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投稿者 スレッド
webadm
投稿日時: 2007-8-26 16:16
Webmaster
登録日: 2004-11-7
居住地:
投稿: 2965
Maxima超便利
消去法では限界が出てきたので連立方程式を解くのに以前紹介されたMaximaを使ってみることに。

まず連立方程式をたてる。R1を流れる電流をI1,検流計(Rg)を流れる電流をIgとして全体を流れる電流をIとした場合、キルヒホッフの法則から以下が成り立つ。

E = I1*R1 + (I1 - Ig)*R2
E = (I - I1)*R3 + (I - I1 + Ig)*R4
E = I1*R1 + Ig*Rg + (I - I1 + Ig)*R4
E = (I - I1)*R3 - Ig*Rg + (I1 - Ig)*R2

この4つの式をMaximaに与えI,I1,Igを一気に解いてみる。

(%i34) e1: E=I1*R1+(I1-Ig)*R2;
(%o34) E=(I1-Ig)*R2+I1*R1
(%i35) e2: E=(I-I1)*R3+(I-I1+Ig)*R4;
(%o35) E=(-I1+I+Ig)*R4+(I-I1)*R3
(%i36) e3: E=I1*R1+Ig*Rg+(I-I1+Ig)*R4;
(%o36) E=(-I1+I+Ig)*R4+I1*R1+Ig*Rg
(%i37) e4: E=(I-I1)*R3-Ig*Rg+(I1-Ig)*R2;
(%o37) E=(I-I1)*R3+(I1-Ig)*R2-Ig*Rg
(%i38) solve([e1,e2,e3,e4],[I,I1,Ig]);
Dependent equations eliminated: (2)
(%o38) [[I=(E*((R3+Rg)*R4+Rg*R3)+E*R1*(R4+Rg)+R2*(E*(R3+Rg)+E*R1))/(R2*(R1*(R4+R3)+(R3+Rg)*R4+Rg*R3)+R1*((R3+Rg)*R4+Rg*R3)),I1=
(E*((R3+Rg)*R4+Rg*R3)+E*R2*R3)/(R2*(R1*(R4+R3)+(R3+Rg)*R4+Rg*R3)+R1*((R3+Rg)*R4+Rg*R3)),Ig=
(E*R2*R3-E*R1*R4)/(R2*(R1*(R4+R3)+(R3+Rg)*R4+Rg*R3)+R1*((R3+Rg)*R4+Rg*R3))]]

著者の解と分母の記述が異なる。著者の解は綺麗にRgの共通項が因数分解されて見やすいがMaximaで自動で因数分解すると分子は因数分解されるが分母は逆に展開され、おまけに負号が付いてしまっている。

factor(%);
(%o39) [[I=(E*(R3*R4+R1*R4+Rg*R4+R2*R3+Rg*R3+R1*R2+Rg*R2+Rg*R1))/(R2*R3*R4+R1*R3*R4+R1*R2*R4+Rg*R2*R4+Rg*R1*R4+R1*R2*R3+Rg*R2*R3+Rg*R1*R3),I1=
(E*(R3*R4+Rg*R4+R2*R3+Rg*R3))/(R2*R3*R4+R1*R3*R4+R1*R2*R4+Rg*R2*R4+Rg*R1*R4+R1*R2*R3+Rg*R2*R3+Rg*R1*R3),Ig=-
(E*(R1*R4-R2*R3))/(R2*R3*R4+R1*R3*R4+R1*R2*R4+Rg*R2*R4+Rg*R1*R4+R1*R2*R3+Rg*R2*R3+Rg*R1*R3)]]

これをRg, R1*R2, R3*R4の共通項を手で因数分解すると

Ig = (R2*R3 - R1*R4)*E/(R2*R3*R4 + R1*R3*R4 + R1*R2*R4 + R1*R2*R3 + Rg*(R2*R4 + R1*R4 +R2*R3 + R1*R3))
= (R2*R3 - R1*R4)*E/(Rg*(R1 + R2)*(R3 + R4) + R1*R2*(R3 + R4) + R3*R4*(R1 + R2))

と著者の解と同じ結果が得られていることがわかる。

ちなみに著者の解は網目電流法で3つの閉回路について方程式をたててそれを解いている。同様にMaximaでその方程式を解かせてみると、

(%i53) e1: (R1+R3+Rg)*I1+(R1+R3)*I2-R3*I3=0;
(%o53) I1*(R3+R1+Rg)+I2*(R3+R1)-I3*R3=0
(%i54) e2: (R1+R3)*I1+(R1+R2+R3+R4)*I2-(R3+R4)*I3=0;
(%o54) I2*(R4+R3+R2+R1)+I3*(-R4-R3)+I1*(R3+R1)=0
(%i55) e3: -R3*I1-(R3+R4)*I2+(R3+R4)*I3=E;
(%o55) I3*(R4+R3)-I2*(R4+R3)-I1*R3=E
(%i56) solve([e1,e2,e3],[I1,I2,I3]);
(%o56) [[I1=(E*R2*R3-E*R1*R4)/(R3*(R1*(R4+R2+Rg)+R2*R4+Rg*R2)+R1*(R2*R4+Rg*R4)+Rg*R2*R4),I2=
(R3*(E*R4+Rg*E)+E*R1*R4+Rg*E*R4)/(R3*(R1*(R4+R2+Rg)+R2*R4+Rg*R2)+R1*(R2*R4+Rg*R4)+Rg*R2*R4),I3=
(R3*(E*R4+E*R2+Rg*E)+R1*(E*R4+E*R2+Rg*E)+Rg*(E*R4+E*R2))/(R3*(R1*(R4+R2+Rg)+R2*R4+Rg*R2)+R1*(R2*R4+Rg*R4)+Rg*R2*R4)]]

Ig=I1なので

Ig = I1 =(E*R2*R3-E*R1*R4)/(R3*(R1*(R4+R2+Rg)+R2*R4+Rg*R2)+R1*(R2*R4+Rg*R4)+Rg*R2*R4)

これも同様に手で因数分解すると

Ig = (R2*R3 - R1*R4)*E/(R3*R1*R4 + R3*R1*R2 + Rg*R3*R1 + R3*R2*R4 + Rg*R2*R3 + R1*R2*R4 + Rg*R4*R1 + Rg*R2*R4)
= (R2*R3 - R1*R4)*E/(Rg*(R1 + R2)*(R3 + R4) + R1*R2*(R3 + R4) + R3*R4*(R1 + R2))

とこれも著者の解と同じ結果が得られていることがわかる。

Maximaを使えば面倒な連立方程式を解くのも一瞬だし、行列式とかも扱う必要は無い。連立方程式はたてて、あと見やすいように手で因数分解して整理すればよい。

今回Maximaは以前本屋で見つけたブルーバックスの「はじめての数式処理ソフト」に付録で付いているCD-ROMからインストール。



かなりオススメ。

これでかなり後続の問題を解くのが楽しみに。著者の解答例だとすべからく連立方程式の解を行列式で解いている。行列式も今後当たり前に教科書に出てくるのでいずれ習得しないといけないが。慣れの問題である。今から行列式に慣れていれば後の回路解析理論についていけるとの配慮だろう。

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題名 投稿者 日時
   直流回路の演習と実験 webadm 2007-2-1 21:14
     分流回路の実験 webadm 2007-2-2 2:20
       分圧回路の実験 webadm 2007-2-2 16:35
         別の測定器でやってみた webadm 2007-2-3 4:47
           さらなる考察 webadm 2007-2-3 13:41
             そのほか考慮すべき点 webadm 2007-2-14 20:09
     直流回路の演習問題を見て webadm 2007-2-3 20:12
       初等の数学をすっかり忘れている webadm 2007-2-4 2:43
         なんとか出来た webadm 2007-2-4 3:33
     格子状抵抗ネットワークの意味 webadm 2007-2-4 12:13
       インピーダンスの思い出 webadm 2007-2-11 14:25
         dBとか webadm 2007-2-11 20:14
           Re: dBとか webadm 2007-4-1 20:29
             Boonton Radio CorpのThe Notebook webadm 2007-4-10 3:53
       早くも躓く webadm 2007-2-13 2:02
         まだ11ページしか進んでいない webadm 2007-2-14 18:51
           実学と理論 webadm 2007-2-21 21:56
             網目電流法の演習 webadm 2007-2-24 13:47
               やっと解けた webadm 2007-2-28 12:21
                 黒板かホワイトボードが欲しい webadm 2007-3-1 15:13
                   やっとp12問題9 webadm 2007-3-2 18:53
                     問題9完了 webadm 2007-3-3 18:44
                       問題10:正三角錐型ワイヤーフレーム webadm 2007-3-4 4:05
                         問題11:3x3格子状ワイヤーネットワーク webadm 2007-3-6 10:36
                           問題12:正立方体格子ワイヤー網 webadm 2007-3-8 1:37
                             問題14:3つのΔ接続ワイヤー網 webadm 2007-3-8 20:54
                               問題15:アッテネーターぽい回路 webadm 2007-3-23 2:59
                                 問題16:Δ接続・Y接続 webadm 2007-3-24 3:40
                                   問題17、18、19 webadm 2007-3-24 13:55
                                     問題20,21:枝電流法と網目電流法を誤解していた疑い webadm 2007-3-25 3:57
                                       問題22:電流計 webadm 2007-3-26 2:39
                                         問題23:電流計と電圧計 webadm 2007-3-26 3:03
                                           問題24:T型アッテネーター webadm 2007-3-27 2:19
                                             問題25:パイ型アッテネーター webadm 2007-3-30 10:49
                                               Re: 問題25:パイ型アッテネーター webadm 2007-4-12 2:16
                                                 Re: 問題25:パイ型アッテネーター webadm 2007-4-12 12:44
                                               問題26:検流計の内部抵抗を求める webadm 2007-5-4 4:34
                                                 問題27:抵抗計 webadm 2007-5-4 6:28
                                                   問題28:アッテネーター webadm 2007-8-18 0:57
                                                     問題29:ボリューム(アッテネーター) webadm 2007-8-22 9:26
                                                       問題30:ブリッジ回路解析 webadm 2007-8-26 3:05
                                                         Re: 問題30:ブリッジ回路解析 webadm 2007-8-26 13:25
                                                         » Maxima超便利 webadm 2007-8-26 16:16
                                                             数式処理システムの問題点 webadm 2007-8-28 1:19
                                                               問題31:ブリッジの逆接続 webadm 2007-8-30 11:03
                                                                 問題32:剣流計の内部抵抗が0のブリッジ回路 webadm 2007-8-30 11:21
                                                                   問題33:平衡ブリッジによる抵抗測定 webadm 2007-9-1 2:16
                                                                     問題34:ひねったブリッジ回路 webadm 2007-9-5 5:27
                                                                       問題35:内部抵抗がない剣流計を流れる電流 webadm 2007-9-5 23:42
                                                                         問題36:バイパスされたT型アッテネーター webadm 2007-9-14 18:54
                                                                           問題37:Kelvinのダブルブリッジ webadm 2007-9-15 9:10
                                                                             問題38:地絡距離測定 webadm 2007-9-15 10:36
                                                                               問題39:キャパシタの並列接続 webadm 2007-9-21 10:33
                                                                                 問題40:キャパシタの直列接続 webadm 2007-9-21 10:53
                                                                                   問題41:コンデンサの直列接続計算 webadm 2007-9-21 18:54
                                                                                     問題42:コンデンサの直・並列組み合わせ回路計算 webadm 2007-9-22 10:26
                                                                                       問題43:コンデンサのY接続 webadm 2007-9-22 18:31
                                                                                         問題44:コンデンサーのラダー回路 webadm 2007-9-22 19:42
                                                                                           問題45:充電済みコンデンサの並列接続 webadm 2007-9-22 20:21
                                                                                             問題46:充電済みコンデンサの並列接続(その2) webadm 2007-9-23 23:27
                                                                                               問題47:充電済みコンデンサの並列接続(その3) webadm 2007-9-24 0:41
                                                                                                 問題48:変わり種ブリッジ回路 webadm 2007-9-24 1:18
                                                                                                   問題49:電圧を殺す webadm 2007-9-24 20:26
                                                                                                     問題50:偏微分を使った回路解析 webadm 2007-9-27 12:39
                                                                                                       Re: 問題50:偏微分を使った回路解析 webadm 2007-9-27 20:47
                                                                                                       問題51:偏微分を使った回路解析(その2) webadm 2007-9-28 0:45
                                                                                                         問題52:偏微分を使った回路解析(その3) webadm 2007-9-28 2:56
                                                                                                           問題53:ちょっとひねった回路解析 webadm 2007-9-28 21:01
                                                                                                             問題54:ちょっとひねった回路解析(その2) webadm 2007-9-29 10:30
                                                                                                               問題55:ちょっとひねった回路解析(その3) webadm 2007-9-29 17:12
                                                                                                                 問題56:ちょっとひねった回路解析(その4) webadm 2007-9-30 13:24
                                                                                                                   問題57:無限に続く抵抗ラダー回路の解析 webadm 2007-10-3 3:16
                                                                                                                     Re: 問題57:無限に続く抵抗ラダー回路の解析 webadm 2007-10-4 23:30
                                                                                                                       Re: 問題57:無限に続く抵抗ラダー回路の解析 webadm 2007-10-6 18:17
                                                                                                                         Re: 問題57:無限に続く抵抗ラダー回路の解析 webadm 2007-10-6 18:30
                                                                                                                           Re: 問題57:無限に続く抵抗ラダー回路の解析 webadm 2007-10-7 13:30
                                                                                                                             Re: 問題57:無限に続く抵抗ラダー回路の解析 webadm 2007-10-7 20:26
                                                                                                                               問題58:n箇所つながった銅線回路 webadm 2007-10-8 2:12
                                                                                                                                 Re: 問題58:n箇所つながった銅線回路 webadm 2007-10-8 13:24
                                                                                                                                   Re: 問題58:n箇所つながった銅線回路 webadm 2007-10-9 1:27
                                                                                                                                     Re: 問題58:n箇所つながった銅線回路 webadm 2007-10-9 13:18
                                                                                                                                       Re: 問題58:n箇所つながった銅線回路 webadm 2007-10-9 19:19
                                                                                                                                         問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-9 20:33
                                                                                                                                           Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-10 18:06
                                                                                                                                             Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-10 19:12
                                                                                                                                               Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-10 20:02
                                                                                                                                                 Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-16 9:24
                                                                                                                                                 Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-17 0:39
                                                                                                                                                   Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-21 15:12
                                                                                                                                                     Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-25 13:12
                                                                                                                                                       Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-25 19:13
                                                                                                                                                         Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-25 19:28
                                                                                                                                                           Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-26 11:24
                                                                                                                                                             Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-26 11:49
                                                                                                                                                               Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-27 2:05
                                                                                                                                                                 Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-27 11:53
                                                                                                                                                                   Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-28 16:45
                                                                                                                                                                     Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-29 1:07
                                                                                                                                                                       Re: 問題59:漏洩抵抗のある架線を流れる電流 webadm 2007-10-29 19:48
                                                                                                                                                                         問題60:短くした電熱線の消費電力 webadm 2007-10-30 0:38
                                                                                                                                                                           問題61:細く短くした電熱線の消費電力 webadm 2007-10-30 1:18
                                                                                                                                                                             問題62:2つの異なる電球を直列につなぐ webadm 2007-10-31 1:39
                                                                                                                                                                               問題63:電力伝送と整合 webadm 2007-10-31 2:07
                                                                                                                                                                                 問題64:電力伝送と整合(その2) webadm 2007-11-1 1:26
                                                                                                                                                                                   問題65:電力伝送と整合(その3) webadm 2007-11-1 1:56
                                                                                                                                                                                     問題66:電池の起電力と内部抵抗を求める webadm 2007-11-1 2:25
                                                                                                                                                                                       問題67:配線抵抗のある電力伝送回路 webadm 2007-11-7 0:49
                                                                                                                                                                                         問題68:電力伝送とマッチング(電源側) webadm 2007-11-7 1:17
                                                                                                                                                                                           問題69:カロリーと電力 webadm 2007-11-7 1:49
                                                                                                                                                                                             問題70:電力量と比熱と融点の応用(フューズ) webadm 2007-11-8 1:54
                                                                                                                                                                                               問題71:電力量と比熱と融点の応用(フューズ)の続き webadm 2007-11-8 2:02
                                                                                                                                                                                                 問題72:合成温度係数 webadm 2007-11-8 2:20
                                                                                                                                                                                                   問題73:抵抗値による温度測定 webadm 2007-11-8 2:40
                                                                                                                                                                                                     問題74:温度係数 webadm 2007-11-8 3:45
                                                                                                                                                                                                       Re: 問題74:温度係数 webadm 2007-11-8 5:07
                                                                                                                                                                                                         Re: 問題74:温度係数 webadm 2007-11-8 5:38
                                                                                                                                                                                                           問題75:膨張係数と温度係数 webadm 2007-11-10 2:29
                                                                                                                                                                                                             問題76:導線の温度上昇 webadm 2007-11-11 4:15
                                                                                                                                                                                                               問題77:温度補償抵抗回路 webadm 2007-11-11 14:25
                                                                             Re: 問題37:Kelvinのダブルブリッジ webadm 2007-9-25 22:48
                                                 Re: 問題26:検流計の内部抵抗を求める Anonymous 2007-5-25 17:32
                                                   Re: 問題26:検流計の内部抵抗を求める webadm 2007-5-26 12:00
                                     誤植発見 webadm 2007-3-25 4:07

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