暑すぎて中断していた演習問題を再開

前回は網目電流法を使って解いたのを枝電流法で解いてみる。



出力端2-2'をオープンにした場合に回路全体を流れる電流をIoとして片方の抵抗ラダーネットワークに流れる電流をIo1として順番にリターン電流をIo2,Io3,Io4,Io5,Io6,Io7と定義して枝電流の式を決めてみた。このやり方には決まった方法は無い。

あとは閉ループ回路内の電圧降下の合計が０となるのと、ひとつのノードに流入する電流と流出する電流の合計は０になるキルヒホッフの法則を利用して以下の回路方程式をたてる。

(%i1) e1:Io2*R1=E;
(%o1) Io2*R1=E
(%i2) e2:(Io1-Io2)*R1+Io3*R1=Io2*R1;
(%o2) Io3*R1+(Io1-Io2)*R1=Io2*R1
(%i3) e3:(Io1-Io2-Io3)*R2+Io4*R2=Io3*R1;
(%o3) Io4*R2+(-Io3-Io2+Io1)*R2=Io3*R1
(%i4) e4:(Io1-Io2-Io3-Io4)*(R2+R3)+Io7*R4=Io4*R2;
(%o4) Io7*R4+(-Io4-Io3-Io2+Io1)*(R3+R2)=Io4*R2
(%i5) e5:Io7*R4+(Io-Io1-Io5-Io6)*R3=Io6*R4;
(%o5) Io7*R4+(-Io6-Io5-Io1+Io)*R3=Io6*R4
(%i6) e6:Io6*R4+(Io-Io1-Io5)*R3=Io5*R4;
(%o6) Io6*R4+(-Io5-Io1+Io)*R3=Io5*R4
(%i7) e7:(Io-Io1)*R3+Io5*R4=E;
(%o7) Io5*R4+(Io-Io1)*R3=E
(%i10) e8:Io=Io2+Io3+Io4+Io5+Io6+Io7;
(%o10) Io=Io7+Io6+Io5+Io4+Io3+Io2

回路全体のインピーダンスをZoとしてそれに関する方程式を追加する。

(%i11) e9:Io*Zo=E;
(%o11) Io*Zo=E

これをIo,Io1,Io2,Io3,Io4,Io5,Io6,Io7,Zに関する９元連立方程式として解けばZoの式を求めることが出来る。

(%i163) solve([e1,e2,e3,e4,e5,e6,e7,e8,e9],[Io,Io1,Io2,Io3,Io4,Io5,Io6,Io7,Zo]);
(%o163) [[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
=E/R1,Io3=((8*E*R2*R3+3*E*R2^2+E*R1*R2)*R4^3+(20*E*R2*R3^2+18*E*R2^2*R3)*R4^2+(12*E*R2*R3^3+15*E*R2^2*R3^2)*
R4+2*E*R2*R3^4+3*E*R2^2*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=((4*E*R3+3*E*R2+E*R1)*R4^3+(10*E*R3^2+6*E*R2*R3)*
R4^2+(6*E*R3^3+5*E*R2*R3^2)*R4+E*R3^4+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),Io5=(((13*E*R2+3*E*R1)*R3+6*E*R2^2+2*E*R1*R2)*R4^2+
((16*E*R2+4*E*R1)*R3^2+(18*E*R2^2+6*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),Io6=(
((10*E*R2+2*E*R1)*R3+6*E*R2^2+2*E*R1*R2)*R4^2+((5*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),Io7=(
((7*E*R2+E*R1)*R3+6*E*R2^2+2*E*R1*R2)*R4^2+4*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)]]

Zoの式にR1,R2,R3,R4の値を代入すると

(%i164)
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));
(%o164) 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)
(%i165) 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));
(%o165) 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)
(%i166)
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));
(%o166) Zo=(8700*R4^3+75000*R4^2+115200*R4+44400)/(1380*R4^3+28500*R4^2+59880*R4+27360)
(%i167) subst(4, R4, Zo=(8700*R4^3+75000*R4^2+115200*R4+44400)/(1380*R4^3+28500*R4^2+59880*R4
+27360));
(%o167) Zo=145/52
(%i168) float(Zo=145/52), numer;
(%o168) Zo=2.788461538461538

網目電流法と同じ結果が得られた。

従って

Zo=2.79 [Ω]

ということになる。

同様に端子2-2'を短絡した回路について方程式をたてると

(%i20) e10:Ic2*R1=E;
(%o20) Ic2*R1=E
(%i21) e11:(Ic1-Ic2)*R1+Ic3*R1=Ic2*R1;
(%o21) Ic3*R1+(Ic1-Ic2)*R1=Ic2*R1
(%i22) e12:(Ic1-Ic2-Ic3)*R2+Ic4*R2=Ic3*R1;
(%o22) Ic4*R2+(-Ic3-Ic2+Ic1)*R2=Ic3*R1
(%i23) e13:(Ic1-Ic2-Ic3-Ic4)*R2=Ic4*R2;
(%o23) (-Ic4-Ic3-Ic2+Ic1)*R2=Ic4*R2
(%i24) e14:(Ic-Ic1-Ic5-Ic6-Ic7)*R3=Ic7*R4;
(%o24) (-Ic7-Ic6-Ic5-Ic1+Ic)*R3=Ic7*R4
(%i25) e15:Ic7*R4+(Ic-Ic1-Ic5-Ic6)*R3=Ic6*R4;
(%o25) Ic7*R4+(-Ic6-Ic5-Ic1+Ic)*R3=Ic6*R4
(%i26) e16:Ic6*R4+(Ic-Ic1-Ic5)*R3=Ic5*R4;
(%o26) Ic6*R4+(-Ic5-Ic1+Ic)*R3=Ic5*R4
(%i27) e17:Ic5*R4+(Ic-Ic1)*R3=E;
(%o27) Ic5*R4+(Ic-Ic1)*R3=E
(%i28) e18:Ic*Zc=E;
(%o28) Ic*Zc=E

これをIc,Ic1,Ic2,Ic3,Ic4,Ic5,Ic6,Ic7,Zcに関する９元連立方程式として解くと

(%i29) solve([e10,e11,e12,e13,e14,e15,e16,e17,e18],[Ic,Ic1,Ic2,Ic3,Ic4,Ic5,Ic6,Ic7,Zc]);
(%o29) [[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/R1,Ic3=(3*E*R2)/(6*R1*R2+2*R1^2),Ic4=
E/(6*R2+2*R1),Ic5=(3*E*R4^2+4*E*R3*R4+E*R3^2)/(4*R4^3+10*R3*R4^2+6*R3^2*R4+R3^3),Ic6=(E*R4)/(2*R4^2+4*R3*R4+R3^2),Ic7=(E*R4^2)/(4*R4^3+10*R3*R4^2+6*R3^2*R4+R3^3),
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)]]

得られたZcの式にR1,R2,R3,R4の値をそれぞれ代入すると

(%i30) subst(15, R1, 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));
(%o30) Zc=((360*R2+1800)*R3*R4^3+(900*R2+4500)*R3^2*R4^2+(540*R2+2700)*R3^3*R4+(90*R2+450)*R3^4)/
(((36*R2+240)*R3+90*R2+450)*R4^3+((90*R2+600)*R3^2+(540*R2+2700)*R3)*R4^2+
((54*R2+360)*R3^3+(450*R2+2250)*R3^2)*R4+(9*R2+60)*R3^4+(90*R2+450)*R3^3)
(%i31)
subst(5, R2, Zc=((360*R2+1800)*R3*R4^3+(900*R2+4500)*R3^2*R4^2+(540*R2+2700)*R3^3*R4
+(90*R2+450)*R3^4)/(((36*R2+240)*R3+90*R2+450)*R4^3+((90*R2+600)*R3^2+(540*R2+2700)*R3)*R4^2
+((54*R2+360)*R3^3+(450*R2+2250)*R3^2)*R4+(9*R2+60)*R3^4+(90*R2+450)*R3^3));
(%o31) Zc=(3600*R3*R4^3+9000*R3^2*R4^2+5400*R3^3*R4+900*R3^4)/((420*R3+900)*R4^3+(1050*R3^2+5400*R3)*R4^2+(630*R3^3+4500*R3^2)*R4+105*R3^4+900*R3^3)
(%i32) subst(2, R3, Zc=(3600*R3*R4^3+9000*R3^2*R4^2+5400*R3^3*R4+900*R3^4)/((420*R3+900)*R4^3
+(1050*R3^2+5400*R3)*R4^2+(630*R3^3+4500*R3^2)*R4+105*R3^4+900*R3^3));
(%o32) Zc=(7200*R4^3+36000*R4^2+43200*R4+14400)/(1740*R4^3+15000*R4^2+23040*R4+8880)
(%i33) subst(4, R4, Zc=(7200*R4^3+36000*R4^2+43200*R4+14400)/(1740*R4^3+15000*R4^2+23040*R4
+8880));
(%o33) Zc=1020/377
(%i34) float(%), numer;
(%o34) Zc=2.705570291777188

と網目電流法と同じ結果が得られた。

従って

Zc=2.71 [Ω]

ということになる。

更に図の下の等価回路に関して

Ic-Ic2-Ic3-Ic4-Ic6-Ic7=Ic-Ica

であることが明らかなので

Ica=Ic2+Ic3+Ic4+Ic6+Ic7

となり未知数はRa,Rb,Rc,Ioaの４つだけとなる。従って以下の４つの方程式を追加すればすべてが解けることになる。

(%i35) e19:Ioa*Ra=E;
(%o35) Ioa*Ra=E
(%i36) e20:(Io-Ioa)*(Rb+Rc)=Ioa*Ra;
(%o36) (Io-Ioa)*(Rc+Rb)=Ioa*Ra
(%i40) e21:(Ic2+Ic3+Ic4+Ic5+Ic6+Ic7)*Ra=E;
(%o40) (Ic7+Ic6+Ic5+Ic4+Ic3+Ic2)*Ra=E
(%i41) e22:(Ic2+Ic3+Ic4+Ic5+Ic6+Ic7)*Ra=(Ic-Ic2-Ic3-Ic4-Ic5-Ic6-Ic7)*Rb;
(%o41) (Ic7+Ic6+Ic5+Ic4+Ic3+Ic2)*Ra=(-Ic7-Ic6-Ic5-Ic4-Ic3-Ic2+Ic)*Rb


これらをIo,Io1,Io2,Io3,Io4,Io5,Io6,Io7,Zo,Ic,Ic1,Ic2,Ic3,Ic4,Ic5,Ic6,Ic7,Zc,Ioa,Ra,Rb,Rcに関する二十二元連立方程式として解くと

(%i169)
solve([e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12,e13,e14,e15,e16,e17,e18,e19,e20,e21,e22],[Io,Io1,Io2,Io3,Io4,Io5,Io6,Io7,Zo,Ic,Ic1,Ic2,Ic3,Ic4,Ic5,Ic6,Ic7,Zc,Ioa,Ra,Rb,Rc]);
(%o169) [[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
=E/R1,Io3=((8*E*R2*R3+3*E*R2^2+E*R1*R2)*R4^3+(20*E*R2*R3^2+18*E*R2^2*R3)*R4^2+(12*E*R2*R3^3+15*E*R2^2*R3^2)*
R4+2*E*R2*R3^4+3*E*R2^2*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=((4*E*R3+3*E*R2+E*R1)*R4^3+(10*E*R3^2+6*E*R2*R3)*
R4^2+(6*E*R3^3+5*E*R2*R3^2)*R4+E*R3^4+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),Io5=(((13*E*R2+3*E*R1)*R3+6*E*R2^2+2*E*R1*R2)*R4^2+
((16*E*R2+4*E*R1)*R3^2+(18*E*R2^2+6*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),Io6=(
((10*E*R2+2*E*R1)*R3+6*E*R2^2+2*E*R1*R2)*R4^2+((5*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),Io7=(
((7*E*R2+E*R1)*R3+6*E*R2^2+2*E*R1*R2)*R4^2+4*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/R1,Ic3=(3*E*R2)/(6*R1*R2+2*R1^2),Ic4=E/(6*R2+2*R1),Ic5=(3*E*R4^2+4*E*R3*R4+E*R3^2)/(4*R4^3+10*R3*R4^2+6*R3^2*R4+R3^3),Ic6=
(E*R4)/(2*R4^2+4*R3*R4+R3^2),Ic7=(E*R4^2)/(4*R4^3+10*R3*R4^2+6*R3^2*R4+R3^3),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),Ioa=(6*E*R4^2+(12*E*R3+6*E*R1)*R4+3*E*R3^2+2*E*R1*R3)/(4*R1*R4^2+8*R1*R3*R4+2*R1*R3^2),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)]]

得られたRa,Rb,Rcの式にR1,R2,R3,R4の値をそれぞれ代入すると

(%i170)
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)]);
(%o170) [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)]
(%i171)
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)]);
(%o171) [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)]
(%i172)
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)]);
(%o172) [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)]
(%i173) 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)]);
(%o173) [Ra=85/26,Rb=204/13,Rc=85/26]
(%i174) float(%), numer;
(%o174) [Ra=3.269230769230769,Rb=15.69230769230769,Rc=3.269230769230769]

これも網目電流法と同じ結果が得られた。

従って

Ra=3.27 [Ω]
Rb=15.7 [Ω]
Rc=3.27 [Ω]

ということになる。

P.S

実際には最初の端子2-2'オープンのケースで端子２を流れる電流の式を間違えてしまって網目電流法と微妙に違う結果が出てしまっていた。