A capacitor with capacitance `C` and a coil with active resistance `R` and inductance `L` are connected in parallel to a source of sinusoidal voltage of frequency `omega`. Find the phase difference between the current fed to the circuit and the source voltage.

Explanation

We use the method of complex voltage
`V=V_(0)e^(iomegat)`
Then `I_(C)=(V_(0)e^(iomegat))/( (1)/( iomegaC))=iomegaCV_(0)e^(iomegat)`
`I_(L,R)=(V_(0)e^(iomegat))/( R+iomegaL)`
`I=I_(C)+I_(L,R)=V_(0)(R-iomegaL+i omegaC(R^(2)+omega^(2)L^(2)))/(R^(2)+omega^(2)L^(2))e^(iomegat)`
Then taking the real part
`I=(V_(0)sqrt(R^(2)+{omegaC(R^(2)+omega^(2)L^(2))-omegaL}^(2)))/(R^(2)+omega^(2)L^(2))cos(omegat-varphi)`
where` tan varphi=( omegaL-omegaC(R^(2)+omega^(2)L^(2)))/(R)`