三角形网络与星型网络变换公式

1、对于三角网络：1. Ia+Ib+Ic=02. Uab+Ubc+Uca=03. Ia=Uab/Rab+Uac/Rac4. Ib=Uba/Rab+Ubc/Rbc由2式可以推出：5. Uab=Uac-Ubc6. Uba=Ubc-Uac将5.6式代入3.4式可得：7. Ia=(Uac-Ubc)/Rab+Uac/Rab=Uac/Rab-Ubc/Rab+Uac/Rac8. Ib=(Ubc-Uac)/Rab+Ubc/Rbc=Ubc/Rab-Uac/Rab+Ubc/Rbc即是：9. Ia=(1/Rab+1/Rac)Uac-(1/Rab)Ubc=(Rab+Rac)/RabRac)Uac-(1/Rab)Ubc10

2、.Ib=(1/Rab+1/Rbc)Ubc-(1/Rab)Uac=(Rab+Rbc)/RabRbc)Ubc-(1/Rab)Uac将9式两边同时乘以(Rab+Rbc)/Rbc可得：11.(Rab+Rbc)/Rbc)Ia=(Rab+Rac)/RabRac)(Rab+Rbc)/Rbc)Uac-(Rab+Rbc)/RabRbc)Ubc将10式+11式化掉Ubc可得：12.(Rab+Rbc)/Rbc)Ia+Ib=(Rab+Rac)/RabRac)(Rab+Rbc)/Rbc)Uac-(1/Rab)Uac =(Rab+Rac)/RabRac)(Rab+Rbc)/Rbc)-(1/Rab)Uac =(Rab+Rb

3、c+Rac)/RacRbc)Uac将12式整理可得：13.Uac=(Rab+Rbc)Rac/(Rab+Rac+Rbc)Ia+(RacRbc/(Rab+Rac+Rbc)Ib将10式两边同时乘以(Rac+Rab)/Rac可得：14.(Rac+Rab)/Rac)Ib=(Rab+Rbc)/RabRbc)(Rac+Rab)/Rac)Ubc-(Rab+Rac)/RacRab)Uac将9式+14式化掉Uac可得：15.(Rac+Rab)/Rac)Ib+Ia=(Rab+Rbc)/RabRbc)(Rac+Rab)/Rac)Ubc-(1/Rab)Ubc =(Rab+Rbc)/RabRbc)(Rac+Rab)/Ra

4、c)Ubc-(1/Rab)Ubc =(Rab+Rbc+Rac)/RacRbc)Ubc将15式整理可得：16.Ubc=( RacRbc/(Rab+Rac+Rbc)Ia+(Rab+Rac)Rbc/(Rab+Rac+Rbc)Ib对于星型网络：17. Ic=Ia+Ib18.Uac=IaRa+IcRc19.Ubc=IbRb+IcRc将1式代入2.3式可得：20.Uac=IaRa=(Ia+Ib)Rc=(Ra+Rc)Ia+RcIb21.Ubc=IbRb=(Ia+Ib)Rc=(Rb+Rc)Ib+RcIa三角型网络与星型网络等效时，可推导：Rc=RacRbc/Rab+Rac+RbcRa+Rc=(Rab+Rbc)Rac/Rab+Rac+RbcRb+Rc=(Rab+Rac)Rbc/Rab+Rac+Rbc整理可得(对应点三角形相邻两电阻的乘积/三角形三边电阻之和)：Ra=RacRab/Rab+Rac+RbcRb=RbcRab/Rab+Rac+RbcRc=RbcRac/Rab+Rac+Rbc同理可得(星型每相邻两电阻乘积之和/星型对角电阻)：Rab=RaRb+RbRc+RaRc/RcRbc=RaRb+RbRc+RaRc/RaRac=RaRb+RbRc+RaR