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webadm | 投稿日時: 2008-7-28 0:01 |
Webmaster 登録日: 2004-11-7 居住地: 投稿: 3068 |
Re: 【33】Δ-Y変換(その3) 最後に出題の意図に沿ってΔ-Y変換の公式を使って解いてみよう。
最終的に単一のπ型(Δ型)抵抗ネットワークに合成するためにまず、Y接続を先にΔ接続に置き換えていく。 Y-Δ変換の公式により R5=(R2*R2+R2*R2+R2*R2)/R2 =3*R2 R6=(R3*R4+R3*R4+R3*R3)/R4 =2*R3+R3*R3/R4 R7=(R3*R4+R3*R4+R3*R3/R3 =(2*R4+R3) 次ぎに中央部の並列接続抵抗をひとつに合成する。 R8=1/(1/R1+1/R5) =R1*R5/(R1+R5) R9=1/(1/R7+1/R4+1/R7) =R4*R7/(2*R4+R7) 次ぎに中央に出来たY接続をΔ接続に変換する。 R10=(R1*R8+R1*R5+R8*R5)/R8 =(R1+R5)+R1*R5/R8 R11=(R1*R8+R1*R5+R8*R5)/R5 =(R1+R8)+R1*R8/R5 R12=(R1*R8+R1*R5+R8*R5)/R1 =(R8+R5)+R8*R5/R1 R13=(R6*R6+R6*R9+R6*R9)/R9 =2*R6+R6*R6/R9 R14=(R6*R6+R6*R9+R6*R9)/R6 =R6+2*R9 次ぎに並列接続抵抗をひとつに合成する。 R15=1/(1/R1+1/R11) =R1*R11/(R1+R11) R16=1/(1/R12+1/R5) =R12*R5/(R12+R5) R17=1/(1/R7+1/R14) =R7*R14/(R7+R14) 最後に残った2つの並列接続されたΔ接続ネットワークを合成し一つにする。 Ra=1/(1/R15+1/R17) =R15*R17/(R15+R17) Rb=1/(1/R10+1/R13) =R10*R13/(R10+R13) Rc=1/(1/R16+1/R17) =R16*R17/(R16+R17) ということになる。 Ra,Rb,RcをR1,R2,R3,R4で表すように書き換えると (%i175) [Ra=1/(1/R15+1/R17),Rb=1/(1/R10+1/R13),Rc=1/(1/R16+1/R17)]; (%o175) [Ra=1/(1/R17+1/R15),Rb=1/(1/R13+1/R10),Rc=1/(1/R17+1/R16)] (%i176) subst(1/(1/R7+1/R14), R17, [Ra=1/(1/R17+1/R15),Rb=1/(1/R13+1/R10),Rc=1/(1/R17+1/R16)]); (%o176) [Ra=1/(1/R7+1/R15+1/R14),Rb=1/(1/R13+1/R10),Rc=1/(1/R7+1/R16+1/R14)] (%i177) subst(1/(1/R12+1/R5), R16, [Ra=1/(1/R7+1/R15+1/R14),Rb=1/(1/R13+1/R10),Rc=1/(1/R7 +1/R16+1/R14)]); (%o177) [Ra=1/(1/R7+1/R15+1/R14),Rb=1/(1/R13+1/R10),Rc=1/(1/R7+1/R5+1/R14+1/R12)] (%i178) subst(1/(1/R1+1/R11), R15, [Ra=1/(1/R7+1/R15+1/R14),Rb=1/(1/R13+1/R10),Rc=1/(1/R7 +1/R5+1/R14+1/R12)]); (%o178) [Ra=1/(1/R7+1/R14+1/R11+1/R1),Rb=1/(1/R13+1/R10),Rc=1/(1/R7+1/R5+1/R14+1/R12)] (%i179) subst((R6*R6+R6*R9+R6*R9)/R6, R14, [Ra=1/(1/R7+1/R14+1/R11+1/R1),Rb=1/(1/R13+1/R10),Rc=1/(1/R7 +1/R5+1/R14+1/R12)]); (%o179) [Ra=1/(R6/(2*R6*R9+R6^2)+1/R7+1/R11+1/R1),Rb=1/(1/R13+1/R10),Rc=1/(R6/(2*R6*R9+R6^2)+1/R7+1/R5+1/R12)] (%i180) subst((R6*R6+R6*R9+R6*R9)/R9, R13, [Ra=1/(R6/(2*R6*R9+R6^2)+1/R7+1/R11+1/R1),Rb=1/(1/R13 +1/R10),Rc=1/(R6/(2*R6*R9+R6^2)+1/R7+1/R5+1/R12)]); (%o180) [Ra=1/(R6/(2*R6*R9+R6^2)+1/R7+1/R11+1/R1),Rb=1/(R9/(2*R6*R9+R6^2)+1/R10),Rc=1/(R6/(2*R6*R9+R6^2)+1/R7+1/R5+1/R12)] (%i181) subst((R1*R8+R1*R5+R8*R5)/R1, R12, [Ra=1/(R6/(2*R6*R9+R6^2)+1/R7+1/R11+1/R1),Rb=1/(R9/(2*R6*R9 +R6^2)+1/R10),Rc=1/(R6/(2*R6*R9+R6^2)+1/R7+1/R5+1/R12)]); (%o181) [Ra=1/(R6/(2*R6*R9+R6^2)+1/R7+1/R11+1/R1),Rb=1/(R9/(2*R6*R9+R6^2)+1/R10),Rc=1/(R6/(2*R6*R9+R6^2)+R1/(R5*R8+R1*R8+R1*R5)+1/R7+1/R5)] (%i182) subst((R1*R8+R1*R5+R8*R5)/R5, R11, [Ra=1/(R6/(2*R6*R9+R6^2)+1/R7+1/R11+1/R1),Rb=1/(R9/(2*R6*R9 +R6^2)+1/R10),Rc=1/(R6/(2*R6*R9+R6^2)+R1/(R5*R8+R1*R8+R1*R5)+1/R7+1/R5)]); (%o182) [Ra=1/(R6/(2*R6*R9+R6^2)+R5/(R5*R8+R1*R8+R1*R5)+1/R7+1/R1),Rb=1/(R9/(2*R6*R9+R6^2)+1/R10),Rc= 1/(R6/(2*R6*R9+R6^2)+R1/(R5*R8+R1*R8+R1*R5)+1/R7+1/R5)] (%i183) subst((R1*R8+R1*R5+R8*R5)/R8, R10, [Ra=1/(R6/(2*R6*R9+R6^2)+R5/(R5*R8+R1*R8+R1*R5) +1/R7+1/R1),Rb=1/(R9/(2*R6*R9+R6^2)+1/R10),Rc=1/(R6/(2*R6*R9+R6^2)+R1/(R5*R8+R1*R8 +R1*R5)+1/R7+1/R5)]); (%o183) [Ra=1/(R6/(2*R6*R9+R6^2)+R5/(R5*R8+R1*R8+R1*R5)+1/R7+1/R1),Rb=1/(R9/(2*R6*R9+R6^2)+R8/(R5*R8+R1*R8+R1*R5)),Rc= 1/(R6/(2*R6*R9+R6^2)+R1/(R5*R8+R1*R8+R1*R5)+1/R7+1/R5)] (%i184) subst(1/(1/R7+1/R4+1/R7), R9, [Ra=1/(R6/(2*R6*R9+R6^2)+R5/(R5*R8+R1*R8+R1*R5)+1/R7 +1/R1),Rb=1/(R9/(2*R6*R9+R6^2)+R8/(R5*R8+R1*R8+R1*R5)),Rc=1/(R6/(2*R6*R9+R6^2)+R1/(R5*R8 +R1*R8+R1*R5)+1/R7+1/R5)]); (%o184) [Ra=1/(R5/(R5*R8+R1*R8+R1*R5)+1/R7+R6/((2*R6)/(2/R7+1/R4)+R6^2)+1/R1),Rb=1/(R8/(R5*R8+R1*R8+R1*R5)+1/(((2*R6)/(2/R7+1/R4)+R6^2)*(2/R7+1/R4))),Rc= 1/(R1/(R5*R8+R1*R8+R1*R5)+1/R7+R6/((2*R6)/(2/R7+1/R4)+R6^2)+1/R5)] (%i185) subst(1/(1/R1+1/R5), R8, [Ra=1/(R5/(R5*R8+R1*R8+R1*R5)+1/R7+R6/((2*R6)/(2/R7+1/R4) +R6^2)+1/R1),Rb=1/(R8/(R5*R8+R1*R8+R1*R5)+1/(((2*R6)/(2/R7+1/R4)+R6^2)*(2/R7+1/R4))),Rc=1/(R1/(R5*R8 +R1*R8+R1*R5)+1/R7+R6/((2*R6)/(2/R7+1/R4)+R6^2)+1/R5)]); (%o185) [Ra=1/(1/R7+R6/((2*R6)/(2/R7+1/R4)+R6^2)+R5/(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1))+1/R1),Rb=1/(1/(((2*R6)/(2/R7+1/R4)+R6^2)*(2/R7+1/R4))+1/((1/R5+1/R1)*(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1)))), Rc=1/(1/R7+R6/((2*R6)/(2/R7+1/R4)+R6^2)+R1/(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1))+1/R5)] (%i186) subst((R3*R4+R3*R4+R3*R3)/R3, R7, [Ra=1/(1/R7+R6/((2*R6)/(2/R7+1/R4)+R6^2)+R5/(R5/(1/R5 +1/R1)+R1*R5+R1/(1/R5+1/R1))+1/R1),Rb=1/(1/(((2*R6)/(2/R7+1/R4)+R6^2)*(2/R7+1/R4)) +1/((1/R5+1/R1)*(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1)))),Rc=1/(1/R7+R6/((2*R6)/(2/R7 +1/R4)+R6^2)+R1/(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1))+1/R5)]); (%o186) [Ra=1/(R6/(R6^2+(2*R6)/((2*R3)/(2*R3*R4+R3^2)+1/R4))+R5/(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1))+R3/(2*R3*R4+R3^2)+1/R1),Rb= 1/(1/(((2*R3)/(2*R3*R4+R3^2)+1/R4)*(R6^2+(2*R6)/((2*R3)/(2*R3*R4+R3^2)+1/R4)))+1/((1/R5+1/R1)*(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1)))),Rc= 1/(R6/(R6^2+(2*R6)/((2*R3)/(2*R3*R4+R3^2)+1/R4))+R1/(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1))+1/R5+R3/(2*R3*R4+R3^2))] (%i187) subst((R3*R4+R3*R4+R3*R3)/R4, R6, [Ra=1/(R6/(R6^2+(2*R6)/((2*R3)/(2*R3*R4+R3^2)+1/R4)) +R5/(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1))+R3/(2*R3*R4+R3^2)+1/R1),Rb=1/(1/(((2*R3)/(2*R3*R4 +R3^2)+1/R4)*(R6^2+(2*R6)/((2*R3)/(2*R3*R4+R3^2)+1/R4)))+1/((1/R5+1/R1)*(R5/(1/R5 +1/R1)+R1*R5+R1/(1/R5+1/R1)))),Rc=1/(R6/(R6^2+(2*R6)/((2*R3)/(2*R3*R4+R3^2)+1/R4)) +R1/(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1))+1/R5+R3/(2*R3*R4+R3^2))]); (%o187) [Ra=1/(R5/(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1))+(2*R3*R4+R3^2)/(R4*((2*(2*R3*R4+R3^2))/(R4*((2*R3)/(2*R3*R4+R3^2)+1/R4))+(2*R3*R4+R3^2)^2/R4^2))+R3/(2*R3*R4+R3^2)+1/R1),Rb= 1/(1/((1/R5+1/R1)*(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1)))+1/(((2*R3)/(2*R3*R4+R3^2)+1/R4)*((2*(2*R3*R4+R3^2))/(R4*((2*R3)/(2*R3*R4+R3^2)+1/R4))+(2*R3*R4+R3^2)^2/R4^2))),Rc= 1/(R1/(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1))+1/R5+(2*R3*R4+R3^2)/(R4*((2*(2*R3*R4+R3^2))/(R4*((2*R3)/(2*R3*R4+R3^2)+1/R4))+(2*R3*R4+R3^2)^2/R4^2))+R3/(2*R3*R4+R3^2))] (%i188) subst((R2*R2+R2*R2+R2*R2)/R2, R5, [Ra=1/(R5/(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1)) +(2*R3*R4+R3^2)/(R4*((2*(2*R3*R4+R3^2))/(R4*((2*R3)/(2*R3*R4+R3^2)+1/R4))+(2*R3*R4 +R3^2)^2/R4^2))+R3/(2*R3*R4+R3^2)+1/R1),Rb=1/(1/((1/R5+1/R1)*(R5/(1/R5+1/R1)+R1*R5 +R1/(1/R5+1/R1)))+1/(((2*R3)/(2*R3*R4+R3^2)+1/R4)*((2*(2*R3*R4+R3^2))/(R4*((2*R3)/(2*R3*R4 +R3^2)+1/R4))+(2*R3*R4+R3^2)^2/R4^2))),Rc=1/(R1/(R5/(1/R5+1/R1)+R1*R5+R1/(1/R5+1/R1)) +1/R5+(2*R3*R4+R3^2)/(R4*((2*(2*R3*R4+R3^2))/(R4*((2*R3)/(2*R3*R4+R3^2)+1/R4))+(2*R3*R4 +R3^2)^2/R4^2))+R3/(2*R3*R4+R3^2))]); (%o188) [Ra=1/((2*R3*R4+R3^2)/(R4*((2*(2*R3*R4+R3^2))/(R4*((2*R3)/(2*R3*R4+R3^2)+1/R4))+(2*R3*R4+R3^2)^2/R4^2))+R3/(2*R3*R4+R3^2)+(3*R2)/((3*R2)/(1/(3*R2)+1/R1)+3*R1*R2+R1/(1/(3*R2)+1/R1))+1/R1),Rb= 1/(1/(((2*R3)/(2*R3*R4+R3^2)+1/R4)*((2*(2*R3*R4+R3^2))/(R4*((2*R3)/(2*R3*R4+R3^2)+1/R4))+(2*R3*R4+R3^2)^2/R4^2))+1/((1/(3*R2)+1/R1)*((3*R2)/(1/(3*R2)+1/R1)+3*R1*R2+R1/(1/(3*R2)+1/R1)))),Rc= 1/((2*R3*R4+R3^2)/(R4*((2*(2*R3*R4+R3^2))/(R4*((2*R3)/(2*R3*R4+R3^2)+1/R4))+(2*R3*R4+R3^2)^2/R4^2))+R3/(2*R3*R4+R3^2)+R1/((3*R2)/(1/(3*R2)+1/R1)+3*R1*R2+R1/(1/(3*R2)+1/R1))+1/(3*R2))] (%i189) factor(%); (%o189) [Ra=(2*R1*(2*R4^2+4*R3*R4+R3^2))/(6*R4^2+12*R3*R4+6*R1*R4+3*R3^2+2*R1*R3),Rb=(2*(3*R2+R1)*R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2))/(4*R3*R4^3+6*R2*R4^3+2*R1*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4) ,Rc=(2*R2*(2*R4^2+4*R3*R4+R3^2))/(2*R4^2+4*R3*R4+6*R2*R4+R3^2+2*R2*R3)] 最終的に得られたRa,Rb,Rcの式にR1,R2,R3,R4の値をそれぞれ代入すると (%i190) subst(15, R1, [Ra=(2*R1*(2*R4^2+4*R3*R4+R3^2))/(6*R4^2+12*R3*R4+6*R1*R4+3*R3^2+2*R1*R3),Rb=(2*(3*R2 +R1)*R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2))/(4*R3*R4^3+6*R2*R4^3+2*R1*R4^3+10*R3^2*R4^2 +6*R3^3*R4+R3^4),Rc=(2*R2*(2*R4^2+4*R3*R4+R3^2))/(2*R4^2+4*R3*R4+6*R2*R4+R3^2+2*R2*R3)]); (%o190) [Ra=(30*(2*R4^2+4*R3*R4+R3^2))/(6*R4^2+12*R3*R4+90*R4+3*R3^2+30*R3),Rb=(2*(3*R2+15)*R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2))/(4*R3*R4^3+6*R2*R4^3+30*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Rc =(2*R2*(2*R4^2+4*R3*R4+R3^2))/(2*R4^2+4*R3*R4+6*R2*R4+R3^2+2*R2*R3)] (%i191) subst(5, R2, [Ra=(30*(2*R4^2+4*R3*R4+R3^2))/(6*R4^2+12*R3*R4+90*R4+3*R3^2+30*R3),Rb=(2*(3*R2 +15)*R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2))/(4*R3*R4^3+6*R2*R4^3+30*R4^3+10*R3^2*R4^2 +6*R3^3*R4+R3^4),Rc=(2*R2*(2*R4^2+4*R3*R4+R3^2))/(2*R4^2+4*R3*R4+6*R2*R4+R3^2+2*R2*R3)]); (%o191) [Ra=(30*(2*R4^2+4*R3*R4+R3^2))/(6*R4^2+12*R3*R4+90*R4+3*R3^2+30*R3),Rb=(60*R3*(2*R4+R3)*(2*R4^2+4*R3*R4+R3^2))/(4*R3*R4^3+60*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Rc= (10*(2*R4^2+4*R3*R4+R3^2))/(2*R4^2+4*R3*R4+30*R4+R3^2+10*R3)] (%i192) subst(2, R3, [Ra=(30*(2*R4^2+4*R3*R4+R3^2))/(6*R4^2+12*R3*R4+90*R4+3*R3^2+30*R3),Rb=(60*R3*(2*R4 +R3)*(2*R4^2+4*R3*R4+R3^2))/(4*R3*R4^3+60*R4^3+10*R3^2*R4^2+6*R3^3*R4+R3^4),Rc=(10*(2*R4^2 +4*R3*R4+R3^2))/(2*R4^2+4*R3*R4+30*R4+R3^2+10*R3)]); (%o192) [Ra=(30*(2*R4^2+8*R4+4))/(6*R4^2+114*R4+72),Rb=(120*(2*R4+2)*(2*R4^2+8*R4+4))/(68*R4^3+40*R4^2+48*R4+16),Rc=(10*(2*R4^2+8*R4+4))/(2*R4^2+38*R4+24)] (%i193) subst(4, R4, [Ra=(30*(2*R4^2+8*R4+4))/(6*R4^2+114*R4+72),Rb=(120*(2*R4+2)*(2*R4^2 +8*R4+4))/(68*R4^3+40*R4^2+48*R4+16),Rc=(10*(2*R4^2+8*R4+4))/(2*R4^2+38*R4+24)]); (%o193) [Ra=85/26,Rb=204/13,Rc=85/26] (%i194) float(%), numer; (%o194) [Ra=3.269230769230769,Rb=15.69230769230769,Rc=3.269230769230769] 従って Ra=3.27 [Ω] Rb=15.7 [Ω] Rc=3.27 [Ω] 一方端子2-2'がオープンの時とクローズの時の回路インピーダンスはそれぞれ Zo=1/(1/Ra+1/(Rb+Rc)) =1/(1/3.27+1/(15.7+3.27)) =2.79 [Ω] Zc=1/(1/Ra+1/Rb) =1/(1/3.27+1/15.7) =2.71 [Ω] ということになる。 他の解法によるものと同じ結果が得られた。 |
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