次ぎもΔ-Y変換の応用。



図の上部の回路網が下部の回路網と等価となるRa,Rb,Rcを求めよというもの。また端子2-2'間を開放した場合と短絡した場合のそれぞれの端子1-1'間のインピーダンスを求めよというもの。

著者の図では抵抗値のみ記載されていて、下部の回路の抵抗名もR1,R2,R3としているが、回路方程式を立てて解けるように同一の抵抗値を持つものには同一の抵抗名を割り当てた。それによって下部の回路図の抵抗名を便宜上区別するためにRa,Rb,Rcとした。

ストラテジーとしては上部の回路と下部の回路からRa,Rb,Rcに関する連立方程式を立ててそれを解く方法をとってみることにする。

著者のようにΔ-Y変換を利用して最終的にπ型ネットワーク回路にして数値を求めるのではなく、数式を導出して最後に抵抗値を代入して同じ結果を得る方法ととってみようと思う。

これには端子1-1'間に流れる電流が上部と下部の回路で等しいことを利用する。また端子2-2'が開放された回路と閉じられた回路についてもそれぞれ方程式をたてることによってRa,Rb,Rcとそれぞれのケースにおける端子1-1'間に流れる電流を同時に解いてしまおうという大胆な試み。端子1-1'間に流れる電流がわかれば端子1-1'にくわえられる電圧とからインピーダンスが決まるのでそれも同時に解くことが出来る。一石二鳥、いや一石三鳥である。

そんなうまいこといくのだろうか？



著者の図を書き直して二次元のネットリストとして書き直すと上のようになる。左が2-2'端子が開放、右が2-2'端子が短絡したもの。

それぞれ網目電流法で回路方程式をたてれば回路全体に流れる電流が求まりそこからインピーダンスも求めることができる。

(%i53) e1;
(%o53) Io=Io2+Io1
(%i54) e2;
(%o54) (Io1-Io3)*R1=E
(%i55) e3;
(%o55) (Io3-Io5)*R1+(Io3-Io1)*R1+Io3*R1=0
(%i56) e4;
(%o56) (Io5-Io7)*R2+Io5*R2+(Io5-Io3)*R1=0
(%i57) e5;
(%o57) (Io7+Io6)*R4+Io7*R3+(Io7-Io5)*R2+Io7*R2=0
(%i58) e6;
(%o58) (Io7+Io6)*R4+(Io6-Io4)*R4+Io6*R3=0
(%i59) e7;
(%o59) (Io4-Io6)*R4+(Io4-Io2)*R4+Io4*R3=0
(%i60) e8;
(%o60) (Io2-Io4)*R4+Io2*R3=E
(%i61) e9;
(%o61) Zo=E/Io

という回路方程式が左上の回路で成り立つので、これをIo,Io1,Io2,Io3,Io4,Io5,Io6,Io7,Zoに関する９元連立方程式として解くと

(%i62) solve([e1,e2,e3,e4,e5,e6,e7,e8,e9],[Io,Io1,Io2,Io3,Io4,Io5,Io6,Io7,Zo]);
(%o62) [[Io=(((24*E*R2+8*E*R1)*R3+9*E*R2^2+6*E*R1*R2+E*R1^2)*R4^3+
((60*E*R2+20*E*R1)*R3^2+(54*E*R2^2+48*E*R1*R2+6*E*R1^2)*R3+18*E*R1*R2^2+6*E*R1^2*R2)*R4^2+
((36*E*R2+12*E*R1)*R3^3+(45*E*R2^2+40*E*R1*R2+5*E*R1^2)*R3^2+(24*E*R1*R2^2+8*E*R1^2*R2)*R3)*R4+
(6*E*R2+2*E*R1)*R3^4+(9*E*R2^2+8*E*R1*R2+E*R1^2)*R3^3+(6*E*R1*R2^2+2*E*R1^2*R2)*R3^2)/(
((16*R1*R2+4*R1^2)*R3+6*R1*R2^2+2*R1^2*R2)*R4^3+((40*R1*R2+10*R1^2)*R3^2+(36*R1*R2^2+12*R1^2*R2)*R3)*R4^2+
((24*R1*R2+6*R1^2)*R3^3+(30*R1*R2^2+10*R1^2*R2)*R3^2)*R4+(4*R1*R2+R1^2)*R3^4+(6*R1*R2^2+2*R1^2*R2)*R3^3),Io1
=(((24*E*R2+8*E*R1)*R3+9*E*R2^2+3*E*R1*R2)*R4^3+((60*E*R2+20*E*R1)*R3^2+(54*E*R2^2+24*E*R1*R2)*R3)*R4^2+
((36*E*R2+12*E*R1)*R3^3+(45*E*R2^2+20*E*R1*R2)*R3^2)*R4+(6*E*R2+2*E*R1)*R3^4+(9*E*R2^2+4*E*R1*R2)*R3^3)/(
((16*R1*R2+4*R1^2)*R3+6*R1*R2^2+2*R1^2*R2)*R4^3+((40*R1*R2+10*R1^2)*R3^2+(36*R1*R2^2+12*R1^2*R2)*R3)*R4^2+
((24*R1*R2+6*R1^2)*R3^3+(30*R1*R2^2+10*R1^2*R2)*R3^2)*R4+(4*R1*R2+R1^2)*R3^4+(6*R1*R2^2+2*R1^2*R2)*R3^3),Io2
=((3*E*R2+E*R1)*R4^3+((24*E*R2+6*E*R1)*R3+18*E*R2^2+6*E*R1*R2)*R4^2+
((20*E*R2+5*E*R1)*R3^2+(24*E*R2^2+8*E*R1*R2)*R3)*R4+(4*E*R2+E*R1)*R3^3+(6*E*R2^2+2*E*R1*R2)*R3^2)/(
((16*R2+4*R1)*R3+6*R2^2+2*R1*R2)*R4^3+((40*R2+10*R1)*R3^2+(36*R2^2+12*R1*R2)*R3)*R4^2+
((24*R2+6*R1)*R3^3+(30*R2^2+10*R1*R2)*R3^2)*R4+(4*R2+R1)*R3^4+(6*R2^2+2*R1*R2)*R3^3),Io3=(
((8*E*R2+4*E*R1)*R3+3*E*R2^2+E*R1*R2)*R4^3+((20*E*R2+10*E*R1)*R3^2+(18*E*R2^2+12*E*R1*R2)*R3)*R4^2+
((12*E*R2+6*E*R1)*R3^3+(15*E*R2^2+10*E*R1*R2)*R3^2)*R4+(2*E*R2+E*R1)*R3^4+(3*E*R2^2+2*E*R1*R2)*R3^3)/(
((16*R1*R2+4*R1^2)*R3+6*R1*R2^2+2*R1^2*R2)*R4^3+((40*R1*R2+10*R1^2)*R3^2+(36*R1*R2^2+12*R1^2*R2)*R3)*R4^2+
((24*R1*R2+6*R1^2)*R3^3+(30*R1*R2^2+10*R1^2*R2)*R3^2)*R4+(4*R1*R2+R1^2)*R3^4+(6*R1*R2^2+2*R1^2*R2)*R3^3),Io4
=((3*E*R2+E*R1)*R4^3+((11*E*R2+3*E*R1)*R3+12*E*R2^2+4*E*R1*R2)*R4^2+
((4*E*R2+E*R1)*R3^2+(6*E*R2^2+2*E*R1*R2)*R3)*R4)/(((16*R2+4*R1)*R3+6*R2^2+2*R1*R2)*R4^3+
((40*R2+10*R1)*R3^2+(36*R2^2+12*R1*R2)*R3)*R4^2+((24*R2+6*R1)*R3^3+(30*R2^2+10*R1*R2)*R3^2)*R4+(4*R2+R1)*
R3^4+(6*R2^2+2*R1*R2)*R3^3),Io5=(4*E*R3*R4^3+(10*E*R3^2+12*E*R2*R3)*R4^2+(6*E*R3^3+10*E*R2*R3^2)*R4+E*
R3^4+2*E*R2*R3^3)/(((16*R2+4*R1)*R3+6*R2^2+2*R1*R2)*R4^3+((40*R2+10*R1)*R3^2+(36*R2^2+12*R1*R2)*R3)*R4^2
+((24*R2+6*R1)*R3^3+(30*R2^2+10*R1*R2)*R3^2)*R4+(4*R2+R1)*R3^4+(6*R2^2+2*R1*R2)*R3^3),Io6=(
(3*E*R2+E*R1)*R4^3+((E*R2+E*R1)*R3+6*E*R2^2+2*E*R1*R2)*R4^2-E*R2*R3^2*R4)/(
((16*R2+4*R1)*R3+6*R2^2+2*R1*R2)*R4^3+((40*R2+10*R1)*R3^2+(36*R2^2+12*R1*R2)*R3)*R4^2+
((24*R2+6*R1)*R3^3+(30*R2^2+10*R1*R2)*R3^2)*R4+(4*R2+R1)*R3^4+(6*R2^2+2*R1*R2)*R3^3),Io7=-(
(3*E*R2+E*R1)*R4^3-6*E*R2*R3*R4^2-5*E*R2*R3^2*R4-E*R2*R3^3)/(((16*R2+4*R1)*R3+6*R2^2+2*R1*R2)*R4^3+
((40*R2+10*R1)*R3^2+(36*R2^2+12*R1*R2)*R3)*R4^2+((24*R2+6*R1)*R3^3+(30*R2^2+10*R1*R2)*R3^2)*R4+(4*R2+R1)*
R3^4+(6*R2^2+2*R1*R2)*R3^3),Zo=(((16*R1*R2+4*R1^2)*R3+6*R1*R2^2+2*R1^2*R2)*R4^3+
((40*R1*R2+10*R1^2)*R3^2+(36*R1*R2^2+12*R1^2*R2)*R3)*R4^2+((24*R1*R2+6*R1^2)*R3^3+(30*R1*R2^2+10*R1^2*R2)*R3^2)*
R4+(4*R1*R2+R1^2)*R3^4+(6*R1*R2^2+2*R1^2*R2)*R3^3)/(((24*R2+8*R1)*R3+9*R2^2+6*R1*R2+R1^2)*R4^3+
((60*R2+20*R1)*R3^2+(54*R2^2+48*R1*R2+6*R1^2)*R3+18*R1*R2^2+6*R1^2*R2)*R4^2+
((36*R2+12*R1)*R3^3+(45*R2^2+40*R1*R2+5*R1^2)*R3^2+(24*R1*R2^2+8*R1^2*R2)*R3)*R4+(6*R2+2*R1)*R3^4+
(9*R2^2+8*R1*R2+R1^2)*R3^3+(6*R1*R2^2+2*R1^2*R2)*R3^2)]]
(%i63)

Zoの式にR1,R2,R3,R4の値を代入すると

subst(15, R1, Zo=(((16*R1*R2+4*R1^2)*R3+6*R1*R2^2+2*R1^2*R2)*R4^3+((40*R1*R2+10*R1^2)*R3^2
+(36*R1*R2^2+12*R1^2*R2)*R3)*R4^2+((24*R1*R2+6*R1^2)*R3^3+(30*R1*R2^2+10*R1^2*R2)*R3^2)*R4
+(4*R1*R2+R1^2)*R3^4+(6*R1*R2^2+2*R1^2*R2)*R3^3)/(((24*R2+8*R1)*R3+9*R2^2+6*R1*R2
+R1^2)*R4^3+((60*R2+20*R1)*R3^2+(54*R2^2+48*R1*R2+6*R1^2)*R3+18*R1*R2^2+6*R1^2*R2)*R4^2
+((36*R2+12*R1)*R3^3+(45*R2^2+40*R1*R2+5*R1^2)*R3^2+(24*R1*R2^2+8*R1^2*R2)*R3)*R4
+(6*R2+2*R1)*R3^4+(9*R2^2+8*R1*R2+R1^2)*R3^3+(6*R1*R2^2+2*R1^2*R2)*R3^2));
(%o63) Zo=(((240*R2+900)*R3+90*R2^2+450*R2)*R4^3+((600*R2+2250)*R3^2+(540*R2^2+2700*R2)*R3)*R4^2+
((360*R2+1350)*R3^3+(450*R2^2+2250*R2)*R3^2)*R4+(60*R2+225)*R3^4+(90*R2^2+450*R2)*R3^3)/(
((24*R2+120)*R3+9*R2^2+90*R2+225)*R4^3+
((60*R2+300)*R3^2+(54*R2^2+720*R2+1350)*R3+270*R2^2+1350*R2)*R4^2+
((36*R2+180)*R3^3+(45*R2^2+600*R2+1125)*R3^2+(360*R2^2+1800*R2)*R3)*R4+(6*R2+30)*R3^4+
(9*R2^2+120*R2+225)*R3^3+(90*R2^2+450*R2)*R3^2)
(%i64) subst(5, R2, Zo=(((240*R2+900)*R3+90*R2^2+450*R2)*R4^3+((600*R2+2250)*R3^2+(540*R2^2
+2700*R2)*R3)*R4^2+((360*R2+1350)*R3^3+(450*R2^2+2250*R2)*R3^2)*R4+(60*R2+225)*R3^4
+(90*R2^2+450*R2)*R3^3)/(((24*R2+120)*R3+9*R2^2+90*R2+225)*R4^3+((60*R2+300)*R3^2
+(54*R2^2+720*R2+1350)*R3+270*R2^2+1350*R2)*R4^2+((36*R2+180)*R3^3+(45*R2^2+600*R2
+1125)*R3^2+(360*R2^2+1800*R2)*R3)*R4+(6*R2+30)*R3^4+(9*R2^2+120*R2+225)*R3^3+(90*R2^2
+450*R2)*R3^2));
(%o64) Zo=
((2100*R3+4500)*R4^3+(5250*R3^2+27000*R3)*R4^2+(3150*R3^3+22500*R3^2)*R4+525*R3^4+4500*R3^3)/((240*R3+900)*R4^3+(600*R3^2+6300*R3+13500)*R4^2+(360*R3^3+5250*R3^2+18000*R3)*R4+60*R3^4+1050*R3^3+4500*R3^2)
(%i65)
subst(2, R3, Zo=((2100*R3+4500)*R4^3+(5250*R3^2+27000*R3)*R4^2+(3150*R3^3+22500*R3^2)*R4
+525*R3^4+4500*R3^3)/((240*R3+900)*R4^3+(600*R3^2+6300*R3+13500)*R4^2+(360*R3^3+5250*R3^2
+18000*R3)*R4+60*R3^4+1050*R3^3+4500*R3^2));
(%o65) Zo=(8700*R4^3+75000*R4^2+115200*R4+44400)/(1380*R4^3+28500*R4^2+59880*R4+27360)
(%i66) subst(4, R4, Zo=(8700*R4^3+75000*R4^2+115200*R4+44400)/(1380*R4^3+28500*R4^2+59880*R4
+27360));
(%o66) Zo=145/52
(%i67) float(%), numer;
(%o67) Zo=2.788461538461538

従って2-2'端子開放時のインピーダンスは

Zo=2.79 [Ω]

ということになる。同様に2-2'端末が短絡された回路についても解くことができる。

右上の回路に関して方程式を立てると

(%i49) e11:Ic=Ic1+Ic2;
(%o49) Ic=Ic2+Ic1
(%i50) e12:(Ic1-Ic3)*R1=E;
(%o50) (Ic1-Ic3)*R1=E
(%i51) e13:(Ic3-Ic1)*R1+Ic3*R1+(Ic3-Ic5)*R1=0;
(%o51) (Ic3-Ic5)*R1+(Ic3-Ic1)*R1+Ic3*R1=0
(%i52) e14:(Ic5-Ic3)*R1+Ic5*R2+(Ic5-Ic7)*R2=0;
(%o52) (Ic5-Ic7)*R2+Ic5*R2+(Ic5-Ic3)*R1=0
(%i53) e15:(Ic7-Ic5)*R2+Ic7*R2=0;
(%o53) (Ic7-Ic5)*R2+Ic7*R2=0
(%i54) e16:(Ic8-Ic6)*R4+Ic8*R3=0;
(%o54) (Ic8-Ic6)*R4+Ic8*R3=0
(%i55) e17:(Ic6-Ic4)*R4+Ic6*R3+(Ic6-Ic8)*R4=0;
(%o55) (Ic6-Ic8)*R4+(Ic6-Ic4)*R4+Ic6*R3=0
(%i56) e18:(Ic4-Ic2)*R4+Ic4*R3+(Ic4-Ic6)*R4=0;
(%o56) (Ic4-Ic6)*R4+(Ic4-Ic2)*R4+Ic4*R3=0
(%i57) e19:(Ic2-Ic4)*R4+Ic2*R3=E;
(%o57) (Ic2-Ic4)*R4+Ic2*R3=E
(%i58) e20:Zc=E/Ic;
(%o58) Zc=E/Ic

これらをIc,Ic1,Ic2,Ic3,Ic4,Ic5,Ic6,Ic7,Ic8,Zcに関する１０元連立方程式として解くと

(%i59) solve([e11,e12,e13,e14,e15,e16,e17,e18,e19,e20],[Ic,Ic1,Ic2,Ic3,Ic4,Ic5,Ic6,Ic7,Ic8,Zc]);
(%o59) [[Ic=(((36*E*R2+16*E*R1)*R3+6*E*R1*R2+2*E*R1^2)*R4^3+
((90*E*R2+40*E*R1)*R3^2+(36*E*R1*R2+12*E*R1^2)*R3)*R4^2+((54*E*R2+24*E*R1)*R3^3+(30*E*R1*R2+10*E*R1^2)*R3^2)*
R4+(9*E*R2+4*E*R1)*R3^4+(6*E*R1*R2+2*E*R1^2)*R3^3)/((24*R1*R2+8*R1^2)*R3*R4^3+(60*R1*R2+20*R1^2)*R3^2*R4^2
+(36*R1*R2+12*R1^2)*R3^3*R4+(6*R1*R2+2*R1^2)*R3^4),Ic1=(9*E*R2+4*E*R1)/(6*R1*R2+2*R1^2),Ic2=
(E*R4^3+6*E*R3*R4^2+5*E*R3^2*R4+E*R3^3)/(4*R3*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Ic3=(3*E*R2+2*E*R1)/(6*R1*R2+2*R1^2),Ic4=(E*R4^3+3*E*R3*R4^2+E*R3^2*R4)/(4*R3*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Ic5=E/(3*R2+R1),
Ic6=(E*R4^3+E*R3*R4^2)/(4*R3*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Ic7=E/(6*R2+2*R1),Ic8=(E*R4^3)/(4*R3*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Zc=(
(24*R1*R2+8*R1^2)*R3*R4^3+(60*R1*R2+20*R1^2)*R3^2*R4^2+(36*R1*R2+12*R1^2)*R3^3*R4+(6*R1*R2+2*R1^2)*R3^4)/(
((36*R2+16*R1)*R3+6*R1*R2+2*R1^2)*R4^3+((90*R2+40*R1)*R3^2+(36*R1*R2+12*R1^2)*R3)*R4^2+
((54*R2+24*R1)*R3^3+(30*R1*R2+10*R1^2)*R3^2)*R4+(9*R2+4*R1)*R3^4+(6*R1*R2+2*R1^2)*R3^3)]]
(%i60) factor(%);
(%o60) [[Ic=(E*(36*R2*R3*R4^3+16*R1*R3*R4^3+6*R1*R2*R4^3+2*R1^2*R4^3+90*R2*R3^2*R4^2+40*R1*R3^2*R4^2+36*
R1*R2*R3*R4^2+12*R1^2*R3*R4^2+54*R2*R3^3*R4+24*R1*R3^3*R4+30*R1*R2*R3^2*R4+10*R1^2*R3^2*R4+9*R2*R3^4+4*R1*
R3^4+6*R1*R2*R3^3+2*R1^2*R3^3))/(2*R1*(3*R2+R1)*R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2)),Ic1=(E*(9*R2+4*R1))/(2*R1*(3*R2+R1))
,Ic2=(E*(R4^3+6*R3*R4^2+5*R3^2*R4+R3^3))/(R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2)),Ic3=(E*(3*R2+2*R1))/(2*R1*(3*R2+R1)),Ic4=(E*R4*(R4^2+3*R3*R4+R3^2))/(R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2)),Ic5=
E/(3*R2+R1),Ic6=(E*R4^2*(R4+R3))/(R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2)),Ic7=E/(2*(3*R2+R1)),Ic8=(E*R4^3)/(R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2)),Zc=
(2*R1*(3*R2+R1)*R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2))/(36*R2*R3*R4^3+16*R1*R3*R4^3+6*R1*R2*R4^3+2*R1^2*R4^3
+90*R2*R3^2*R4^2+40*R1*R3^2*R4^2+36*R1*R2*R3*R4^2+12*R1^2*R3*R4^2+54*R2*R3^3*R4+24*R1*R3^3*R4+30*R1*R2*R3^2*
R4+10*R1^2*R3^2*R4+9*R2*R3^4+4*R1*R3^4+6*R1*R2*R3^3+2*R1^2*R3^3)]]

Zcの式にR1,R2,R3,R4をそれぞれ代入すると

(%i61) subst(15, R1, Zc=(2*R1*(3*R2+R1)*R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2))/(36*R2*R3*R4^3
+16*R1*R3*R4^3+6*R1*R2*R4^3+2*R1^2*R4^3+90*R2*R3^2*R4^2+40*R1*R3^2*R4^2+36*R1*R2*R3*R4^2
+12*R1^2*R3*R4^2+54*R2*R3^3*R4+24*R1*R3^3*R4+30*R1*R2*R3^2*R4+10*R1^2*R3^2*R4+9*R2*R3^4
+4*R1*R3^4+6*R1*R2*R3^3+2*R1^2*R3^3));
(%o61) Zc=(30*(3*R2+15)*R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2))/(36*R2*R3*R4^3+240*R3*R4^3+90*R2*R4^3+
450*R4^3+90*R2*R3^2*R4^2+600*R3^2*R4^2+540*R2*R3*R4^2+2700*R3*R4^2+54*R2*R3^3*R4+360*R3^3*R4+450*R2*R3^2*R4
+2250*R3^2*R4+9*R2*R3^4+60*R3^4+90*R2*R3^3+450*R3^3)
(%i62)
subst(5, R2, Zc=(30*(3*R2+15)*R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2))/(36*R2*R3*R4^3+240*R3*R4^3
+90*R2*R4^3+450*R4^3+90*R2*R3^2*R4^2+600*R3^2*R4^2+540*R2*R3*R4^2+2700*R3*R4^2+54*R2*R3^3*R4
+360*R3^3*R4+450*R2*R3^2*R4+2250*R3^2*R4+9*R2*R3^4+60*R3^4+90*R2*R3^3+450*R3^3));
(%o62) Zc=(900*R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2))/(420*R3*R4^3+900*R4^3+1050*R3^2*R4^2+5400*R3*R4^2+630*R3^3*R4+4500*R3^2*R4+105*R3^4+900*R3^3)
(%i63)
subst(2, R3, Zc=(900*R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2))/(420*R3*R4^3+900*R4^3+1050*R3^2*R4^2
+5400*R3*R4^2+630*R3^3*R4+4500*R3^2*R4+105*R3^4+900*R3^3));
(%o63) Zc=(1800*(2*R4+2)*(2*R4^2+8*R4+4))/(1740*R4^3+15000*R4^2+23040*R4+8880)
(%i64) subst(4, R4, Zc=(1800*(2*R4+2)*(2*R4^2+8*R4+4))/(1740*R4^3+15000*R4^2+23040*R4+8880));
(%o64) Zc=1020/377
(%i65) float(%), numer;
(%o65) Zc=2.705570291777188

Zc=2.71 [Ω]

ということになる。

残るは下半分の等価回路について解くだけ。

上半分の実際の回路と下半分の等価回路を見比べると

Io=Ioa
Ic=Ica
Icb=Ic7+Ic8

であることは自明なので、未知数はIob,Ra,Rb,Rcの４つということになる。

２つの等価回路に関する以下の４つの追加の方程式を追加する。

(%i115) e21;
(%o115) (Io-Iob)*Ra=E
(%i116) e22;
(%o116) Iob*Rc+Iob*Rb+(Iob-Io)*Ra=0
(%i117) e24;
(%o117) (-Ic8-Ic7+Ic)*Ra=E
(%i118) e25;
(%o118) (Ic8+Ic7)*Rb+(Ic8+Ic7-Ic)*Ra=0

これで全部について解くと

(%i119)
solve([e1,e2,e3,e4,e5,e6,e7,e8,e9,e11,e12,e13,e14,e15,e16,e17,e18,e19,e20,e21,e22,e24,e25],[Io,Io1,Io2,Io3,Io4,Io5,Io6,Io7,Zo,Ic,Ic1,Ic2,Ic3,Ic4,Ic5,Ic6,Ic7,Ic8,Zc,Ra,Rb,Rc,Iob]);
(%o119) [[Io=(((24*E*R2+8*E*R1)*R3+9*E*R2^2+6*E*R1*R2+E*R1^2)*R4^3+((60*E*R2+20*E*R1)*R3^2+(54*E*R2^2+48*E*R1*R2+6*E*R1^2)*R3+18*E*R1*R2^2+6*E*R1^2*R2)*R4^2+
((36*E*R2+12*E*R1)*R3^3+(45*E*R2^2+40*E*R1*R2+5*E*R1^2)*R3^2+(24*E*R1*R2^2+8*E*R1^2*R2)*R3)*R4+(6*E*R2+2*E*R1)*R3^4+(9*E*R2^2+8*E*R1*R2+E*R1^2)*R3^3+(6*E*R1*R2^2+2*E*R1^2*R2)*R3^2)/(
((16*R1*R2+4*R1^2)*R3+6*R1*R2^2+2*R1^2*R2)*R4^3+((40*R1*R2+10*R1^2)*R3^2+(36*R1*R2^2+12*R1^2*R2)*R3)*R4^2+((24*R1*R2+6*R1^2)*R3^3+(30*R1*R2^2+10*R1^2*R2)*R3^2)*R4+(4*R1*R2+R1^2)*R3^4+
(6*R1*R2^2+2*R1^2*R2)*R3^3),Io1=
(((24*E*R2+8*E*R1)*R3+9*E*R2^2+3*E*R1*R2)*R4^3+((60*E*R2+20*E*R1)*R3^2+(54*E*R2^2+24*E*R1*R2)*R3)*R4^2+((36*E*R2+12*E*R1)*R3^3+(45*E*R2^2+20*E*R1*R2)*R3^2)*R4+(6*E*R2+2*E*R1)*R3^4+(9*E*R2^2+4*E*R1*R2)*R3^3)/(((16*R1*R2+4*R1^2)*R3+6*R1*R2^2+2*R1^2*R2)*R4^3+((40*R1*R2+10*R1^2)*R3^2+(36*R1*R2^2+12*R1^2*R2)*R3)*R4^2+((24*R1*R2+6*R1^2)*R3^3+(30*R1*R2^2+10*R1^2*R2)*R3^2)*R4+(4*R1*R2+R1^2)*R3^4+(6*R1*R2^2+2*R1^2*R2)*R3^3),Io2=
((3*E*R2+E*R1)*R4^3+((24*E*R2+6*E*R1)*R3+18*E*R2^2+6*E*R1*R2)*R4^2+((20*E*R2+5*E*R1)*R3^2+(24*E*R2^2+8*E*R1*R2)*R3)*R4+(4*E*R2+E*R1)*R3^3+(6*E*R2^2+2*E*R1*R2)*R3^2)/(((16*R2+4*R1)*R3+6*R2^2+2*R1*R2)*R4^3+((40*R2+10*R1)*R3^2+(36*R2^2+12*R1*R2)*R3)*R4^2+((24*R2+6*R1)*R3^3+(30*R2^2+10*R1*R2)*R3^2)*R4+(4*R2+R1)*R3^4+(6*R2^2+2*R1*R2)*R3^3),Io3=
(((8*E*R2+4*E*R1)*R3+3*E*R2^2+E*R1*R2)*R4^3+((20*E*R2+10*E*R1)*R3^2+(18*E*R2^2+12*E*R1*R2)*R3)*R4^2+((12*E*R2+6*E*R1)*R3^3+(15*E*R2^2+10*E*R1*R2)*R3^2)*R4+(2*E*R2+E*R1)*R3^4+(3*E*R2^2+2*E*R1*R2)*R3^3)/(((16*R1*R2+4*R1^2)*R3+6*R1*R2^2+2*R1^2*R2)*R4^3+((40*R1*R2+10*R1^2)*R3^2+(36*R1*R2^2+12*R1^2*R2)*R3)*R4^2+((24*R1*R2+6*R1^2)*R3^3+(30*R1*R2^2+10*R1^2*R2)*R3^2)*R4+(4*R1*R2+R1^2)*R3^4+(6*R1*R2^2+2*R1^2*R2)*R3^3),Io4=
((3*E*R2+E*R1)*R4^3+((11*E*R2+3*E*R1)*R3+12*E*R2^2+4*E*R1*R2)*R4^2+((4*E*R2+E*R1)*R3^2+(6*E*R2^2+2*E*R1*R2)*R3)*R4)/(((16*R2+4*R1)*R3+6*R2^2+2*R1*R2)*R4^3+((40*R2+10*R1)*R3^2+(36*R2^2+12*R1*R2)*R3)*R4^2+((24*R2+6*R1)*R3^3+(30*R2^2+10*R1*R2)*R3^2)*R4+(4*R2+R1)*R3^4+(6*R2^2+2*R1*R2)*R3^3),Io5=
(4*E*R3*R4^3+(10*E*R3^2+12*E*R2*R3)*R4^2+(6*E*R3^3+10*E*R2*R3^2)*R4+E*R3^4+2*E*R2*R3^3)/(((16*R2+4*R1)*R3+6*R2^2+2*R1*R2)*R4^3+((40*R2+10*R1)*R3^2+(36*R2^2+12*R1*R2)*R3)*R4^2+((24*R2+6*R1)*R3^3+(30*R2^2+10*R1*R2)*R3^2)*R4+(4*R2+R1)*R3^4+(6*R2^2+2*R1*R2)*R3^3),Io6=
((3*E*R2+E*R1)*R4^3+((E*R2+E*R1)*R3+6*E*R2^2+2*E*R1*R2)*R4^2-E*R2*R3^2*R4)/(((16*R2+4*R1)*R3+6*R2^2+2*R1*R2)*R4^3+((40*R2+10*R1)*R3^2+(36*R2^2+12*R1*R2)*R3)*R4^2+((24*R2+6*R1)*R3^3+(30*R2^2+10*R1*R2)*R3^2)*R4+(4*R2+R1)*R3^4+(6*R2^2+2*R1*R2)*R3^3),Io7=-
((3*E*R2+E*R1)*R4^3-6*E*R2*R3*R4^2-5*E*R2*R3^2*R4-E*R2*R3^3)/(((16*R2+4*R1)*R3+6*R2^2+2*R1*R2)*R4^3+((40*R2+10*R1)*R3^2+(36*R2^2+12*R1*R2)*R3)*R4^2+((24*R2+6*R1)*R3^3+(30*R2^2+10*R1*R2)*R3^2)*R4+(4*R2+R1)*R3^4+(6*R2^2+2*R1*R2)*R3^3),Zo=(
((16*R1*R2+4*R1^2)*R3+6*R1*R2^2+2*R1^2*R2)*R4^3+((40*R1*R2+10*R1^2)*R3^2+(36*R1*R2^2+12*R1^2*R2)*R3)*R4^2+((24*R1*R2+6*R1^2)*R3^3+(30*R1*R2^2+10*R1^2*R2)*R3^2)*R4+(4*R1*R2+R1^2)*R3^4+
(6*R1*R2^2+2*R1^2*R2)*R3^3)/(((24*R2+8*R1)*R3+9*R2^2+6*R1*R2+R1^2)*R4^3+((60*R2+20*R1)*R3^2+(54*R2^2+48*R1*R2+6*R1^2)*R3+18*R1*R2^2+6*R1^2*R2)*R4^2+
((36*R2+12*R1)*R3^3+(45*R2^2+40*R1*R2+5*R1^2)*R3^2+(24*R1*R2^2+8*R1^2*R2)*R3)*R4+(6*R2+2*R1)*R3^4+(9*R2^2+8*R1*R2+R1^2)*R3^3+(6*R1*R2^2+2*R1^2*R2)*R3^2),Ic=
(((36*E*R2+16*E*R1)*R3+6*E*R1*R2+2*E*R1^2)*R4^3+((90*E*R2+40*E*R1)*R3^2+(36*E*R1*R2+12*E*R1^2)*R3)*R4^2+((54*E*R2+24*E*R1)*R3^3+(30*E*R1*R2+10*E*R1^2)*R3^2)*R4+(9*E*R2+4*E*R1)*R3^4+(6*E*R1*R2+2*E*R1^2)*R3^3)/((24*R1*R2+8*R1^2)*R3*R4^3+(60*R1*R2+20*R1^2)*R3^2*R4^2+(36*R1*R2+12*R1^2)*R3^3*R4+(6*R1*R2+2*R1^2)*R3^4),Ic1=
(9*E*R2+4*E*R1)/(6*R1*R2+2*R1^2),Ic2=(E*R4^3+6*E*R3*R4^2+5*E*R3^2*R4+E*R3^3)/(4*R3*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Ic3=(3*E*R2+2*E*R1)/(6*R1*R2+2*R1^2),Ic4=(E*R4^3+3*E*R3*R4^2+E*R3^2*R4)/(4*R3*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Ic5=E/(3*R2+R1),Ic6=(E*R4^3+E*R3*R4^2)/(4*R3*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Ic7=E/(6*R2+2*R1),
Ic8=(E*R4^3)/(4*R3*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Zc=
((24*R1*R2+8*R1^2)*R3*R4^3+(60*R1*R2+20*R1^2)*R3^2*R4^2+(36*R1*R2+12*R1^2)*R3^3*R4+(6*R1*R2+2*R1^2)*R3^4)/(((36*R2+16*R1)*R3+6*R1*R2+2*R1^2)*R4^3+((90*R2+40*R1)*R3^2+(36*R1*R2+12*R1^2)*R3)*R4^2+((54*R2+24*R1)*R3^3+(30*R1*R2+10*R1^2)*R3^2)*R4+(9*R2+4*R1)*R3^4+(6*R1*R2+2*R1^2)*R3^3),Ra=
(4*R1*R4^2+8*R1*R3*R4+2*R1*R3^2)/(6*R4^2+(12*R3+6*R1)*R4+3*R3^2+2*R1*R3),Rb=((24*R2+8*R1)*R3*R4^3+(60*R2+20*R1)*R3^2*R4^2+(36*R2+12*R1)*R3^3*R4+(6*R2+2*R1)*R3^4)/((4*R3+6*R2+2*R1)*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Rc=(4*R2*R4^2+8*R2*R3*R4+2*R2*R3^2)/(2*R4^2+(4*R3+6*R2)*R4+R3^2+2*R2*R3),Iob=((8*E*R3+12*E*R2+4*E*R1)*
R4^5+(36*E*R3^2+(48*E*R2+8*E*R1)*R3+36*E*R2^2+12*E*R1*R2)*R4^4+(56*E*R3^3+(74*E*R2+2*E*R1)*R3^2+(12*E*R2^2+4*E*R1*R2)*R3)*R4^3+(36*E*R3^4+56*E*R2*R3^3)*R4^2+(10*E*R3^5+18*E*R2*R3^4)*R4+
E*R3^6+2*E*R2*R3^5)/(((64*R2+16*R1)*R3+24*R2^2+8*R1*R2)*R4^5+((288*R2+72*R1)*R3^2+(192*R2^2+64*R1*R2)*R3)*R4^4+((448*R2+112*R1)*R3^3+(420*R2^2+140*R1*R2)*R3^2)*R4^3+
((288*R2+72*R1)*R3^4+(336*R2^2+112*R1*R2)*R3^3)*R4^2+((80*R2+20*R1)*R3^5+(108*R2^2+36*R1*R2)*R3^4)*R4+(8*R2+2*R1)*R3^6+(12*R2^2+4*R1*R2)*R3^5)]]

と見事にすべての解が得られる。ここでRa,Rb,Rcの式にR1,R2,R3,R4の値を代入すると

(%i121) subst(15, R1, [Ra=(4*R1*R4^2+8*R1*R3*R4+2*R1*R3^2)/(6*R4^2+(12*R3+6*R1)*R4+3*R3^2+2*R1*R3),Rb=((24*R2
+8*R1)*R3*R4^3+(60*R2+20*R1)*R3^2*R4^2+(36*R2+12*R1)*R3^3*R4+(6*R2+2*R1)*R3^4)/((4*R3
+6*R2+2*R1)*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Rc=(4*R2*R4^2+8*R2*R3*R4+2*R2*R3^2)/(2*R4^2
+(4*R3+6*R2)*R4+R3^2+2*R2*R3)]);
(%o121) [Ra=(60*R4^2+120*R3*R4+30*R3^2)/(6*R4^2+(12*R3+90)*R4+3*R3^2+30*R3),Rb=((24*R2+120)*R3*R4^3+(60*R2+300)*R3^2*R4^2+(36*R2+180)*R3^3*R4+(6*R2+30)*R3^4)/((4*R3+6*R2+30)*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Rc=(4*R2*R4^2+8*R2*R3*R4+2*R2*R3^2)/(2*R4^2+(4*R3+6*R2)*R4+R3^2+2*R2*R3)]
(%i122) subst(5, R2, [Ra=(60*R4^2+120*R3*R4+30*R3^2)/(6*R4^2+(12*R3+90)*R4+3*R3^2+30*R3),Rb=((24*R2
+120)*R3*R4^3+(60*R2+300)*R3^2*R4^2+(36*R2+180)*R3^3*R4+(6*R2+30)*R3^4)/((4*R3+6*R2
+30)*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Rc=(4*R2*R4^2+8*R2*R3*R4+2*R2*R3^2)/(2*R4^2
+(4*R3+6*R2)*R4+R3^2+2*R2*R3)]);
(%o122) [Ra=(60*R4^2+120*R3*R4+30*R3^2)/(6*R4^2+(12*R3+90)*R4+3*R3^2+30*R3),Rb=(240*R3*R4^3+600*R3^2*R4^2+360*R3^3*R4+60*R3^4)/((4*R3+60)*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Rc=(20*R4^2+40*R3*R4+10*R3^2)/(2*R4^2+(4*R3+30)*R4+R3^2+10*R3)]
(%i123) subst(2, R3, [Ra=(60*R4^2+120*R3*R4+30*R3^2)/(6*R4^2+(12*R3+90)*R4+3*R3^2+30*R3),Rb=(240*R3*R4^3
+600*R3^2*R4^2+360*R3^3*R4+60*R3^4)/((4*R3+60)*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Rc=(20*R4^2
+40*R3*R4+10*R3^2)/(2*R4^2+(4*R3+30)*R4+R3^2+10*R3)]);
(%o123) [Ra=(60*R4^2+240*R4+120)/(6*R4^2+114*R4+72),Rb=(480*R4^3+2400*R4^2+2880*R4+960)/(68*R4^3+40*R4^2+48*R4+16),Rc=(20*R4^2+80*R4+40)/(2*R4^2+38*R4+24)]
(%i124) subst(4, R4, [Ra=(60*R4^2+240*R4+120)/(6*R4^2+114*R4+72),Rb=(480*R4^3+2400*R4^2+2880*R4
+960)/(68*R4^3+40*R4^2+48*R4+16),Rc=(20*R4^2+80*R4+40)/(2*R4^2+38*R4+24)]);
(%o124) [Ra=85/26,Rb=204/13,Rc=85/26]
(%i125) float(%), numer;
(%o125) [Ra=3.269230769230769,Rb=15.69230769230769,Rc=3.269230769230769]

Ra=3.27 [Ω]
Rb=15.7 [Ω]
Rc=3.27 [Ω]

ということになる。これは著者の解と一致する。

方程式を立てるだけで回路図を教えたわけではないのに数学的に解けてしまうというのは気持ちが良い。これだけの回路になるとちょっと手で式を操作するのは大変だけど、後に学ぶマトリックスを使うと楽になる。