直列回路に続いて並列回路の合成複素インピーダンスとアドミッタンスを求めてみる

並列抵抗回路の場合



Z=1/(G1+G2)=1/(1/R1+1/R2)=R1*R2/(R1+R2)
Y=(G1+G2)=(1/R1+1/R2)=(R1+R2)/(R1*R2)
|Z|=R1*R2/(R1+R2)
φ=0

並列インダクタンス回路の場合



Z=1/(-jBL1-jBL2))=j/(1/ωL1+1/ωL2)=jω(L1*L2/(L1+L2))=jωL=jXL
Y=-jBL1-jBL2)=-j(1/ωL1+1/ωL2))=-(j/ω)*((L1+L2)/(L1*L2))
=-j(L1+L2)/(ω(L1*L2))
|Z|=sqrt((ω(L1*L2/(L1+L2)))^2)=ω(L1*L2/(L1+L2))
φ=atan(XL/R)=atan(∞)=π/2
L=L1*L2/(L1+L2)

並列キャパシタンス回路の場合



Z=1/(jBC1+jBC2)=1/(jωC1+jωC2))=-j/ω(C1+C2)=-j/ωC=-jXC
Y=jBC1+jBC2=jωC1+jωC2=jω(C1+C2)=jωC=j/XC
|Z|=sqrt((-1/ω(C1+C2))^2)=sqrt(1/(ω(C1+C2))^2)=1/ω(C1+C2)
φ=atan(-XC/R)=-π/2
C=C1+C2

RL並列回路の場合



Z=1/(G-jBL)=1/(1/R-j/ωL))=1/((ωL-jR)/(ωLR))
=ωLR/(ωL-jR)
=ωLR*(ωL+jR)/((ωL+jR)*(ωL-jR))
=(ω^2L^2*R+jωLR^2)/((ωL)^2+R^2)
=ω^2L^2*R/((ωL)^2+R^2)+j(ωLR^2/((ωl)^2+R^2))
Y=G-jBL=1/R-j/ωL=(ωL-R)/(ωLR)
|Z|=sqrt((ω^2L^2*R/((ωL)^2+R^2))^2+(ωLR^2/((ωL)^2+R^2))^2)
=sqrt((ω^4L^4*R^2+ω^2*L^2*R^4)/((ωL)^2+R^2)^2)
=sqrt(ω^2*L^2*R^2*(ω^2*L^2+R^2)/((ωL)^2+R^2)^2)
=ωLR/sqrt((ωL)^2+R^2)
φ=atan((ωLR^2/((ωL)^2+R^2))/(ω^2L^2*R/((ωL)^2+R^2)))
=atan(ωLR^2/(ω^2L^2*R))
=atan(R/(ωL))

RC並列回路の場合



Z=1/(G+jBC)=1/(1/R+jωC)=R/(1+jωCR)
=(1-jωCR)*R/((1-jωCR)*(1+jωCR))
=(R-jωCR^2)/(1+(ωCR)^2)
Y=G+jBC=1/R+jωC
|Z|=sqrt((R/(1+(ωCR)^2))^2+(ωCR^2/(1+(ωCR)^2))^2)
=sqrt((R^2+(ωCR^2)^2)/(1+(ωCR)^2)^2)
=sqrt(R^2*(1+(ωCR)^2)/(1+(ωCR)^2)^2)
=R/sqrt(1+(ωCR)^2)
φ=atan((-ωCR^2/(1+(ωCR)^2))/(R/(1+(ωCR)^2))
=atan(-ωCR^2/R)
=-atan(ωCR)

LC並列回路の場合



Z=1/(jBC-jBL)=1/(jωC-j/ωL)=1/j(ωC-1/ωL)=-j/(ωC-1/ωL)
=-j/((ω^2LC-1)/ωL)
=jωL/(1-ω^2LC)
Y=jBC-jBL=jωC-j/ωL=j(ωC-1/ωL)
φ=atan((ωL/(1-ω^2LC))/R)=atan(±∞)=±π/2

RLC並列回路の場合



Z=1/(G+j(BC-BL))=1/(1/R+j(ωC-1/ωL))
=(1/R-j(ωC-1/ωL))/((1/R-j(ωC-1/ωL))*(1/R+j(ωC-1/ωL)))
=(1/R-j(ωC-1/ωL))/((1/R)^2+(ωC-1/ωL)^2)
Y=G+j(BC-BL)=1/R+j(ωC-1/ωL)
|Z|=sqrt(((1/R)/((1/R)^2+(ωC-1/ωL)^2))^2+(-(ωC-1/ωL)/((1/R)^2+(ωC-1/ωL)^2))^2)
=sqrt(((1/R)^2+(ωC-1/ωL)^2)/((1/R)^2+(ωC-1/ωL)^2)^2)
=1/sqrt((1/R)^2+(ωC-1/ωL)^2)
=1/sqrt((1/R)^2*(1+R^2*(ωC-1/ωL)^2)
=R/sqrt(1+R^2*(ωC-1/ωL)^2)
φ=atan((-(ωC-1/ωL)/((1/R)^2+(ωC-1/ωL)^2))/((1/R)/((1/R)^2+(ωC-1/ωL)^2)))
=atan(-(ωC-1/ωL)/(1/R))
=atan(R(1/ωL-ωC))