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webadm | 投稿日時: 2008-9-5 7:40 |
Webmaster 登録日: 2004-11-7 居住地: 投稿: 3086 |
【81】タップ付き可変抵抗のあるブリッジ これが交流ブリッジの最後の問題。幸いにして相互誘導回路は無いが今まで無かったようなタップのある可変抵抗回路を含む。
以下の関係が成り立つ。 (R1+jωL)*I1+R3*(I1-Ig)+R*(I1+I2-Ic)=E (R2-Rx)*I2+Rx*(I2-Ic)+R4*(I2-Ic+Ig)+R*(I1+I2-Ic)=E (R2-Rx)*I2+(1/jωC)*Ic=E (R1+jωL)*I1+R4*(I2-Ic+Ig)+R*(I1+I2-Ic)=E これをI1,I2,Ic,Igについて解くと (%i66) e1:(R1+%i*o*L)*I1+R3*(I1-Ig)+R*(I1+I2-Ic)=E; (%o66) (I1-Ig)*R3+I1*(R1+%i*o*L)+(I2+I1-Ic)*R=E (%i67) e2:(R2-Rx)*I2+Rx*(I2-Ic)+R4*(I2-Ic+Ig)+R*(I1+I2-Ic)=E; (%o67) (I2+Ig-Ic)*R4+I2*(R2-Rx)+(I2+I1-Ic)*R+Rx*(I2-Ic)=E (%i68) e3:(R2-Rx)*I2+(-%i/(o*C))*Ic=E; (%o68) I2*(R2-Rx)-(%i*Ic)/(o*C)=E (%i69) e4:(R1+%i*o*L)*I1+R4*(I2-Ic+Ig)+R*(I1+I2-Ic)=E; (%o69) (I2+Ig-Ic)*R4+I1*(R1+%i*o*L)+(I2+I1-Ic)*R=E (%i70) solve([e1,e2,e3,e4],[I1,I2,Ic,Ig]); (%o70) [[I1=(R3*(E*(o*((C*R2-Rx*C)*R4+Rx*C*R2-Rx^2*C)-%i*R2)+o*E*R*(C*R2-Rx*C))+E* (o*(Rx*C*R2-Rx^2*C)*R4-%i*R2*R4)+o*E*R*(C*R2-Rx*C)*R4)/(R3*(L* (%i*o^2*((C*R2-Rx*C)*R4+Rx*C*R2-Rx^2*C)+o*(R4+R2))+R1* (o*((C*R2-Rx*C)*R4+Rx*C*R2-Rx^2*C)+%i*(-R4-R2)+R*(o*(C*R2-Rx*C)-%i))+o*(Rx*C*R2-Rx^2*C)*R4-%i*R2*R4 +R*(L*(%i*o^2*(C*R2-Rx*C)+o)+o*(Rx*C*R2-Rx^2*C)-%i*R2))+R* (L*(%i*o^2*(C*R2-Rx*C)*R4+o*R4)+o*(Rx*C*R2-Rx^2*C)*R4-%i*R2*R4)+R1* (R*(o*(C*R2-Rx*C)*R4-%i*R4)+o*(Rx*C*R2-Rx^2*C)*R4-%i*R2*R4)+L*(%i*o^2*(Rx*C*R2-Rx^2*C)*R4+o*R2*R4)),I2= (R3*(R1*(E*(o*(C*R4+Rx*C)-%i)+o*C*E*R)+E*L*(%i*o^2*(C*R4+Rx*C)+o)+o*Rx*C*E*R4+(%i*o^2*C*E*L+o*Rx*C*E)*R)+ R1*(E*(o*Rx*C*R4-%i*R4)+o*C*E*R*R4)+R*(%i*o^2*C*E*L*R4+o*Rx*C*E*R4)+E*L*(%i*o^2*Rx*C*R4+o*R4))/(R3*(L* (%i*o^2*((C*R2-Rx*C)*R4+Rx*C*R2-Rx^2*C)+o*(R4+R2))+R1* (o*((C*R2-Rx*C)*R4+Rx*C*R2-Rx^2*C)+%i*(-R4-R2)+R*(o*(C*R2-Rx*C)-%i))+o*(Rx*C*R2-Rx^2*C)*R4-%i*R2*R4 +R*(L*(%i*o^2*(C*R2-Rx*C)+o)+o*(Rx*C*R2-Rx^2*C)-%i*R2))+R* (L*(%i*o^2*(C*R2-Rx*C)*R4+o*R4)+o*(Rx*C*R2-Rx^2*C)*R4-%i*R2*R4)+R1* (R*(o*(C*R2-Rx*C)*R4-%i*R4)+o*(Rx*C*R2-Rx^2*C)*R4-%i*R2*R4)+L*(%i*o^2*(Rx*C*R2-Rx^2*C)*R4+o*R2*R4)),Ic= (R3*(R1*(o*E*(C*R4+Rx*C)+o*C*E*R)+%i*o^2*E*L*(C*R4+Rx*C)+o*C*E*R2*R4+R*(o*C*E*R2+%i*o^2*C*E*L))+R* (o*C*E*R2*R4+%i*o^2*C*E*L*R4)+R1*(o*C*E*R*R4+o*Rx*C*E*R4)+%i*o^2*Rx*C*E*L*R4)/(R3*(L* (%i*o^2*((C*R2-Rx*C)*R4+Rx*C*R2-Rx^2*C)+o*(R4+R2))+R1* (o*((C*R2-Rx*C)*R4+Rx*C*R2-Rx^2*C)+%i*(-R4-R2)+R*(o*(C*R2-Rx*C)-%i))+o*(Rx*C*R2-Rx^2*C)*R4-%i*R2*R4 +R*(L*(%i*o^2*(C*R2-Rx*C)+o)+o*(Rx*C*R2-Rx^2*C)-%i*R2))+R* (L*(%i*o^2*(C*R2-Rx*C)*R4+o*R4)+o*(Rx*C*R2-Rx^2*C)*R4-%i*R2*R4)+R1* (R*(o*(C*R2-Rx*C)*R4-%i*R4)+o*(Rx*C*R2-Rx^2*C)*R4-%i*R2*R4)+L*(%i*o^2*(Rx*C*R2-Rx^2*C)*R4+o*R2*R4)),Ig= (R3*(E*(o*((C*R2-Rx*C)*R4+Rx*C*R2-Rx^2*C)-%i*R2)+o*E*R*(C*R2-Rx*C))+o*E*R*(C*R2-Rx*C)*R4+%i*E*R1*R4-o* E*L*R4)/(R3*(L*(%i*o^2*((C*R2-Rx*C)*R4+Rx*C*R2-Rx^2*C)+o*(R4+R2))+R1* (o*((C*R2-Rx*C)*R4+Rx*C*R2-Rx^2*C)+%i*(-R4-R2)+R*(o*(C*R2-Rx*C)-%i))+o*(Rx*C*R2-Rx^2*C)*R4-%i*R2*R4 +R*(L*(%i*o^2*(C*R2-Rx*C)+o)+o*(Rx*C*R2-Rx^2*C)-%i*R2))+R* (L*(%i*o^2*(C*R2-Rx*C)*R4+o*R4)+o*(Rx*C*R2-Rx^2*C)*R4-%i*R2*R4)+R1* (R*(o*(C*R2-Rx*C)*R4-%i*R4)+o*(Rx*C*R2-Rx^2*C)*R4-%i*R2*R4)+L*(%i*o^2*(Rx*C*R2-Rx^2*C)*R4+o*R2*R4))]] (%i71) factor(%); (%o71) [[I1=(E*(o*C*R2*R3*R4-o*Rx*C*R3*R4+o*C*R*R2*R4+o*Rx*C*R2*R4-%i*R2*R4-o*Rx*C*R*R4-o*Rx^2*C*R4 +o*C*R*R2*R3+o*Rx*C*R2*R3-%i*R2*R3-o*Rx*C*R*R3-o*Rx^2*C*R3))/(o*C*R1*R2*R3*R4+%i*o^2*C*L*R2*R3*R4+o*Rx* C*R2*R3*R4-%i*R2*R3*R4-o*Rx*C*R1*R3*R4-%i*R1*R3*R4-%i*o^2*Rx*C*L*R3*R4+o*L*R3*R4-o*Rx^2*C*R3*R4+o*C*R*R1* R2*R4+o*Rx*C*R1*R2*R4-%i*R1*R2*R4+%i*o^2*C*L*R*R2*R4+o*Rx*C*R*R2*R4-%i*R*R2*R4+%i*o^2*Rx*C*L*R2*R4+o*L*R2* R4-o*Rx*C*R*R1*R4-%i*R*R1*R4-o*Rx^2*C*R1*R4-%i*o^2*Rx*C*L*R*R4+o*L*R*R4-o*Rx^2*C*R*R4-%i*o^2*Rx^2*C*L*R4+o* C*R*R1*R2*R3+o*Rx*C*R1*R2*R3-%i*R1*R2*R3+%i*o^2*C*L*R*R2*R3+o*Rx*C*R*R2*R3-%i*R*R2*R3+%i*o^2*Rx*C*L*R2*R3+ o*L*R2*R3-o*Rx*C*R*R1*R3-%i*R*R1*R3-o*Rx^2*C*R1*R3-%i*o^2*Rx*C*L*R*R3+o*L*R*R3-o*Rx^2*C*R*R3-%i*o^2*Rx^2*C*L* R3),I2=(E*(o*C*R1*R3*R4+%i*o^2*C*L*R3*R4+o*Rx*C*R3*R4+o*C*R*R1*R4+o*Rx*C*R1*R4-%i*R1*R4+%i*o^2*C*L*R* R4+o*Rx*C*R*R4+%i*o^2*Rx*C*L*R4+o*L*R4+o*C*R*R1*R3+o*Rx*C*R1*R3-%i*R1*R3+%i*o^2*C*L*R*R3+o*Rx*C*R*R3+%i* o^2*Rx*C*L*R3+o*L*R3))/(o*C*R1*R2*R3*R4+%i*o^2*C*L*R2*R3*R4+o*Rx*C*R2*R3*R4-%i*R2*R3*R4-o*Rx*C*R1*R3*R4- %i*R1*R3*R4-%i*o^2*Rx*C*L*R3*R4+o*L*R3*R4-o*Rx^2*C*R3*R4+o*C*R*R1*R2*R4+o*Rx*C*R1*R2*R4-%i*R1*R2*R4+%i* o^2*C*L*R*R2*R4+o*Rx*C*R*R2*R4-%i*R*R2*R4+%i*o^2*Rx*C*L*R2*R4+o*L*R2*R4-o*Rx*C*R*R1*R4-%i*R*R1*R4-o*Rx^2*C* R1*R4-%i*o^2*Rx*C*L*R*R4+o*L*R*R4-o*Rx^2*C*R*R4-%i*o^2*Rx^2*C*L*R4+o*C*R*R1*R2*R3+o*Rx*C*R1*R2*R3-%i*R1*R2* R3+%i*o^2*C*L*R*R2*R3+o*Rx*C*R*R2*R3-%i*R*R2*R3+%i*o^2*Rx*C*L*R2*R3+o*L*R2*R3-o*Rx*C*R*R1*R3-%i*R*R1*R3- o*Rx^2*C*R1*R3-%i*o^2*Rx*C*L*R*R3+o*L*R*R3-o*Rx^2*C*R*R3-%i*o^2*Rx^2*C*L*R3),Ic=(o*C*E*(R2*R3*R4+R1*R3*R4 +%i*o*L*R3*R4+R*R2*R4+R*R1*R4+Rx*R1*R4+%i*o*L*R*R4+%i*o*Rx*L*R4+R*R2*R3+R*R1*R3+Rx*R1*R3+%i*o*L*R* R3+%i*o*Rx*L*R3))/(o*C*R1*R2*R3*R4+%i*o^2*C*L*R2*R3*R4+o*Rx*C*R2*R3*R4-%i*R2*R3*R4-o*Rx*C*R1*R3*R4-%i* R1*R3*R4-%i*o^2*Rx*C*L*R3*R4+o*L*R3*R4-o*Rx^2*C*R3*R4+o*C*R*R1*R2*R4+o*Rx*C*R1*R2*R4-%i*R1*R2*R4+%i*o^2*C* L*R*R2*R4+o*Rx*C*R*R2*R4-%i*R*R2*R4+%i*o^2*Rx*C*L*R2*R4+o*L*R2*R4-o*Rx*C*R*R1*R4-%i*R*R1*R4-o*Rx^2*C*R1*R4 -%i*o^2*Rx*C*L*R*R4+o*L*R*R4-o*Rx^2*C*R*R4-%i*o^2*Rx^2*C*L*R4+o*C*R*R1*R2*R3+o*Rx*C*R1*R2*R3-%i*R1*R2*R3+%i* o^2*C*L*R*R2*R3+o*Rx*C*R*R2*R3-%i*R*R2*R3+%i*o^2*Rx*C*L*R2*R3+o*L*R2*R3-o*Rx*C*R*R1*R3-%i*R*R1*R3-o*Rx^2*C* R1*R3-%i*o^2*Rx*C*L*R*R3+o*L*R*R3-o*Rx^2*C*R*R3-%i*o^2*Rx^2*C*L*R3),Ig=(E*(o*C*R2*R3*R4-o*Rx*C*R3*R4+o*C* R*R2*R4+%i*R1*R4-o*Rx*C*R*R4-o*L*R4+o*C*R*R2*R3+o*Rx*C*R2*R3-%i*R2*R3-o*Rx*C*R*R3-o*Rx^2*C*R3))/(o* C*R1*R2*R3*R4+%i*o^2*C*L*R2*R3*R4+o*Rx*C*R2*R3*R4-%i*R2*R3*R4-o*Rx*C*R1*R3*R4-%i*R1*R3*R4-%i*o^2*Rx*C*L*R3* R4+o*L*R3*R4-o*Rx^2*C*R3*R4+o*C*R*R1*R2*R4+o*Rx*C*R1*R2*R4-%i*R1*R2*R4+%i*o^2*C*L*R*R2*R4+o*Rx*C*R*R2*R4- %i*R*R2*R4+%i*o^2*Rx*C*L*R2*R4+o*L*R2*R4-o*Rx*C*R*R1*R4-%i*R*R1*R4-o*Rx^2*C*R1*R4-%i*o^2*Rx*C*L*R*R4+o*L*R* R4-o*Rx^2*C*R*R4-%i*o^2*Rx^2*C*L*R4+o*C*R*R1*R2*R3+o*Rx*C*R1*R2*R3-%i*R1*R2*R3+%i*o^2*C*L*R*R2*R3+o*Rx*C*R* R2*R3-%i*R*R2*R3+%i*o^2*Rx*C*L*R2*R3+o*L*R2*R3-o*Rx*C*R*R1*R3-%i*R*R1*R3-o*Rx^2*C*R1*R3-%i*o^2*Rx*C*L*R*R3 +o*L*R*R3-o*Rx^2*C*R*R3-%i*o^2*Rx^2*C*L*R3)]] Igについて整理すると Ig=(E*(ω*C*R2*R3*R4-ω*Rx*C*R3*R4+ω*C* R*R2*R4+j*R1*R4-ω*Rx*C*R*R4-ω*L*R4+ω*C*R*R2*R3+ω*Rx*C*R2*R3-j*R2*R3-ω*Rx*C*R*R3-ω*Rx^2*C*R3))/(ω* C*R1*R2*R3*R4+j*ω^2*C*L*R2*R3*R4+ω*Rx*C*R2*R3*R4-j*R2*R3*R4-ω*Rx*C*R1*R3*R4-j*R1*R3*R4-j*ω^2*Rx*C*L*R3* R4+ω*L*R3*R4-ω*Rx^2*C*R3*R4+ω*C*R*R1*R2*R4+ω*Rx*C*R1*R2*R4-j*R1*R2*R4+j*ω^2*C*L*R*R2*R4+ω*Rx*C*R*R2*R4- j*R*R2*R4+j*ω^2*Rx*C*L*R2*R4+ω*L*R2*R4-ω*Rx*C*R*R1*R4-j*R*R1*R4-ω*Rx^2*C*R1*R4-j*ω^2*Rx*C*L*R*R4+ω*L*R* R4-ω*Rx^2*C*R*R4-j*ω^2*Rx^2*C*L*R4+ω*C*R*R1*R2*R3+ω*Rx*C*R1*R2*R3-j*R1*R2*R3+j*ω^2*C*L*R*R2*R3+ω*Rx*C*R* R2*R3-j*R*R2*R3+j*ω^2*Rx*C*L*R2*R3+ω*L*R2*R3-ω*Rx*C*R*R1*R3-j*R*R1*R3-ω*Rx^2*C*R1*R3-j*ω^2*Rx*C*L*R*R3 +ω*L*R*R3-ω*Rx^2*C*R*R3-j*ω^2*Rx^2*C*L*R3) 従ってIg=0となるためには分子が0となる条件 ω*C*R2*R3*R4-ω*Rx*C*R3*R4+ω*C*R*R2*R4+j*R1*R4-ω*Rx*C*R*R4-ω*L*R4+ω*C*R*R2*R3+ω*Rx*C*R2*R3-j*R2*R3-ω*Rx*C*R*R3-ω*Rx^2*C*R3=0 を満たす必要がある。 直交形式に整理すると ω*(C*(R2-Rx)*((R3+R)*R4+(R+Rx)*R3)-L*R4)+j*(R1*R4-R2*R3)=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 C*(R2-Rx)*((R3+R)*R4+(R+Rx)*R3)-L*R4=0 ∴L=C*(R2-Rx)*((R3+R)*R4+(R+Rx)*R3)/R4 虚数部より R1*R4-R2*R3=0 ∴R1*R4=R2*R3 ということになる。 著者はR2-RxをR2'、RxをR2''として網目電流法で方程式をたてて同じ結果を得ている。 |
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