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webadm | 投稿日時: 2008-9-3 10:22 |
Webmaster 登録日: 2004-11-7 居住地: 投稿: 3068 |
【73】Andersonブリッジ(その2) もう一種類のAndersonブリッジの平衡条件に関する問題。
AndersonブリッジはMaxwell-Wienブリッジを6素子に改良したもので、やはり周波数に依存しない平衡条件を持つのが特徴らしい。計測回路の本とかに載っているが、Webで検索するとシンガポールにあるアンダーソン橋のほうがむしろ有名で一杯出てくる。 以下の関係が成り立つ (R2+1/(1/R4+jωC4))*I-R2*I1-(1/jωC4)*I4=E (R1+R2+R)*I1-R2*I-R*I2=0 (R+R3+1/jωC3)*I2-R*I1+(1/jωC3)*I3=0 (1/jωC3+R4)*I3+(1/jωC3)*I2+R4*I4=0 (R4+(1/jωC4))*I4+R4*I3-(1/jωC4)*I=0 Ig=I1+I3 これをI,I1,I2,I3,I4,Igに関して解くと (%i40) e1:(R2+1/(1/R4+%i*o*C4))*I-R2*I1-(-%i/(o*C4))*I3=E; (%o40) I*(1/(1/R4+%i*o*C4)+R2)-I1*R2+(%i*I3)/(o*C4)=E (%i41) e2:(R1+R2+R)*I1-R2*I-R*I2=0; (%o41) I1*(R2+R1+R)-I*R2-I2*R=0 (%i42) e3:(R+R3-%i/(o*C3))*I2-R*I1+(-%i/(o*C3))*I3=0; (%o42) I2*(R3+R-%i/(o*C3))-I1*R-(%i*I3)/(o*C3)=0 (%i43) e4:(-%i/(o*C3)+R4)*I3+(-%i/(o*C3))*I2+R4*I4=0; (%o43) I3*(R4-%i/(o*C3))+I4*R4-(%i*I2)/(o*C3)=0 (%i44) e5:(R4+(-%i/(o*C4)))*I4+R4*I3-(-%i/(o*C4))*I=0; (%o44) I4*(R4-%i/(o*C4))+I3*R4+(%i*I)/(o*C4)=0 (%i45) e6:Ig=I1+I3; (%o45) Ig=I3+I1 (%i46) solve([e1,e2,e3,e4,e5,e6],[I,I1,I2,I3,I4,Ig]); (%o46) [[I=((E*(C4^2*(o^3*(R1*(C3*R3+C3*R)+C3*R*R3)+%i*o^2*(-R1-R))+o^3*C4^3*(R1*(R3+R)+R*R3))+E*R2* (C4^2*(o^3*(C3*R3+C3*R)-%i*o^2)+o^3*C4^3*(R3+R)))*R4^2+(E* (C4*(%i*o^2*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*(-R1-R))+%i*o^2*C4^2*(R1*(-2*R3-2*R)-2*R*R3))+E*R2* (C4*(%i*o^2*(-C3*R3-C3*R)-o)+%i*o^2*C4^2*(-2*R3-2*R)))*R4+o*C4*E*(R1*(-R3-R)-R*R3)+o*C4*E*R2*(-R3-R)) /((R2*(C4^2*(o^3*(R1*(C3*R3+C3*R)+C3*R*R3)+%i*o^2*(-R3-R1-2*R))+C4*(%i*o^2*(-2*C3*R3-2*C3*R)-2*o)+o^3* C4^3*(R1*(R3+R)+R*R3))+C4*(%i*o^2*(R1*(-2*C3*R3-2*C3*R)-2*C3*R*R3)+o*(-2*R1-2*R))+%i*o^2*C4^2* (R1*(-R3-R)-R*R3))*R4^2+(R2*(C4*(%i*o^2*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*(-R3-R1-3*R))+%i*o^2*C4^2* (R1*(-2*R3-2*R)-2*R*R3)+o*(-C3*R3-C3*R)+%i)+o*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*C4*(R1*(-R3-R)-R*R3) +%i*(R1+R))*R4+R2*(o*C4*(R1*(-R3-R)-R*R3)+%i*R)),I1=( (E*R2*(C4^2*(o^3*(C3*R3+C3*R)-%i*o^2)+o^3*C4^3*(R3+R))-%i*o^2*C4^2*E*R)*R4^2+ (E*R2*(C4*(%i*o^2*(-C3*R3-C3*R)-o)+%i*o^2*C4^2*(-2*R3-2*R))-o*C4*E*R)*R4+o*C4*E*R2*(-R3-R))/((R2*(C4^2* (o^3*(R1*(C3*R3+C3*R)+C3*R*R3)+%i*o^2*(-R3-R1-2*R))+C4*(%i*o^2*(-2*C3*R3-2*C3*R)-2*o)+o^3*C4^3* (R1*(R3+R)+R*R3))+C4*(%i*o^2*(R1*(-2*C3*R3-2*C3*R)-2*C3*R*R3)+o*(-2*R1-2*R))+%i*o^2*C4^2* (R1*(-R3-R)-R*R3))*R4^2+(R2*(C4*(%i*o^2*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*(-R3-R1-3*R))+%i*o^2*C4^2* (R1*(-2*R3-2*R)-2*R*R3)+o*(-C3*R3-C3*R)+%i)+o*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*C4*(R1*(-R3-R)-R*R3) +%i*(R1+R))*R4+R2*(o*C4*(R1*(-R3-R)-R*R3)+%i*R)),I2=( (E*(C4^2*(o^3*C3*R-%i*o^2)+o^3*C4^3*R)*R2+%i*o^2*C4^2*E*(-R1-R))*R4^2+ (E*(C4*(-%i*o^2*C3*R-o)-2*%i*o^2*C4^2*R)*R2+o*C4*E*(-R1-R))*R4-o*C4*E*R*R2)/((R2*(C4^2* (o^3*(R1*(C3*R3+C3*R)+C3*R*R3)+%i*o^2*(-R3-R1-2*R))+C4*(%i*o^2*(-2*C3*R3-2*C3*R)-2*o)+o^3*C4^3* (R1*(R3+R)+R*R3))+C4*(%i*o^2*(R1*(-2*C3*R3-2*C3*R)-2*C3*R*R3)+o*(-2*R1-2*R))+%i*o^2*C4^2* (R1*(-R3-R)-R*R3))*R4^2+(R2*(C4*(%i*o^2*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*(-R3-R1-3*R))+%i*o^2*C4^2* (R1*(-2*R3-2*R)-2*R*R3)+o*(-C3*R3-C3*R)+%i)+o*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*C4*(R1*(-R3-R)-R*R3) +%i*(R1+R))*R4+R2*(o*C4*(R1*(-R3-R)-R*R3)+%i*R)),I3=-( (C4^2*E*(o^3*(R1*(C3*R3+C3*R)+C3*R*R3)+%i*o^2*(-R1-R))+E*R2*(C4^2*(o^3*(C3*R3+C3*R)-%i*o^2)+o^3*C4^3*R))*R4^2+ (C4*E*(%i*o^2*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*(-R1-R))+E*R2*(C4*(%i*o^2*(-C3*R3-C3*R)-o)-2*%i*o^2*C4^2*R))*R4 -o*C4*E*R*R2)/((R2*(C4^2*(o^3*(R1*(C3*R3+C3*R)+C3*R*R3)+%i*o^2*(-R3-R1-2*R))+C4* (%i*o^2*(-2*C3*R3-2*C3*R)-2*o)+o^3*C4^3*(R1*(R3+R)+R*R3))+C4* (%i*o^2*(R1*(-2*C3*R3-2*C3*R)-2*C3*R*R3)+o*(-2*R1-2*R))+%i*o^2*C4^2*(R1*(-R3-R)-R*R3))*R4^2+(R2*(C4* (%i*o^2*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*(-R3-R1-3*R))+%i*o^2*C4^2*(R1*(-2*R3-2*R)-2*R*R3)+o* (-C3*R3-C3*R)+%i)+o*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*C4*(R1*(-R3-R)-R*R3)+%i*(R1+R))*R4+R2* (o*C4*(R1*(-R3-R)-R*R3)+%i*R)),I4=( (C4^2*E*(o^3*(R1*(C3*R3+C3*R)+C3*R*R3)+%i*o^2*(-R1-R))+E*R2*(C4^2*(o^3*(C3*R3+C3*R)-%i*o^2)+o^3*C4^3*R))*R4^2+ (E*(C4*(%i*o^2*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*(-R1-R))+%i*o^2*C4^2*(R1*(-R3-R)-R*R3))+E*R2* (C4*(%i*o^2*(-C3*R3-C3*R)-o)+%i*o^2*C4^2*(-R3-2*R)))*R4+o*C4*E*(R1*(-R3-R)-R*R3)+o*C4*E*R2*(-R3-R))/ ((R2*(C4^2*(o^3*(R1*(C3*R3+C3*R)+C3*R*R3)+%i*o^2*(-R3-R1-2*R))+C4*(%i*o^2*(-2*C3*R3-2*C3*R)-2*o)+o^3* C4^3*(R1*(R3+R)+R*R3))+C4*(%i*o^2*(R1*(-2*C3*R3-2*C3*R)-2*C3*R*R3)+o*(-2*R1-2*R))+%i*o^2*C4^2* (R1*(-R3-R)-R*R3))*R4^2+(R2*(C4*(%i*o^2*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*(-R3-R1-3*R))+%i*o^2*C4^2* (R1*(-2*R3-2*R)-2*R*R3)+o*(-C3*R3-C3*R)+%i)+o*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*C4*(R1*(-R3-R)-R*R3) +%i*(R1+R))*R4+R2*(o*C4*(R1*(-R3-R)-R*R3)+%i*R)),Ig=( (C4^2*E*(o^3*(R1*(-C3*R3-C3*R)-C3*R*R3)+%i*o^2*R1)+o^3*C4^3*E*R2*R3)*R4^2+ (C4*E*(%i*o^2*(R1*(C3*R3+C3*R)+C3*R*R3)+o*R1)-2*%i*o^2*C4^2*E*R2*R3)*R4-o*C4*E*R2*R3)/((R2*(C4^2* (o^3*(R1*(C3*R3+C3*R)+C3*R*R3)+%i*o^2*(-R3-R1-2*R))+C4*(%i*o^2*(-2*C3*R3-2*C3*R)-2*o)+o^3*C4^3* (R1*(R3+R)+R*R3))+C4*(%i*o^2*(R1*(-2*C3*R3-2*C3*R)-2*C3*R*R3)+o*(-2*R1-2*R))+%i*o^2*C4^2* (R1*(-R3-R)-R*R3))*R4^2+(R2*(C4*(%i*o^2*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*(-R3-R1-3*R))+%i*o^2*C4^2* (R1*(-2*R3-2*R)-2*R*R3)+o*(-C3*R3-C3*R)+%i)+o*(R1*(-C3*R3-C3*R)-C3*R*R3)+o*C4*(R1*(-R3-R)-R*R3) +%i*(R1+R))*R4+R2*(o*C4*(R1*(-R3-R)-R*R3)+%i*R))]] (%i47) factor(%); (%o47) [[I=(o*C4*E*(o^2*C4^2*R2*R3*R4^2+o^2*C3*C4*R2*R3*R4^2+o^2*C4^2*R1*R3*R4^2+o^2*C3*C4*R1*R3*R4^2+o^2*C4^2* R*R3*R4^2+o^2*C3*C4*R*R3*R4^2+o^2*C4^2*R*R2*R4^2+o^2*C3*C4*R*R2*R4^2-%i*o*C4*R2*R4^2+o^2*C4^2*R*R1*R4^2+o^2*C3*C4*R* R1*R4^2-%i*o*C4*R1*R4^2-%i*o*C4*R*R4^2-2*%i*o*C4*R2*R3*R4-%i*o*C3*R2*R3*R4-2*%i*o*C4*R1*R3*R4-%i*o*C3*R1* R3*R4-2*%i*o*C4*R*R3*R4-%i*o*C3*R*R3*R4-2*%i*o*C4*R*R2*R4-%i*o*C3*R*R2*R4-R2*R4-2*%i*o*C4*R*R1*R4-%i*o* C3*R*R1*R4-R1*R4-R*R4-R2*R3-R1*R3-R*R3-R*R2-R*R1))/(o^3*C4^3*R1*R2*R3*R4^2+o^3*C3*C4^2*R1*R2*R3*R4^2 +o^3*C4^3*R*R2*R3*R4^2+o^3*C3*C4^2*R*R2*R3*R4^2-%i*o^2*C4^2*R2*R3*R4^2-2*%i*o^2*C3*C4*R2*R3*R4^2-%i*o^2*C4^2*R1*R3* R4^2-2*%i*o^2*C3*C4*R1*R3*R4^2-%i*o^2*C4^2*R*R3*R4^2-2*%i*o^2*C3*C4*R*R3*R4^2+o^3*C4^3*R*R1*R2*R4^2+o^3*C3*C4^2*R*R1* R2*R4^2-%i*o^2*C4^2*R1*R2*R4^2-2*%i*o^2*C4^2*R*R2*R4^2-2*%i*o^2*C3*C4*R*R2*R4^2-2*o*C4*R2*R4^2-%i*o^2*C4^2*R*R1*R4^2 -2*%i*o^2*C3*C4*R*R1*R4^2-2*o*C4*R1*R4^2-2*o*C4*R*R4^2-2*%i*o^2*C4^2*R1*R2*R3*R4-%i*o^2*C3*C4*R1*R2*R3*R4-2*%i* o^2*C4^2*R*R2*R3*R4-%i*o^2*C3*C4*R*R2*R3*R4-o*C4*R2*R3*R4-o*C3*R2*R3*R4-o*C4*R1*R3*R4-o*C3*R1*R3*R4-o*C4*R* R3*R4-o*C3*R*R3*R4-2*%i*o^2*C4^2*R*R1*R2*R4-%i*o^2*C3*C4*R*R1*R2*R4-o*C4*R1*R2*R4-3*o*C4*R*R2*R4-o*C3*R*R2* R4+%i*R2*R4-o*C4*R*R1*R4-o*C3*R*R1*R4+%i*R1*R4+%i*R*R4-o*C4*R1*R2*R3-o*C4*R*R2*R3-o*C4*R*R1*R2+%i*R* R2),I1=(o*C4*E*(o^2*C4^2*R2*R3*R4^2+o^2*C3*C4*R2*R3*R4^2+o^2*C4^2*R*R2*R4^2+o^2*C3*C4*R*R2*R4^2-%i*o*C4*R2*R4^2 -%i*o*C4*R*R4^2-2*%i*o*C4*R2*R3*R4-%i*o*C3*R2*R3*R4-2*%i*o*C4*R*R2*R4-%i*o*C3*R*R2*R4-R2*R4-R*R4-R2*R3 -R*R2))/(o^3*C4^3*R1*R2*R3*R4^2+o^3*C3*C4^2*R1*R2*R3*R4^2+o^3*C4^3*R*R2*R3*R4^2+o^3*C3*C4^2*R*R2*R3*R4^2-%i*o^2* C4^2*R2*R3*R4^2-2*%i*o^2*C3*C4*R2*R3*R4^2-%i*o^2*C4^2*R1*R3*R4^2-2*%i*o^2*C3*C4*R1*R3*R4^2-%i*o^2*C4^2*R*R3*R4^2-2* %i*o^2*C3*C4*R*R3*R4^2+o^3*C4^3*R*R1*R2*R4^2+o^3*C3*C4^2*R*R1*R2*R4^2-%i*o^2*C4^2*R1*R2*R4^2-2*%i*o^2*C4^2*R*R2*R4^2-2* %i*o^2*C3*C4*R*R2*R4^2-2*o*C4*R2*R4^2-%i*o^2*C4^2*R*R1*R4^2-2*%i*o^2*C3*C4*R*R1*R4^2-2*o*C4*R1*R4^2-2*o*C4*R*R4^2- 2*%i*o^2*C4^2*R1*R2*R3*R4-%i*o^2*C3*C4*R1*R2*R3*R4-2*%i*o^2*C4^2*R*R2*R3*R4-%i*o^2*C3*C4*R*R2*R3*R4-o*C4*R2*R3* R4-o*C3*R2*R3*R4-o*C4*R1*R3*R4-o*C3*R1*R3*R4-o*C4*R*R3*R4-o*C3*R*R3*R4-2*%i*o^2*C4^2*R*R1*R2*R4-%i*o^2*C3* C4*R*R1*R2*R4-o*C4*R1*R2*R4-3*o*C4*R*R2*R4-o*C3*R*R2*R4+%i*R2*R4-o*C4*R*R1*R4-o*C3*R*R1*R4+%i*R1*R4+ %i*R*R4-o*C4*R1*R2*R3-o*C4*R*R2*R3-o*C4*R*R1*R2+%i*R*R2),I2=(o*C4*E*(o^2*C4^2*R*R2*R4^2+o^2*C3*C4*R*R2* R4^2-%i*o*C4*R2*R4^2-%i*o*C4*R1*R4^2-%i*o*C4*R*R4^2-2*%i*o*C4*R*R2*R4-%i*o*C3*R*R2*R4-R2*R4-R1*R4-R*R4- R*R2))/(o^3*C4^3*R1*R2*R3*R4^2+o^3*C3*C4^2*R1*R2*R3*R4^2+o^3*C4^3*R*R2*R3*R4^2+o^3*C3*C4^2*R*R2*R3*R4^2-%i*o^2*C4^2* R2*R3*R4^2-2*%i*o^2*C3*C4*R2*R3*R4^2-%i*o^2*C4^2*R1*R3*R4^2-2*%i*o^2*C3*C4*R1*R3*R4^2-%i*o^2*C4^2*R*R3*R4^2-2*%i*o^2* C3*C4*R*R3*R4^2+o^3*C4^3*R*R1*R2*R4^2+o^3*C3*C4^2*R*R1*R2*R4^2-%i*o^2*C4^2*R1*R2*R4^2-2*%i*o^2*C4^2*R*R2*R4^2-2*%i*o^2* C3*C4*R*R2*R4^2-2*o*C4*R2*R4^2-%i*o^2*C4^2*R*R1*R4^2-2*%i*o^2*C3*C4*R*R1*R4^2-2*o*C4*R1*R4^2-2*o*C4*R*R4^2-2*%i* o^2*C4^2*R1*R2*R3*R4-%i*o^2*C3*C4*R1*R2*R3*R4-2*%i*o^2*C4^2*R*R2*R3*R4-%i*o^2*C3*C4*R*R2*R3*R4-o*C4*R2*R3*R4-o* C3*R2*R3*R4-o*C4*R1*R3*R4-o*C3*R1*R3*R4-o*C4*R*R3*R4-o*C3*R*R3*R4-2*%i*o^2*C4^2*R*R1*R2*R4-%i*o^2*C3*C4*R* R1*R2*R4-o*C4*R1*R2*R4-3*o*C4*R*R2*R4-o*C3*R*R2*R4+%i*R2*R4-o*C4*R*R1*R4-o*C3*R*R1*R4+%i*R1*R4+%i*R* R4-o*C4*R1*R2*R3-o*C4*R*R2*R3-o*C4*R*R1*R2+%i*R*R2),I3=-(o*C4*E*(o^2*C3*C4*R2*R3*R4^2+o^2*C3*C4*R1*R3* R4^2+o^2*C3*C4*R*R3*R4^2+o^2*C4^2*R*R2*R4^2+o^2*C3*C4*R*R2*R4^2-%i*o*C4*R2*R4^2+o^2*C3*C4*R*R1*R4^2-%i*o*C4*R1* R4^2-%i*o*C4*R*R4^2-%i*o*C3*R2*R3*R4-%i*o*C3*R1*R3*R4-%i*o*C3*R*R3*R4-2*%i*o*C4*R*R2*R4-%i*o*C3*R*R2*R4- R2*R4-%i*o*C3*R*R1*R4-R1*R4-R*R4-R*R2))/(o^3*C4^3*R1*R2*R3*R4^2+o^3*C3*C4^2*R1*R2*R3*R4^2+o^3*C4^3*R*R2*R3* R4^2+o^3*C3*C4^2*R*R2*R3*R4^2-%i*o^2*C4^2*R2*R3*R4^2-2*%i*o^2*C3*C4*R2*R3*R4^2-%i*o^2*C4^2*R1*R3*R4^2-2*%i*o^2*C3*C4* R1*R3*R4^2-%i*o^2*C4^2*R*R3*R4^2-2*%i*o^2*C3*C4*R*R3*R4^2+o^3*C4^3*R*R1*R2*R4^2+o^3*C3*C4^2*R*R1*R2*R4^2-%i*o^2*C4^2* R1*R2*R4^2-2*%i*o^2*C4^2*R*R2*R4^2-2*%i*o^2*C3*C4*R*R2*R4^2-2*o*C4*R2*R4^2-%i*o^2*C4^2*R*R1*R4^2-2*%i*o^2*C3*C4*R*R1* R4^2-2*o*C4*R1*R4^2-2*o*C4*R*R4^2-2*%i*o^2*C4^2*R1*R2*R3*R4-%i*o^2*C3*C4*R1*R2*R3*R4-2*%i*o^2*C4^2*R*R2*R3*R4- %i*o^2*C3*C4*R*R2*R3*R4-o*C4*R2*R3*R4-o*C3*R2*R3*R4-o*C4*R1*R3*R4-o*C3*R1*R3*R4-o*C4*R*R3*R4-o*C3*R*R3*R4 -2*%i*o^2*C4^2*R*R1*R2*R4-%i*o^2*C3*C4*R*R1*R2*R4-o*C4*R1*R2*R4-3*o*C4*R*R2*R4-o*C3*R*R2*R4+%i*R2*R4-o*C4* R*R1*R4-o*C3*R*R1*R4+%i*R1*R4+%i*R*R4-o*C4*R1*R2*R3-o*C4*R*R2*R3-o*C4*R*R1*R2+%i*R*R2),I4=(o*C4*E* (o^2*C3*C4*R2*R3*R4^2+o^2*C3*C4*R1*R3*R4^2+o^2*C3*C4*R*R3*R4^2+o^2*C4^2*R*R2*R4^2+o^2*C3*C4*R*R2*R4^2-%i*o*C4*R2* R4^2+o^2*C3*C4*R*R1*R4^2-%i*o*C4*R1*R4^2-%i*o*C4*R*R4^2-%i*o*C4*R2*R3*R4-%i*o*C3*R2*R3*R4-%i*o*C4*R1*R3*R4- %i*o*C3*R1*R3*R4-%i*o*C4*R*R3*R4-%i*o*C3*R*R3*R4-2*%i*o*C4*R*R2*R4-%i*o*C3*R*R2*R4-R2*R4-%i*o*C4*R*R1*R4 -%i*o*C3*R*R1*R4-R1*R4-R*R4-R2*R3-R1*R3-R*R3-R*R2-R*R1))/(o^3*C4^3*R1*R2*R3*R4^2+o^3*C3*C4^2*R1*R2* R3*R4^2+o^3*C4^3*R*R2*R3*R4^2+o^3*C3*C4^2*R*R2*R3*R4^2-%i*o^2*C4^2*R2*R3*R4^2-2*%i*o^2*C3*C4*R2*R3*R4^2-%i*o^2*C4^2* R1*R3*R4^2-2*%i*o^2*C3*C4*R1*R3*R4^2-%i*o^2*C4^2*R*R3*R4^2-2*%i*o^2*C3*C4*R*R3*R4^2+o^3*C4^3*R*R1*R2*R4^2+o^3*C3*C4^2* R*R1*R2*R4^2-%i*o^2*C4^2*R1*R2*R4^2-2*%i*o^2*C4^2*R*R2*R4^2-2*%i*o^2*C3*C4*R*R2*R4^2-2*o*C4*R2*R4^2-%i*o^2*C4^2*R*R1* R4^2-2*%i*o^2*C3*C4*R*R1*R4^2-2*o*C4*R1*R4^2-2*o*C4*R*R4^2-2*%i*o^2*C4^2*R1*R2*R3*R4-%i*o^2*C3*C4*R1*R2*R3*R4-2* %i*o^2*C4^2*R*R2*R3*R4-%i*o^2*C3*C4*R*R2*R3*R4-o*C4*R2*R3*R4-o*C3*R2*R3*R4-o*C4*R1*R3*R4-o*C3*R1*R3*R4-o* C4*R*R3*R4-o*C3*R*R3*R4-2*%i*o^2*C4^2*R*R1*R2*R4-%i*o^2*C3*C4*R*R1*R2*R4-o*C4*R1*R2*R4-3*o*C4*R*R2*R4-o*C3* R*R2*R4+%i*R2*R4-o*C4*R*R1*R4-o*C3*R*R1*R4+%i*R1*R4+%i*R*R4-o*C4*R1*R2*R3-o*C4*R*R2*R3-o*C4*R*R1*R2+ %i*R*R2),Ig=(o*C4*E*(o^2*C4^2*R2*R3*R4^2-o^2*C3*C4*R1*R3*R4^2-o^2*C3*C4*R*R3*R4^2-o^2*C3*C4*R*R1*R4^2+%i*o*C4* R1*R4^2-2*%i*o*C4*R2*R3*R4+%i*o*C3*R1*R3*R4+%i*o*C3*R*R3*R4+%i*o*C3*R*R1*R4+R1*R4-R2*R3))/(o^3*C4^3*R1* R2*R3*R4^2+o^3*C3*C4^2*R1*R2*R3*R4^2+o^3*C4^3*R*R2*R3*R4^2+o^3*C3*C4^2*R*R2*R3*R4^2-%i*o^2*C4^2*R2*R3*R4^2-2*%i*o^2* C3*C4*R2*R3*R4^2-%i*o^2*C4^2*R1*R3*R4^2-2*%i*o^2*C3*C4*R1*R3*R4^2-%i*o^2*C4^2*R*R3*R4^2-2*%i*o^2*C3*C4*R*R3*R4^2+o^3* C4^3*R*R1*R2*R4^2+o^3*C3*C4^2*R*R1*R2*R4^2-%i*o^2*C4^2*R1*R2*R4^2-2*%i*o^2*C4^2*R*R2*R4^2-2*%i*o^2*C3*C4*R*R2*R4^2-2* o*C4*R2*R4^2-%i*o^2*C4^2*R*R1*R4^2-2*%i*o^2*C3*C4*R*R1*R4^2-2*o*C4*R1*R4^2-2*o*C4*R*R4^2-2*%i*o^2*C4^2*R1*R2*R3*R4 -%i*o^2*C3*C4*R1*R2*R3*R4-2*%i*o^2*C4^2*R*R2*R3*R4-%i*o^2*C3*C4*R*R2*R3*R4-o*C4*R2*R3*R4-o*C3*R2*R3*R4-o*C4* R1*R3*R4-o*C3*R1*R3*R4-o*C4*R*R3*R4-o*C3*R*R3*R4-2*%i*o^2*C4^2*R*R1*R2*R4-%i*o^2*C3*C4*R*R1*R2*R4-o*C4*R1* R2*R4-3*o*C4*R*R2*R4-o*C3*R*R2*R4+%i*R2*R4-o*C4*R*R1*R4-o*C3*R*R1*R4+%i*R1*R4+%i*R*R4-o*C4*R1*R2*R3- o*C4*R*R2*R3-o*C4*R*R1*R2+%i*R*R2)]] Igに関して整理すると Ig=(ω*C4*E*(ω^2*C4^2*R2*R3*R4^2-ω^2*C3*C4*R1*R3*R4^2-ω^2*C3*C4*R*R3*R4^2-ω^2*C3*C4*R*R1*R4^2+j*ω*C4* R1*R4^2-2*j*ω*C4*R2*R3*R4+j*ω*C3*R1*R3*R4+j*ω*C3*R*R3*R4+j*ω*C3*R*R1*R4+R1*R4-R2*R3))/(ω^3*C4^3*R1* R2*R3*R4^2+ω^3*C3*C4^2*R1*R2*R3*R4^2+ω^3*C4^3*R*R2*R3*R4^2+ω^3*C3*C4^2*R*R2*R3*R4^2-j*ω^2*C4^2*R2*R3*R4^2-2*j*ω^2* C3*C4*R2*R3*R4^2-j*ω^2*C4^2*R1*R3*R4^2-2*j*ω^2*C3*C4*R1*R3*R4^2-j*ω^2*C4^2*R*R3*R4^2-2*j*ω^2*C3*C4*R*R3*R4^2+ω^3* C4^3*R*R1*R2*R4^2+ω^3*C3*C4^2*R*R1*R2*R4^2-j*ω^2*C4^2*R1*R2*R4^2-2*j*ω^2*C4^2*R*R2*R4^2-2*j*ω^2*C3*C4*R*R2*R4^2-2* ω*C4*R2*R4^2-j*ω^2*C4^2*R*R1*R4^2-2*j*ω^2*C3*C4*R*R1*R4^2-2*ω*C4*R1*R4^2-2*ω*C4*R*R4^2-2*j*ω^2*C4^2*R1*R2*R3*R4 -j*ω^2*C3*C4*R1*R2*R3*R4-2*j*ω^2*C4^2*R*R2*R3*R4-j*ω^2*C3*C4*R*R2*R3*R4-ω*C4*R2*R3*R4-ω*C3*R2*R3*R4-ω*C4* R1*R3*R4-ω*C3*R1*R3*R4-ω*C4*R*R3*R4-ω*C3*R*R3*R4-2*j*ω^2*C4^2*R*R1*R2*R4-j*ω^2*C3*C4*R*R1*R2*R4-ω*C4*R1* R2*R4-3*ω*C4*R*R2*R4-ω*C3*R*R2*R4+j*R2*R4-ω*C4*R*R1*R4-ω*C3*R*R1*R4+j*R1*R4+j*R*R4-ω*C4*R1*R2*R3- ω*C4*R*R2*R3-ω*C4*R*R1*R2+j*R*R2) 平衡条件ではIg=0となるため、Igの分子が0となる条件 ω^2*C4^2*R2*R3*R4^2-ω^2*C3*C4*R1*R3*R4^2-ω^2*C3*C4*R*R3*R4^2-ω^2*C3*C4*R*R1*R4^2+j*ω*C4* R1*R4^2-2*j*ω*C4*R2*R3*R4+j*ω*C3*R1*R3*R4+j*ω*C3*R*R3*R4+j*ω*C3*R*R1*R4+R1*R4-R2*R3=0 を満たす必要がある。この式を整理すると ω^2*R4^2*C4*(C4*R2*R3-C3*(R1*R3+R*(R3+R1))+R1*R4-R2*R3+j*ω*R4*(C4*(R1*R4-2*R2*R3)+C3*(R1*R3+R*(R3+R1)))=0 実数部と虚数部がそれぞれ0でなければならないので 実数部より C4*R2*R3-C3*(R1*R3+R*(R3+R1))=0 ∴C4=C3*(R1*R3+R*(R3+R1))/(R2*R3) R1*R4-R2*R3=0 ∴R1*R4=R2*R3 虚数部より C4*(R1*R4-2*R2*R3)+C3*(R1*R3+R*(R3+R1))=0 C4=-C3*(R1*R3+R*(R3+R1))/(R1*R4-2*R2*R3) ここで R1*R4=R2*R3 を代入すると C4=-C3*(R1*R3+R*(R3+R1))/(R2*R3-2*R2*R3) =C3*(R1*R3+R*(R3+R1))/(R2*R3) ということになる。 一瞬Igの式が複雑過ぎて間違ったかと思ったが、良く手で整理すると実数部と虚数部で同じような条件式が浮かび上がってきて光が見えた。 |
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