*pi*f, R1: 10^4, R2: 10^4, R3:10^3, C1:10^-7, C2: 10^-8, C3:5*10^-8}\n", "\n", "h(f) = H.subs(vals)\n", "\n", "# 電気ではデジベルで表示するため、toDb関数を定義する\n", "def toDb(v):\n", " return 20*log(abs(v), 10) \n", "\n", "# 位相は以下の様になる\n", "Phi(f) = arctan(imaginary(h(f))/real(h(f)))\n", "\n", "# 位相を度で表示\n", "def toDeg(v):\n", " return v*180/pi " ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false, "scrolled": false }, "outputs": [ { "data": { "image/png": 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Kpk+HhQtNb3QVYgkl+nEWn+jd2zTpf/1120kkVBUUwEMPQdu25kMklKgYi09U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\noEePHng8HsaOHXvK53IcB4/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"text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot(toDb(h(f)), [f, 50, 10^5], scale=\"semilogx\", plot_points=1024, figsize=5)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# プロットに少し時間がかかります。\n", "plot(toDeg(Phi(f)), [f, 50, 10^5], scale=\"semilogx\", plot_points=300, figsize=5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### フィルタ計算ツールとのクロスチェック\n", "以下のサイトのフィルタの伝達関数とのクロスチェックし、一致することを確認しました。\n", "- http://sim.okawa-denshi.jp/TwinTCRtool.php" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "s |--> (s^3 + 2000*s^2 + 12000000*s + 200000000000)/(s^3 + 114000*s^2 + 412000000*s + 200000000000)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "vals = {R1: 10^4, R2: 10^4, R3:10^3, C1:10^-7, C2: 10^-8, C3:5*10^-8}\n", "\n", "hs(s) = H.subs(vals)\n", "show(hs)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "## ツインT型ノッチフィルタの伝達関数の求め方\n", "昔の人はどうやってこんな複雑な伝達関数を計算できたのでしょう？\n", "\n", "Sageを使って、伝達関数の導出を確認してみましょう。\n", "\n", "以下のサイトを参考にしました。\n", "- [ツインＴノッチフィルタの解析](https://note.chiebukuro.yahoo.co.jp/detail/n327729?fr=pc_fb_share_n)\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "次に重ねの理を使って、Vo = Vo1 + Vo2に分解します。ここがポイントです。\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "上部のVo1を以下のようにVT, ZT, ZAに整理します。\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "R1とC1の間の電圧は、テブナンの定理から以下のようになります。\n", "\n", "$$\n", "V_T = V_i \\frac{\\frac{1}{s C_1}}{R_1 + \\frac{1}{s C_1}}\n", "$$\n", "\n", "$$\n", "Z_T = R_2 + \\frac{\\frac{1}{s C_1}}{R_1 + \\frac{1}{s C_1}}\n", "$$\n", "\n", "$$\n", "Z_A = \\frac{1}{s C_3} + \\frac{\\frac{R_3}{s C_2}}{R_3 + \\frac{1}{s C_2}}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "これをSageで定義します。" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Vi/(C1*(R1 + 1/(C1*s))*s)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "R2 + R1/(C1*(R1 + 1/(C1*s))*s)" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "1/(C3*s) + R3/(C2*(R3 + 1/(C2*s))*s)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "s, R1, R2, R3, C1, C2, C3 = var('s R1 R2 R3 C1 C2 C3')\n", "VT, Vi, Vo, Vo1, Vo2, ZT, ZA = var('VT Vi Vo Vo1 Vo2 ZT ZA')\n", "\n", "VT = Vi * (1/(s*C1)/(R1 + 1/(s*C1)))\n", "ZT = R2 + (R1/(s*C1))/(R1+1/(s*C1))\n", "ZA = 1/(s*C3) + (R3/(s*C2))/(R3 + 1/(s*C2))\n", "show(VT); show(ZT); show(ZA)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Vo1は、VTをZTとZAで分圧しているので、以下のように求まります。" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "Vi*(1/(C3*s) + R3/(C2*(R3 + 1/(C2*s))*s))/(C1*(R1 + 1/(C1*s))*(R2 + 1/(C3*s) + R1/(C1*(R1 + 1/(C1*s))*s) + R3/(C2*(R3 + 1/(C2*s))*s))*s)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Vo1 = VT*ZA/(ZT+ZA)\n", "show(Vo1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Vo2は、vo1に以下の置換をすると求まります。\n", "\n", "$$\n", "R1 → 1/sC2, R2 → 1/sC3, R3 → 1/sC1\n", "$$\n", "\n", "$$\n", "C1 → 1/sR3, C2 → 1/sR1, C3→ 1/sR2\n", "$$" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "(R22 + R11/(C11*(R11 + 1/(C11*s))*s))*R33*Vi/((R22 + 1/(C33*s) + R11/(C11*(R11 + 1/(C11*s))*s) + R33/(C22*(R33 + 1/(C22*s))*s))*(R33 + 1/(C22*s)))" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "(R2 + R1/(C1*(R1 + 1/(C1*s))*s))*R3*Vi/((R2 + 1/(C3*s) + R1/(C1*(R1 + 1/(C1*s))*s) + R3/(C2*(R3 + 1/(C2*s))*s))*(R3 + 1/(C2*s)))" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# 循環変換を避けるため、一旦R11, R22, R33, C11, C22, C33で置換\n", "R11, R22, R33, C11, C22, C33 = var('R11 R22 R33 C11 C22 C33')\n", "rel1= {R1: 1/(s*C22), R2: 1/(s*C33), R3: 1/(s*C11), C1: 1/(s*R33), C2: 1/(s*R11), C3 : 1/(s*R22)}\n", "Vo2 = Vo1.subs(rel1)\n", "show(Vo2)\n", "# R1, R2, R3, C1, C2, C3に戻す\n", "rel2 = {R11: R1, R22: R2, R33: R3, C11: C1, C22: C2, C33:C3}\n", "Vo2 = Vo2.subs(rel2)\n", "show(Vo2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "重ね合わせの理より、Voは、Vo1+Vo2で求まります。" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "(R2 + R1/(C1*(R1 + 1/(C1*s))*s))*R3*Vi/((R2 + 1/(C3*s) + R1/(C1*(R1 + 1/(C1*s))*s) + R3/(C2*(R3 + 1/(C2*s))*s))*(R3 + 1/(C2*s))) + Vi*(1/(C3*s) + R3/(C2*(R3 + 1/(C2*s))*s))/(C1*(R1 + 1/(C1*s))*(R2 + 1/(C3*s) + R1/(C1*(R1 + 1/(C1*s))*s) + R3/(C2*(R3 + 1/(C2*s))*s))*s)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "Vo = Vo1 + Vo2\n", "show(Vo)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "伝達関数 $H(s) = \\frac{V_o}{V_i}$を求めると、以下のようになります。" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "((R2 + R1/(C1*(R1 + 1/(C1*s))*s))*R3*Vi/((R2 + 1/(C3*s) + R1/(C1*(R1 + 1/(C1*s))*s) + R3/(C2*(R3 + 1/(C2*s))*s))*(R3 + 1/(C2*s))) + Vi*(1/(C3*s) + R3/(C2*(R3 + 1/(C2*s))*s))/(C1*(R1 + 1/(C1*s))*(R2 + 1/(C3*s) + R1/(C1*(R1 + 1/(C1*s))*s) + R3/(C2*(R3 + 1/(C2*s))*s))*s))/Vi" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "h_hat = Vo/Vi\n", "show(h_hat)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "h_hatに抵抗、コンデンサーの値を代入し、式を整理すると求める伝達関数となります。" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "(s^3 + 2000*s^2 + 12000000*s + 200000000000)/(s^3 + 114000*s^2 + 412000000*s + 200000000000)" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(h_hat.subs(vals).simplify_full())" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "このようにSageを使うと式のまま計算を進め、グラフの表示や値の計算ができます。" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 7.5.1", "language": "", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.13" } }, "nbformat": 4, "nbformat_minor": 2 }